https://xenreference.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=Ground 2026-10-29T20:59:38Z User contributions MediaWiki 1.46.2 https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8582 2026-10-29T16:45:57Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> == Ground's perspective ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The primary &amp; specialist solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a '''primary tuning system''', which is the default choice most of the time. This is the one that you use as much like 12edo as possible; the one you keep coming back to, that feels like it can express anything, that you know better than any other. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) fills the gaps in intervals and allows it to approximate the structures of other tunings (31+43 and 53+58). For good candidates for a primary tuning, see [[tuning recommendations for beginners]].<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned, which I call a '''specialist tuning'''. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.<br /> <br /> The fundamental structure is what sets these apart from a large edo with the same options. 111edo may have a good tuning of Cohemimabila, but ignoring it is trivial. In 25edo, it's possibly the most important structure in the entire tuning. It doubles as Sixix and virtually every harmonic interpretation runs through it. If the intention when starting a piece is to use something generated by a supraminor third, that should be the focus. And if that's the focus, the choice is obvious. If there isn't a clear concept, the focus should be on what the main tuning is capable of. Edo multiplication and other ways of obtaining extra notes can be used in specialist tunings, but it risks missing the point of them.<br /> <br /> This system actually opens up more tuning options for me. I like 21edo for oneirotonic/Slendric/Llywelyn/Sidewalk, but the very sharp 5 and flat 3 felt wrong. Now, I can confidently disregard them because they aren't part of what 21edo's purpose is to me.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8581 2026-10-29T16:42:58Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> == Ground's perspective ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The primary &amp; specialist solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a '''primary tuning system''', which is the default choice most of the time. This is the one that you use as much like 12edo as possible; the one you keep coming back to, that feels like it can express anything, that you know better than any other. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing. For good candidates for a primary tuning, see [[tuning recommendations for beginners]].<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned, which I call a '''specialist tuning'''. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.<br /> <br /> The fundamental structure is what sets these apart from a large edo with the same options. 111edo may have a good tuning of Cohemimabila, but ignoring it is trivial. In 25edo, it's possibly the most important structure in the entire tuning. It doubles as Sixix and virtually every harmonic interpretation runs through it. If the intention when starting a piece is to use something generated by a supraminor third, that should be the focus. And if that's the focus, the choice is obvious. If there isn't a clear concept, the focus should be on what the main tuning is capable of. Edo multiplication and other ways of obtaining extra notes can be used in specialist tunings, but it risks missing the point of them.<br /> <br /> This system actually opens up more tuning options for me. I like 21edo for oneirotonic/Slendric/Llywelyn/Sidewalk, but the very sharp 5 and flat 3 felt wrong. Now, I can confidently disregard them because they aren't part of what 21edo's purpose is to me.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8580 2026-10-29T16:40:36Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> == Ground's perspective ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The primary &amp; specialist solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a '''primary tuning system''', which is the default choice most of the time. This is the one that you use as much like 12edo as possible; the one you keep coming back to, that feels like it can express anything, that you know better than any other. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing.<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned, which I call a '''specialist tuning'''. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.<br /> <br /> The fundamental structure is what sets these apart from a large edo with the same options. 111edo may have a good tuning of Cohemimabila, but ignoring it is trivial. In 25edo, it's possibly the most important structure in the entire tuning. It doubles as Sixix and virtually every harmonic interpretation runs through it. If the intention when starting a piece is to use something generated by a supraminor third, that should be the focus. And if that's the focus, the choice is obvious. If there isn't a clear concept, the focus should be on what the main tuning is capable of. Edo multiplication and other ways of obtaining extra notes can be used in specialist tunings, but it risks missing the point of them.<br /> <br /> This system actually opens up more tuning options for me. I like 21edo for oneirotonic/Slendric/Llywelyn/Sidewalk, but the very sharp 5 and flat 3 felt wrong. Now, I can confidently disregard them because they aren't part of what 21edo's purpose is to me.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8575 2026-10-29T14:31:35Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a '''primary tuning system''', which is the default choice most of the time. This is the one that you use as much like 12edo as possible; the one you keep coming back to, that feels like it can express anything, that you know better than any other. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing.<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned, which I call a '''specialist tuning'''. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.<br /> <br /> The fundamental structure is what sets these apart from a large edo with the same options. 111edo may have a good tuning of Cohemimabila, but ignoring it is trivial. In 25edo, it's possibly the most important structure in the entire tuning. It doubles as Sixix and virtually every harmonic interpretation runs through it. If the intention when starting a piece is to use something generated by a supraminor third, that should be the focus. And if that's the focus, the choice is obvious. If there isn't a clear concept, the focus should be on what the main tuning is capable of. Edo multiplication and other ways of obtaining extra notes can be used in specialist tunings, but it risks missing the point of them.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8574 2026-10-29T14:22:03Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a main default tuning system. This is the one that you use as much like 12edo as possible, the one you keep coming back to, the one that feels like it can express anything. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing.<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.<br /> <br /> The fundamental structure is what sets these apart from a large edo with the same options. 111edo may have a good tuning of Cohemimabila, but ignoring it is trivial. In 25edo, it's possibly the most important structure in the entire tuning. It doubles as Sixix and virtually every harmonic interpretation runs through it. If the intention when starting a piece is to use something generated by a supraminor third, that should be the focus. And if that's the focus, the choice is obvious. If there isn't a clear concept, the focus should be on what the main tuning is capable of.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8573 2026-10-29T13:50:22Z

<p>Ground: </p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> === The solution ===<br /> For a composer with sufficient experience in microtuning, my solution is to have a main default tuning system. This is the one that you use as much like 12edo as possible, the one you keep coming back to, the one that feels like it can express anything. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing.<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_choice_paralysis&diff=8572 2026-10-29T13:49:11Z

<p>Ground: Created page with &quot;&#039;&#039;&#039;Tuning choice paralysis&#039;&#039;&#039; or just &#039;&#039;&#039;tuning paralysis&#039;&#039;&#039; is a term for the effects of having too many tuning systems and intervals available. {{UTBox|g_|Ground|7766ff}} I was a 12edo composer with a similar style for years before getting into microtuning. I&#039;ve noticed that my music has stagnated, while nearly all the progress I&#039;ve made in the past several years is tuning-related. For a composer with sufficient experience in microtuning, my solution is to have a mai...&quot;</p> <hr /> <div>'''Tuning choice paralysis''' or just '''tuning paralysis''' is a term for the effects of having too many tuning systems and intervals available.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> I was a 12edo composer with a similar style for years before getting into microtuning. I've noticed that my music has stagnated, while nearly all the progress I've made in the past several years is tuning-related.<br /> <br /> For a composer with sufficient experience in microtuning, my solution is to have a main default tuning system. This is the one that you use as much like 12edo as possible, the one you keep coming back to, the one that feels like it can express anything. Mine is 37edo; it's complicated, but I could find a good ternary scale for just about anything once I knew how. Edo multiplication (74 and 111) adds just about anything I'm missing.<br /> <br /> I'm also not advocating for using one tuning forever. No matter how many ternary scales I have or how large I multiply my main edo, other tunings have two massive advantages: simplicity and specific generator ranges. I suggest using a reasonably small edo or similar when you have a clear and specific concept for how a piece should be tuned. For example, I use 25edo for primarily antidiatonic pieces. It simplifies the interactions compared to the ternary equivalent in 37edo and requires fewer notes per octave at the expense of versatility, but in this case versatility is not necessary. My usage of 18edo takes advantage of how restrictive it is, forcing everything into oneirotonic and smitonic frameworks.</div>

Ground https://xenreference.com/wiki/index.php?title=Aberrisma&diff=8528 2026-10-28T02:59:39Z

<p>Ground: </p> <hr /> <div>{{Proposed}}<br /> <br /> An '''aberrisma''' is an interval representing some comma as an additional type of melodic step smaller than a semitone. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Although they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas about the size of a diesis, the aberrisma is a melodic function and not an interval size. Larger and smaller sizes can function as aberrismas depending on the scale tuning. '''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas. See [[straddle primes]] for aberrismas equated to 1/1 without being tempered out.<br /> <br /> Ground's '''aberrismic hypothesis''' is that intervals an audible distance apart can be broadly used together in the same scale framework while merely registering as detunings of the same interval. {{adv|In a specific test case, trained and untrained 12edo listeners may notice aberrismic steps in a 27edo blackdye melody but consider them to correspond to the same diatonic degree.}} Straddle theory applies the hypothesis to definite ratios, whereas in the original aberrismic scales, it's applied to diatonic scale degrees. Commas have been described as a type of unison in other contexts, but rarely as distinct steps vital to the structure of a scale, or as intervals that are represented by the same ratio. An aberrisma is categorically a unison but doesn't sound like one when isolated, which represents both observed RTT commas and the difference between straddled intervals; it is a unison widened into a melodic step. Aberrismic theory is a collection of the effects of applying the aberrismic hypothesis to scale theory and RTT.<br /> <br /> The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals. Aberrismas may also appear in MOS scales, such as Garibaldi[17]. Superpyth[12] is a good example of the distinction between interval function and size. The diatonic semitone, a deflated 256/243, is equated to the inflated 81/80. Despite being the same size, the former represents a narrow semitone, the latter is an aberrisma, and a stack of the two is a classic semitone.<br /> <br /> == The &quot;why&quot; of aberrismic theory ==<br /> === Adding comma steps to Pyth diatonic ===<br /> 81/80 and 64/63 are the canonical choices for 2.3.5 and 2.3.7 formal commas for modifications to Pythagorean (2.3) JI ratios — 81/80 separates 5/4 from 81/64, 10/9 from 9/8, and so on; and 64/63 separates 7/4 from 16/9, 9/7 from 81/64, and so on. Aberrismic theory asks: what if we made 81/80 or 64/63 into scale steps, so that we can use both of the intervals separated by the formal comma in one scale? Indeed, doing this to 2.3 diatonic to make a ternary scale, in a way that best preserves the fifth-generated structure, results in (2.3.5) blackdye and (2.3.7) diasem, respectively.<br /> <br /> ==== Example: Blackdye ====<br /> The Zarlino diatonic is chiral — there are two different, equally valid second degrees of the Ionian mode. Both are useful, as the sharp one forms a perfect fifth with the fifth degree but a wolf fifth with the sixth degree, and the flat one forms a perfect fifth with the sixth degree but a wolf fifth with the fifth degree.<br /> [[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]<br /> One way to make it achiral is to temper out 81/80, the difference between these two steps, resulting in [[Meantone]] diatonic; intuitively this requires flattening the fifth and sharpening the sixth somewhat. However, an alternative way, if you wish to observe 81/80 or to use just intonation, is to include both varieties of whole tone over the unison, treating 81/80 as a melodic step between them. This can be thought of as dividing up a 9/8 into a 10/9 and an 81/80. It is then reasonable to extend this action to all instances of 9/8 in the scale (as, for instance, the Didymic diatonic has 27/16 as opposed to 5/3). The result is a 10-note ternary scale called '''blackdye'''.<br /> <br /> === Stylistic compositional features for aberrismic theory ===<br /> This section will attempt to describe the principles and stylistic features of a specific style of music that justify using aberrismic scales in a composition. It is not an attempt to present aberrismic theory as absolute truth.<br /> * A style of music that is melodic and heavy in modulations benefits from<br /> ** Multiple step sizes for melodic interest, for example diesis-sized steps that are below conventional semitones, specifically ones large enough to be melodically distinct but small enough to represent intonational changes.<br /> ** A set of modulatory intervals, including fifths.<br /> ** A system that allows unlimited modulation. <br /> * It is widely agreed that lower primes are more robust to detuning. Hence for approximating JI with edos, we use lower prime temperaments, and which also represent either 81/80 or 64/63 steps for greater accuracy.<br /> The above suggests temperaments, in particular edos, that use tempered lower primes, and edos large enough to have small diesis-sized steps. In the context of fifth-based modulation, scales also benefit from having offset arcs of fifths. One simple way to have this is to detemper MOS scales into ternary scales with an additional smaller melodic step size, which have a generator arc with fifths or a generator arc that stacks to fifths via a detempered generator chain.<br /> <br /> {{UserTag|g_|Ground|7766ff|I assert that ~40¢ is the ideal point between &quot;categorical unison&quot; and &quot;clear step&quot;; context pushes it one way or the other, like in 27edo Superpyth diatonic and blackdye.}}<br /> <br /> == List of aberrismic scales ==<br /> * {{Adv|&quot;GS(...)[n]&quot; is [[generator sequence]] notation.}}<br /> * {{adv|&quot;subst ax(bycz)&quot; denotes [[MOS substitution]].}}<br /> * {{adv|&quot;Almost&quot; a cross-set means that one or two notes may be missing from the full cross-set and one note may have been added. Exact cross-sets are italicized.}}<br /> * {{adv|Under &quot;Patterns&quot;, &quot;C&quot; is [[achiral]], and &quot;R&quot; and &quot;L&quot; denote two [[chiral]]ities of a chiral pair.}}<br /> === Quasi-diatonic aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !|pinedye / dia1s&lt;br/&gt;(5L2m1s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m1s)}}}}<br /> ||1sC: LLsLLmLm&lt;br/&gt;1sR: LLmLLmLs&lt;br/&gt;1sL: LLmLLsLm<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 27/25, 81/80]<br /> ||{{adv|GS(3/2)[3] and GS(10/9)[3]}}<br /> ||1sC has 4 fifths and 1sR/1sL have 5<br /> |-<br /> !class=&quot;thl&quot;|diasem / dia2s&lt;br/&gt;(5L2m2s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m2s)}}}}<br /> ||2sR: LmLsLmLsL&lt;br/&gt;2sL: LsLmLsLmL<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 28/27, 64/63]<br /> ||{{adv|GS(3/2)[5] and 7/6}}<br /> ||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.<br /> ''See also: [[Chthonic harmony#Diasem|Diasem]]''<br /> |-<br /> !class=&quot;thl&quot;|blackdye / dia3s&lt;br/&gt;(5L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m3s)}}}}<br /> ||sLmLsLmLsL<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 16/15, 81/80]<br /> ||{{adv|''GS(3/2)[5] and 10/9''}}<br /> ||Two interleaved 3-limit pentatonics<br /> ''See also: [[10-form#Blackdye|Blackdye]]''<br /> |-<br /> !|diaslen / dia4s&lt;br/&gt;(5L2m4s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m4s)}}}}<br /> ||4sC: LmLsLsLmLss&lt;br/&gt;4sR: LsLmLsLsLms&lt;br/&gt;4sL: LsLsLmLsLsm<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 49/48, 64/63]<br /> ||{{adv|GS(3/2)[4] and GS(8/7)[3]}}<br /> ||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]].&lt;br/&gt;{{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}<br /> |-<br /> !|diachrome / chromedye / dia5s&lt;br/&gt;(5L2m5s)<br /> ||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}}&lt;br/&gt;5sR: LmsLsLsLmsLs&lt;br/&gt;5sL: LsLsLsmLsLsm<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 256/243, 81/80]<br /> ||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}<br /> ||Fifth-generated but with a 6-step offset<br /> |-<br /> !|whitedye / dia7s&lt;br/&gt;(5L2m7s)&lt;br/&gt;{{adv|{{nowrap|subst 7s(5L2m)}}}}<br /> ||LsLsLsmsLsLsms<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 28/27, 81/80]<br /> ||{{adv|''GS(3/2)[7] and 81/80''}}<br /> ||Two interleaved diatonics<br /> |}<br /> See [[Ground's composition theory#Quasi-diatonic scales]] for more tunings of diasem and blackdye.<br /> <br /> === Other aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI/[[erac]] interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !class=&quot;thl&quot;|[[penslen]] / slen5m&lt;br/&gt;(5L5m6s)&lt;br/&gt;{{adv|{{nowrap|subst 6s(5L5m)}}}}<br /> ||LmsLmsLsmLsmLsms<br /> ||2.3.5.7.11[41 &amp; 46]&lt;br/&gt;[L, m, s] = [12/11, 33/32, 64/63]<br /> ||{{adv|''GS(8/7)[8] and 11/8''}}<br /> || Has two aberrisma sizes, s and m.<br /> |-<br /> !|prejubilic / citro3s&lt;br/&gt;(4L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 2m(4L3s)}}}}<br /> ||C: LLsmLsLms&lt;br/&gt;R: LmLsLmsLs&lt;br/&gt;L: LmLsLsmLs<br /> ||2.5.7&lt;br/&gt;[L, m, s] = [28/25, 35/32, 50/49]<br /> ||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}<br /> ||Didacus tempering makes L = m + s.<br /> |-<br /> !|arm5s&lt;br/&gt;(7L2m5s)&lt;br/&gt;{{adv|{{nowrap|subst 7L(2m5s)}}}}<br /> ||LmLsLsLmLsLsLs<br /> ||2.x&lt;3.5.7.11.13[37edo] (4:2:1)<br /> ||{{adv|''GS(&lt;&lt;3/2)[7] and 14/13''}}<br /> ||An interleaving of two antidiatonic scales.<br /> |-<br /> !|mosh3s&lt;br/&gt;(3L4m3s)&lt;br/&gt;{{adv|{{nowrap|subst 4m(3L3s)}}}}<br /> ||LmsLmsmLsm<br /> ||2.x&lt;3.7.11.13[37edo] (5:4:2)<br /> ||{{adv|''GS(16/13)[5] and 11/8''}}<br /> ||<br /> |-<br /> !|smi2s&lt;br/&gt;(4L3m2s)&lt;br/&gt;{{adv|{{nowrap|subst 2s(4L3m)}}}}<br /> ||C: LLmsLmLsm&lt;br/&gt;R: LmLmsLmLs&lt;br/&gt;L: LmLsLmLsm<br /> ||2.9.7.11.17[46edo] (8:4:1)<br /> ||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}<br /> ||<br /> |}<br /> <br /> === Systematic naming ===<br /> Basic systematic names for aberrismic scales are of the form<br /> <br /> [mos_prefix]n[added_step_size] (e.g. dia2s for diasem).<br /> <br /> Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.<br /> <br /> This is subject to change as aberrismic theory nomenclature is updated in the future.<br /> <br /> == Notation ==<br /> The [[generator gronut]] is an aberrismic extension of the circle of fifths.<br /> <br /> === Quasi-diatonic ===<br /> Quasi-diatonic aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.<br /> <br /> Aberrismic/straddle-prime accidentals:<br /> * Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. &lt;!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. --&gt;<br /> * Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).<br /> * [[Erac|Eracs]] may be used to represent the inflating of commas typically found in aberrismic scales, although they represent a specific unit of error.<br /> <br /> == Aberrismic theory and RTT ==<br /> Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under Ground's use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to &lt;!--simultaneously achieve sufficient accuracy to LCJI and --&gt;improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.<br /> <br /> At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.<br /> <br /> There are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.<br /> <br /> === Example: Blackdye ===<br /> The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.<br /> * ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.<br /> * [[Ultrapyth]], 2.3.5.7.11.13[32 &amp; 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. Ground once termed the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.<br /> <br /> {| class=&quot;wikitable right-2 right-3 right-4 right-5&quot;<br /> |+ style=&quot;font-size: 105%;&quot; | Blackdye intervals in two temperaments<br /> |-<br /> ! Interval class<br /> ! Sizes<br /> ! Untempered 2.3.5<br /> ! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)<br /> |-<br /> ! 1-step<br /> | s&lt;br/&gt;m&lt;br/&gt;L <br /> | 81/80&lt;br/&gt;16/15&lt;br/&gt;10/9<br /> | 143/140&lt;br/&gt;22/21&lt;br/&gt;160/143<br /> |-<br /> ! 2-step<br /> | L + s&lt;br/&gt;L + m <br /> | 9/8&lt;br/&gt;32/27<br /> | 8/7, 9/8&lt;br/&gt;7/6<br /> |- <br /> ! 3-step<br /> | L + 2s&lt;br/&gt;L + m + s&lt;br/&gt;2L + s&lt;br/&gt;2L + m<br /> | 729/640&lt;br/&gt;6/5&lt;br/&gt;5/4&lt;br/&gt;320/243<br /> | 7/6&lt;br/&gt;13/11&lt;br/&gt;14/11&lt;br/&gt;13/10<br /> |- <br /> ! 4-step<br /> | 2L + 2s&lt;br/&gt;2L + m + s<br /> | 81/64&lt;br/&gt;4/3<br /> | 13/10&lt;br/&gt;4/3<br /> |-<br /> ! 5-step<br /> | 2L + m + 2s&lt;br/&gt;2L + 2m + s&lt;br/&gt;3L + 2s&lt;br/&gt;3L + m + s<br /> | 27/20&lt;br/&gt;64/45&lt;br/&gt;45/32&lt;br/&gt;40/27<br /> | 66/49&lt;br/&gt;11/8&lt;br/&gt;16/11&lt;br/&gt;49/33<br /> |-<br /> ! 6-step<br /> | 3L + m + 2s&lt;br/&gt;3L + 2m + s<br /> | 3/2&lt;br/&gt;128/81<br /> | 3/2&lt;br/&gt;20/13<br /> |- <br /> ! 7-step<br /> | 3L + m + 3s&lt;br/&gt;3L + 2m + 2s&lt;br/&gt;4L + m + 2s&lt;br/&gt;4L + 2m + s<br /> | 243/160&lt;br/&gt;8/5&lt;br/&gt;5/3&lt;br/&gt;1280/729<br /> | 20/13&lt;br/&gt;11/7&lt;br/&gt;22/13&lt;br/&gt;12/7<br /> |- <br /> ! 8-step<br /> | 4L + m + 3s&lt;br/&gt;4L + 2m + 2s<br /> | 27/16&lt;br/&gt;16/9<br /> | 12/7&lt;br/&gt;7/4, 16/9<br /> |-<br /> ! 9-step<br /> | 5L + 2m + s&lt;br/&gt;5L + m + 2s&lt;br/&gt;4L + 2m + 2s<br /> | 9/5&lt;br/&gt;15/8&lt;br/&gt;160/81<br /> | 143/80&lt;br/&gt;21/11&lt;br/&gt;280/143<br /> |}<br /> <br /> == Glossary ==<br /> * '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.<br /> * '''Aberrisma''': The smallest interval region that melodically sounds like a step.<br /> * '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.<br /> * '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.<br /> * '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.<br /> * '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.<br /> <br /> == Extension to higher ranks ==<br /> Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a [[GS]] with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.<br /> <br /> {{UserTag|g_|Ground|7766ff|I propose that we call this type of scale '''bicatenary'''.}}<br /> <br /> However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 41 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28] with a JI tuning of 35/32 : 21/20 : 36/35 : 64/63. [[Ground's composition theory#Interval logic deviation theory|ILD]] is a simple way to access notes outside of a scale with a higher rank.<br /> <br /> A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. Possibly the most common ratio representation of an aberrisma is 36/35, but few ternary scales use it without tempering out 81/80 or 64/63.<br /> <br /> (81/80)^2 = 43¢ is notably in the middle of the usual aberrisma size range as well. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.<br /> <br /> == Compositional examples ==<br /> Some compositional snippets using aberrismic scales:<br /> <br /> * The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]<br /> * A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]<br /> * A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]<br /> <br /> == External links ==<br /> * [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]<br /> {{cat|<br /> Terms<br /> Aberrismic terms<br /> Ternary scales<br /> }}</div>

Ground https://xenreference.com/wiki/index.php?title=User:Ground&diff=8527 2026-10-28T02:56:01Z

<p>Ground: </p> <hr /> <div>* I'm Skyler 'Sky' &quot;Ground&quot; Talbott-Settle &quot;Trebuchet-Skittle&quot;.<br /> ** This may be where the old name Skytset came from, but the truth is unknown.<br /> * I made the wiki.<br /> ** I've been doing work in an attempt to guide the wiki towards a more collaborative culture.<br /> *** [[Template:UserTag]]<br /> * My music project is called groundFault Corporation for reasons I don't feel like getting into.<br /> ** [[List of xenharmonic musicians]]<br /> * I compose and theorize.<br /> ** A good portion of my theory work has been with [[User:Inthar]] and he's done a great job at documentation.<br /> ** Bad theories:<br /> *** [[Aberrisma]] including [[Straddle primes]]<br /> *** [[Ground's composition theory]]<br /> ** Even worse temperaments:<br /> *** [[Pentagoth|Pentagoth and Sidewalk]]<br /> *** [[Hibernal]]<br /> *** [[Intergan]]<br /> *** [[Hemiseven]] and my interpretations<br /> *** [[Monarch|Monarch and Interarch]]<br /> *** [[Straddle cluster temperaments]]<br /> *** [[Diaschismic straddle temperaments]]<br /> *** [[86edo]] (TODO: make this 43edo)<br /> *** Many others without articles or as sections of other articles<br /> ** Future video scripts:<br /> *** [[Ground's intro to tuning diversity]]<br /> *** Music theory from the perspective of tuning theory<br /> * I'm a social construct.<br /> ** There is nothing useful to know about me.</div>

Ground https://xenreference.com/wiki/index.php?title=Ground%27s_composition_theory&diff=8526 2026-10-28T02:44:35Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> This article is about composition and the practical considerations of tuning. I always have more content to add and revise.<br /> <br /> I write in the context of my unified [[Aberrisma|aberrismic theory]]. An aberrisma is categorically a unison but doesn't sound like one when isolated. It could separate different ratios or two versions of the same ratio as in the subset [[Straddle primes|straddle theory]].<br /> <br /> It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the [[Levels of tempering|mesotempered category]]. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.<br /> <br /> == Interval logic deviation theory ==<br /> <br /> Much of my music has had a distinctively shifting tonality since 2018 or earlier, which started in 12edo. This article is an attempt to explain how it works. I'm introducing a placeholder term for it, '''interval logic deviation''' theory (ILD), which can be replaced if this turns out to be something already described.<br /> <br /> === Local tonality ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic example of probabilities (up to word length 3)<br /> !Sequence<br /> !Next step probability<br /> |-<br /> |s<br /> |L 1/1<br /> |-<br /> |L<br /> |L 5/7, s 2/7<br /> |-<br /> |sL<br /> |L 1/1<br /> |-<br /> |Ls<br /> |L 1/1<br /> |-<br /> |sLL<br /> |s 1/2, L 1/2<br /> |-<br /> |LsL<br /> |L 1/1<br /> |-<br /> |LLs<br /> |L 1/1<br /> |-<br /> |LLL<br /> |s 2/3, L 1/3<br /> |}<br /> &lt;/div&gt;<br /> I have a simultaneous regard and disregard for standard Western tonality. This is because I view it as an important but strictly local property, meaning it fundamentally only applies on the scope of a single path from tension to release, however long that is. Thus, modulations are only generally uncommon because phrases usually resolve to the same key center they started from, but changing tonality is just as much of a choice as not changing it. Modulation flows just like any other melodic or harmonic movement.<br /> <br /> This flow is facilitated by ILD, in which scales aren't a fixed set of notes, but a template for interval logic to be rearranged and deviated from. As such, their main features are probabilities in an interval sequence and &quot;bubble deviations&quot; from that interval sequence. ILD is best for music with a strong melodic focus, such as mine, where the melody informs the harmony instead of the reverse. Other concepts may be used instead with the same general goal.<br /> <br /> Melodic interval sequences, or &quot;words&quot; of step sizes, are the most minimal expression of tonal tension and release. For example, if a diatonic melody were to play C then B, the listener is likely to expect A to be next and feel a small resolution upon hearing it. This is the descending sL word. Melodies are full of small sequences like this based on the scale that they are in. It's possible to use sequences with notes outside the scale while still feeling like they belong, and how much they belong can be predicted. The longer the word and the greater probability of occurring indicates that it's more likely to sound like it belongs in the scale.<br /> <br /> === Axes of deviation ===<br /> <br /> &quot;Bubble deviations&quot; in ILD are named after the bubble sort algorithm, which repeatedly swaps adjacent items in an array. Steps in a base scale can be swapped in the same way to modify the scale with no requirement for a clear structure like a generator chain or lattice splotch, which is my term for a collection of generator chains in aberrismic theory. Suppose you want the scale word sLs in diatonic. This would require only one bubble deviation from the expected step order, turning sLLs to sLsL. While the probability of encountering it in the base diatonic scale is zero, it sounds more &quot;probable&quot; (less unexpected) than something like ssL.<br /> <br /> This explains why I used Dimininished[8] in 12edo more often than Augmented[6], because its stepwise interval logic has less deviation from diatonic. Diminished[8] is made of a repeating sequence of 1\12 and 2\12, common in diatonic, whereas Augmented[6]'s steps of 1\12 and 3\12 do not occur in diatonic at all. As a result, melodies in Diminished sound less exotic.<br /> <br /> Bubble deviations are only one axis of deviation. There is another axis which I've found to be exclusively useful in tuning systems with aberrismic-sized steps or smaller: the axis of microtonal deviation from expected intervals. This involves changing the pitch of an expected interval only slightly, so it is heard as a variation of the expected interval rather than a different interval entirely. This axis interacts with the base scale by introducing or modifying an aberrismic offset, for example diatonic being diasem or blackdye with the offset removed, 2.3.7 diasem having a larger offset than 2.3.23, or 2.3.5 blackdye having a smaller offset than 2.3.17/7. The intervals affected by the offset, usually thirds and sixths, differ microtonally when the offset is changed.<br /> <br /> Straddling intervals that are stacked the most (usually 3/2) introduce a third axis that can be simplified into a combination of the other two. It's possible to have bubble deviation from a scale that isn't even in the tuning system being used, like how alternating &lt;&lt;3 and &gt;3 in 37edo straddles 74edo meantone and results in trackdye.<br /> <br /> {{UserTag|KC|Inthar|000000|[[Stretching and compression]] constitute yet another axis of deviation separate from straddling. Diatonic-based example: This is important in 4L3s and 5L3s which are warped-diatonic MOSes. Note that dual-3 diatonic 5L1m1s is a subset of interleaved diatonic 7s(5L2m), tens-interleaved diatonic 6s(5L2m) ''and'' tract-interleaved diatonic 8s(5L2m).}}<br /> <br /> === Vague interval logic ===<br /> <br /> Scales are useful for ILD, but not technically necessary. One may pick the desired step sizes and find just a few arrangements leading to useful intervals, like the perfect fifth. This should be especially useful when trying to avoid making quasi-diatonic music in any tuning system.<br /> <br /> Take 3\24 s and 5\24 L for example. The words LL and sssL make 10\24 and 14\24 respectively, so the two can together be used to infer vague probabilities. Edos that straddle important intervals are also useful for this because they increase the chances of landing on those intervals. If s and L were replaced with 11\86 and 18\86, LL becomes &gt;4/3 and sssL becomes &gt;3/2.<br /> <br /> === Aberrismic theory ===<br /> <br /> The above concepts also apply to ternary scales, where they may be even more useful. Conventional aberrismic scales feature an alternating generator sequence that creates a 2-dimensional lattice, which can be used similarly to a generator sequence, but is less intuitive due to its complexity. Thus, ILD proves to be a practical alternative when modulating. This is how I write 2.3.5 and 2.3.7 music in tunings that aren't Meantone or Archy.<br /> <br /> Quasi-diatonic aberrismic scales are the ideal case due to the general importance of stacking prime 3 and internalized diatonic logic of Western music. They typically have four main step sizes: '''aberrisma''' (s), '''semitone''' (m), and the two whole tones '''solitone''' (L) and '''magnitone''' (L+s). Other scales or ILD can introduce more sizes such as the '''magnisemitone''' (m+s). The decision of whether to use a solitone or magnitone depends on which intervals in a chord are preferred and which sounds more melodically to the composer in a given situation. The aberrisma is a new class so its theory is less well defined, but it works best as a passing tone. It provides the ability to change the length of scale runs without repeating any notes.<br /> <br /> I find it most generally useful to define the size range of an aberrisma as being between 81/80 and half of 16/15, but my personal usage has ranged from about 12¢ (the size of 81/80 in 5-limit CWE Negri and around one step of the largest full edos I use) to about 100¢ (81/80 in some tunings of Blackwood with a flat 5). I provide 40¢ as the general ideal, but the actual ideal varies. Faster music prefers a larger aberrisma because it is more audible. Scales with a large magnitone can handle a larger aberrisma without it sounding entirely like a semitone, such as 37edo pental blackdye. '''Subaberrismas''', aberrismas small enough that they are not reliably recognizable as a melodic step, have a unique sound that may be desirable in certain situations.<br /> <br /> Because they are often passing tones when used melodically, it can be relatively easy to swap an aberrisma for one of a more preferable size. For example, I may swap 1\43 in 43edo diasem for the semiquartal 2\43 to increase its audibility.<br /> <br /> I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.<br /> <br /> == Diatonic analogues ==<br /> <br /> === 9-note &quot;diatygic&quot; ===<br /> <br /> ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.<br /> <br /> When it comes to extending the diatonic scale's circle of fifths as seen in ILD, there's no alteration more important than the modes of the melodic minor scale, which I'll be referring to together as '''melodiatonic'''. It's the only non-diatonic scale that's possible to construct by making a single bubble deviation from diatonic, and it makes sense looking at the circle of fifths. Assume that some altered diatonic scale must contain only L and s steps to preserve as much step logic as possible. To do this, only ordinals following a semitone may be sharpened (C and F when the unaltered notes are all naturals), and only ordinals preceding a semitone may be flattened (B and E). Sharpening F or flattening B just changes the key signature and the result is still diatonic. Sharpening C or flattening E results in melodiatonic. I have found melodiatonic to be a remarkably reliable way to make diatonic tunings (including 12edo) sound exotic yet familiar enough to hit the same emotional beats diatonic can provide. After diatonic, it has the smallest span across the chain of fifths without duplicating a diatonic ordinal, meaning that it contains no pairs of notes that are 7 generators apart. Just like how diatonic is the framework for all quasi-diatonic scales, melodiatonic can be used as the framework for certain alterations of those quasi-diatonic scales. {{adv|Unconventional uses of melodiatonic I can think of off the top of my head are parts of Undertale's Heartache in Bb Dorian b2 and the main theme of Animusic's Harmonic Voltage in C Mixolydian b6.}}<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Quasi-Diatygic Scales<br /> !Altered&lt;br&gt;Blackdye<br /> !Altered&lt;br&gt;Diasem<br /> |-<br /> |10/9&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;5/4&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;5/3&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |9/8&lt;br&gt;8/7&lt;br&gt;7/6&lt;br&gt;9/7&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;14/9&lt;br&gt;12/7&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |}<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic Tertian Triads in D Gadygic<br /> !Root<br /> !Triads<br /> |-<br /> |D<br /> |minor, Major<br /> |-<br /> |E<br /> |dim., minor<br /> |-<br /> |F<br /> |Major<br /> |-<br /> |F♯<br /> |dim.<br /> |-<br /> |G<br /> |minor, Major<br /> |-<br /> |A<br /> |minor<br /> |-<br /> |B♭<br /> |Major<br /> |-<br /> |B<br /> |dim., minor<br /> |-<br /> |C<br /> |Major<br /> |}<br /> &lt;/div&gt;<br /> The gaps in the fifth chain can be filled to result in a contiguous generator chain of 9 notes rather than 7, allowing some modes to freely switch between major and minor thirds and sixths. This is probably the earliest music theory I've ever consciously developed, dating to 2014 or so ([https://soundcloud.com/groundfault/doodle-62 example]). It has been scale #1 on the [https://allthescales.org/scales.php?n=9 9-note page of AllTheScales] for at least as long, and I tentatively propose calling the scale '''diatygic''' using the seemingly arbitrary -ygic suffix from AllTheScales since it seems like a term I would have chosen at the time. It has stood the test of time and I find it to be the natural extension of the diatonic framework that doesn't require microtuning, so it remains my favorite 12edo-compatible scale. The scale word for the symmetrical mode is LdcdLdcdL, with d and c for diatonic and chromatic semitones. It is called '''Gadygic''' by AllTheScales. I use plenty of extended interval chains when I compose, but diatygic is intended to be a proper albitonic scale with simple self-contained rules, prescriptive rather than descriptive in the sense that it is the new template to be extended and deviated from. {{adv|For some reason, the first examples of this scale I can think of (besides my own music) all come from the brony fandom. Morning in Ponyville, Awesome as I Wanna Be, and Awoken. It should leave no guesswork as to what I was doing in 2014. All of them are coincidentally in the key of D.}}<br /> <br /> I derived diatygic from my own observation of my favorite music, intuition, and the methods above, including that 9 notes is the closest thing to a MOS between 7 and 12 notes. For one thing, MOS scales with generators approaching half an octave will have sizes that are successive odd numbers; the diatonic range jumps from 7 to 12, but the adjacent antidiatonic range follows 7 with 9. Stacking 9 fifths results in a scale with groups of 3 semitones in a row, which is substantially easier to navigate than the 10-note &quot;supermode&quot; in [https://mtosmt.org/issues/mto.11.17.4/mto.11.17.4.temperley.html Temperley 2011], with a group of 5 semitones. I've found it more useful to use various modes of diatygic than to add additional notes, since needing more than two non-diatonic notes at a time is very rare. This is so that the scale can contain the diatonic modes Aeolian to Ionian as subsets, whereas diatygic only manages Aeolian to Mixolydian. I've never used the major seventh degree much myself, despite its high apparent importance in Western music. Even back when I used major modes more than minor, I preferred Mixolydian to Ionian. It's about as important to me as the minor second degree, which is easy to achieve through using modes other than the symmetrical one.<br /> <br /> This extension of diatonic also works for quasi-diatonic scales. {{adv|Pental blackdye is two chains of 5 of 3/2 separated by 10/9. Extending both of those chains by 2 is analogous to what I'm doing to diatonic, but separating them by 81/80 instead makes 5L2m7s. Separating by 10/9 results in a quaternary scale that is very nearly ternary; temper out the Schisma and you get 3L6m5s with a non-standard step order to what I've established in my aberrismic theory.}} The altered pental blackdye is almost the 3sA Aeolian mode with 5/4 and 5/3 added. The retroversion of this can be obtained by adding 6/5 and 8/5 to 3sG Mixolydian, but it's inferior because it lacks the perfect fifth over 6/5. The one additional change is 81/80 to 10/9 to fix the wolf fifth from 9/8 to 5/3. All 12 notes are required to make a chain of pure 3/2 sometimes altered by 81/80 just as blackdye does, except 4/3-16/9 and 27/20-9/5 are redundant. The extra note is kept to keep the scale to a minimum of four unique step sizes, similar to what is done with blackdye. A similar process may be done to septal diasem.<br /> <br /> One of my favorite parts of the scale is a major third on D, a minor third on E, and a major third on F, using notes from D Gadygic.<br /> {| class=&quot;wikitable&quot;<br /> |+Interval Matrix of Gadygic, DR 3:4:5 Meantone<br /> !<br /> !0<br /> !1<br /> !2<br /> !3<br /> !4<br /> !5<br /> !6<br /> !7<br /> !8<br /> !(9)<br /> |-<br /> !0.<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !194.6<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1200<br /> |-<br /> !308.2<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !389.1<br /> |0<br /> |113.6<br /> |308.2<br /> |421.8<br /> |502.7<br /> |616.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !502.7<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |-<br /> !697.3<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !810.9<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |583.7<br /> |697.3<br /> |778.2<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !891.8<br /> |0<br /> |113.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !1005.4<br /> |0<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |}<br /> <br /> === Neutral diatonic extensions ===<br /> <br /> Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.<br /> <br /> A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.<br /> <br /> I believe xenharmony involves a balance of being unusual without being jarring. Neutral seconds around 150¢ are excellent for the former, but are more difficult to fit into a diatonic framework. When given the choice, I prefer to have neutral seconds around 135¢ to fit in place of a semitone and 165¢ to fit in place of a whole tone. Temperaments in small edos are Flattone for the former and Porcupine for the latter.<br /> <br /> When converting a simple subgroup to contain these intervals, it makes sense to use 2.3.5 because it's closer to neutral whereas 2.3.7 is closer to interordinal.<br /> * Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.<br /> * Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.<br /> This is an excellent use case for straddle-3 temperaments. 37edo has mild Mavila and extreme Archy fifths, but their distance from 3/2 is somewhat masked in 5-limit chords due to the 5/4 anchor concordance.<br /> <br /> == Adaptive diatonic modification system ==<br /> <br /> This section is incomplete.<br /> <br /> In order to make modified diatonic scales easier to use and quickly understand the intervals I use in my head, I have a proposal for an interval scheme based on splitting diatonic or quasi-diatonic intervals, somewhat similar to [[Adaptive diatonic interval names|ADIN]].<br /> <br /> First specify which is the default diatonic or quasi-diatonic quality. This is usually pental (4:5:6), pyth, or septal (6:7:9).<br /> <br /> Neutral seconds are about 3/7 to 4/7 of the minor third.<br /> <br /> == Structural tuning selection ==<br /> <br /> Edos are the most common type of tuning system, and the one I use exclusively. They're a simple and discrete set, which makes them very easy to compare to one another and list in the titles of music made with them. This creates a common assumption that they sound more different from each other than they actually do, which I suffer from myself sometimes. Small edos really do have distinct personalities, but as they get larger and more flexible, those differences gradually disappear. I try to guess the tuning system of a piece whenever it isn't stated, and I'm very often wrong.<br /> <br /> I propose that the most important distinction between tuning systems of sufficient flexibility is not how they sound, but how their unique combinations of structures guide the composer. I like to believe that I have sole control over what I write, but my choice of tuning system prioritizes certain scales, chords, and intervals over others by making them easier to use in practice.<br /> <br /> I use large edos in an unusual way. I like to have multiple octaves of range directly in the piano roll, so I access large edos by multiplying smaller ones using midi channels and my custom tuning scripts. This creates a more complex process of tuning selection, involving the combination of two considerations:<br /> # Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?<br /> # Which of those can be multiplied by 2 or 3 to achieve a superset that's enjoyable for the same reasons, but also complements the subset edo well?<br /> {| class=&quot;wikitable&quot;<br /> |+A selection of large edos I use<br /> !Edo<br /> !Justification<br /> |-<br /> | style=&quot;text-align: right;&quot; |27x3&amp;nbsp;=&amp;nbsp;81<br /> |81 is an [[Monarch|Interarch]] edo, which I find useful. 27x2 = 54 is also uniquely bad, an example of poor LCJI approximations providing insufficient advantages to be worth using.<br /> |-<br /> | style=&quot;text-align: right;&quot; |29x3&amp;nbsp;=&amp;nbsp;87<br /> |I don't like 29 enough to use it much on its own, but 87 is my favorite highly accurate LCJI edo, the intersection of Aberschismic and a variant of [[Hemiseven]].<br /> |-<br /> | style=&quot;text-align: right;&quot; |37x2&amp;nbsp;=&amp;nbsp;74<br /> |Note that this is not 37x3 = 111, a much better LCJI edo. 111 offers little to complement the existing structure of 37edo that 74 doesn't already do.<br /> |-<br /> | style=&quot;text-align: right;&quot; |43x2&amp;nbsp;=&amp;nbsp;86<br /> |I found 86 to be the best Interarch edo for my edo multiplication approach. It clearly straddles every prime up to 19 with the possible exception of 13, but it's large enough that their accuracy is not a major issue.<br /> |-<br /> | style=&quot;text-align: right;&quot; |67x2ed4&amp;nbsp;=&amp;nbsp;67<br /> |Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.<br /> |}<br /> <br /> == Edo multiplication ==<br /> <br /> I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the '''base edo''' to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.<br /> <br /> Example tier list of 37edo multiples:<br /> * Base ([[37edo|37]]): S (One of a kind.)<br /> * x2 ([[74edo|74]]): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)<br /> * x3 ([[111edo|111]]): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)<br /> * x6 ([[222edo|222]]): D (Has extra approximations and generators, but too cumbersome to do much.)<br /> This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:<br /> * Base: remainder<br /> * x2: 12%<br /> * x3: 8%<br /> * x6: 4%<br /> <br /> This should show just how important the base edo is. No matter how many intervals you have, large edos or edo multiplication will never be as efficient to write in as an elegant mid-sized edo that does what you want the most. Everything is filtered through the base edo's structure. Because of this, there's no such thing as the perfect edo. At this point, I use whatever I feel like at the time and multiply it by 6 to fill any gaps that might come up.<br /> <br /> When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.<br /> <br /> Edos that are multiples of 6 usually have more than one reasonable way to split them. I will use 144edo to explain my thought process in deciding whether to use 24x6 or 36x4.<br /> * The most important consideration is which base edo the composer enjoys the most. Both options have features that are important to me, but I've written a good amount in 24 because I believe its lack of a good 5 is less important, so it's been proven by time.<br /> * In terms of the number of useful scales available, 36 is superior. To some extent, this comes with having more notes in general.<br /> * x6 has more channels overall, but contains both the x2 and x3 subsets. x4 requires a larger base edo and has only the option to use x2 and x4 notes. Because of the distribution of the number of channels I typically use, I expect x6 to have more practical options.<br /> * x6 is more of a standard for me, so I'm set up better for that.<br /> * I mentioned how I like to make clear switches between x2 and x3 to make it easier to read my piano roll notes. x6 is better for this because x4 has only one distinct subset besides the base edo, x2.<br /> * 36x2 makes accessing 72edo especially convenient because fewer channels is easier. It wins for 11-limit stuff.<br /> * Two important subsets of 144 are 16 and 18. 24x6 has them as x2 and x3, but 36x4 has them as x4 and x1. This is a tie.<br /> * I use 5-form structures a lot, and some of my favorite edos are 43 and 67. Add 5 to bring their 3/2 closer to just, resulting in 48 and 72, 24x2 and x3. This is an elegant way to approximate both 43 and 67.<br /> * Using a smaller edo gets more range in the piano roll, which is useful for quick creative writing.<br /> * 36 has clearer concordances because it's so good at 2.3.7. It's good to have some anchor concordance in the base edo, but 24 also has 3/2. Having 2.3.7 outside the base edo naturally guides composition towards using other channels and makes it hard to forget about them. There's a nice symmetry with 24x6 putting 7/6 and 6/5 on the same channel as well as 9/7 and 5/4.<br /> With all this in mind, 24x6 is clearly the better choice to start with, although it's close.<br /> <br /> === Channel management ===<br /> <br /> Blackdye and diasem structures have a generator of a perfect fifth and some offset for certain notes. Because all of the offsets are in the same direction, using an offset that is part of a superset of the generator doubles the number of distinct channels required. This most notably results in scales where the generator is from the x2 edo and the offset is from the x3 or x6 edo only requiring 4 channels.<br /> <br /> == Anchor concordances ==<br /> <br /> We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an '''anchor concordance''' because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.<br /> <br /> For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.<br /> <br /> I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.<br /> <br /> In 37edo, both 4:5:&gt;6 and 4:5:&lt;&lt;6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides &lt;&lt;6:8:&lt;&lt;&lt;&lt;9 = 6:&gt;&gt;8:&lt;&lt;9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:<br /> * Using the very flat &lt;&lt;3 results in &gt;&gt;16/15, quite audibly wider while keeping some semblance of 4:5:6.<br /> * &lt;&lt;6/5 is found in the [[Pentagoth]] triad, which uses the exaggerated quality of the minor third and the stability of the major third.<br /> <br /> Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:<br /> * ~69edo: DR 4:5:6, 10:12:15<br /> * ~31edo: 5/4 anchor concordance<br /> * ~74edo: DR 3:4:5<br /> * ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)<br /> <br /> == Quartertone composition ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;<br /> |+Island Rastmic with Straddle Primes<br /> !Gens<br /> !949¢<br /> !349¢<br /> ! colspan=&quot;2&quot; |&lt;nowiki&gt;Proposed Names (semi- | hemi-)&lt;/nowiki&gt;<br /> |-<br /> | -12<br /> | colspan=&quot;2&quot; |612<br /> | colspan=&quot;2&quot; |Diminished Fifth<br /> |-<br /> | -11<br /> |361<br /> | class=&quot;thl&quot; |961<br /> |&quot;Semififth&quot;<br /> |&quot;Hemidim.&quot; Seventh<br /> |-<br /> | -10<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |110<br /> | colspan=&quot;2&quot; |Minor Second<br /> |-<br /> | -9<br /> |1059<br /> |459<br /> |&quot;Semithirteenth&quot;<br /> |&quot;Hemidim.&quot; Fourth<br /> |-<br /> | -8<br /> | colspan=&quot;2&quot; |808<br /> | colspan=&quot;2&quot; |Minor Sixth<br /> |-<br /> | -7<br /> | class=&quot;thl&quot; |557<br /> |1157<br /> |&quot;Semiseventh&quot;<br /> |&quot;Hemidim.&quot; Octave<br /> |-<br /> | -6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |306<br /> | colspan=&quot;2&quot; |Minor Third<br /> |-<br /> | -5<br /> |55<br /> |655<br /> |&quot;Semisecond&quot;<br /> |&quot;Hemidim.&quot; Fifth<br /> |-<br /> | -4<br /> | colspan=&quot;2&quot; |1004<br /> | colspan=&quot;2&quot; |Minor Seventh<br /> |-<br /> | -3<br /> |753<br /> |153<br /> |Semitenth<br /> |Neutral Second<br /> |-<br /> | -2<br /> | colspan=&quot;2&quot; |502<br /> | colspan=&quot;2&quot; |Perfect Fourth<br /> |-<br /> | -1<br /> |251<br /> |851<br /> |Semifourth<br /> |Neutral Sixth<br /> |-<br /> |0<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |0<br /> | colspan=&quot;2&quot; |Unison<br /> |-<br /> |1<br /> |949<br /> |349<br /> |Semitwelfth<br /> |Neutral Third<br /> |-<br /> |2<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |698 ''(98)''<br /> | colspan=&quot;2&quot; |Perfect Fifth<br /> |-<br /> |3<br /> |447<br /> |1047<br /> |Semisixth<br /> |Neutral Seventh<br /> |-<br /> |4<br /> | colspan=&quot;2&quot; |196<br /> | colspan=&quot;2&quot; |Major Second<br /> |-<br /> |5<br /> |1145<br /> | class=&quot;thl&quot; |545<br /> |&quot;Semifourteenth&quot;<br /> |&quot;Hemiaug.&quot; Fourth<br /> |-<br /> |6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |894 ''(294)''<br /> | colspan=&quot;2&quot; |Major Sixth<br /> |-<br /> |7<br /> |643<br /> |43<br /> |&quot;Semininth&quot;<br /> |&quot;Hemiaug.&quot; Unison<br /> |-<br /> |8<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |392<br /> | colspan=&quot;2&quot; |Major Third<br /> |-<br /> |9<br /> |141<br /> |741<br /> |&quot;Semithird&quot;<br /> |&quot;Hemiaug.&quot; Fifth<br /> |-<br /> |10<br /> | colspan=&quot;2&quot; |1090<br /> | colspan=&quot;2&quot; |Major Seventh<br /> |-<br /> |11<br /> | class=&quot;thl&quot; |839<br /> |239<br /> |&quot;Semieleventh&quot;<br /> |&quot;Hemiaug.&quot; Second<br /> |-<br /> |12<br /> | colspan=&quot;2&quot; |588<br /> | colspan=&quot;2&quot; |Augmented Fourth<br /> |}<br /> &lt;/div&gt;<br /> This topic isn't directly related to the other sections, but I don't have anywhere else to put it. I've long been interested in what I'm calling '''quartertone composition''', which is composition based on the 24-form. It offers both familiarity to 12edo and all the intervals that are the most alien. The most important generator chains are the ones that split 4/3 in half (semiquartal, aesthetically my favorite, see [[Intergan]]) and split 3/2 in half (mosh or dicoid). Together, they can be analyzed as a diatonic generator chain that can be deviated from by either a semifourth or hemififth. Quartertone tunings may have one chain or both.<br /> <br /> I fond it very useful to examine tertian functions by stacking different combinations inframinor/minor/neutral/major/ultramajor thirds.<br /> <br /> TODO: List<br /> <br /> === Scale theory ===<br /> <br /> One of the best ways I've found to introduce quartertone intervals into the diatonic framework is by altering either B or F (assuming all natural notes) by a quartertone. This results in a scale with only one fewer perfect fifth, since the other fifth containing the altered note is diminished. This gives four options:<br /> * A Bd C D E F G A (-&gt; mosh)<br /> * A Bt C D E F G A (-&gt; sq)<br /> * A B C D E Fd G A (-&gt; sq)<br /> * A B C D E Ft G A (-&gt; mosh)<br /> These introduce &quot;dimimajor&quot;, &quot;diminimor&quot;, and neutral chords for the mosh-leaning scales and &quot;small/large subdim&quot; and inframinor/ultramajor chords for the semiquartal-leaning scales.<br /> <br /> === Scale tempering ===<br /> <br /> Having two separate generator chains is ideal, but it can be inconvenient. This can be fixed by using a half-octave period instead of an octave, where the semitwelfth and hemififth differ by half an octave. {{adv|I would argue that the most important quartertone temperament overall is '''Island Rastmic''', a half-octave 24&amp;34 2.3.11.13/5 subgroup temperament that splits 4/3 into two 15/13s and 3/2 into two 11/9s. Like in Intergan, all the quartertone intervals don't leave much room to introduce comma steps if prime 5 is desired, so a mild Meantone tempering is the best option. In addition, I find this tuning range around Mohajira to have the most pleasant-sounding neutral triads. 17/12 may be equated to half an octave and 19/17 to 9/8 as in Intergan. The final result is 2.3.5.11.13.17.19 24&amp;62.}}<br /> <br /> In the accompanying generator table, diatonic intervals are aligned, while adjacent quartertone intervals always differ by 600¢. Primes are highlighted. {{adv|The top half of the table shows an optional 7 (the simplest mapping is in the hemififth chain due to the semitwelfth already being so close to 7/4, but it's in the wrong direction) and an alternate sharp 11, 17, and 19 all found in Intergan, but none of these were included in 2.3.5.11.13.17.19 24&amp;62. The flat 17 and 19 are shown in parenthesis because they differ from diatonic intervals by half an octave.}} My proposed interval names are extrapolated from existing names in order to be as unambiguous as possible.<br /> <br /> This way of displaying intervals in a half-octave temperament, having two full-octave generators that differ by half an octave, is mostly similar to the standard way. The columns don't correspond to which of the two periods the intervals are in, but it's simple enough to guess because the smaller one is first period and the larger one is second period. The major difference is that intervals an even number of generators from unison only show one interval in one period. For temperament reasons, I included some of the ones not shown in parenthesis. This matters a lot because it ignores an important Diaschismic equivalence: sqrt(2) / (9/8) ≈ 5/4.<br /> <br /> This may result in a new way of generating scales. An Aeolian diatonic scale ranges from -4 to 2 generators. This temperament divides the generator in half, so take all intervals in the table from -8 to 4 generators, ignoring anything in parenthesis. The resulting scale has 19 notes: 55 153 196 251 306 349 447 502 557 655 698 753 808 851 949 1004 1047 1157 1200. No more notes can be added without introducing some extremely small steps to the scale, which would be tempered out in 24edo. The scale has four unique step sizes: 43 55 98 110. There is only one step of 110¢, which can be removed by replacing 1047 or 1157 (the most extreme two intervals unique to the 349¢ column) with 1059 or 1145 (the next two intervals unique 949¢ column). This results in the two chiralities of 5L9m5s. Sharpening the two generators to 950¢ and 350¢ results in 5L14s, and further sharpening them flips m and s to 5L5m9s.<br /> <br /> === Melodic intervals ===<br /> <br /> The diatonic half of quartertone composition does not need an explanation. Semiquartal and dicoid have their own compositional practices, much of which can be inherited from diasem and blackdye respectively due to being degenerate cases. What's left to explain is how this all fits together.<br /> * The quartertone has function similar to a wide aberrisma and a narrow semitone, since semiquartal is just diasem with the two equated. It's useful for altering intervals like an aberrisma and has a distinctive &quot;metallic&quot; sound as a semitone.<br /> * The neutral second is best known for occurring in dicoid and antidiatonic, which it can take most of its function from. It has a distinctive &quot;sour&quot; sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.<br /> * The semifourth is most useful as an inframinor third, but it also works as an ultramajor second. It pairs best with wider minor thirds which make it sound more like a second in comparison. It's useful for shrinking the semitone between the second and third without shrinking the third.<br /> <br /> == Tertian categorization ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Thirds categorized as a fraction of a diatonic fifth.<br /> !Fraction f<br /> !*7/12 =<br /> !4\7 (¢)<br /> !7\12 (¢)<br /> !3/2 (¢)<br /> !3\5 (¢)<br /> !Lower bound of:<br /> |-<br /> !1/3<br /> |7/36<br /> |228.6<br /> |233.3<br /> |234.0<br /> |240.0<br /> |&quot;Horiminor&quot;<br /> |-<br /> !12/35<br /> |1/5<br /> |235.1<br /> |240.0<br /> |240.7<br /> |246.9<br /> |Inframinor<br /> |-<br /> !18/49<br /> |3/14<br /> |251.9<br /> |257.1<br /> |257.9<br /> |264.5<br /> |Subminor<br /> |-<br /> !11/28<br /> |11/48<br /> |269.4<br /> |275.0<br /> |275.8<br /> |282.9<br /> |Neominor<br /> |-<br /> !59/140<br /> |59/240<br /> |289.0<br /> |295.0<br /> |295.8<br /> |303.4<br /> |Grave Minor<br /> |-<br /> !61/140<br /> |61/240<br /> |298.8<br /> |305.0<br /> |305.9<br /> |313.7<br /> |Classic Minor<br /> |-<br /> !23/49<br /> |23/84<br /> |321.9<br /> |328.6<br /> |329.5<br /> |338.0<br /> |Supraminor<br /> |-<br /> !24/49<br /> |2/7<br /> |335.9<br /> |342.9<br /> |343.8<br /> |352.7<br /> |Neutral<br /> |-<br /> !25/49<br /> |25/84<br /> |349.9<br /> |357.1<br /> |358.1<br /> |367.3<br /> |Submajor<br /> |-<br /> !26/49<br /> |13/42<br /> |363.8<br /> |371.4<br /> |372.5<br /> |382.0<br /> |Classic Major<br /> |-<br /> !79/140<br /> |79/240<br /> |386.9<br /> |395.0<br /> |396.1<br /> |406.3<br /> |Acute Major<br /> |-<br /> !81/140<br /> |81/240<br /> |396.7<br /> |405.0<br /> |406.1<br /> |416.6<br /> |Neomajor<br /> |-<br /> !17/28<br /> |17/48<br /> |416.3<br /> |425.0<br /> |426.2<br /> |437.1<br /> |Supermajor<br /> |-<br /> !31/49<br /> |31/84<br /> |433.8<br /> |442.9<br /> |444.1<br /> |455.5<br /> |Ultramajor<br /> |-<br /> !23/35<br /> |23/60<br /> |450.6<br /> |460.0<br /> |461.3<br /> |473.1<br /> |&quot;Horimajor&quot;<br /> |-<br /> !2/3<br /> |7/18<br /> |457.1<br /> |466.7<br /> |468.0<br /> |480.0<br /> |[end of range]<br /> |}<br /> &lt;/div&gt;<br /> The accompanying table shows my own way of categorizing triads and tetrads bounded by ~3/2, with the third being a logarithmic fraction of the fifth. {{adv|Note that the thirds are presented as a logarithmic fraction of the fifth, which scales the major and minor thirds by the same ratio to fit the fifth. Delta-rational logarithmically scales the lower third in the triad slightly more than the upper third.}} This is not an exhaustive list of qualities, but rather groups of my major use cases.<br /> <br /> * I've proposed the terms &quot;horiminor&quot; and &quot;horimajor&quot; after the same root in the word &quot;horizon&quot;. These are the chords that are technically possible to hear as containing thirds, but it requires special care in the melody to make it convincing, such as splitting the horiminor third. I find this class especially useful in 10edo, which is often considered to only have neutral chords.<br /> * Inframinor and ultramajor chords share the metallic quality with the quartertone, as mentioned in the previous section. They are similar in function to the subminor and supermajor chords based on 6:7:9, but have a harsher and exaggerated sound, usually interpreted as based on 10:13:15.<br /> * Subminor and supermajor chords are in the vicinity of 6:7:9 and 14:18:21. Subminor especially has a pure sound like 4:5:6, but these chords may sound too different from Western tuning to be desirable in many cases.<br /> * Neominor and neomajor are used in this case to refer to any thirds not in the concordance wells of 6:7:9 or 4:5:6. This gives them a sound that I would describe as impure but even. They're the most versatile for transcribing modern 12edo music while remaining somewhat xenharmonic. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.<br /> * Grave minor and acute major chords are approximately the ones found in 12edo, around 16:19:24. I consider these to be a separate category less often than any other, but this presence of this otonal chord justifies them. A grave minor third is also characteristic of [[Hibernal]].<br /> * Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15, so they may also be called pental. They are considered to be the default in xenharmonic circles, and have been used in Western music for hundreds of years at least, largely falling out of consideration once 12edo became the standard. I think it's always good to have these to some extent because they provide a consonance that can anchor everything else.<br /> * Supraminor and submajor chords lean towards, but I find them to work better as a consonance because they still resemble minor and major functions. They're usually interpreted as based on 14:17:21.<br /> * Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered largely concordant, between 24edo and 31edo. This is where two slightly sharp 11/9s stack to make a slightly flat 3/2. In this range, they sound like ambiguous minor/major chords, which may be useful for modulation or when unsure of which quality is better for a given chord. More broadly, they function as an extremely versatile mild dissonance. People tend to see them as a cross between major and minor, but I actually see them as a cross between major, minor, diminished, and augmented. They're weird as a consonance in a quasi-diatonic context, not because of the complexity of 18:22:27, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.<br /> ** I call this technique a &quot;neutral substitution&quot;. It involves replacing a tertian chord with a neutral one in order to modulate. The easiest way to do this is line up a neutral chord with the root or fifth of where a diminished chord would be, then use whichever of the two is changed to alter the key.<br /> <br /> === Grave fifth chords and mainline fifths ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Successive fifths from 22/15 to 40/27,&lt;br&gt;thirds between logarithmic 12/35 and 23/35 of the fifth<br /> ! style=&quot;border-right: double;&quot; |D<br /> ! colspan=&quot;5&quot; style=&quot;border-right: double;&quot; |N<br /> !&lt;3/7f<br /> !&lt;1/2f<br /> !&gt;1/2f<br /> !&gt;4/7f<br /> ! style=&quot;border-right: double;&quot; |Fifth<br /> ! colspan=&quot;4&quot; |Fifth Complement<br /> |-<br /> ! style=&quot;border-right: double;&quot; |15<br /> |<br /> |18<br /> |<br /> | style=&quot;border-right: double;&quot; |19<br /> | style=&quot;border-right: double;&quot; |22<br /> |<br /> |315.6<br /> |<br /> | style=&quot;border-right: double;&quot; |409.2<br /> | style=&quot;border-right: double;&quot; |663.0<br /> |<br /> |347.4<br /> |<br /> |253.8<br /> |-<br /> ! style=&quot;border-right: double;&quot; |17<br /> |<br /> | class=&quot;thl&quot; |20<br /> |21<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |25<br /> |<br /> | class=&quot;thl&quot; |281.4<br /> |365.8<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |667.7<br /> |<br /> | class=&quot;thl&quot; |386.3<br /> |301.8<br /> |<br /> |-<br /> ! style=&quot;border-right: double;&quot; |19<br /> |22<br /> |23<br /> |<br /> | style=&quot;border-right: double;&quot; |24<br /> | style=&quot;border-right: double;&quot; |28<br /> |253.8<br /> |330.8<br /> |<br /> | style=&quot;border-right: double;&quot; |404.4<br /> | style=&quot;border-right: double;&quot; |671.3<br /> |417.5<br /> |340.6<br /> |<br /> |266.9<br /> |-<br /> ! style=&quot;border-right: double;&quot; |21<br /> |24<br /> | class=&quot;thl&quot; |25<br /> | class=&quot;thl&quot; |26<br /> | style=&quot;border-right: double;&quot; |27<br /> | style=&quot;border-right: double;&quot; |31<br /> |231.2<br /> | class=&quot;thl&quot; |301.8<br /> | class=&quot;thl&quot; |369.7<br /> | style=&quot;border-right: double;&quot; |435.1<br /> | style=&quot;border-right: double;&quot; |674.3<br /> |443.1<br /> | class=&quot;thl&quot; |372.4<br /> | class=&quot;thl&quot; |304.5<br /> |239.2<br /> |-<br /> ! style=&quot;border-right: double;&quot; |23<br /> |<br /> |27<br /> |28<br /> | style=&quot;border-right: double;&quot; |29<br /> | style=&quot;border-right: double;&quot; |34<br /> |<br /> |277.6<br /> |340.6<br /> | style=&quot;border-right: double;&quot; |401.3<br /> | style=&quot;border-right: double;&quot; |676.7<br /> |<br /> |399.1<br /> |336.1<br /> |275.4<br /> |-<br /> ! style=&quot;border-right: double;&quot; |25<br /> |29<br /> |30<br /> |31<br /> | style=&quot;border-right: double;&quot; |32<br /> | style=&quot;border-right: double;&quot; |37<br /> |256.9<br /> |315.6<br /> |372.4<br /> | style=&quot;border-right: double;&quot; |427.4<br /> | style=&quot;border-right: double;&quot; |678.7<br /> |421.8<br /> |363.1<br /> |306.3<br /> |251.3<br /> |-<br /> ! style=&quot;border-right: double;&quot; |27<br /> |31<br /> | class=&quot;thl&quot; |32<br /> |33<br /> | style=&quot;border-right: double;&quot; |34<br /> | style=&quot;border-right: double;&quot; |40<br /> |239.2<br /> | class=&quot;thl&quot; |294.1<br /> |347.4<br /> | style=&quot;border-right: double;&quot; |399.1<br /> | style=&quot;border-right: double;&quot; |680.4<br /> |441.3<br /> | class=&quot;thl&quot; |386.3<br /> |333.0<br /> |281.4<br /> |}<br /> &lt;/div&gt;<br /> I use a similar system for antidiatonic tertian harmony, but there's no set JI approximation for a flat fifth, so I use multiple to get the simplest ratios possible. [[Pentagoth]] equates many similar intervals in this table.<br /> <br /> I refer to the range from about 23\40 to 25\42 as '''mainline fifths'''. The full diatonic range includes other fifths that generate scales I have found over time to resemble the usual understanding of diatonic less than they resemble 7 or 5edo, and resemble 3/2 less than sqrt(20/9) and sqrt(16/7). The 25\42 boundary is clearer to me because I prefer very hard scales to very soft ones and thus I have more experience. But conveniently, these are the points where an alternate diatonic fifth can be generated at a low complexity. Analogously, I could refer to the range from about 11\20 to 17\30 as '''mainline antidiatonic fifths''' based on my personal experience.<br /> <br /> I've found 25 and 39edo to be the two regions of the armotonic tuning range that sound better than others. {{adv|29edo sounds more pure than both, and is the basis for an excellent 29&amp;67 straddle temperament I've called Gravel, but it involves the 15:19:22 chord with a fifth at my limit for resembling 3/2 which does not generate armotonic. 2.15.11.19 9&amp;56 also generates this, and the subgroup can be extended to 2.3.5.11.19 while hardly changing the edo list. 11:13:16 and 13:16:19 are proper subfifth triads and are not counted in the range. They are also nearly retroversions of each other.}}<br /> <br /> 25edo represents the two otonal chords on the table with 5/4, with flatter fifths like 34 and 43edo being better for 17:20:25 and sharper fifths like 41 and 57edo being better for 27:32:40. Both of these are found in Pentagoth, so I propose calling this range '''pentagothic'''. 43edo is especially notable for how well it also approximates 17:21:25.<br /> <br /> 39edo represents 21:26:31, which is notably a +1+1 chord, and 21:25:31, very close to its retroversion. I propose calling this range '''mavilic''' after Erv Wilson's original Meta-Mavila, the &quot;proper&quot; tuning range for Mavila. {{adv|Mavila possibly has more variants of its name than any other temperament. I'm genuinely surprised that Mavilic isn't a temperament name.}} [[Pentagoth#Otomavila|Otomavila]] is my temperament using these chords.<br /> <br /> For the most part, antidiatonic tertian harmony works like diatonic tertian harmony, but with the reversed qualities of the antidiatonic scale. The pentagothic neutral triad generates Cohemimabila temperament and has a similar function to the neutral triad explained above.<br /> <br /> === Quasi-diatonic scales ===<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Blackdye and diasem JI tunings, sorted from flat to sharp minor third, denominators up to 20<br /> !Tertian Triad<br /> !16:21:24<br /> !20:23:30<br /> !10:13:15<br /> !6:7:9&lt;br&gt;(14:18:21)<br /> !18:23:27<br /> !16:19:24<br /> !4:5:6&lt;br&gt;(10:12:15)<br /> !14:17:21<br /> !18:22:27<br /> |-<br /> |Subgroup<br /> |2.3.7<br /> |2.3.23/5<br /> |2.3.13/5<br /> |2.3.7<br /> |2.3.23<br /> |2.3.19<br /> |2.3.5<br /> |2.3.17/7<br /> |2.3.11<br /> |-<br /> |Name<br /> |&quot;Septal&quot;<br /> |?<br /> |?<br /> |&quot;Septal&quot;<br /> |&quot;Eridial&quot;<br /> |&quot;Novemdecal&quot;<br /> |Pental<br /> |?<br /> |&quot;Undecal&quot;<br /> |-<br /> |Aberrisma<br /> |28/27<br /> |640/621<br /> |416/405<br /> |64/63<br /> |736/729<br /> |513/512<br /> |81/80<br /> |459/448<br /> |33/32<br /> |-<br /> |Quasi-&lt;br&gt;Aeolian<br /> |9/8&lt;br&gt;8/7&lt;br&gt;9/7&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;12/7&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;23/20&lt;br&gt;207/160&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;23/15&lt;br&gt;69/40&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;15/13&lt;br&gt;135/104&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;20/13&lt;br&gt;45/26&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;7/6&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;14/9&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |513/512&lt;br&gt;9/8&lt;br&gt;19/16&lt;br&gt;4/3&lt;br&gt;171/128&lt;br&gt;3/2&lt;br&gt;19/12&lt;br&gt;16/9&lt;br&gt;57/32&lt;br&gt;2/1<br /> |81/80&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |459/448&lt;br&gt;9/8&lt;br&gt;17/14&lt;br&gt;4/3&lt;br&gt;153/112&lt;br&gt;3/2&lt;br&gt;34/21&lt;br&gt;16/9&lt;br&gt;51/28&lt;br&gt;2/1<br /> |33/32&lt;br&gt;9/8&lt;br&gt;11/9&lt;br&gt;4/3&lt;br&gt;11/8&lt;br&gt;3/2&lt;br&gt;33/20&lt;br&gt;16/9&lt;br&gt;11/6&lt;br&gt;2/1<br /> |-<br /> |Similar&lt;br&gt;Unlisted&lt;br&gt;Aberrismas<br /> | colspan=&quot;2&quot; |17/13: 1088/1053&lt;br&gt;(swap m/s)<br /> |17/11: 1408/1377&lt;br&gt;19/11: 304/297<br /> |<br /> |25: 2048/2025&lt;br&gt;11/7: 896/891&lt;br&gt;13/11: 352/351&lt;br&gt;17/5: 136/135&lt;br&gt;23/13: 208/207<br /> |25/7: 225/224&lt;br&gt;17: 4131/4096&lt;br&gt;19/5: 1216/1215&lt;br&gt;(diasem)<br /> |<br /> |23/7: 189/184&lt;br&gt;23/19: 621/608<br /> |13: 1053/1024<br /> |}<br /> The diatonic scale and its tertian harmony are the framework of all Western music. It makes sense that using the lowest prime as the period and second lowest prime as the generator would produce a versatile scale, and the fact that it clusters around 12edo rather than 3edo in the case of prime 5 produces more useful step sizes. Modifying this into an aberrismic scale containing the simplest tertian triad 4:5:6 and its retroversion in as many places as possible produces blackdye, and doing the same with 6:7:9 produces diasem. These are the main scales I use to represent my take on Western music: melodically interesting, harmonically pure, and not far from diatonic. However, there are other tunings of them that are useful in different places. These are generalized forms and their aberrismas may be tempered out for a true diatonic scale. 64/63 and 81/80 are common, but 736/729 and especially 513/512 are smaller and make more sense to temper out. If not, inflating them is recommended to keep the melodic significance of the aberrisma.<br /> <br /> '''Eridial''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named after &quot;eridian&quot;, the demonym for Eris, with the -al suffix of subgroups. Eris is probably one of the most goth deities of Ancient Greece and is also associated with the number 23 by Discordians. It targets 18:23:27, a little-known and surprisingly concordant neogothic tertian triad. Many microtonalists contend that 12edo pop music, while historically tracing back to Meantone, tends to sound better in Pythagorean tuning. I can really hear it after comparing different quasi-diatonic options for retuning my melodies. I've taken a particular liking to gentle/neogothic diatonic scales in the past for this reason, but I believe eridian diasem is a further refinement. As with diasem in general, I prefer this with the 3 tuned slightly flat to inflate the aberrisma, although it is especially important in this case because 736/729 has a size of only 17¢. When inflated in tunings such as 43edo, the aberrisma is subtle enough to not disturb the Pythagorean sound much, but impactful in cases like the subminor seventh, which is tuned close to 7/4. Its melody is not particularly xenharmonic, but it's one of my default scales when available because it works for almost everything. It works well as an extension to [[Septimal meantone]]. {{adv|The difference between eridian diasem's analogous 2.3.25 and 2.3.17/5 thirds (e.g. 75/64 and 20/17) is 256/255, also the difference between 16/15 and 17/16, supporting it as one of the most important 17-limit commas.}}<br /> <br /> The default set of qualities to consider is septal and pental due to their simplicity, possibly with a Pythagorean diatonic scale in the middle as found in Aberschismic edos. But I've found that I particularly like shifting both septal and pental towards neutral, to eridian and exaggerated pental. This is probably because it puts both of them about an equal distance from diatonic when using a prime 3 within a few cents of just. Eridian provides a more versatile sound and pental sounds more interesting while still working about the same. In some edos that do this such as 48 and 86, the pental tuning is Negri, meaning that semiquartal is also available.<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Cursed Eridian Scales<br /> !Standard&lt;br&gt;(23/18)(27/23)<br /> !Wolf → Perfect&lt;br&gt;(23/18)(32/27)<br /> !the other one&lt;br&gt;(81/64)(27/23)<br /> |-<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |398688256/387420489&lt;br&gt;67712/59049&lt;br&gt;32/27&lt;br&gt;243/184&lt;br&gt;2166784/1594323&lt;br&gt;368/243&lt;br&gt;36/23&lt;br&gt;59049/33856&lt;br&gt;11776/6561&lt;br&gt;2/1<br /> |4782969/4333568&lt;br&gt;27/23&lt;br&gt;129140163/99672064&lt;br&gt;2944/2187&lt;br&gt;2187/1472&lt;br&gt;128/81&lt;br&gt;59049/33856&lt;br&gt;8667136/4782969&lt;br&gt;2/1<br /> |}<br /> &lt;/div&gt;<br /> 27/23 is close enough to 32/27 that it can be replaced and result in another 2.3.23 blackdye scale with 32/27*23/18 = 368/243 triads. The fifth is the same as the existing Eridian wolf fifth, which is barely diatonic in JI. I might call this '''hiberidial''' because it applies the principle of [[Hibernal]] to the eridian subgroup. 82edo is the near-perfect 2.3.23 tuning, but 20 and 37edo have a close approximation of the scale despite not being good for the subgroup. Switching thirds to 32/27 or 81/64 to remain in the subgroup with a wolf fifth as the main one can be applied to other scales in the table.<br /> <br /> I have a lot more to write about how blackdye and diasem work harmonically.<br /> <br /> == Aberrismic details ==<br /> <br /> Note that this section is incomplete.<br /> <br /> As described in the [[Monarch]] article, I like to tune pental blackdye (3sA Aeolian: tempered 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1) with a sharp 3 and thus inflated 81/80. When extreme, more interesting intervals show up along with new problems. The first is what is seen in 11-limit Superpyth and Porcupine (55/54), turning 27/20 into 11/8, although it could be interpreted as any other superfourth. I like this interval a lot because the 1/1-9/8-6/5-27/20 tetrachord is a natural-sounding stretched version of the diatonic minor tetrachord. In fact, it may sound better melodically to use the superfourth rather than the perfect fourth to avoid the very deflated 10/9 between 6/5 and 4/3. It would be beneficial to have a triad on 27/20 to replace the one on 4/3, and this makes use of the 81/80 in the next octave, the least harmonically useful note in the scale. 32/27 is available as the more conventional quasi-diatonic third, but also 320/243, one of the least recognizeable intervals that all correspond to septal intervals in Aberschismic temperament (5120/5103). 320/243 is 476.539¢ in JI and 729/640 is 225.416¢, but these approach each other with a sharper 3. Most notably in this tuning range, tempering out 2187/2240 equates them to 9/7 and 7/6, different from the same ratios that would normally be generated by Archy. Interpreting 320/243 as 14/11 allows for 8:11:14. The exact tempering is irrelevant as long as they're some sort of supermajor and subminor thirds.<br /> <br /> The size of the aberrisma presents other issues.<br /> <br /> == Song structure ==<br /> <br /> This is my system for categorizing the songs I write by structure. Something similar can be applied to other people's music, but the standards will be different due to the unusual complexity of my music.<br /> <br /> * '''straightforward''': The simplest type, being in a similar style throughout. Always works.<br /> * '''opposing sections''': Two straightforward parts that are very different.<br /> * '''medium''': Somewhat straightforward, somewhat disjointed.<br /> * '''episodic''': Containing many different sections with themes and a progression throughout the song. The most desirable when done well and in moderation.<br /> * '''disjointed''': Containing many different sections with no clear pattern. The least desirable.<br /> <br /> I've noticed that I write more &quot;disjointed&quot; songs than I would like, and fewer &quot;straightforward&quot; and &quot;episodic&quot;. This system is intended to keep track of what I'm doing and help me work toward a better result.<br /> <br /> {| class=&quot;wikitable sortable&quot;<br /> |+Song Structure Types of The Gralbums<br /> !Track<br /> !Title<br /> !Type<br /> |-<br /> |g1t1<br /> |The Lake Reflects a Black Sky<br /> |'''disjointed'''<br /> |-<br /> |g1t2<br /> |The Art of It All<br /> |'''episodic'''<br /> |-<br /> |g1t3<br /> |Rail Twinge<br /> |'''straightforward'''<br /> |-<br /> |g1t4<br /> |Back Stalk<br /> |'''medium'''<br /> |-<br /> |g1t5<br /> |Superior Intermedial<br /> |'''episodic'''<br /> |-<br /> |g1t6<br /> |Snowboard Index<br /> |'''straightforward'''<br /> |-<br /> |g1t7<br /> |Winter's Mortal Hope<br /> |'''medium'''<br /> |-<br /> |g1t8<br /> |Transpiration<br /> |'''opposing sections'''<br /> |-<br /> |g1t9<br /> |Revelation of Your Forever<br /> |'''medium'''<br /> |-<br /> |g2t1<br /> |The Life Unreachable<br /> |'''disjointed'''<br /> |-<br /> |g2t2<br /> |Monolithium<br /> |'''opposing sections'''<br /> |-<br /> |g2t3<br /> |Nocturne Paranoia<br /> |'''disjointed'''<br /> |-<br /> |g2t4<br /> |Glimmer Extrication<br /> |'''episodic'''<br /> |-<br /> |g2t5<br /> |Not This Time<br /> |'''straightforward'''<br /> |-<br /> |g2t6<br /> |Sakura Blade Minivan<br /> |'''medium'''<br /> |-<br /> |g2t7<br /> |Resolute Prelude<br /> |'''straightforward'''<br /> |-<br /> |g2t8<br /> |Life and Limb<br /> |'''straightforward'''<br /> |-<br /> |g2t9<br /> |Residual Soliloquy<br /> |'''disjointed'''<br /> |}</div>

Ground https://xenreference.com/wiki/index.php?title=Aberrisma&diff=8525 2026-10-28T02:42:04Z

<p>Ground: </p> <hr /> <div>{{Proposed}}<br /> <br /> An '''aberrisma''' is an interval representing some comma as an additional type of melodic step smaller than a semitone. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Although they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas about the size of a diesis, the aberrisma is a melodic function and not an interval size. Larger and smaller sizes can function as aberrismas depending on the scale tuning. '''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas. See [[straddle primes]] for aberrismas equated to 1/1 without being tempered out.<br /> <br /> Ground's '''aberrismic hypothesis''' is that intervals an audible distance apart can be used together in the same scale framework while merely registering as detunings of the same interval. Straddle theory applies the hypothesis to definite ratios, whereas in the original aberrismic scales, it's applied to diatonic scale degrees. Commas have been described as a type of unison in other contexts, but rarely as distinct steps vital to the structure of a scale, or as intervals that are represented by the same ratio. An aberrisma is categorically a unison but doesn't sound like one when isolated, which represents both observed RTT commas and the difference between straddled intervals; it is a unison widened into a melodic step. Aberrismic theory is a collection of the effects of applying the aberrismic hypothesis to scale theory and RTT.<br /> <br /> The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals. Aberrismas may also appear in MOS scales, such as Garibaldi[17]. Superpyth[12] is a good example of the distinction between interval function and size. The diatonic semitone, a deflated 256/243, is equated to the inflated 81/80. Despite being the same size, the former represents a narrow semitone, the latter is an aberrisma, and a stack of the two is a classic semitone.<br /> <br /> == The &quot;why&quot; of aberrismic theory ==<br /> === Adding comma steps to Pyth diatonic ===<br /> 81/80 and 64/63 are the canonical choices for 2.3.5 and 2.3.7 formal commas for modifications to Pythagorean (2.3) JI ratios — 81/80 separates 5/4 from 81/64, 10/9 from 9/8, and so on; and 64/63 separates 7/4 from 16/9, 9/7 from 81/64, and so on. Aberrismic theory asks: what if we made 81/80 or 64/63 into scale steps, so that we can use both of the intervals separated by the formal comma in one scale? Indeed, doing this to 2.3 diatonic to make a ternary scale, in a way that best preserves the fifth-generated structure, results in (2.3.5) blackdye and (2.3.7) diasem, respectively.<br /> <br /> ==== Example: Blackdye ====<br /> The Zarlino diatonic is chiral — there are two different, equally valid second degrees of the Ionian mode. Both are useful, as the sharp one forms a perfect fifth with the fifth degree but a wolf fifth with the sixth degree, and the flat one forms a perfect fifth with the sixth degree but a wolf fifth with the fifth degree.<br /> [[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]<br /> One way to make it achiral is to temper out 81/80, the difference between these two steps, resulting in [[Meantone]] diatonic; intuitively this requires flattening the fifth and sharpening the sixth somewhat. However, an alternative way, if you wish to observe 81/80 or to use just intonation, is to include both varieties of whole tone over the unison, treating 81/80 as a melodic step between them. This can be thought of as dividing up a 9/8 into a 10/9 and an 81/80. It is then reasonable to extend this action to all instances of 9/8 in the scale (as, for instance, the Didymic diatonic has 27/16 as opposed to 5/3). The result is a 10-note ternary scale called '''blackdye'''.<br /> <br /> === Stylistic compositional features for aberrismic theory ===<br /> This section will attempt to describe the principles and stylistic features of a specific style of music that justify using aberrismic scales in a composition. It is not an attempt to present aberrismic theory as absolute truth.<br /> * A style of music that is melodic and heavy in modulations benefits from<br /> ** Multiple step sizes for melodic interest, for example diesis-sized steps that are below conventional semitones, specifically ones large enough to be melodically distinct but small enough to represent intonational changes.<br /> ** A set of modulatory intervals, including fifths.<br /> ** A system that allows unlimited modulation. <br /> * It is widely agreed that lower primes are more robust to detuning. Hence for approximating JI with edos, we use lower prime temperaments, and which also represent either 81/80 or 64/63 steps for greater accuracy.<br /> The above suggests temperaments, in particular edos, that use tempered lower primes, and edos large enough to have small diesis-sized steps. In the context of fifth-based modulation, scales also benefit from having offset arcs of fifths. One simple way to have this is to detemper MOS scales into ternary scales with an additional smaller melodic step size, which have a generator arc with fifths or a generator arc that stacks to fifths via a detempered generator chain.<br /> <br /> {{UserTag|g_|Ground|7766ff|I assert that ~40¢ is the ideal point between &quot;categorical unison&quot; and &quot;clear step&quot;; context pushes it one way or the other, like in 27edo Superpyth diatonic and blackdye.}}<br /> <br /> == List of aberrismic scales ==<br /> * {{Adv|&quot;GS(...)[n]&quot; is [[generator sequence]] notation.}}<br /> * {{adv|&quot;subst ax(bycz)&quot; denotes [[MOS substitution]].}}<br /> * {{adv|&quot;Almost&quot; a cross-set means that one or two notes may be missing from the full cross-set and one note may have been added. Exact cross-sets are italicized.}}<br /> * {{adv|Under &quot;Patterns&quot;, &quot;C&quot; is [[achiral]], and &quot;R&quot; and &quot;L&quot; denote two [[chiral]]ities of a chiral pair.}}<br /> === Quasi-diatonic aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !|pinedye / dia1s&lt;br/&gt;(5L2m1s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m1s)}}}}<br /> ||1sC: LLsLLmLm&lt;br/&gt;1sR: LLmLLmLs&lt;br/&gt;1sL: LLmLLsLm<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 27/25, 81/80]<br /> ||{{adv|GS(3/2)[3] and GS(10/9)[3]}}<br /> ||1sC has 4 fifths and 1sR/1sL have 5<br /> |-<br /> !class=&quot;thl&quot;|diasem / dia2s&lt;br/&gt;(5L2m2s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m2s)}}}}<br /> ||2sR: LmLsLmLsL&lt;br/&gt;2sL: LsLmLsLmL<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 28/27, 64/63]<br /> ||{{adv|GS(3/2)[5] and 7/6}}<br /> ||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.<br /> ''See also: [[Chthonic harmony#Diasem|Diasem]]''<br /> |-<br /> !class=&quot;thl&quot;|blackdye / dia3s&lt;br/&gt;(5L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m3s)}}}}<br /> ||sLmLsLmLsL<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 16/15, 81/80]<br /> ||{{adv|''GS(3/2)[5] and 10/9''}}<br /> ||Two interleaved 3-limit pentatonics<br /> ''See also: [[10-form#Blackdye|Blackdye]]''<br /> |-<br /> !|diaslen / dia4s&lt;br/&gt;(5L2m4s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m4s)}}}}<br /> ||4sC: LmLsLsLmLss&lt;br/&gt;4sR: LsLmLsLsLms&lt;br/&gt;4sL: LsLsLmLsLsm<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 49/48, 64/63]<br /> ||{{adv|GS(3/2)[4] and GS(8/7)[3]}}<br /> ||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]].&lt;br/&gt;{{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}<br /> |-<br /> !|diachrome / chromedye / dia5s&lt;br/&gt;(5L2m5s)<br /> ||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}}&lt;br/&gt;5sR: LmsLsLsLmsLs&lt;br/&gt;5sL: LsLsLsmLsLsm<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 256/243, 81/80]<br /> ||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}<br /> ||Fifth-generated but with a 6-step offset<br /> |-<br /> !|whitedye / dia7s&lt;br/&gt;(5L2m7s)&lt;br/&gt;{{adv|{{nowrap|subst 7s(5L2m)}}}}<br /> ||LsLsLsmsLsLsms<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 28/27, 81/80]<br /> ||{{adv|''GS(3/2)[7] and 81/80''}}<br /> ||Two interleaved diatonics<br /> |}<br /> See [[Ground's composition theory#Quasi-diatonic scales]] for more tunings of diasem and blackdye.<br /> <br /> === Other aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI/[[erac]] interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !class=&quot;thl&quot;|[[penslen]] / slen5m&lt;br/&gt;(5L5m6s)&lt;br/&gt;{{adv|{{nowrap|subst 6s(5L5m)}}}}<br /> ||LmsLmsLsmLsmLsms<br /> ||2.3.5.7.11[41 &amp; 46]&lt;br/&gt;[L, m, s] = [12/11, 33/32, 64/63]<br /> ||{{adv|''GS(8/7)[8] and 11/8''}}<br /> || Has two aberrisma sizes, s and m.<br /> |-<br /> !|prejubilic / citro3s&lt;br/&gt;(4L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 2m(4L3s)}}}}<br /> ||C: LLsmLsLms&lt;br/&gt;R: LmLsLmsLs&lt;br/&gt;L: LmLsLsmLs<br /> ||2.5.7&lt;br/&gt;[L, m, s] = [28/25, 35/32, 50/49]<br /> ||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}<br /> ||Didacus tempering makes L = m + s.<br /> |-<br /> !|arm5s&lt;br/&gt;(7L2m5s)&lt;br/&gt;{{adv|{{nowrap|subst 7L(2m5s)}}}}<br /> ||LmLsLsLmLsLsLs<br /> ||2.x&lt;3.5.7.11.13[37edo] (4:2:1)<br /> ||{{adv|''GS(&lt;&lt;3/2)[7] and 14/13''}}<br /> ||An interleaving of two antidiatonic scales.<br /> |-<br /> !|mosh3s&lt;br/&gt;(3L4m3s)&lt;br/&gt;{{adv|{{nowrap|subst 4m(3L3s)}}}}<br /> ||LmsLmsmLsm<br /> ||2.x&lt;3.7.11.13[37edo] (5:4:2)<br /> ||{{adv|''GS(16/13)[5] and 11/8''}}<br /> ||<br /> |-<br /> !|smi2s&lt;br/&gt;(4L3m2s)&lt;br/&gt;{{adv|{{nowrap|subst 2s(4L3m)}}}}<br /> ||C: LLmsLmLsm&lt;br/&gt;R: LmLmsLmLs&lt;br/&gt;L: LmLsLmLsm<br /> ||2.9.7.11.17[46edo] (8:4:1)<br /> ||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}<br /> ||<br /> |}<br /> <br /> === Systematic naming ===<br /> Basic systematic names for aberrismic scales are of the form<br /> <br /> [mos_prefix]n[added_step_size] (e.g. dia2s for diasem).<br /> <br /> Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.<br /> <br /> This is subject to change as aberrismic theory nomenclature is updated in the future.<br /> <br /> == Notation ==<br /> The [[generator gronut]] is an aberrismic extension of the circle of fifths.<br /> <br /> === Quasi-diatonic ===<br /> Quasi-diatonic aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.<br /> <br /> Aberrismic/straddle-prime accidentals:<br /> * Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. &lt;!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. --&gt;<br /> * Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).<br /> * [[Erac|Eracs]] may be used to represent the inflating of commas typically found in aberrismic scales, although they represent a specific unit of error.<br /> <br /> == Aberrismic theory and RTT ==<br /> Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under Ground's use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to &lt;!--simultaneously achieve sufficient accuracy to LCJI and --&gt;improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.<br /> <br /> At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.<br /> <br /> There are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.<br /> <br /> === Example: Blackdye ===<br /> The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.<br /> * ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.<br /> * [[Ultrapyth]], 2.3.5.7.11.13[32 &amp; 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. Ground once termed the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.<br /> <br /> {| class=&quot;wikitable right-2 right-3 right-4 right-5&quot;<br /> |+ style=&quot;font-size: 105%;&quot; | Blackdye intervals in two temperaments<br /> |-<br /> ! Interval class<br /> ! Sizes<br /> ! Untempered 2.3.5<br /> ! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)<br /> |-<br /> ! 1-step<br /> | s&lt;br/&gt;m&lt;br/&gt;L <br /> | 81/80&lt;br/&gt;16/15&lt;br/&gt;10/9<br /> | 143/140&lt;br/&gt;22/21&lt;br/&gt;160/143<br /> |-<br /> ! 2-step<br /> | L + s&lt;br/&gt;L + m <br /> | 9/8&lt;br/&gt;32/27<br /> | 8/7, 9/8&lt;br/&gt;7/6<br /> |- <br /> ! 3-step<br /> | L + 2s&lt;br/&gt;L + m + s&lt;br/&gt;2L + s&lt;br/&gt;2L + m<br /> | 729/640&lt;br/&gt;6/5&lt;br/&gt;5/4&lt;br/&gt;320/243<br /> | 7/6&lt;br/&gt;13/11&lt;br/&gt;14/11&lt;br/&gt;13/10<br /> |- <br /> ! 4-step<br /> | 2L + 2s&lt;br/&gt;2L + m + s<br /> | 81/64&lt;br/&gt;4/3<br /> | 13/10&lt;br/&gt;4/3<br /> |-<br /> ! 5-step<br /> | 2L + m + 2s&lt;br/&gt;2L + 2m + s&lt;br/&gt;3L + 2s&lt;br/&gt;3L + m + s<br /> | 27/20&lt;br/&gt;64/45&lt;br/&gt;45/32&lt;br/&gt;40/27<br /> | 66/49&lt;br/&gt;11/8&lt;br/&gt;16/11&lt;br/&gt;49/33<br /> |-<br /> ! 6-step<br /> | 3L + m + 2s&lt;br/&gt;3L + 2m + s<br /> | 3/2&lt;br/&gt;128/81<br /> | 3/2&lt;br/&gt;20/13<br /> |- <br /> ! 7-step<br /> | 3L + m + 3s&lt;br/&gt;3L + 2m + 2s&lt;br/&gt;4L + m + 2s&lt;br/&gt;4L + 2m + s<br /> | 243/160&lt;br/&gt;8/5&lt;br/&gt;5/3&lt;br/&gt;1280/729<br /> | 20/13&lt;br/&gt;11/7&lt;br/&gt;22/13&lt;br/&gt;12/7<br /> |- <br /> ! 8-step<br /> | 4L + m + 3s&lt;br/&gt;4L + 2m + 2s<br /> | 27/16&lt;br/&gt;16/9<br /> | 12/7&lt;br/&gt;7/4, 16/9<br /> |-<br /> ! 9-step<br /> | 5L + 2m + s&lt;br/&gt;5L + m + 2s&lt;br/&gt;4L + 2m + 2s<br /> | 9/5&lt;br/&gt;15/8&lt;br/&gt;160/81<br /> | 143/80&lt;br/&gt;21/11&lt;br/&gt;280/143<br /> |}<br /> <br /> == Glossary ==<br /> * '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.<br /> * '''Aberrisma''': The smallest interval region that melodically sounds like a step.<br /> * '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.<br /> * '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.<br /> * '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.<br /> * '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.<br /> <br /> == Extension to higher ranks ==<br /> Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a [[GS]] with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.<br /> <br /> {{UserTag|g_|Ground|7766ff|I propose that we call this type of scale '''bicatenary'''.}}<br /> <br /> However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 41 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28] with a JI tuning of 35/32 : 21/20 : 36/35 : 64/63. [[Ground's composition theory#Interval logic deviation theory|ILD]] is a simple way to access notes outside of a scale with a higher rank.<br /> <br /> A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. Possibly the most common ratio representation of an aberrisma is 36/35, but few ternary scales use it without tempering out 81/80 or 64/63.<br /> <br /> (81/80)^2 = 43¢ is notably in the middle of the usual aberrisma size range as well. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.<br /> <br /> == Compositional examples ==<br /> Some compositional snippets using aberrismic scales:<br /> <br /> * The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]<br /> * A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]<br /> * A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]<br /> <br /> == External links ==<br /> * [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]<br /> {{cat|<br /> Terms<br /> Aberrismic terms<br /> Ternary scales<br /> }}</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8524 2026-10-28T02:34:06Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly.--&gt;]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 of a turn clockwise in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside. This is the shear in the gluing map that produces the torus.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows [[Aberrisma|aberrismic]] pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.<br /> <br /> This process shows that aberrismic and [[Straddle primes|straddle]] theories are fundamentally the same. The aberrismic pitch drift occurs regardless of how much the target generator is straddled.<br /> <br /> See [[{{FULLPAGENAME}}/Mathematics]] for a mathematical treatment.<br /> <br /> {{Cat|<br /> Aberrismic terms<br /> Atypical ratios<br /> Notation<br /> }}</div>

Ground https://xenreference.com/wiki/index.php?title=Straddle_primes&diff=8523 2026-10-28T02:28:41Z

<p>Ground: </p> <hr /> <div>{{proposed}}<br /> <br /> '''Straddle primes''' are functional primes in a temperament that straddle their respective just interval by having at least one flat and one sharp approximation available. Dual-fifth systems are often reasonably interpreted as straddle-3. The ratio between straddle approximations is categorically 1/1, making straddle theory a type of [[Aberrisma|aberrismic theory]].<br /> <br /> The ''straddle ratio'' ''m'':''n'' denotes the ratio of the flatter approximation (''m'') to the sharper approximation (''n''). A straddle ratio of approximately 1:1 is called ''equistraddle''.<br /> <br /> == Motivation ==<br /> <br /> Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:<br /> * [[37edo]] has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.<br /> * [[86edo]] is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.<br /> <br /> With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:<br /> * Straddling primes without using a more accurate approximation in the middle widens the smallest step size a tuning must have to represent the straddle. 37edo would have to be tripled to 111edo in order to get an accurate 3, which is totally different in character and size. Larger tuning systems are less practically convenient and make decisions harder.<br /> * Using the same ratio to describe two slightly different intervals drastically simplifies the ratios produced by stacking it, and less care is needed to ensure that error cancellation works out mathematically.<br /> <br /> Straddle-primes differ significantly from the typical way ratios are understood in RTT. They do not work for everyone, but their high level of flexibility makes more unusual approaches possible.<br /> <br /> {{UserTag|g_|Ground|7766ff|I have a philosophy where I limit edo size to the minimum required to do whatever I'm trying to do, in order to make composition easier. When using a tuning with two diatonic fifths, the smallest interval I actually need always ending up being the difference between them. 3.5¢ is an often mentioned value for the just-noticeable difference (JND), but I find that it can be much larger depending on context. My lower bound for the typical size of an [[Aberrisma|aberrisma]], around 25¢, is the limit where it stops sounding like a clear melodic step in many contexts of the music I write, like a fast melody competing with other instruments, slightly detuned synths, and reverb. This also makes error less noticeable, limiting the effect of extreme concordance. Most of the time in my music, there's hardly a difference between a just 4:5:6 and one tempered with Meantone. In more ideal contexts where notes are more clear, my lower bound is about half that size, not coincidentally the step size of the largest edos I use. Even then, being audible enough as a step is a genuine concern.}}<br /> <br /> == Eracs ==<br /> <br /> '''Eracs''' (short for error accidentals) are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for groups involving straddle primes.<br /> <br /> Eracs provide a more complete picture of error cancellation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's [[JI group]] might be 2.5/3.15.7.11 in standard notation, which is a subgroup of the erac group 2.x3.x5.7.11. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancellation in standard notation creates a very long JI group.<br /> <br /> Example erac groups for edos can be found in [[EDO#List of Edos]].<br /> <br /> === Symbols ===<br /> {| class=&quot;wikitable&quot;<br /> |+Erac Meanings (intervals represented by an underscore)<br /> |-<br /> |&lt;_<br /> |Flat by a set arbitrary amount.<br /> |-<br /> |&gt;_<br /> |Sharp by a set arbitrary amount.<br /> |-<br /> |x_<br /> |Critically flat/sharp. Shorthand for &lt;_.&gt;_.<br /> |-<br /> |&lt;x_ and &gt;x_<br /> |Shorthand for &lt;&lt;_.&gt;_ and &lt;_.&gt;&gt;_ respectively.<br /> |-<br /> |{ and }<br /> |Partial eracs, indicating error that may be ignored.<br /> |-<br /> |~_<br /> |Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply.<br /> |}<br /> <br /> Eracs are placed before their respective numbers. This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, &gt;5/3 is sharp of 5/3, but 5/&gt;3 is flat and may simplify to &lt;5/3.<br /> <br /> === Erac simplification ===<br /> <br /> Deciding how eracs should be assigned to elements of a subgroup is not an exact science. In addition to the amount of precision the composer finds most useful, the exact ways the errors cancel are important to consider.<br /> <br /> Assume a 2.3.5 subgroup with a slightly inaccurate 3 which is still good enough to not need eracs, but a straddle 5 that's somewhere between 1:1 and 2:1 error. The direction errors tend in determines the best erac subgroup. The following calculations assume that the most important part of 2.3.5 is 4:5:6 and its inversions and retroversions, with the accuracy of 15 being less of a concern.<br /> <br /> &lt;pre&gt;<br /> Just:<br /> 3/2 702 0<br /> 5/4 386 0<br /> 5/3 884 0<br /> &lt;/pre&gt;<br /> &lt;pre&gt;<br /> Effectively just flat 3, straddle-5 tends flat:<br /> 3/2 699 -3<br /> &lt;5/4 377 -9<br /> &gt;5/4 392 +6<br /> &lt;5/3 878 -6<br /> &gt;5/3 893 +9<br /> &lt;/pre&gt;<br /> Here, it makes the most sense to call the subgroup 2.3.&lt;5.&gt;5. The 3 is flat, but not by enough to be worth keeping track of in many cases. Even though the sharp 5 is more accurate, the slightly flat 3 adds to this error and makes the sharp 5/3 less accurate than the flat one. Therefore on average, the two versions of 5 result in about equal errors.<br /> <br /> &lt;pre&gt;<br /> Effectively just sharp 3, straddle-5 tends flat:<br /> 3/2 705 +3<br /> &lt;5/4 377 -9<br /> &gt;5/4 392 +6<br /> &lt;5/3 872 -12<br /> &gt;5/3 887 +3<br /> &lt;/pre&gt;<br /> This is the same as above, but now the 3 is slightly sharp. In this case, the subgroup should be 2.3.&lt;&lt;5.&gt;5. The 3 cancels out some error on the sharp 5, so the sharp 5 results in about half the error of the flat 5 on average.<br /> &lt;/pre&gt;<br /> <br /> == Straddle temperaments ==<br /> <br /> Straddle temperaments use inaccuracy to distort intervals in interesting ways while maintaining their lattice structure.<br /> <br /> Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as [https://spoogly.website/tools/gto.html GTO]. Straddle-prime temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy. Straddle-prime temperaments may also be found as (strong or weak) extensions of subgroup temperaments with prime powers (such as 2.9.21).<br /> <br /> Examples of straddle-prime temperaments include:<br /> * Straddle-3 Meantone (sometimes called dual-3 Meantone or Greentone), where &gt;3/2 * &lt;3/2 = 9/4, and (9/4)^2 = 5/1. This is supported by 18edo and 37edo.<br /> * [[Monarch|Monarch and Interarch]]<br /> * [[Straddle cluster temperaments]]<br /> <br /> == SRTT ==<br /> <br /> '''SRTT''', short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable ''r'' to represent the erac alteration, the erac subgroup 2.&lt;3.&gt;3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in [[Monarch]].<br /> <br /> === Cluster temperaments ===<br /> <br /> In temperaments with a generator very close to some equal temperament, intervals can be optimized by altering the generator in either direction. This flips the direction of eracs, which may be confusing. SRTT alterations may be positive or negative, and may also be stacked. For example, 4\5 is flat lf 7/4. Buzzard and Hemiseven are two well-known 5edo cluster temperaments, with the Buzzard generator flat of 2\5 and the Hemiseven generator sharp of 2\5.<br /> <br /> a and b are arbitrary. It doesn't matter whether they are multiplied or divided, just that both instances of a have the same operation and both instances of b have different operations.<br /> * -3 generators: 7/4 a b<br /> * 2 generatora: 7/4 a/b<br /> At 5edo, b is assumed to be about 1, so a is less than 1 and accounts for the flat tuning of 7/4.<br /> * In Buzzard, the accurate 7/4 is at -3 generators. This is where a and b are about reciprocal, cancelling their errors via multiplication.<br /> * In Hemiseven, the accurate 7/4 is at 2 generators. This is where a and b are about equal, cancelling their errors via division.<br /> * This also means that each cycle of 5 generators ends up at a pitch that differs by a ratio of (7/4 a b)/(7/4 a/b) = b^2.<br /> In temperaments further from optimal, the size of the offset a becomes less significant relative to the ratio between b and 1/b. This is the concept behind straddle cluster temperaments.<br /> <br /> == Straddle-2 ==<br /> <br /> Because edos have a justly tuned 2, straddling it requires a '''triple-straddle''', that is a '''central''' tuning close to the intended ratio and two '''fringe''' tunings on either side. These terms have not been standardized yet.<br /> <br /> Triple-diatonic edos (those with three diatonic generators) may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than standard straddle-3 because the central 3 is too accurate to really straddle 3 at all, creating a lack of balance. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2. If this were used, it would be more intuitive to triple-straddle both 2 and 3, both to pair versions of primes with error in the same direction and allow 3/1 or 3/2 to be the &quot;anchor consonance&quot; allowing 2/1 to be further off. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.<br /> <br /> SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):<br /> &lt;pre&gt;Just-3/2 tunings:<br /> <br /> (2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br /> (2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢<br /> <br /> Just-4/3 tunings are also viable be replacing 3/2 in these equations with 2x / (4/3) = 3x/2, although doing this flips Meantone and Archy.<br /> <br /> These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.<br /> <br /> This is distinct from something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.&lt;/pre&gt;<br /> <br /> Edos large enough to triple-straddle 2 are especially useful for containing stuctures from adjacent edos stretched or compressed. For example, [[82edo]] lacks Meantone and Archy due to triple-straddling 3, but both are possible by using an alternate octave as shown above. This extends to other structures, such as containing a compressed 27edo inherited from 81edo, known to be more optimal for various ratios, as well as compressed Meantone diasem, whose flat 3 benefits from the flat 2. When triple-straddle-2 is used, it generally sounds better to use pure octaves whenever possible to prevent the stretch or compression from accumulating over the full pitch range.<br /> <br /> == Erac-derived temperaments ==<br /> <br /> These were a concept derived by [[User:2^67-1|Cole]] to be able to quickly create conventional temperaments.<br /> <br /> His procedure:<br /> <br /> * For a flat harmonic and a sharp harmonic:<br /> ** take the flat harmonic and take it to the power of how many eracs the sharp harmonic has<br /> ** take the sharp harmonic and take it to the power of how many eracs the flat harmonic has<br /> ** multiply them together<br /> ** take it to the power of how many steps there are in an EDO<br /> ** octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)<br /> <br /> {{Cat|<br /> Aberrismic terms<br /> Atypical ratios<br /> Notation<br /> }}</div>

Ground https://xenreference.com/wiki/index.php?title=Aberrisma&diff=8522 2026-10-28T02:27:49Z

<p>Ground: </p> <hr /> <div>{{Proposed}}<br /> <br /> An '''aberrisma''' is an interval representing some comma as an additional type of melodic step smaller than a semitone. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Although they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas about the size of a diesis, the aberrisma is a melodic function and not an interval size. Larger and smaller sizes can function as aberrismas depending on the scale tuning. '''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.<br /> <br /> Ground's '''aberrismic hypothesis''' is that intervals an audible distance apart can be used together in the same scale framework while merely registering as detunings of the same interval. [[Straddle primes|Straddle theory]] applies the hypothesis to definite ratios, whereas in the original aberrismic scales, it's applied to diatonic scale degrees. Commas have been described as a type of unison in other contexts, but rarely as distinct steps vital to the structure of a scale, or as intervals that are represented by the same ratio. An aberrisma is categorically a unison but doesn't sound like one when isolated, which represents both observed RTT commas and the difference between straddled intervals; it is a unison widened into a melodic step.<br /> <br /> The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals. Aberrismas may also appear in MOS scales, such as Garibaldi[17]. Superpyth[12] is a good example of the distinction between interval function and size. The diatonic semitone, a deflated 256/243, is equated to the inflated 81/80. Despite being the same size, the former represents a narrow semitone, the latter is an aberrisma, and a stack of the two is a classic semitone.<br /> <br /> == The &quot;why&quot; of aberrismic theory ==<br /> === Adding comma steps to Pyth diatonic ===<br /> 81/80 and 64/63 are the canonical choices for 2.3.5 and 2.3.7 formal commas for modifications to Pythagorean (2.3) JI ratios — 81/80 separates 5/4 from 81/64, 10/9 from 9/8, and so on; and 64/63 separates 7/4 from 16/9, 9/7 from 81/64, and so on. Aberrismic theory asks: what if we made 81/80 or 64/63 into scale steps, so that we can use both of the intervals separated by the formal comma in one scale? Indeed, doing this to 2.3 diatonic to make a ternary scale, in a way that best preserves the fifth-generated structure, results in (2.3.5) blackdye and (2.3.7) diasem, respectively.<br /> <br /> ==== Example: Blackdye ====<br /> The Zarlino diatonic is chiral — there are two different, equally valid second degrees of the Ionian mode. Both are useful, as the sharp one forms a perfect fifth with the fifth degree but a wolf fifth with the sixth degree, and the flat one forms a perfect fifth with the sixth degree but a wolf fifth with the fifth degree.<br /> [[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]<br /> One way to make it achiral is to temper out 81/80, the difference between these two steps, resulting in [[Meantone]] diatonic; intuitively this requires flattening the fifth and sharpening the sixth somewhat. However, an alternative way, if you wish to observe 81/80 or to use just intonation, is to include both varieties of whole tone over the unison, treating 81/80 as a melodic step between them. This can be thought of as dividing up a 9/8 into a 10/9 and an 81/80. It is then reasonable to extend this action to all instances of 9/8 in the scale (as, for instance, the Didymic diatonic has 27/16 as opposed to 5/3). The result is a 10-note ternary scale called '''blackdye'''.<br /> <br /> === Stylistic compositional features for aberrismic theory ===<br /> This section will attempt to describe the principles and stylistic features of a specific style of music that justify using aberrismic scales in a composition. It is not an attempt to present aberrismic theory as absolute truth.<br /> * A style of music that is melodic and heavy in modulations benefits from<br /> ** Multiple step sizes for melodic interest, for example diesis-sized steps that are below conventional semitones, specifically ones large enough to be melodically distinct but small enough to represent intonational changes.<br /> ** A set of modulatory intervals, including fifths.<br /> ** A system that allows unlimited modulation. <br /> * It is widely agreed that lower primes are more robust to detuning. Hence for approximating JI with edos, we use lower prime temperaments, and which also represent either 81/80 or 64/63 steps for greater accuracy.<br /> The above suggests temperaments, in particular edos, that use tempered lower primes, and edos large enough to have small diesis-sized steps. In the context of fifth-based modulation, scales also benefit from having offset arcs of fifths. One simple way to have this is to detemper MOS scales into ternary scales with an additional smaller melodic step size, which have a generator arc with fifths or a generator arc that stacks to fifths via a detempered generator chain.<br /> <br /> == List of aberrismic scales ==<br /> * {{Adv|&quot;GS(...)[n]&quot; is [[generator sequence]] notation.}}<br /> * {{adv|&quot;subst ax(bycz)&quot; denotes [[MOS substitution]].}}<br /> * {{adv|&quot;Almost&quot; a cross-set means that one or two notes may be missing from the full cross-set and one note may have been added. Exact cross-sets are italicized.}}<br /> * {{adv|Under &quot;Patterns&quot;, &quot;C&quot; is [[achiral]], and &quot;R&quot; and &quot;L&quot; denote two [[chiral]]ities of a chiral pair.}}<br /> === Quasi-diatonic aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !|pinedye / dia1s&lt;br/&gt;(5L2m1s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m1s)}}}}<br /> ||1sC: LLsLLmLm&lt;br/&gt;1sR: LLmLLmLs&lt;br/&gt;1sL: LLmLLsLm<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 27/25, 81/80]<br /> ||{{adv|GS(3/2)[3] and GS(10/9)[3]}}<br /> ||1sC has 4 fifths and 1sR/1sL have 5<br /> |-<br /> !class=&quot;thl&quot;|diasem / dia2s&lt;br/&gt;(5L2m2s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m2s)}}}}<br /> ||2sR: LmLsLmLsL&lt;br/&gt;2sL: LsLmLsLmL<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 28/27, 64/63]<br /> ||{{adv|GS(3/2)[5] and 7/6}}<br /> ||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.<br /> ''See also: [[Chthonic harmony#Diasem|Diasem]]''<br /> |-<br /> !class=&quot;thl&quot;|blackdye / dia3s&lt;br/&gt;(5L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m3s)}}}}<br /> ||sLmLsLmLsL<br /> ||2.3.5&lt;br/&gt;[L, m, s] = [10/9, 16/15, 81/80]<br /> ||{{adv|''GS(3/2)[5] and 10/9''}}<br /> ||Two interleaved 3-limit pentatonics<br /> ''See also: [[10-form#Blackdye|Blackdye]]''<br /> |-<br /> !|diaslen / dia4s&lt;br/&gt;(5L2m4s)&lt;br/&gt;{{adv|{{nowrap|subst 5L(2m4s)}}}}<br /> ||4sC: LmLsLsLmLss&lt;br/&gt;4sR: LsLmLsLsLms&lt;br/&gt;4sL: LsLsLmLsLsm<br /> ||2.3.7&lt;br/&gt;[L, m, s] = [9/8, 49/48, 64/63]<br /> ||{{adv|GS(3/2)[4] and GS(8/7)[3]}}<br /> ||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]].&lt;br/&gt;{{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}<br /> |-<br /> !|diachrome / chromedye / dia5s&lt;br/&gt;(5L2m5s)<br /> ||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}}&lt;br/&gt;5sR: LmsLsLsLmsLs&lt;br/&gt;5sL: LsLsLsmLsLsm<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 256/243, 81/80]<br /> ||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}<br /> ||Fifth-generated but with a 6-step offset<br /> |-<br /> !|whitedye / dia7s&lt;br/&gt;(5L2m7s)&lt;br/&gt;{{adv|{{nowrap|subst 7s(5L2m)}}}}<br /> ||LsLsLsmsLsLsms<br /> ||5120/5103-tempered 2.3.5.7&lt;br/&gt;[L, m, s] = [10/9, 28/27, 81/80]<br /> ||{{adv|''GS(3/2)[7] and 81/80''}}<br /> ||Two interleaved diatonics<br /> |}<br /> See [[Ground's composition theory#Quasi-diatonic scales]] for more tunings of diasem and blackdye.<br /> <br /> === Other aberrismic scales ===<br /> {|class=&quot;wikitable&quot;<br /> |-<br /> !|Name / Signature<br /> !|Pattern(s)<br /> !|Possible JI/[[erac]] interp.<br /> !|{{adv|Almost a [[cross-set]] of...&lt;br/&gt;(interpreted)}}<br /> !|Notes<br /> |-<br /> !class=&quot;thl&quot;|[[penslen]] / slen5m&lt;br/&gt;(5L5m6s)&lt;br/&gt;{{adv|{{nowrap|subst 6s(5L5m)}}}}<br /> ||LmsLmsLsmLsmLsms<br /> ||2.3.5.7.11[41 &amp; 46]&lt;br/&gt;[L, m, s] = [12/11, 33/32, 64/63]<br /> ||{{adv|''GS(8/7)[8] and 11/8''}}<br /> || Has two aberrisma sizes, s and m.<br /> |-<br /> !|prejubilic / citro3s&lt;br/&gt;(4L2m3s)&lt;br/&gt;{{adv|{{nowrap|subst 2m(4L3s)}}}}<br /> ||C: LLsmLsLms&lt;br/&gt;R: LmLsLmsLs&lt;br/&gt;L: LmLsLsmLs<br /> ||2.5.7&lt;br/&gt;[L, m, s] = [28/25, 35/32, 50/49]<br /> ||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}<br /> ||Didacus tempering makes L = m + s.<br /> |-<br /> !|arm5s&lt;br/&gt;(7L2m5s)&lt;br/&gt;{{adv|{{nowrap|subst 7L(2m5s)}}}}<br /> ||LmLsLsLmLsLsLs<br /> ||2.x&lt;3.5.7.11.13[37edo] (4:2:1)<br /> ||{{adv|''GS(&lt;&lt;3/2)[7] and 14/13''}}<br /> ||An interleaving of two antidiatonic scales.<br /> |-<br /> !|mosh3s&lt;br/&gt;(3L4m3s)&lt;br/&gt;{{adv|{{nowrap|subst 4m(3L3s)}}}}<br /> ||LmsLmsmLsm<br /> ||2.x&lt;3.7.11.13[37edo] (5:4:2)<br /> ||{{adv|''GS(16/13)[5] and 11/8''}}<br /> ||<br /> |-<br /> !|smi2s&lt;br/&gt;(4L3m2s)&lt;br/&gt;{{adv|{{nowrap|subst 2s(4L3m)}}}}<br /> ||C: LLmsLmLsm&lt;br/&gt;R: LmLmsLmLs&lt;br/&gt;L: LmLsLmLsm<br /> ||2.9.7.11.17[46edo] (8:4:1)<br /> ||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}<br /> ||<br /> |}<br /> <br /> === Systematic naming ===<br /> Basic systematic names for aberrismic scales are of the form<br /> <br /> [mos_prefix]n[added_step_size] (e.g. dia2s for diasem).<br /> <br /> Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.<br /> <br /> This is subject to change as aberrismic theory nomenclature is updated in the future.<br /> <br /> == Notation ==<br /> Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.<br /> <br /> Aberrismic/straddle-prime accidentals:<br /> * Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. &lt;!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. --&gt;<br /> * Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).<br /> * [[Erac|Eracs]] may be used to represent the inflating of commas typically found in aberrismic scales, although they represent a specific unit of error.<br /> <br /> == Aberrismic theory and RTT ==<br /> Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under Ground's use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to &lt;!--simultaneously achieve sufficient accuracy to LCJI and --&gt;improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.<br /> <br /> At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.<br /> <br /> There are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.<br /> <br /> === Example: Blackdye ===<br /> The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.<br /> * ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.<br /> * [[Ultrapyth]], 2.3.5.7.11.13[32 &amp; 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. Ground once termed the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.<br /> <br /> {| class=&quot;wikitable right-2 right-3 right-4 right-5&quot;<br /> |+ style=&quot;font-size: 105%;&quot; | Blackdye intervals in two temperaments<br /> |-<br /> ! Interval class<br /> ! Sizes<br /> ! Untempered 2.3.5<br /> ! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)<br /> |-<br /> ! 1-step<br /> | s&lt;br/&gt;m&lt;br/&gt;L <br /> | 81/80&lt;br/&gt;16/15&lt;br/&gt;10/9<br /> | 143/140&lt;br/&gt;22/21&lt;br/&gt;160/143<br /> |-<br /> ! 2-step<br /> | L + s&lt;br/&gt;L + m <br /> | 9/8&lt;br/&gt;32/27<br /> | 8/7, 9/8&lt;br/&gt;7/6<br /> |- <br /> ! 3-step<br /> | L + 2s&lt;br/&gt;L + m + s&lt;br/&gt;2L + s&lt;br/&gt;2L + m<br /> | 729/640&lt;br/&gt;6/5&lt;br/&gt;5/4&lt;br/&gt;320/243<br /> | 7/6&lt;br/&gt;13/11&lt;br/&gt;14/11&lt;br/&gt;13/10<br /> |- <br /> ! 4-step<br /> | 2L + 2s&lt;br/&gt;2L + m + s<br /> | 81/64&lt;br/&gt;4/3<br /> | 13/10&lt;br/&gt;4/3<br /> |-<br /> ! 5-step<br /> | 2L + m + 2s&lt;br/&gt;2L + 2m + s&lt;br/&gt;3L + 2s&lt;br/&gt;3L + m + s<br /> | 27/20&lt;br/&gt;64/45&lt;br/&gt;45/32&lt;br/&gt;40/27<br /> | 66/49&lt;br/&gt;11/8&lt;br/&gt;16/11&lt;br/&gt;49/33<br /> |-<br /> ! 6-step<br /> | 3L + m + 2s&lt;br/&gt;3L + 2m + s<br /> | 3/2&lt;br/&gt;128/81<br /> | 3/2&lt;br/&gt;20/13<br /> |- <br /> ! 7-step<br /> | 3L + m + 3s&lt;br/&gt;3L + 2m + 2s&lt;br/&gt;4L + m + 2s&lt;br/&gt;4L + 2m + s<br /> | 243/160&lt;br/&gt;8/5&lt;br/&gt;5/3&lt;br/&gt;1280/729<br /> | 20/13&lt;br/&gt;11/7&lt;br/&gt;22/13&lt;br/&gt;12/7<br /> |- <br /> ! 8-step<br /> | 4L + m + 3s&lt;br/&gt;4L + 2m + 2s<br /> | 27/16&lt;br/&gt;16/9<br /> | 12/7&lt;br/&gt;7/4, 16/9<br /> |-<br /> ! 9-step<br /> | 5L + 2m + s&lt;br/&gt;5L + m + 2s&lt;br/&gt;4L + 2m + 2s<br /> | 9/5&lt;br/&gt;15/8&lt;br/&gt;160/81<br /> | 143/80&lt;br/&gt;21/11&lt;br/&gt;280/143<br /> |}<br /> <br /> == Glossary ==<br /> * '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.<br /> * '''Aberrisma''': The smallest interval region that melodically sounds like a step.<br /> * '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.<br /> * '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.<br /> * '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.<br /> * '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.<br /> <br /> == Extension to higher ranks ==<br /> Aberrismic theory is typically concerned with ternary scales, especially those with two chains of generators of the same length or differing by one, meaning that they have a [[GS]] with only two intervals. This is because they are not much harder to navigate than an MOS while offering significantly more flexibility.<br /> <br /> {{UserTag|g_|Ground|7766ff|I propose that we call this type of scale '''bicatenary'''.}}<br /> <br /> However, this is often not enough to accurately represent a larger JI subgroup. Portent penslen (5L5m6s) is a somewhat unique counterexample, being bicatenary yet microtempered in the 11-limit. The solution is to add more step sizes while maintaining as much lower-rank structure as possible. For example, while penslen is found in edos like 41 and 46, the equivalent in the similarly LCJI-accurate 53 and 58 is 5L2m3s4t, a quaternary subset of Buzzard[28] with a JI tuning of 35/32 : 21/20 : 36/35 : 64/63. [[Ground's composition theory#Interval logic deviation theory|ILD]] is a simple way to access notes outside of a scale with a higher rank.<br /> <br /> A common tempering of blackdye involves a sharper 3 to significantly inflate 81/80, both for melodic utility and to equate 27/20 to a superfourth like 15/11 or 11/8. It also becomes natural to temper out 64/63. Possibly the most common ratio representation of an aberrisma is 36/35, but few ternary scales use it without tempering out 81/80 or 64/63.<br /> <br /> (81/80)^2 = 43¢ is notably in the middle of the usual aberrisma size range as well. This does not exist in blackdye, but becomes accessible in certain variants of 5L2m8s as well as numerous quaternary scales with even more options for available intervals.<br /> <br /> == Compositional examples ==<br /> Some compositional snippets using aberrismic scales:<br /> <br /> * The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]<br /> * A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]<br /> * A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]<br /> <br /> == External links ==<br /> * [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]<br /> {{cat|<br /> Terms<br /> Aberrismic terms<br /> Ternary scales<br /> }}</div>

Ground https://xenreference.com/wiki/index.php?title=Straddle_primes&diff=8521 2026-10-28T02:19:10Z

<p>Ground: </p> <hr /> <div>{{proposed}}<br /> <br /> '''Straddle primes''' are functional primes in a temperament that straddle their respective just interval by having at least one flat and one sharp approximation available. Dual-fifth systems are often reasonably interpreted as straddle-3. The ratio between straddle approximations is categorically 1/1, making straddle theory a type of [[Aberrisma|aberrismic theory]].<br /> <br /> The ''straddle ratio'' ''m'':''n'' denotes the ratio of the flatter approximation (''m'') to the sharper approximation (''n''). A straddle ratio of approximately 1:1 is called ''equistraddle''.<br /> <br /> == Motivation ==<br /> <br /> Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:<br /> * [[37edo]] has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.<br /> * [[86edo]] is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.<br /> <br /> With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:<br /> * Straddling primes without using a more accurate approximation in the middle widens the smallest step size a tuning must have to represent the straddle. 37edo would have to be tripled to 111edo in order to get an accurate 3, which is totally different in character and size. Larger tuning systems are less practically convenient and make decisions harder.<br /> * Using the same ratio to describe two slightly different intervals drastically simplifies the ratios produced by stacking it, and less care is needed to ensure that error cancellation works out mathematically.<br /> <br /> Straddle-primes differ significantly from the typical way ratios are understood in RTT. They do not work for everyone, but their high level of flexibility makes more unusual approaches possible.<br /> <br /> {{UserTag|g_|Ground|7766ff|I have a philosophy where I limit edo size to the minimum required to do whatever I'm trying to do, in order to make composition easier. When using a tuning with two diatonic fifths, the smallest interval I actually need always ending up being the difference between them. 3.5¢ is an often mentioned value for the just-noticeable difference (JND), but I find that it can be much larger depending on context. My lower bound for the typical size of an [[Aberrisma|aberrisma]], around 25¢, is the limit where it stops sounding like a clear melodic step in many contexts of the music I write, like a fast melody competing with other instruments, slightly detuned synths, and reverb. This also makes error less noticeable, limiting the effect of extreme concordance. Most of the time in my music, there's hardly a difference between a just 4:5:6 and one tempered with Meantone. In more ideal contexts where notes are more clear, my lower bound is about half that size, not coincidentally the step size of the largest edos I use. Even then, being audible enough as a step is a genuine concern.}}<br /> <br /> == Eracs ==<br /> <br /> '''Eracs''' (short for error accidentals) are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for groups involving straddle primes.<br /> <br /> Eracs provide a more complete picture of error cancellation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's [[JI group]] might be 2.5/3.15.7.11 in standard notation, which is a subgroup of the erac group 2.x3.x5.7.11. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancellation in standard notation creates a very long JI group.<br /> <br /> Example erac groups for edos can be found in [[EDO#List of Edos]].<br /> <br /> === Symbols ===<br /> {| class=&quot;wikitable&quot;<br /> |+Erac Meanings (intervals represented by an underscore)<br /> |-<br /> |&lt;_<br /> |Flat by a set arbitrary amount.<br /> |-<br /> |&gt;_<br /> |Sharp by a set arbitrary amount.<br /> |-<br /> |x_<br /> |Critically flat/sharp. Shorthand for &lt;_.&gt;_.<br /> |-<br /> |&lt;x_ and &gt;x_<br /> |Shorthand for &lt;&lt;_.&gt;_ and &lt;_.&gt;&gt;_ respectively.<br /> |-<br /> |{ and }<br /> |Partial eracs, indicating error that may be ignored.<br /> |-<br /> |~_<br /> |Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply.<br /> |}<br /> <br /> Eracs are placed before their respective numbers. This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, &gt;5/3 is sharp of 5/3, but 5/&gt;3 is flat and may simplify to &lt;5/3.<br /> <br /> === Erac simplification ===<br /> <br /> Deciding how eracs should be assigned to elements of a subgroup is not an exact science. In addition to the amount of precision the composer finds most useful, the exact ways the errors cancel are important to consider.<br /> <br /> Assume a 2.3.5 subgroup with a slightly inaccurate 3 which is still good enough to not need eracs, but a straddle 5 that's somewhere between 1:1 and 2:1 error. The direction errors tend in determines the best erac subgroup. The following calculations assume that the most important part of 2.3.5 is 4:5:6 and its inversions and retroversions, with the accuracy of 15 being less of a concern.<br /> <br /> &lt;pre&gt;<br /> Just:<br /> 3/2 702 0<br /> 5/4 386 0<br /> 5/3 884 0<br /> &lt;/pre&gt;<br /> &lt;pre&gt;<br /> Effectively just flat 3, straddle-5 tends flat:<br /> 3/2 699 -3<br /> &lt;5/4 377 -9<br /> &gt;5/4 392 +6<br /> &lt;5/3 878 -6<br /> &gt;5/3 893 +9<br /> &lt;/pre&gt;<br /> Here, it makes the most sense to call the subgroup 2.3.&lt;5.&gt;5. The 3 is flat, but not by enough to be worth keeping track of in many cases. Even though the sharp 5 is more accurate, the slightly flat 3 adds to this error and makes the sharp 5/3 less accurate than the flat one. Therefore on average, the two versions of 5 result in about equal errors.<br /> <br /> &lt;pre&gt;<br /> Effectively just sharp 3, straddle-5 tends flat:<br /> 3/2 705 +3<br /> &lt;5/4 377 -9<br /> &gt;5/4 392 +6<br /> &lt;5/3 872 -12<br /> &gt;5/3 887 +3<br /> &lt;/pre&gt;<br /> This is the same as above, but now the 3 is slightly sharp. In this case, the subgroup should be 2.3.&lt;&lt;5.&gt;5. The 3 cancels out some error on the sharp 5, so the sharp 5 results in about half the error of the flat 5 on average.<br /> &lt;/pre&gt;<br /> <br /> == Straddle temperaments ==<br /> <br /> Straddle temperaments use inaccuracy to distort intervals in interesting ways while maintaining their lattice structure.<br /> <br /> Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as [https://spoogly.website/tools/gto.html GTO]. Straddle-prime temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy. Straddle-prime temperaments may also be found as (strong or weak) extensions of subgroup temperaments with prime powers (such as 2.9.21).<br /> <br /> Examples of straddle-prime temperaments include:<br /> * Straddle-3 Meantone (sometimes called dual-3 Meantone or Greentone), where &gt;3/2 * &lt;3/2 = 9/4, and (9/4)^2 = 5/1. This is supported by 18edo and 37edo.<br /> * [[Monarch|Monarch and Interarch]]<br /> * [[Straddle cluster temperaments]]<br /> <br /> == SRTT ==<br /> <br /> '''SRTT''', short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable ''r'' to represent the erac alteration, the erac subgroup 2.&lt;3.&gt;3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in [[Monarch]].<br /> <br /> === Cluster temperaments ===<br /> <br /> In temperaments with a generator very close to some equal temperament, intervals can be optimized by altering the generator in either direction. This flips the direction of eracs, which may be confusing. SRTT alterations may be positive or negative, and may also be stacked. For example, 4\5 is flat lf 7/4. Buzzard and Hemiseven are two well-known 5edo cluster temperaments, with the Buzzard generator flat of 2\5 and the Hemiseven generator sharp of 2\5.<br /> <br /> a and b are arbitrary. It doesn't matter whether they are multiplied or divided, just that both instances of a have the same operation and both instances of b have different operations.<br /> * -3 generators: 7/4 a b<br /> * 2 generatora: 7/4 a/b<br /> At 5edo, b is assumed to be about 1, so a is less than 1 and accounts for the flat tuning of 7/4.<br /> * In Buzzard, the accurate 7/4 is at -3 generators. This is where a and b are about reciprocal, cancelling their errors via multiplication.<br /> * In Hemiseven, the accurate 7/4 is at 2 generators. This is where a and b are about equal, cancelling their errors via division.<br /> * This also means that each cycle of 5 generators ends up at a pitch that differs by a ratio of (7/4 a b)/(7/4 a/b) = b^2.<br /> In temperaments further from optimal, the size of the offset a becomes less significant relative to the ratio between b and 1/b. This is the concept behind straddle cluster temperaments.<br /> <br /> == Straddle-2 ==<br /> <br /> Because edos have a justly tuned 2, straddling it requires a '''triple-straddle''', that is a '''central''' tuning close to the intended ratio and two '''fringe''' tunings on either side. These terms have not been standardized yet.<br /> <br /> Triple-diatonic edos (those with three diatonic generators) may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than standard straddle-3 because the central 3 is too accurate to really straddle 3 at all, creating a lack of balance. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2. If this were used, it would be more intuitive to triple-straddle both 2 and 3, both to pair versions of primes with error in the same direction and allow 3/1 or 3/2 to be the &quot;anchor consonance&quot; allowing 2/1 to be further off. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.<br /> <br /> SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):<br /> &lt;pre&gt;Just-3/2 tunings:<br /> <br /> (2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢<br /> (2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢<br /> <br /> Just-4/3 tunings are also viable be replacing 3/2 in these equations with 2x / (4/3) = 3x/2, although doing this flips Meantone and Archy.<br /> <br /> These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.<br /> <br /> This is distinct from something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.&lt;/pre&gt;<br /> <br /> Edos large enough to triple-straddle 2 are especially useful for containing stuctures from adjacent edos stretched or compressed. For example, [[82edo]] lacks Meantone and Archy due to triple-straddling 3, but both are possible by using an alternate octave as shown above. This extends to other structures, such as containing a compressed 27edo inherited from 81edo, known to be more optimal for various ratios, as well as compressed Meantone diasem, whose flat 3 benefits from the flat 2. When triple-straddle-2 is used, it generally sounds better to use pure octaves whenever possible to prevent the stretch or compression from accumulating over the full pitch range.<br /> <br /> == Erac-derived temperaments ==<br /> <br /> These were a concept derived by [[User:2^67-1|Cole]] to be able to quickly create conventional temperaments.<br /> <br /> His procedure:<br /> <br /> * For a flat harmonic and a sharp harmonic:<br /> ** take the flat harmonic and take it to the power of how many eracs the sharp harmonic has<br /> ** take the sharp harmonic and take it to the power of how many eracs the flat harmonic has<br /> ** multiply them together<br /> ** take it to the power of how many steps there are in an EDO<br /> ** octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)<br /> <br /> {{Cat|Atypical ratios<br /> Notation}}</div>

Ground https://xenreference.com/wiki/index.php?title=Concordance&diff=8498 2026-10-27T03:05:05Z

<p>Ground: </p> <hr /> <div>'''Concordance''' (opposed to discordance) is a measure of how well harmonics of different notes in a chord line up. It is often equated to consonance (opposed to dissonance), although consonance is cultural rather than physical.<br /> <br /> == Conventional models ==<br /> In order of complexity:<br /> * Tenney norm[https://en.xen.wiki/w/Tenney_norm]<br /> * Wilson norm[https://en.xen.wiki/w/Wilson_norm]<br /> * Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy]<br /> <br /> == Harmonic Cross-Beating ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> [[File:Harmonic cross-beating saw 20-6.png|thumb|Harmonic cross-beating diagram of two saw waves from 55 to 880 Hz. The beating confidence curve has a mean of 20 Hz and a standard deviation of 6 Hz.]]<br /> Harmonic entropy is the accepted standard for quantifying concordance, but it does not explicitly take register and timbre into account. It also uses a probability distribution for how likely an interval is to be perceived as a certain integer ratio, which may not be the most perceptually significant metric. I created '''harmonic cross-beating''' to address these issues. This model doesn't use integer ratios at all, and the probability distribution is a &quot;perceptual beating curve&quot; that approximates the strength of audible beating in a difference between harmonics. I originally assumed a normal distribution with a mean of 20 Hz and a standard deviation of 6 Hz, which is about right as far as my experimentation goes.<br /> <br /> === Algorithm ===<br /> Calculate the frequency difference between each harmonic of each waveform and each harmonic of the other waveforms to get a set of all possible beating frequencies. Multiply them by the relative strength of the current two harmonics being evaluated. Then multiply them by a perceptual beating curve. Finally, sum everything.<br /> <br /> === Psychoacoustics ===<br /> My hypothesis is that the perceived dominant beating frequency in an interval with a given register and timbre is similar between listeners and can be mathematically modeled to high confidence, which can be used to approximate the vague consensus of concordance using a perceptual probability distribution.<br /> <br /> Having the axes in frequency space rather than logarithmic space shows how the concordance wells have a constant width in terms of frequency, which is what I predicted, and new concordance wells gradually show up as the frequencies increase. The cross-beating values become unusually large as the waveforms approach the assumed perceptual beating frequency.</div>

Ground https://xenreference.com/wiki/index.php?title=Concordance&diff=8497 2026-10-27T02:35:32Z

<p>Ground: Created page with &quot;&#039;&#039;&#039;Concordance&#039;&#039;&#039; (opposed to discordance) is a measure of how well harmonics of different notes in a chord line up. It is often equated to consonance (opposed to dissonance), although consonance is cultural rather than physical. == Conventional models == In order of complexity: * Tenney norm[https://en.xen.wiki/w/Tenney_norm] * Wilson norm[https://en.xen.wiki/w/Wilson_norm] * Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy] == Harmonic Cross-Beating == {{UTBox...&quot;</p> <hr /> <div>'''Concordance''' (opposed to discordance) is a measure of how well harmonics of different notes in a chord line up. It is often equated to consonance (opposed to dissonance), although consonance is cultural rather than physical.<br /> <br /> == Conventional models ==<br /> In order of complexity:<br /> * Tenney norm[https://en.xen.wiki/w/Tenney_norm]<br /> * Wilson norm[https://en.xen.wiki/w/Wilson_norm]<br /> * Harmonic entropy[https://en.xen.wiki/w/Harmonic_entropy]<br /> <br /> == Harmonic Cross-Beating ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> [[File:Harmonic cross-beating saw 20-6.png|thumb|Harmonic cross-beating diagram of two saw waves from 55 to 880 Hz. The beating confidence curve has a mean of 20 Hz and a standard deviation of 6 Hz.]]<br /> Harmonic entropy is the accepted standard for quantifying concordance, but it does not explicitly take register and timbre into account. It uses a probability distribution for how likely an interval is to be perceived as a certain ratio, which is a choice I contend with. I created '''harmonic cross-beating''' to address these issues. This model doesn't use ratios at all, and the probability distribution is for whether a difference between harmonics is heard as beating.<br /> <br /> === Algorithm ===<br /> Calculate the frequency difference between each harmonic of each waveform and each harmonic of the other waveforms to get a set of all possible beating frequencies. Multiply them by the relative strength of the current two harmonics being evaluated. Then multiply them by a &quot;beating confidence curve&quot; that approximates the strength of audible beating. Finally, sum everything.<br /> <br /> === Psychoacoustics ===<br /> My hypothesis is that perceived dominant beating frequency in an interval with a given register and timbre is similar between listeners and can be mathematically modeled to high confidence, which can be used to approximate the vague consensus of concordance using a perceptual probability distribution.<br /> <br /> Having the axes in frequency space rather than logarithmic space really shows how the concordance wells have a constant width in terms of frequency, which is what I predicted, and new concordance wells gradually show up as the frequencies increase. The cross-beating values become unusually large as the waveforms approach the assumed audible beating frequency.</div>

Ground https://xenreference.com/wiki/index.php?title=File:Harmonic_cross-beating_saw_20-6.png&diff=8496 2026-10-27T02:19:40Z

<p>Ground: </p> <hr /> <div>Harmonic cross-beating diagram of two saw waves from 55 to 880 Hz. The beating confidence curve has a mean of 20 Hz and a standard deviation of 6 Hz.</div>

Ground https://xenreference.com/wiki/index.php?title=Ground%27s_composition_theory&diff=8493 2026-10-26T17:21:13Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> This article is about composition and the practical considerations of tuning. I always have more content to add and revise.<br /> <br /> I write in the context of my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]], which are both based on my concept of an '''altered unison''', something that is categorically a unison but doesn't sound like one when isolated. An aberrisma is generally defined as an altered unison, and straddled intervals are separated by one.<br /> <br /> It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the [[Levels of tempering|mesotempered category]]. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.<br /> <br /> == Interval logic deviation theory ==<br /> <br /> Much of my music has had a distinctively shifting tonality since 2018 or earlier, which started in 12edo. This article is an attempt to explain how it works. I'm introducing a placeholder term for it, '''interval logic deviation''' theory (ILD), which can be replaced if this turns out to be something already described.<br /> <br /> === Local tonality ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic example of probabilities (up to word length 3)<br /> !Sequence<br /> !Next step probability<br /> |-<br /> |s<br /> |L 1/1<br /> |-<br /> |L<br /> |L 5/7, s 2/7<br /> |-<br /> |sL<br /> |L 1/1<br /> |-<br /> |Ls<br /> |L 1/1<br /> |-<br /> |sLL<br /> |s 1/2, L 1/2<br /> |-<br /> |LsL<br /> |L 1/1<br /> |-<br /> |LLs<br /> |L 1/1<br /> |-<br /> |LLL<br /> |s 2/3, L 1/3<br /> |}<br /> &lt;/div&gt;<br /> I have a simultaneous regard and disregard for standard Western tonality. This is because I view it as an important but strictly local property, meaning it fundamentally only applies on the scope of a single path from tension to release, however long that is. Thus, modulations are only generally uncommon because phrases usually resolve to the same key center they started from, but changing tonality is just as much of a choice as not changing it. Modulation flows just like any other melodic or harmonic movement.<br /> <br /> This flow is facilitated by ILD, in which scales aren't a fixed set of notes, but a template for interval logic to be rearranged and deviated from. As such, their main features are probabilities in an interval sequence and &quot;bubble deviations&quot; from that interval sequence. ILD is best for music with a strong melodic focus, such as mine, where the melody informs the harmony instead of the reverse. Other concepts may be used instead with the same general goal.<br /> <br /> Melodic interval sequences, or &quot;words&quot; of step sizes, are the most minimal expression of tonal tension and release. For example, if a diatonic melody were to play C then B, the listener is likely to expect A to be next and feel a small resolution upon hearing it. This is the descending sL word. Melodies are full of small sequences like this based on the scale that they are in. It's possible to use sequences with notes outside the scale while still feeling like they belong, and how much they belong can be predicted. The longer the word and the greater probability of occurring indicates that it's more likely to sound like it belongs in the scale.<br /> <br /> === Axes of deviation ===<br /> <br /> &quot;Bubble deviations&quot; in ILD are named after the bubble sort algorithm, which repeatedly swaps adjacent items in an array. Steps in a base scale can be swapped in the same way to modify the scale with no requirement for a clear structure like a generator chain or lattice splotch, which is my term for a collection of generator chains in aberrismic theory. Suppose you want the scale word sLs in diatonic. This would require only one bubble deviation from the expected step order, turning sLLs to sLsL. While the probability of encountering it in the base diatonic scale is zero, it sounds more &quot;probable&quot; (less unexpected) than something like ssL.<br /> <br /> This explains why I used Dimininished[8] in 12edo more often than Augmented[6], because its stepwise interval logic has less deviation from diatonic. Diminished[8] is made of a repeating sequence of 1\12 and 2\12, common in diatonic, whereas Augmented[6]'s steps of 1\12 and 3\12 do not occur in diatonic at all. As a result, melodies in Diminished sound less exotic.<br /> <br /> Bubble deviations are only one axis of deviation. There is another axis which I've found to be exclusively useful in tuning systems with aberrismic-sized steps or smaller: the axis of microtonal deviation from expected intervals. This involves changing the pitch of an expected interval only slightly, so it is heard as a variation of the expected interval rather than a different interval entirely. This axis interacts with the base scale by introducing or modifying an aberrismic offset, for example diatonic being diasem or blackdye with the offset removed, 2.3.7 diasem having a larger offset than 2.3.23, or 2.3.5 blackdye having a smaller offset than 2.3.17/7. The intervals affected by the offset, usually thirds and sixths, differ microtonally when the offset is changed.<br /> <br /> Straddling intervals that are stacked the most (usually 3/2) introduce a third axis that can be simplified into a combination of the other two. It's possible to have bubble deviation from a scale that isn't even in the tuning system being used, like how alternating &lt;&lt;3 and &gt;3 in 37edo straddles 74edo meantone and results in trackdye.<br /> <br /> {{UserTag|KC|Inthar|000000|[[Stretching and compression]] constitute yet another axis of deviation separate from straddling. Diatonic-based example: This is important in 4L3s and 5L3s which are warped-diatonic MOSes. Note that dual-3 diatonic 5L1m1s is a subset of interleaved diatonic 7s(5L2m), tens-interleaved diatonic 6s(5L2m) ''and'' tract-interleaved diatonic 8s(5L2m).}}<br /> <br /> === Vague interval logic ===<br /> <br /> Scales are useful for ILD, but not technically necessary. One may pick the desired step sizes and find just a few arrangements leading to useful intervals, like the perfect fifth. This should be especially useful when trying to avoid making quasi-diatonic music in any tuning system.<br /> <br /> Take 3\24 s and 5\24 L for example. The words LL and sssL make 10\24 and 14\24 respectively, so the two can together be used to infer vague probabilities. Edos that straddle important intervals are also useful for this because they increase the chances of landing on those intervals. If s and L were replaced with 11\86 and 18\86, LL becomes &gt;4/3 and sssL becomes &gt;3/2.<br /> <br /> === Aberrismic theory ===<br /> <br /> The above concepts also apply to ternary scales, where they may be even more useful. Conventional aberrismic scales feature an alternating generator sequence that creates a 2-dimensional lattice, which can be used similarly to a generator sequence, but is less intuitive due to its complexity. Thus, ILD proves to be a practical alternative when modulating. This is how I write 2.3.5 and 2.3.7 music in tunings that aren't Meantone or Archy.<br /> <br /> Quasi-diatonic aberrismic scales are the ideal case due to the general importance of stacking prime 3 and internalized diatonic logic of Western music. They typically have four main step sizes: '''aberrisma''' (s), '''semitone''' (m), and the two whole tones '''solitone''' (L) and '''magnitone''' (L+s). Other scales or ILD can introduce more sizes such as the '''magnisemitone''' (m+s). The decision of whether to use a solitone or magnitone depends on which intervals in a chord are preferred and which sounds more melodically to the composer in a given situation. The aberrisma is a new class so its theory is less well defined, but it works best as a passing tone. It provides the ability to change the length of scale runs without repeating any notes.<br /> <br /> I find it most generally useful to define the size range of an aberrisma as being between 81/80 and half of 16/15, but my personal usage has ranged from about 12¢ (the size of 81/80 in 5-limit CWE Negri and around one step of the largest full edos I use) to about 100¢ (81/80 in some tunings of Blackwood with a flat 5). I provide 40¢ as the general ideal, but the actual ideal varies. Faster music prefers a larger aberrisma because it is more audible. Scales with a large magnitone can handle a larger aberrisma without it sounding entirely like a semitone, such as 37edo pental blackdye. '''Subaberrismas''', aberrismas small enough that they are not reliably recognizable as a melodic step, have a unique sound that may be desirable in certain situations.<br /> <br /> Because they are often passing tones when used melodically, it can be relatively easy to swap an aberrisma for one of a more preferable size. For example, I may swap 1\43 in 43edo diasem for the semiquartal 2\43 to increase its audibility.<br /> <br /> I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.<br /> <br /> == Diatonic analogues ==<br /> <br /> === 9-note &quot;diatygic&quot; ===<br /> <br /> ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.<br /> <br /> When it comes to extending the diatonic scale's circle of fifths as seen in ILD, there's no alteration more important than the modes of the melodic minor scale, which I'll be referring to together as '''melodiatonic'''. It's the only non-diatonic scale that's possible to construct by making a single bubble deviation from diatonic, and it makes sense looking at the circle of fifths. Assume that some altered diatonic scale must contain only L and s steps to preserve as much step logic as possible. To do this, only ordinals following a semitone may be sharpened (C and F when the unaltered notes are all naturals), and only ordinals preceding a semitone may be flattened (B and E). Sharpening F or flattening B just changes the key signature and the result is still diatonic. Sharpening C or flattening E results in melodiatonic. I have found melodiatonic to be a remarkably reliable way to make diatonic tunings (including 12edo) sound exotic yet familiar enough to hit the same emotional beats diatonic can provide. After diatonic, it has the smallest span across the chain of fifths without duplicating a diatonic ordinal, meaning that it contains no pairs of notes that are 7 generators apart. Just like how diatonic is the framework for all quasi-diatonic scales, melodiatonic can be used as the framework for certain alterations of those quasi-diatonic scales. {{adv|Unconventional uses of melodiatonic I can think of off the top of my head are parts of Undertale's Heartache in Bb Dorian b2 and the main theme of Animusic's Harmonic Voltage in C Mixolydian b6.}}<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Quasi-Diatygic Scales<br /> !Altered&lt;br&gt;Blackdye<br /> !Altered&lt;br&gt;Diasem<br /> |-<br /> |10/9&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;5/4&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;5/3&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |9/8&lt;br&gt;8/7&lt;br&gt;7/6&lt;br&gt;9/7&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;14/9&lt;br&gt;12/7&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |}<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic Tertian Triads in D Gadygic<br /> !Root<br /> !Triads<br /> |-<br /> |D<br /> |minor, Major<br /> |-<br /> |E<br /> |dim., minor<br /> |-<br /> |F<br /> |Major<br /> |-<br /> |F♯<br /> |dim.<br /> |-<br /> |G<br /> |minor, Major<br /> |-<br /> |A<br /> |minor<br /> |-<br /> |B♭<br /> |Major<br /> |-<br /> |B<br /> |dim., minor<br /> |-<br /> |C<br /> |Major<br /> |}<br /> &lt;/div&gt;<br /> The gaps in the fifth chain can be filled to result in a contiguous generator chain of 9 notes rather than 7, allowing some modes to freely switch between major and minor thirds and sixths. This is probably the earliest music theory I've ever consciously developed, dating to 2014 or so ([https://soundcloud.com/groundfault/doodle-62 example]). It has been scale #1 on the [https://allthescales.org/scales.php?n=9 9-note page of AllTheScales] for at least as long, and I tentatively propose calling the scale '''diatygic''' using the seemingly arbitrary -ygic suffix from AllTheScales since it seems like a term I would have chosen at the time. It has stood the test of time and I find it to be the natural extension of the diatonic framework that doesn't require microtuning, so it remains my favorite 12edo-compatible scale. The scale word for the symmetrical mode is LdcdLdcdL, with d and c for diatonic and chromatic semitones. It is called '''Gadygic''' by AllTheScales. I use plenty of extended interval chains when I compose, but diatygic is intended to be a proper albitonic scale with simple self-contained rules, prescriptive rather than descriptive in the sense that it is the new template to be extended and deviated from. {{adv|For some reason, the first examples of this scale I can think of (besides my own music) all come from the brony fandom. Morning in Ponyville, Awesome as I Wanna Be, and Awoken. It should leave no guesswork as to what I was doing in 2014. All of them are coincidentally in the key of D.}}<br /> <br /> I derived diatygic from my own observation of my favorite music, intuition, and the methods above, including that 9 notes is the closest thing to a MOS between 7 and 12 notes. For one thing, MOS scales with generators approaching half an octave will have sizes that are successive odd numbers; the diatonic range jumps from 7 to 12, but the adjacent antidiatonic range follows 7 with 9. Stacking 9 fifths results in a scale with groups of 3 semitones in a row, which is substantially easier to navigate than the 10-note &quot;supermode&quot; in [https://mtosmt.org/issues/mto.11.17.4/mto.11.17.4.temperley.html Temperley 2011], with a group of 5 semitones. I've found it more useful to use various modes of diatygic than to add additional notes, since needing more than two non-diatonic notes at a time is very rare. This is so that the scale can contain the diatonic modes Aeolian to Ionian as subsets, whereas diatygic only manages Aeolian to Mixolydian. I've never used the major seventh degree much myself, despite its high apparent importance in Western music. Even back when I used major modes more than minor, I preferred Mixolydian to Ionian. It's about as important to me as the minor second degree, which is easy to achieve through using modes other than the symmetrical one.<br /> <br /> This extension of diatonic also works for quasi-diatonic scales. {{adv|Pental blackdye is two chains of 5 of 3/2 separated by 10/9. Extending both of those chains by 2 is analogous to what I'm doing to diatonic, but separating them by 81/80 instead makes 5L2m7s. Separating by 10/9 results in a quaternary scale that is very nearly ternary; temper out the Schisma and you get 3L6m5s with a non-standard step order to what I've established in my aberrismic theory.}} The altered pental blackdye is almost the 3sA Aeolian mode with 5/4 and 5/3 added. The retroversion of this can be obtained by adding 6/5 and 8/5 to 3sG Mixolydian, but it's inferior because it lacks the perfect fifth over 6/5. The one additional change is 81/80 to 10/9 to fix the wolf fifth from 9/8 to 5/3. All 12 notes are required to make a chain of pure 3/2 sometimes altered by 81/80 just as blackdye does, except 4/3-16/9 and 27/20-9/5 are redundant. The extra note is kept to keep the scale to a minimum of four unique step sizes, similar to what is done with blackdye. A similar process may be done to septal diasem.<br /> <br /> One of my favorite parts of the scale is a major third on D, a minor third on E, and a major third on F, using notes from D Gadygic.<br /> {| class=&quot;wikitable&quot;<br /> |+Interval Matrix of Gadygic, DR 3:4:5 Meantone<br /> !<br /> !0<br /> !1<br /> !2<br /> !3<br /> !4<br /> !5<br /> !6<br /> !7<br /> !8<br /> !(9)<br /> |-<br /> !0.<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !194.6<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1200<br /> |-<br /> !308.2<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !389.1<br /> |0<br /> |113.6<br /> |308.2<br /> |421.8<br /> |502.7<br /> |616.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !502.7<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |-<br /> !697.3<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !810.9<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |583.7<br /> |697.3<br /> |778.2<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !891.8<br /> |0<br /> |113.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !1005.4<br /> |0<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |}<br /> <br /> === Neutral diatonic extensions ===<br /> <br /> Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.<br /> <br /> A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.<br /> <br /> I believe xenharmony involves a balance of being unusual without being jarring. Neutral seconds around 150¢ are excellent for the former, but are more difficult to fit into a diatonic framework. When given the choice, I prefer to have neutral seconds around 135¢ to fit in place of a semitone and 165¢ to fit in place of a whole tone. Temperaments in small edos are Flattone for the former and Porcupine for the latter.<br /> <br /> When converting a simple subgroup to contain these intervals, it makes sense to use 2.3.5 because it's closer to neutral whereas 2.3.7 is closer to interordinal.<br /> * Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.<br /> * Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.<br /> This is an excellent use case for straddle-3 temperaments. 37edo has mild Mavila and extreme Archy fifths, but their distance from 3/2 is somewhat masked in 5-limit chords due to the 5/4 anchor concordance.<br /> <br /> == Adaptive diatonic modification system ==<br /> <br /> This section is incomplete.<br /> <br /> In order to make modified diatonic scales easier to use and quickly understand the intervals I use in my head, I have a proposal for an interval scheme based on splitting diatonic or quasi-diatonic intervals, somewhat similar to [[Adaptive diatonic interval names|ADIN]].<br /> <br /> First specify which is the default diatonic or quasi-diatonic quality. This is usually pental (4:5:6), pyth, or septal (6:7:9).<br /> <br /> Neutral seconds are about 3/7 to 4/7 of the minor third.<br /> <br /> == Structural tuning selection ==<br /> <br /> Edos are the most common type of tuning system, and the one I use exclusively. They're a simple and discrete set, which makes them very easy to compare to one another and list in the titles of music made with them. This creates a common assumption that they sound more different from each other than they actually do, which I suffer from myself sometimes. Small edos really do have distinct personalities, but as they get larger and more flexible, those differences gradually disappear. I try to guess the tuning system of a piece whenever it isn't stated, and I'm very often wrong.<br /> <br /> I propose that the most important distinction between tuning systems of sufficient flexibility is not how they sound, but how their unique combinations of structures guide the composer. I like to believe that I have sole control over what I write, but my choice of tuning system prioritizes certain scales, chords, and intervals over others by making them easier to use in practice.<br /> <br /> I use large edos in an unusual way. I like to have multiple octaves of range directly in the piano roll, so I access large edos by multiplying smaller ones using midi channels and my custom tuning scripts. This creates a more complex process of tuning selection, involving the combination of two considerations:<br /> # Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?<br /> # Which of those can be multiplied by 2 or 3 to achieve a superset that's enjoyable for the same reasons, but also complements the subset edo well?<br /> {| class=&quot;wikitable&quot;<br /> |+A selection of large edos I use<br /> !Edo<br /> !Justification<br /> |-<br /> | style=&quot;text-align: right;&quot; |27x3&amp;nbsp;=&amp;nbsp;81<br /> |81 is an [[Monarch|Interarch]] edo, which I find useful. 27x2 = 54 is also uniquely bad, an example of poor LCJI approximations providing insufficient advantages to be worth using.<br /> |-<br /> | style=&quot;text-align: right;&quot; |29x3&amp;nbsp;=&amp;nbsp;87<br /> |I don't like 29 enough to use it much on its own, but 87 is my favorite highly accurate LCJI edo, the intersection of Aberschismic and a variant of [[Hemiseven]].<br /> |-<br /> | style=&quot;text-align: right;&quot; |37x2&amp;nbsp;=&amp;nbsp;74<br /> |Note that this is not 37x3 = 111, a much better LCJI edo. 111 offers little to complement the existing structure of 37edo that 74 doesn't already do.<br /> |-<br /> | style=&quot;text-align: right;&quot; |43x2&amp;nbsp;=&amp;nbsp;86<br /> |I found 86 to be the best Interarch edo for my edo multiplication approach. It clearly straddles every prime up to 19 with the possible exception of 13, but it's large enough that their accuracy is not a major issue.<br /> |-<br /> | style=&quot;text-align: right;&quot; |67x2ed4&amp;nbsp;=&amp;nbsp;67<br /> |Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.<br /> |}<br /> <br /> == Edo multiplication ==<br /> <br /> I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the '''base edo''' to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.<br /> <br /> Example tier list of 37edo multiples:<br /> * Base ([[37edo|37]]): S (One of a kind.)<br /> * x2 ([[74edo|74]]): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)<br /> * x3 ([[111edo|111]]): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)<br /> * x6 ([[222edo|222]]): D (Has extra approximations and generators, but too cumbersome to do much.)<br /> This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:<br /> * Base: remainder<br /> * x2: 12%<br /> * x3: 8%<br /> * x6: 4%<br /> <br /> This should show just how important the base edo is. No matter how many intervals you have, large edos or edo multiplication will never be as efficient to write in as an elegant mid-sized edo that does what you want the most. Everything is filtered through the base edo's structure. Because of this, there's no such thing as the perfect edo. At this point, I use whatever I feel like at the time and multiply it by 6 to fill any gaps that might come up.<br /> <br /> When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.<br /> <br /> Edos that are multiples of 6 usually have more than one reasonable way to split them. I will use 144edo to explain my thought process in deciding whether to use 24x6 or 36x4.<br /> * The most important consideration is which base edo the composer enjoys the most. Both options have features that are important to me, but I've written a good amount in 24 because I believe its lack of a good 5 is less important, so it's been proven by time.<br /> * In terms of the number of useful scales available, 36 is superior. To some extent, this comes with having more notes in general.<br /> * x6 has more channels overall, but contains both the x2 and x3 subsets. x4 requires a larger base edo and has only the option to use x2 and x4 notes. Because of the distribution of the number of channels I typically use, I expect x6 to have more practical options.<br /> * x6 is more of a standard for me, so I'm set up better for that.<br /> * I mentioned how I like to make clear switches between x2 and x3 to make it easier to read my piano roll notes. x6 is better for this because x4 has only one distinct subset besides the base edo, x2.<br /> * 36x2 makes accessing 72edo especially convenient because fewer channels is easier. It wins for 11-limit stuff.<br /> * Two important subsets of 144 are 16 and 18. 24x6 has them as x2 and x3, but 36x4 has them as x4 and x1. This is a tie.<br /> * I use 5-form structures a lot, and some of my favorite edos are 43 and 67. Add 5 to bring their 3/2 closer to just, resulting in 48 and 72, 24x2 and x3. This is an elegant way to approximate both 43 and 67.<br /> * Using a smaller edo gets more range in the piano roll, which is useful for quick creative writing.<br /> * 36 has clearer concordances because it's so good at 2.3.7. It's good to have some anchor concordance in the base edo, but 24 also has 3/2. Having 2.3.7 outside the base edo naturally guides composition towards using other channels and makes it hard to forget about them. There's a nice symmetry with 24x6 putting 7/6 and 6/5 on the same channel as well as 9/7 and 5/4.<br /> With all this in mind, 24x6 is clearly the better choice to start with, although it's close.<br /> <br /> === Channel management ===<br /> <br /> Blackdye and diasem structures have a generator of a perfect fifth and some offset for certain notes. Because all of the offsets are in the same direction, using an offset that is part of a superset of the generator doubles the number of distinct channels required. This most notably results in scales where the generator is from the x2 edo and the offset is from the x3 or x6 edo only requiring 4 channels.<br /> <br /> == Anchor concordances ==<br /> <br /> We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an '''anchor concordance''' because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.<br /> <br /> For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.<br /> <br /> I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.<br /> <br /> In 37edo, both 4:5:&gt;6 and 4:5:&lt;&lt;6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides &lt;&lt;6:8:&lt;&lt;&lt;&lt;9 = 6:&gt;&gt;8:&lt;&lt;9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:<br /> * Using the very flat &lt;&lt;3 results in &gt;&gt;16/15, quite audibly wider while keeping some semblance of 4:5:6.<br /> * &lt;&lt;6/5 is found in the [[Pentagoth]] triad, which uses the exaggerated quality of the minor third and the stability of the major third.<br /> <br /> Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:<br /> * ~69edo: DR 4:5:6, 10:12:15<br /> * ~31edo: 5/4 anchor concordance<br /> * ~74edo: DR 3:4:5<br /> * ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)<br /> <br /> == Quartertone composition ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;<br /> |+Island Rastmic with Straddle Primes<br /> !Gens<br /> !949¢<br /> !349¢<br /> ! colspan=&quot;2&quot; |&lt;nowiki&gt;Proposed Names (semi- | hemi-)&lt;/nowiki&gt;<br /> |-<br /> | -12<br /> | colspan=&quot;2&quot; |612<br /> | colspan=&quot;2&quot; |Diminished Fifth<br /> |-<br /> | -11<br /> |361<br /> | class=&quot;thl&quot; |961<br /> |&quot;Semififth&quot;<br /> |&quot;Hemidim.&quot; Seventh<br /> |-<br /> | -10<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |110<br /> | colspan=&quot;2&quot; |Minor Second<br /> |-<br /> | -9<br /> |1059<br /> |459<br /> |&quot;Semithirteenth&quot;<br /> |&quot;Hemidim.&quot; Fourth<br /> |-<br /> | -8<br /> | colspan=&quot;2&quot; |808<br /> | colspan=&quot;2&quot; |Minor Sixth<br /> |-<br /> | -7<br /> | class=&quot;thl&quot; |557<br /> |1157<br /> |&quot;Semiseventh&quot;<br /> |&quot;Hemidim.&quot; Octave<br /> |-<br /> | -6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |306<br /> | colspan=&quot;2&quot; |Minor Third<br /> |-<br /> | -5<br /> |55<br /> |655<br /> |&quot;Semisecond&quot;<br /> |&quot;Hemidim.&quot; Fifth<br /> |-<br /> | -4<br /> | colspan=&quot;2&quot; |1004<br /> | colspan=&quot;2&quot; |Minor Seventh<br /> |-<br /> | -3<br /> |753<br /> |153<br /> |Semitenth<br /> |Neutral Second<br /> |-<br /> | -2<br /> | colspan=&quot;2&quot; |502<br /> | colspan=&quot;2&quot; |Perfect Fourth<br /> |-<br /> | -1<br /> |251<br /> |851<br /> |Semifourth<br /> |Neutral Sixth<br /> |-<br /> |0<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |0<br /> | colspan=&quot;2&quot; |Unison<br /> |-<br /> |1<br /> |949<br /> |349<br /> |Semitwelfth<br /> |Neutral Third<br /> |-<br /> |2<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |698 ''(98)''<br /> | colspan=&quot;2&quot; |Perfect Fifth<br /> |-<br /> |3<br /> |447<br /> |1047<br /> |Semisixth<br /> |Neutral Seventh<br /> |-<br /> |4<br /> | colspan=&quot;2&quot; |196<br /> | colspan=&quot;2&quot; |Major Second<br /> |-<br /> |5<br /> |1145<br /> | class=&quot;thl&quot; |545<br /> |&quot;Semifourteenth&quot;<br /> |&quot;Hemiaug.&quot; Fourth<br /> |-<br /> |6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |894 ''(294)''<br /> | colspan=&quot;2&quot; |Major Sixth<br /> |-<br /> |7<br /> |643<br /> |43<br /> |&quot;Semininth&quot;<br /> |&quot;Hemiaug.&quot; Unison<br /> |-<br /> |8<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |392<br /> | colspan=&quot;2&quot; |Major Third<br /> |-<br /> |9<br /> |141<br /> |741<br /> |&quot;Semithird&quot;<br /> |&quot;Hemiaug.&quot; Fifth<br /> |-<br /> |10<br /> | colspan=&quot;2&quot; |1090<br /> | colspan=&quot;2&quot; |Major Seventh<br /> |-<br /> |11<br /> | class=&quot;thl&quot; |839<br /> |239<br /> |&quot;Semieleventh&quot;<br /> |&quot;Hemiaug.&quot; Second<br /> |-<br /> |12<br /> | colspan=&quot;2&quot; |588<br /> | colspan=&quot;2&quot; |Augmented Fourth<br /> |}<br /> &lt;/div&gt;<br /> This topic isn't directly related to the other sections, but I don't have anywhere else to put it. I've long been interested in what I'm calling '''quartertone composition''', which is composition based on the 24-form. It offers both familiarity to 12edo and all the intervals that are the most alien. The most important generator chains are the ones that split 4/3 in half (semiquartal, aesthetically my favorite, see [[Intergan]]) and split 3/2 in half (mosh or dicoid). Together, they can be analyzed as a diatonic generator chain that can be deviated from by either a semifourth or hemififth. Quartertone tunings may have one chain or both.<br /> <br /> I fond it very useful to examine tertian functions by stacking different combinations inframinor/minor/neutral/major/ultramajor thirds.<br /> <br /> TODO: List<br /> <br /> === Scale theory ===<br /> <br /> One of the best ways I've found to introduce quartertone intervals into the diatonic framework is by altering either B or F (assuming all natural notes) by a quartertone. This results in a scale with only one fewer perfect fifth, since the other fifth containing the altered note is diminished. This gives four options:<br /> * A Bd C D E F G A (-&gt; mosh)<br /> * A Bt C D E F G A (-&gt; sq)<br /> * A B C D E Fd G A (-&gt; sq)<br /> * A B C D E Ft G A (-&gt; mosh)<br /> These introduce &quot;dimimajor&quot;, &quot;diminimor&quot;, and neutral chords for the mosh-leaning scales and &quot;small/large subdim&quot; and inframinor/ultramajor chords for the semiquartal-leaning scales.<br /> <br /> === Scale tempering ===<br /> <br /> Having two separate generator chains is ideal, but it can be inconvenient. This can be fixed by using a half-octave period instead of an octave, where the semitwelfth and hemififth differ by half an octave. {{adv|I would argue that the most important quartertone temperament overall is '''Island Rastmic''', a half-octave 24&amp;34 2.3.11.13/5 subgroup temperament that splits 4/3 into two 15/13s and 3/2 into two 11/9s. Like in Intergan, all the quartertone intervals don't leave much room to introduce comma steps if prime 5 is desired, so a mild Meantone tempering is the best option. In addition, I find this tuning range around Mohajira to have the most pleasant-sounding neutral triads. 17/12 may be equated to half an octave and 19/17 to 9/8 as in Intergan. The final result is 2.3.5.11.13.17.19 24&amp;62.}}<br /> <br /> In the accompanying generator table, diatonic intervals are aligned, while adjacent quartertone intervals always differ by 600¢. Primes are highlighted. {{adv|The top half of the table shows an optional 7 (the simplest mapping is in the hemififth chain due to the semitwelfth already being so close to 7/4, but it's in the wrong direction) and an alternate sharp 11, 17, and 19 all found in Intergan, but none of these were included in 2.3.5.11.13.17.19 24&amp;62. The flat 17 and 19 are shown in parenthesis because they differ from diatonic intervals by half an octave.}} My proposed interval names are extrapolated from existing names in order to be as unambiguous as possible.<br /> <br /> This way of displaying intervals in a half-octave temperament, having two full-octave generators that differ by half an octave, is mostly similar to the standard way. The columns don't correspond to which of the two periods the intervals are in, but it's simple enough to guess because the smaller one is first period and the larger one is second period. The major difference is that intervals an even number of generators from unison only show one interval in one period. For temperament reasons, I included some of the ones not shown in parenthesis. This matters a lot because it ignores an important Diaschismic equivalence: sqrt(2) / (9/8) ≈ 5/4.<br /> <br /> This may result in a new way of generating scales. An Aeolian diatonic scale ranges from -4 to 2 generators. This temperament divides the generator in half, so take all intervals in the table from -8 to 4 generators, ignoring anything in parenthesis. The resulting scale has 19 notes: 55 153 196 251 306 349 447 502 557 655 698 753 808 851 949 1004 1047 1157 1200. No more notes can be added without introducing some extremely small steps to the scale, which would be tempered out in 24edo. The scale has four unique step sizes: 43 55 98 110. There is only one step of 110¢, which can be removed by replacing 1047 or 1157 (the most extreme two intervals unique to the 349¢ column) with 1059 or 1145 (the next two intervals unique 949¢ column). This results in the two chiralities of 5L9m5s. Sharpening the two generators to 950¢ and 350¢ results in 5L14s, and further sharpening them flips m and s to 5L5m9s.<br /> <br /> === Melodic intervals ===<br /> <br /> The diatonic half of quartertone composition does not need an explanation. Semiquartal and dicoid have their own compositional practices, much of which can be inherited from diasem and blackdye respectively due to being degenerate cases. What's left to explain is how this all fits together.<br /> * The quartertone has function similar to a wide aberrisma and a narrow semitone, since semiquartal is just diasem with the two equated. It's useful for altering intervals like an aberrisma and has a distinctive &quot;metallic&quot; sound as a semitone.<br /> * The neutral second is best known for occurring in dicoid and antidiatonic, which it can take most of its function from. It has a distinctive &quot;sour&quot; sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.<br /> * The semifourth is most useful as an inframinor third, but it also works as an ultramajor second. It pairs best with wider minor thirds which make it sound more like a second in comparison. It's useful for shrinking the semitone between the second and third without shrinking the third.<br /> <br /> == Tertian categorization ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Thirds categorized as a fraction of a diatonic fifth.<br /> !Fraction f<br /> !*7/12 =<br /> !4\7 (¢)<br /> !7\12 (¢)<br /> !3/2 (¢)<br /> !3\5 (¢)<br /> !Lower bound of:<br /> |-<br /> !1/3<br /> |7/36<br /> |228.6<br /> |233.3<br /> |234.0<br /> |240.0<br /> |&quot;Horiminor&quot;<br /> |-<br /> !12/35<br /> |1/5<br /> |235.1<br /> |240.0<br /> |240.7<br /> |246.9<br /> |Inframinor<br /> |-<br /> !18/49<br /> |3/14<br /> |251.9<br /> |257.1<br /> |257.9<br /> |264.5<br /> |Subminor<br /> |-<br /> !11/28<br /> |11/48<br /> |269.4<br /> |275.0<br /> |275.8<br /> |282.9<br /> |Neominor<br /> |-<br /> !59/140<br /> |59/240<br /> |289.0<br /> |295.0<br /> |295.8<br /> |303.4<br /> |Grave Minor<br /> |-<br /> !61/140<br /> |61/240<br /> |298.8<br /> |305.0<br /> |305.9<br /> |313.7<br /> |Classic Minor<br /> |-<br /> !23/49<br /> |23/84<br /> |321.9<br /> |328.6<br /> |329.5<br /> |338.0<br /> |Supraminor<br /> |-<br /> !24/49<br /> |2/7<br /> |335.9<br /> |342.9<br /> |343.8<br /> |352.7<br /> |Neutral<br /> |-<br /> !25/49<br /> |25/84<br /> |349.9<br /> |357.1<br /> |358.1<br /> |367.3<br /> |Submajor<br /> |-<br /> !26/49<br /> |13/42<br /> |363.8<br /> |371.4<br /> |372.5<br /> |382.0<br /> |Classic Major<br /> |-<br /> !79/140<br /> |79/240<br /> |386.9<br /> |395.0<br /> |396.1<br /> |406.3<br /> |Acute Major<br /> |-<br /> !81/140<br /> |81/240<br /> |396.7<br /> |405.0<br /> |406.1<br /> |416.6<br /> |Neomajor<br /> |-<br /> !17/28<br /> |17/48<br /> |416.3<br /> |425.0<br /> |426.2<br /> |437.1<br /> |Supermajor<br /> |-<br /> !31/49<br /> |31/84<br /> |433.8<br /> |442.9<br /> |444.1<br /> |455.5<br /> |Ultramajor<br /> |-<br /> !23/35<br /> |23/60<br /> |450.6<br /> |460.0<br /> |461.3<br /> |473.1<br /> |&quot;Horimajor&quot;<br /> |-<br /> !2/3<br /> |7/18<br /> |457.1<br /> |466.7<br /> |468.0<br /> |480.0<br /> |[end of range]<br /> |}<br /> &lt;/div&gt;<br /> The accompanying table shows my own way of categorizing triads and tetrads bounded by ~3/2, with the third being a logarithmic fraction of the fifth. {{adv|Note that the thirds are presented as a logarithmic fraction of the fifth, which scales the major and minor thirds by the same ratio to fit the fifth. Delta-rational logarithmically scales the lower third in the triad slightly more than the upper third.}} This is not an exhaustive list of qualities, but rather groups of my major use cases.<br /> <br /> * I've proposed the terms &quot;horiminor&quot; and &quot;horimajor&quot; after the same root in the word &quot;horizon&quot;. These are the chords that are technically possible to hear as containing thirds, but it requires special care in the melody to make it convincing, such as splitting the horiminor third. I find this class especially useful in 10edo, which is often considered to only have neutral chords.<br /> * Inframinor and ultramajor chords share the metallic quality with the quartertone, as mentioned in the previous section. They are similar in function to the subminor and supermajor chords based on 6:7:9, but have a harsher and exaggerated sound, usually interpreted as based on 10:13:15.<br /> * Subminor and supermajor chords are in the vicinity of 6:7:9 and 14:18:21. Subminor especially has a pure sound like 4:5:6, but these chords may sound too different from Western tuning to be desirable in many cases.<br /> * Neominor and neomajor are used in this case to refer to any thirds not in the concordance wells of 6:7:9 or 4:5:6. This gives them a sound that I would describe as impure but even. They're the most versatile for transcribing modern 12edo music while remaining somewhat xenharmonic. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.<br /> * Grave minor and acute major chords are approximately the ones found in 12edo, around 16:19:24. I consider these to be a separate category less often than any other, but this presence of this otonal chord justifies them. A grave minor third is also characteristic of [[Hibernal]].<br /> * Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15, so they may also be called pental. They are considered to be the default in xenharmonic circles, and have been used in Western music for hundreds of years at least, largely falling out of consideration once 12edo became the standard. I think it's always good to have these to some extent because they provide a consonance that can anchor everything else.<br /> * Supraminor and submajor chords lean towards, but I find them to work better as a consonance because they still resemble minor and major functions. They're usually interpreted as based on 14:17:21.<br /> * Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered largely concordant, between 24edo and 31edo. This is where two slightly sharp 11/9s stack to make a slightly flat 3/2. In this range, they sound like ambiguous minor/major chords, which may be useful for modulation or when unsure of which quality is better for a given chord. More broadly, they function as an extremely versatile mild dissonance. People tend to see them as a cross between major and minor, but I actually see them as a cross between major, minor, diminished, and augmented. They're weird as a consonance in a quasi-diatonic context, not because of the complexity of 18:22:27, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.<br /> ** I call this technique a &quot;neutral substitution&quot;. It involves replacing a tertian chord with a neutral one in order to modulate. The easiest way to do this is line up a neutral chord with the root or fifth of where a diminished chord would be, then use whichever of the two is changed to alter the key.<br /> <br /> === Grave fifth chords and mainline fifths ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Successive fifths from 22/15 to 40/27,&lt;br&gt;thirds between logarithmic 12/35 and 23/35 of the fifth<br /> ! style=&quot;border-right: double;&quot; |D<br /> ! colspan=&quot;5&quot; style=&quot;border-right: double;&quot; |N<br /> !&lt;3/7f<br /> !&lt;1/2f<br /> !&gt;1/2f<br /> !&gt;4/7f<br /> ! style=&quot;border-right: double;&quot; |Fifth<br /> ! colspan=&quot;4&quot; |Fifth Complement<br /> |-<br /> ! style=&quot;border-right: double;&quot; |15<br /> |<br /> |18<br /> |<br /> | style=&quot;border-right: double;&quot; |19<br /> | style=&quot;border-right: double;&quot; |22<br /> |<br /> |315.6<br /> |<br /> | style=&quot;border-right: double;&quot; |409.2<br /> | style=&quot;border-right: double;&quot; |663.0<br /> |<br /> |347.4<br /> |<br /> |253.8<br /> |-<br /> ! style=&quot;border-right: double;&quot; |17<br /> |<br /> | class=&quot;thl&quot; |20<br /> |21<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |25<br /> |<br /> | class=&quot;thl&quot; |281.4<br /> |365.8<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |667.7<br /> |<br /> | class=&quot;thl&quot; |386.3<br /> |301.8<br /> |<br /> |-<br /> ! style=&quot;border-right: double;&quot; |19<br /> |22<br /> |23<br /> |<br /> | style=&quot;border-right: double;&quot; |24<br /> | style=&quot;border-right: double;&quot; |28<br /> |253.8<br /> |330.8<br /> |<br /> | style=&quot;border-right: double;&quot; |404.4<br /> | style=&quot;border-right: double;&quot; |671.3<br /> |417.5<br /> |340.6<br /> |<br /> |266.9<br /> |-<br /> ! style=&quot;border-right: double;&quot; |21<br /> |24<br /> | class=&quot;thl&quot; |25<br /> | class=&quot;thl&quot; |26<br /> | style=&quot;border-right: double;&quot; |27<br /> | style=&quot;border-right: double;&quot; |31<br /> |231.2<br /> | class=&quot;thl&quot; |301.8<br /> | class=&quot;thl&quot; |369.7<br /> | style=&quot;border-right: double;&quot; |435.1<br /> | style=&quot;border-right: double;&quot; |674.3<br /> |443.1<br /> | class=&quot;thl&quot; |372.4<br /> | class=&quot;thl&quot; |304.5<br /> |239.2<br /> |-<br /> ! style=&quot;border-right: double;&quot; |23<br /> |<br /> |27<br /> |28<br /> | style=&quot;border-right: double;&quot; |29<br /> | style=&quot;border-right: double;&quot; |34<br /> |<br /> |277.6<br /> |340.6<br /> | style=&quot;border-right: double;&quot; |401.3<br /> | style=&quot;border-right: double;&quot; |676.7<br /> |<br /> |399.1<br /> |336.1<br /> |275.4<br /> |-<br /> ! style=&quot;border-right: double;&quot; |25<br /> |29<br /> |30<br /> |31<br /> | style=&quot;border-right: double;&quot; |32<br /> | style=&quot;border-right: double;&quot; |37<br /> |256.9<br /> |315.6<br /> |372.4<br /> | style=&quot;border-right: double;&quot; |427.4<br /> | style=&quot;border-right: double;&quot; |678.7<br /> |421.8<br /> |363.1<br /> |306.3<br /> |251.3<br /> |-<br /> ! style=&quot;border-right: double;&quot; |27<br /> |31<br /> | class=&quot;thl&quot; |32<br /> |33<br /> | style=&quot;border-right: double;&quot; |34<br /> | style=&quot;border-right: double;&quot; |40<br /> |239.2<br /> | class=&quot;thl&quot; |294.1<br /> |347.4<br /> | style=&quot;border-right: double;&quot; |399.1<br /> | style=&quot;border-right: double;&quot; |680.4<br /> |441.3<br /> | class=&quot;thl&quot; |386.3<br /> |333.0<br /> |281.4<br /> |}<br /> &lt;/div&gt;<br /> I use a similar system for antidiatonic tertian harmony, but there's no set JI approximation for a flat fifth, so I use multiple to get the simplest ratios possible. [[Pentagoth]] equates many similar intervals in this table.<br /> <br /> I refer to the range from about 23\40 to 25\42 as '''mainline fifths'''. The full diatonic range includes other fifths that generate scales I have found over time to resemble the usual understanding of diatonic less than they resemble 7 or 5edo, and resemble 3/2 less than sqrt(20/9) and sqrt(16/7). The 25\42 boundary is clearer to me because I prefer very hard scales to very soft ones and thus I have more experience. But conveniently, these are the points where an alternate diatonic fifth can be generated at a low complexity. Analogously, I could refer to the range from about 11\20 to 17\30 as '''mainline antidiatonic fifths''' based on my personal experience.<br /> <br /> I've found 25 and 39edo to be the two regions of the armotonic tuning range that sound better than others. {{adv|29edo sounds more pure than both, and is the basis for an excellent 29&amp;67 straddle temperament I've called Gravel, but it involves the 15:19:22 chord with a fifth at my limit for resembling 3/2 which does not generate armotonic. 2.15.11.19 9&amp;56 also generates this, and the subgroup can be extended to 2.3.5.11.19 while hardly changing the edo list. 11:13:16 and 13:16:19 are proper subfifth triads and are not counted in the range. They are also nearly retroversions of each other.}}<br /> <br /> 25edo represents the two otonal chords on the table with 5/4, with flatter fifths like 34 and 43edo being better for 17:20:25 and sharper fifths like 41 and 57edo being better for 27:32:40. Both of these are found in Pentagoth, so I propose calling this range '''pentagothic'''. 43edo is especially notable for how well it also approximates 17:21:25.<br /> <br /> 39edo represents 21:26:31, which is notably a +1+1 chord, and 21:25:31, very close to its retroversion. I propose calling this range '''mavilic''' after Erv Wilson's original Meta-Mavila, the &quot;proper&quot; tuning range for Mavila. {{adv|Mavila possibly has more variants of its name than any other temperament. I'm genuinely surprised that Mavilic isn't a temperament name.}} [[Pentagoth#Otomavila|Otomavila]] is my temperament using these chords.<br /> <br /> For the most part, antidiatonic tertian harmony works like diatonic tertian harmony, but with the reversed qualities of the antidiatonic scale. The pentagothic neutral triad generates Cohemimabila temperament and has a similar function to the neutral triad explained above.<br /> <br /> === Quasi-diatonic scales ===<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Blackdye and diasem JI tunings, sorted from flat to sharp minor third, denominators up to 20<br /> !Tertian Triad<br /> !16:21:24<br /> !20:23:30<br /> !10:13:15<br /> !6:7:9&lt;br&gt;(14:18:21)<br /> !18:23:27<br /> !16:19:24<br /> !4:5:6&lt;br&gt;(10:12:15)<br /> !14:17:21<br /> !18:22:27<br /> |-<br /> |Subgroup<br /> |2.3.7<br /> |2.3.23/5<br /> |2.3.13/5<br /> |2.3.7<br /> |2.3.23<br /> |2.3.19<br /> |2.3.5<br /> |2.3.17/7<br /> |2.3.11<br /> |-<br /> |Name<br /> |&quot;Septal&quot;<br /> |?<br /> |?<br /> |&quot;Septal&quot;<br /> |&quot;Eridial&quot;<br /> |&quot;Novemdecal&quot;<br /> |Pental<br /> |?<br /> |&quot;Undecal&quot;<br /> |-<br /> |Aberrisma<br /> |28/27<br /> |640/621<br /> |416/405<br /> |64/63<br /> |736/729<br /> |513/512<br /> |81/80<br /> |459/448<br /> |33/32<br /> |-<br /> |Quasi-&lt;br&gt;Aeolian<br /> |9/8&lt;br&gt;8/7&lt;br&gt;9/7&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;12/7&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;23/20&lt;br&gt;207/160&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;23/15&lt;br&gt;69/40&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;15/13&lt;br&gt;135/104&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;20/13&lt;br&gt;45/26&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;7/6&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;14/9&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |513/512&lt;br&gt;9/8&lt;br&gt;19/16&lt;br&gt;4/3&lt;br&gt;171/128&lt;br&gt;3/2&lt;br&gt;19/12&lt;br&gt;16/9&lt;br&gt;57/32&lt;br&gt;2/1<br /> |81/80&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |459/448&lt;br&gt;9/8&lt;br&gt;17/14&lt;br&gt;4/3&lt;br&gt;153/112&lt;br&gt;3/2&lt;br&gt;34/21&lt;br&gt;16/9&lt;br&gt;51/28&lt;br&gt;2/1<br /> |33/32&lt;br&gt;9/8&lt;br&gt;11/9&lt;br&gt;4/3&lt;br&gt;11/8&lt;br&gt;3/2&lt;br&gt;33/20&lt;br&gt;16/9&lt;br&gt;11/6&lt;br&gt;2/1<br /> |-<br /> |Similar&lt;br&gt;Unlisted&lt;br&gt;Aberrismas<br /> | colspan=&quot;2&quot; |17/13: 1088/1053&lt;br&gt;(swap m/s)<br /> |17/11: 1408/1377&lt;br&gt;19/11: 304/297<br /> |<br /> |25: 2048/2025&lt;br&gt;11/7: 896/891&lt;br&gt;13/11: 352/351&lt;br&gt;17/5: 136/135&lt;br&gt;23/13: 208/207<br /> |25/7: 225/224&lt;br&gt;17: 4131/4096&lt;br&gt;19/5: 1216/1215&lt;br&gt;(diasem)<br /> |<br /> |23/7: 189/184&lt;br&gt;23/19: 621/608<br /> |13: 1053/1024<br /> |}<br /> The diatonic scale and its tertian harmony are the framework of all Western music. It makes sense that using the lowest prime as the period and second lowest prime as the generator would produce a versatile scale, and the fact that it clusters around 12edo rather than 3edo in the case of prime 5 produces more useful step sizes. Modifying this into an aberrismic scale containing the simplest tertian triad 4:5:6 and its retroversion in as many places as possible produces blackdye, and doing the same with 6:7:9 produces diasem. These are the main scales I use to represent my take on Western music: melodically interesting, harmonically pure, and not far from diatonic. However, there are other tunings of them that are useful in different places. These are generalized forms and their aberrismas may be tempered out for a true diatonic scale. 64/63 and 81/80 are common, but 736/729 and especially 513/512 are smaller and make more sense to temper out. If not, inflating them is recommended to keep the melodic significance of the aberrisma.<br /> <br /> '''Eridial''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named after &quot;eridian&quot;, the demonym for Eris, with the -al suffix of subgroups. Eris is probably one of the most goth deities of Ancient Greece and is also associated with the number 23 by Discordians. It targets 18:23:27, a little-known and surprisingly concordant neogothic tertian triad. Many microtonalists contend that 12edo pop music, while historically tracing back to Meantone, tends to sound better in Pythagorean tuning. I can really hear it after comparing different quasi-diatonic options for retuning my melodies. I've taken a particular liking to gentle/neogothic diatonic scales in the past for this reason, but I believe eridian diasem is a further refinement. As with diasem in general, I prefer this with the 3 tuned slightly flat to inflate the aberrisma, although it is especially important in this case because 736/729 has a size of only 17¢. When inflated in tunings such as 43edo, the aberrisma is subtle enough to not disturb the Pythagorean sound much, but impactful in cases like the subminor seventh, which is tuned close to 7/4. Its melody is not particularly xenharmonic, but it's one of my default scales when available because it works for almost everything. It works well as an extension to [[Septimal meantone]]. {{adv|The difference between eridian diasem's analogous 2.3.25 and 2.3.17/5 thirds (e.g. 75/64 and 20/17) is 256/255, also the difference between 16/15 and 17/16, supporting it as one of the most important 17-limit commas.}}<br /> <br /> The default set of qualities to consider is septal and pental due to their simplicity, possibly with a Pythagorean diatonic scale in the middle as found in Aberschismic edos. But I've found that I particularly like shifting both septal and pental towards neutral, to eridian and exaggerated pental. This is probably because it puts both of them about an equal distance from diatonic when using a prime 3 within a few cents of just. Eridian provides a more versatile sound and pental sounds more interesting while still working about the same. In some edos that do this such as 48 and 86, the pental tuning is Negri, meaning that semiquartal is also available.<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Cursed Eridian Scales<br /> !Standard&lt;br&gt;(23/18)(27/23)<br /> !Wolf → Perfect&lt;br&gt;(23/18)(32/27)<br /> !the other one&lt;br&gt;(81/64)(27/23)<br /> |-<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |398688256/387420489&lt;br&gt;67712/59049&lt;br&gt;32/27&lt;br&gt;243/184&lt;br&gt;2166784/1594323&lt;br&gt;368/243&lt;br&gt;36/23&lt;br&gt;59049/33856&lt;br&gt;11776/6561&lt;br&gt;2/1<br /> |4782969/4333568&lt;br&gt;27/23&lt;br&gt;129140163/99672064&lt;br&gt;2944/2187&lt;br&gt;2187/1472&lt;br&gt;128/81&lt;br&gt;59049/33856&lt;br&gt;8667136/4782969&lt;br&gt;2/1<br /> |}<br /> &lt;/div&gt;<br /> 27/23 is close enough to 32/27 that it can be replaced and result in another 2.3.23 blackdye scale with 32/27*23/18 = 368/243 triads. The fifth is the same as the existing Eridian wolf fifth, which is barely diatonic in JI. I might call this '''hiberidial''' because it applies the principle of [[Hibernal]] to the eridian subgroup. 82edo is the near-perfect 2.3.23 tuning, but 20 and 37edo have a close approximation of the scale despite not being good for the subgroup. Switching thirds to 32/27 or 81/64 to remain in the subgroup with a wolf fifth as the main one can be applied to other scales in the table.<br /> <br /> I have a lot more to write about how blackdye and diasem work harmonically.<br /> <br /> == Aberrismic details ==<br /> <br /> Note that this section is incomplete.<br /> <br /> As described in the [[Monarch]] article, I like to tune pental blackdye (3sA Aeolian: tempered 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1) with a sharp 3 and thus inflated 81/80. When extreme, more interesting intervals show up along with new problems. The first is what is seen in 11-limit Superpyth and Porcupine (55/54), turning 27/20 into 11/8, although it could be interpreted as any other superfourth. I like this interval a lot because the 1/1-9/8-6/5-27/20 tetrachord is a natural-sounding stretched version of the diatonic minor tetrachord. In fact, it may sound better melodically to use the superfourth rather than the perfect fourth to avoid the very deflated 10/9 between 6/5 and 4/3. It would be beneficial to have a triad on 27/20 to replace the one on 4/3, and this makes use of the 81/80 in the next octave, the least harmonically useful note in the scale. 32/27 is available as the more conventional quasi-diatonic third, but also 320/243, one of the least recognizeable intervals that all correspond to septal intervals in Aberschismic temperament (5120/5103). 320/243 is 476.539¢ in JI and 729/640 is 225.416¢, but these approach each other with a sharper 3. Most notably in this tuning range, tempering out 2187/2240 equates them to 9/7 and 7/6, different from the same ratios that would normally be generated by Archy. Interpreting 320/243 as 14/11 allows for 8:11:14. The exact tempering is irrelevant as long as they're some sort of supermajor and subminor thirds.<br /> <br /> The size of the aberrisma presents other issues.<br /> <br /> == Song structure ==<br /> <br /> This is my system for categorizing the songs I write by structure. Something similar can be applied to other people's music, but the standards will be different due to the unusual complexity of my music.<br /> <br /> * '''straightforward''': The simplest type, being in a similar style throughout. Always works.<br /> * '''opposing sections''': Two straightforward parts that are very different.<br /> * '''medium''': Somewhat straightforward, somewhat disjointed.<br /> * '''episodic''': Containing many different sections with themes and a progression throughout the song. The most desirable when done well and in moderation.<br /> * '''disjointed''': Containing many different sections with no clear pattern. The least desirable.<br /> <br /> I've noticed that I write more &quot;disjointed&quot; songs than I would like, and fewer &quot;straightforward&quot; and &quot;episodic&quot;. This system is intended to keep track of what I'm doing and help me work toward a better result.<br /> <br /> {| class=&quot;wikitable sortable&quot;<br /> |+Song Structure Types of The Gralbums<br /> !Track<br /> !Title<br /> !Type<br /> |-<br /> |g1t1<br /> |The Lake Reflects a Black Sky<br /> |'''disjointed'''<br /> |-<br /> |g1t2<br /> |The Art of It All<br /> |'''episodic'''<br /> |-<br /> |g1t3<br /> |Rail Twinge<br /> |'''straightforward'''<br /> |-<br /> |g1t4<br /> |Back Stalk<br /> |'''medium'''<br /> |-<br /> |g1t5<br /> |Superior Intermedial<br /> |'''episodic'''<br /> |-<br /> |g1t6<br /> |Snowboard Index<br /> |'''straightforward'''<br /> |-<br /> |g1t7<br /> |Winter's Mortal Hope<br /> |'''medium'''<br /> |-<br /> |g1t8<br /> |Transpiration<br /> |'''opposing sections'''<br /> |-<br /> |g1t9<br /> |Revelation of Your Forever<br /> |'''medium'''<br /> |-<br /> |g2t1<br /> |The Life Unreachable<br /> |'''disjointed'''<br /> |-<br /> |g2t2<br /> |Monolithium<br /> |'''opposing sections'''<br /> |-<br /> |g2t3<br /> |Nocturne Paranoia<br /> |'''disjointed'''<br /> |-<br /> |g2t4<br /> |Glimmer Extrication<br /> |'''episodic'''<br /> |-<br /> |g2t5<br /> |Not This Time<br /> |'''straightforward'''<br /> |-<br /> |g2t6<br /> |Sakura Blade Minivan<br /> |'''medium'''<br /> |-<br /> |g2t7<br /> |Resolute Prelude<br /> |'''straightforward'''<br /> |-<br /> |g2t8<br /> |Life and Limb<br /> |'''straightforward'''<br /> |-<br /> |g2t9<br /> |Residual Soliloquy<br /> |'''disjointed'''<br /> |}</div>

Ground https://xenreference.com/wiki/index.php?title=Ground%27s_composition_theory&diff=8492 2026-10-26T17:18:15Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> This article is about composition and the practical considerations of tuning. I always have more content to add and revise. It has an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]], which are both based on my concept of an '''altered unison''', which is categorically a unison but doesn't sound like one. An aberrisma generally equivalent to an altered unison, and straddled intervals are separated by one.<br /> <br /> It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the [[Levels of tempering|mesotempered category]]. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.<br /> <br /> == Interval logic deviation theory ==<br /> <br /> Much of my music has had a distinctively shifting tonality since 2018 or earlier, which started in 12edo. This article is an attempt to explain how it works. I'm introducing a placeholder term for it, '''interval logic deviation''' theory (ILD), which can be replaced if this turns out to be something already described.<br /> <br /> === Local tonality ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic example of probabilities (up to word length 3)<br /> !Sequence<br /> !Next step probability<br /> |-<br /> |s<br /> |L 1/1<br /> |-<br /> |L<br /> |L 5/7, s 2/7<br /> |-<br /> |sL<br /> |L 1/1<br /> |-<br /> |Ls<br /> |L 1/1<br /> |-<br /> |sLL<br /> |s 1/2, L 1/2<br /> |-<br /> |LsL<br /> |L 1/1<br /> |-<br /> |LLs<br /> |L 1/1<br /> |-<br /> |LLL<br /> |s 2/3, L 1/3<br /> |}<br /> &lt;/div&gt;<br /> I have a simultaneous regard and disregard for standard Western tonality. This is because I view it as an important but strictly local property, meaning it fundamentally only applies on the scope of a single path from tension to release, however long that is. Thus, modulations are only generally uncommon because phrases usually resolve to the same key center they started from, but changing tonality is just as much of a choice as not changing it. Modulation flows just like any other melodic or harmonic movement.<br /> <br /> This flow is facilitated by ILD, in which scales aren't a fixed set of notes, but a template for interval logic to be rearranged and deviated from. As such, their main features are probabilities in an interval sequence and &quot;bubble deviations&quot; from that interval sequence. ILD is best for music with a strong melodic focus, such as mine, where the melody informs the harmony instead of the reverse. Other concepts may be used instead with the same general goal.<br /> <br /> Melodic interval sequences, or &quot;words&quot; of step sizes, are the most minimal expression of tonal tension and release. For example, if a diatonic melody were to play C then B, the listener is likely to expect A to be next and feel a small resolution upon hearing it. This is the descending sL word. Melodies are full of small sequences like this based on the scale that they are in. It's possible to use sequences with notes outside the scale while still feeling like they belong, and how much they belong can be predicted. The longer the word and the greater probability of occurring indicates that it's more likely to sound like it belongs in the scale.<br /> <br /> === Axes of deviation ===<br /> <br /> &quot;Bubble deviations&quot; in ILD are named after the bubble sort algorithm, which repeatedly swaps adjacent items in an array. Steps in a base scale can be swapped in the same way to modify the scale with no requirement for a clear structure like a generator chain or lattice splotch, which is my term for a collection of generator chains in aberrismic theory. Suppose you want the scale word sLs in diatonic. This would require only one bubble deviation from the expected step order, turning sLLs to sLsL. While the probability of encountering it in the base diatonic scale is zero, it sounds more &quot;probable&quot; (less unexpected) than something like ssL.<br /> <br /> This explains why I used Dimininished[8] in 12edo more often than Augmented[6], because its stepwise interval logic has less deviation from diatonic. Diminished[8] is made of a repeating sequence of 1\12 and 2\12, common in diatonic, whereas Augmented[6]'s steps of 1\12 and 3\12 do not occur in diatonic at all. As a result, melodies in Diminished sound less exotic.<br /> <br /> Bubble deviations are only one axis of deviation. There is another axis which I've found to be exclusively useful in tuning systems with aberrismic-sized steps or smaller: the axis of microtonal deviation from expected intervals. This involves changing the pitch of an expected interval only slightly, so it is heard as a variation of the expected interval rather than a different interval entirely. This axis interacts with the base scale by introducing or modifying an aberrismic offset, for example diatonic being diasem or blackdye with the offset removed, 2.3.7 diasem having a larger offset than 2.3.23, or 2.3.5 blackdye having a smaller offset than 2.3.17/7. The intervals affected by the offset, usually thirds and sixths, differ microtonally when the offset is changed.<br /> <br /> Straddling intervals that are stacked the most (usually 3/2) introduce a third axis that can be simplified into a combination of the other two. It's possible to have bubble deviation from a scale that isn't even in the tuning system being used, like how alternating &lt;&lt;3 and &gt;3 in 37edo straddles 74edo meantone and results in trackdye.<br /> <br /> {{UserTag|KC|Inthar|000000|[[Stretching and compression]] constitute yet another axis of deviation separate from straddling. Diatonic-based example: This is important in 4L3s and 5L3s which are warped-diatonic MOSes. Note that dual-3 diatonic 5L1m1s is a subset of interleaved diatonic 7s(5L2m), tens-interleaved diatonic 6s(5L2m) ''and'' tract-interleaved diatonic 8s(5L2m).}}<br /> <br /> === Vague interval logic ===<br /> <br /> Scales are useful for ILD, but not technically necessary. One may pick the desired step sizes and find just a few arrangements leading to useful intervals, like the perfect fifth. This should be especially useful when trying to avoid making quasi-diatonic music in any tuning system.<br /> <br /> Take 3\24 s and 5\24 L for example. The words LL and sssL make 10\24 and 14\24 respectively, so the two can together be used to infer vague probabilities. Edos that straddle important intervals are also useful for this because they increase the chances of landing on those intervals. If s and L were replaced with 11\86 and 18\86, LL becomes &gt;4/3 and sssL becomes &gt;3/2.<br /> <br /> === Aberrismic theory ===<br /> <br /> The above concepts also apply to ternary scales, where they may be even more useful. Conventional aberrismic scales feature an alternating generator sequence that creates a 2-dimensional lattice, which can be used similarly to a generator sequence, but is less intuitive due to its complexity. Thus, ILD proves to be a practical alternative when modulating. This is how I write 2.3.5 and 2.3.7 music in tunings that aren't Meantone or Archy.<br /> <br /> Quasi-diatonic aberrismic scales are the ideal case due to the general importance of stacking prime 3 and internalized diatonic logic of Western music. They typically have four main step sizes: '''aberrisma''' (s), '''semitone''' (m), and the two whole tones '''solitone''' (L) and '''magnitone''' (L+s). Other scales or ILD can introduce more sizes such as the '''magnisemitone''' (m+s). The decision of whether to use a solitone or magnitone depends on which intervals in a chord are preferred and which sounds more melodically to the composer in a given situation. The aberrisma is a new class so its theory is less well defined, but it works best as a passing tone. It provides the ability to change the length of scale runs without repeating any notes.<br /> <br /> I find it most generally useful to define the size range of an aberrisma as being between 81/80 and half of 16/15, but my personal usage has ranged from about 12¢ (the size of 81/80 in 5-limit CWE Negri and around one step of the largest full edos I use) to about 100¢ (81/80 in some tunings of Blackwood with a flat 5). I provide 40¢ as the general ideal, but the actual ideal varies. Faster music prefers a larger aberrisma because it is more audible. Scales with a large magnitone can handle a larger aberrisma without it sounding entirely like a semitone, such as 37edo pental blackdye. '''Subaberrismas''', aberrismas small enough that they are not reliably recognizable as a melodic step, have a unique sound that may be desirable in certain situations.<br /> <br /> Because they are often passing tones when used melodically, it can be relatively easy to swap an aberrisma for one of a more preferable size. For example, I may swap 1\43 in 43edo diasem for the semiquartal 2\43 to increase its audibility.<br /> <br /> I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.<br /> <br /> == Diatonic analogues ==<br /> <br /> === 9-note &quot;diatygic&quot; ===<br /> <br /> ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.<br /> <br /> When it comes to extending the diatonic scale's circle of fifths as seen in ILD, there's no alteration more important than the modes of the melodic minor scale, which I'll be referring to together as '''melodiatonic'''. It's the only non-diatonic scale that's possible to construct by making a single bubble deviation from diatonic, and it makes sense looking at the circle of fifths. Assume that some altered diatonic scale must contain only L and s steps to preserve as much step logic as possible. To do this, only ordinals following a semitone may be sharpened (C and F when the unaltered notes are all naturals), and only ordinals preceding a semitone may be flattened (B and E). Sharpening F or flattening B just changes the key signature and the result is still diatonic. Sharpening C or flattening E results in melodiatonic. I have found melodiatonic to be a remarkably reliable way to make diatonic tunings (including 12edo) sound exotic yet familiar enough to hit the same emotional beats diatonic can provide. After diatonic, it has the smallest span across the chain of fifths without duplicating a diatonic ordinal, meaning that it contains no pairs of notes that are 7 generators apart. Just like how diatonic is the framework for all quasi-diatonic scales, melodiatonic can be used as the framework for certain alterations of those quasi-diatonic scales. {{adv|Unconventional uses of melodiatonic I can think of off the top of my head are parts of Undertale's Heartache in Bb Dorian b2 and the main theme of Animusic's Harmonic Voltage in C Mixolydian b6.}}<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Quasi-Diatygic Scales<br /> !Altered&lt;br&gt;Blackdye<br /> !Altered&lt;br&gt;Diasem<br /> |-<br /> |10/9&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;5/4&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;5/3&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |9/8&lt;br&gt;8/7&lt;br&gt;7/6&lt;br&gt;9/7&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;14/9&lt;br&gt;12/7&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |}<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic Tertian Triads in D Gadygic<br /> !Root<br /> !Triads<br /> |-<br /> |D<br /> |minor, Major<br /> |-<br /> |E<br /> |dim., minor<br /> |-<br /> |F<br /> |Major<br /> |-<br /> |F♯<br /> |dim.<br /> |-<br /> |G<br /> |minor, Major<br /> |-<br /> |A<br /> |minor<br /> |-<br /> |B♭<br /> |Major<br /> |-<br /> |B<br /> |dim., minor<br /> |-<br /> |C<br /> |Major<br /> |}<br /> &lt;/div&gt;<br /> The gaps in the fifth chain can be filled to result in a contiguous generator chain of 9 notes rather than 7, allowing some modes to freely switch between major and minor thirds and sixths. This is probably the earliest music theory I've ever consciously developed, dating to 2014 or so ([https://soundcloud.com/groundfault/doodle-62 example]). It has been scale #1 on the [https://allthescales.org/scales.php?n=9 9-note page of AllTheScales] for at least as long, and I tentatively propose calling the scale '''diatygic''' using the seemingly arbitrary -ygic suffix from AllTheScales since it seems like a term I would have chosen at the time. It has stood the test of time and I find it to be the natural extension of the diatonic framework that doesn't require microtuning, so it remains my favorite 12edo-compatible scale. The scale word for the symmetrical mode is LdcdLdcdL, with d and c for diatonic and chromatic semitones. It is called '''Gadygic''' by AllTheScales. I use plenty of extended interval chains when I compose, but diatygic is intended to be a proper albitonic scale with simple self-contained rules, prescriptive rather than descriptive in the sense that it is the new template to be extended and deviated from. {{adv|For some reason, the first examples of this scale I can think of (besides my own music) all come from the brony fandom. Morning in Ponyville, Awesome as I Wanna Be, and Awoken. It should leave no guesswork as to what I was doing in 2014. All of them are coincidentally in the key of D.}}<br /> <br /> I derived diatygic from my own observation of my favorite music, intuition, and the methods above, including that 9 notes is the closest thing to a MOS between 7 and 12 notes. For one thing, MOS scales with generators approaching half an octave will have sizes that are successive odd numbers; the diatonic range jumps from 7 to 12, but the adjacent antidiatonic range follows 7 with 9. Stacking 9 fifths results in a scale with groups of 3 semitones in a row, which is substantially easier to navigate than the 10-note &quot;supermode&quot; in [https://mtosmt.org/issues/mto.11.17.4/mto.11.17.4.temperley.html Temperley 2011], with a group of 5 semitones. I've found it more useful to use various modes of diatygic than to add additional notes, since needing more than two non-diatonic notes at a time is very rare. This is so that the scale can contain the diatonic modes Aeolian to Ionian as subsets, whereas diatygic only manages Aeolian to Mixolydian. I've never used the major seventh degree much myself, despite its high apparent importance in Western music. Even back when I used major modes more than minor, I preferred Mixolydian to Ionian. It's about as important to me as the minor second degree, which is easy to achieve through using modes other than the symmetrical one.<br /> <br /> This extension of diatonic also works for quasi-diatonic scales. {{adv|Pental blackdye is two chains of 5 of 3/2 separated by 10/9. Extending both of those chains by 2 is analogous to what I'm doing to diatonic, but separating them by 81/80 instead makes 5L2m7s. Separating by 10/9 results in a quaternary scale that is very nearly ternary; temper out the Schisma and you get 3L6m5s with a non-standard step order to what I've established in my aberrismic theory.}} The altered pental blackdye is almost the 3sA Aeolian mode with 5/4 and 5/3 added. The retroversion of this can be obtained by adding 6/5 and 8/5 to 3sG Mixolydian, but it's inferior because it lacks the perfect fifth over 6/5. The one additional change is 81/80 to 10/9 to fix the wolf fifth from 9/8 to 5/3. All 12 notes are required to make a chain of pure 3/2 sometimes altered by 81/80 just as blackdye does, except 4/3-16/9 and 27/20-9/5 are redundant. The extra note is kept to keep the scale to a minimum of four unique step sizes, similar to what is done with blackdye. A similar process may be done to septal diasem.<br /> <br /> One of my favorite parts of the scale is a major third on D, a minor third on E, and a major third on F, using notes from D Gadygic.<br /> {| class=&quot;wikitable&quot;<br /> |+Interval Matrix of Gadygic, DR 3:4:5 Meantone<br /> !<br /> !0<br /> !1<br /> !2<br /> !3<br /> !4<br /> !5<br /> !6<br /> !7<br /> !8<br /> !(9)<br /> |-<br /> !0.<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !194.6<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1200<br /> |-<br /> !308.2<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !389.1<br /> |0<br /> |113.6<br /> |308.2<br /> |421.8<br /> |502.7<br /> |616.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !502.7<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |-<br /> !697.3<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !810.9<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |583.7<br /> |697.3<br /> |778.2<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !891.8<br /> |0<br /> |113.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !1005.4<br /> |0<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |}<br /> <br /> === Neutral diatonic extensions ===<br /> <br /> Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.<br /> <br /> A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.<br /> <br /> I believe xenharmony involves a balance of being unusual without being jarring. Neutral seconds around 150¢ are excellent for the former, but are more difficult to fit into a diatonic framework. When given the choice, I prefer to have neutral seconds around 135¢ to fit in place of a semitone and 165¢ to fit in place of a whole tone. Temperaments in small edos are Flattone for the former and Porcupine for the latter.<br /> <br /> When converting a simple subgroup to contain these intervals, it makes sense to use 2.3.5 because it's closer to neutral whereas 2.3.7 is closer to interordinal.<br /> * Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.<br /> * Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.<br /> This is an excellent use case for straddle-3 temperaments. 37edo has mild Mavila and extreme Archy fifths, but their distance from 3/2 is somewhat masked in 5-limit chords due to the 5/4 anchor concordance.<br /> <br /> == Adaptive diatonic modification system ==<br /> <br /> This section is incomplete.<br /> <br /> In order to make modified diatonic scales easier to use and quickly understand the intervals I use in my head, I have a proposal for an interval scheme based on splitting diatonic or quasi-diatonic intervals, somewhat similar to [[Adaptive diatonic interval names|ADIN]].<br /> <br /> First specify which is the default diatonic or quasi-diatonic quality. This is usually pental (4:5:6), pyth, or septal (6:7:9).<br /> <br /> Neutral seconds are about 3/7 to 4/7 of the minor third.<br /> <br /> == Structural tuning selection ==<br /> <br /> Edos are the most common type of tuning system, and the one I use exclusively. They're a simple and discrete set, which makes them very easy to compare to one another and list in the titles of music made with them. This creates a common assumption that they sound more different from each other than they actually do, which I suffer from myself sometimes. Small edos really do have distinct personalities, but as they get larger and more flexible, those differences gradually disappear. I try to guess the tuning system of a piece whenever it isn't stated, and I'm very often wrong.<br /> <br /> I propose that the most important distinction between tuning systems of sufficient flexibility is not how they sound, but how their unique combinations of structures guide the composer. I like to believe that I have sole control over what I write, but my choice of tuning system prioritizes certain scales, chords, and intervals over others by making them easier to use in practice.<br /> <br /> I use large edos in an unusual way. I like to have multiple octaves of range directly in the piano roll, so I access large edos by multiplying smaller ones using midi channels and my custom tuning scripts. This creates a more complex process of tuning selection, involving the combination of two considerations:<br /> # Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?<br /> # Which of those can be multiplied by 2 or 3 to achieve a superset that's enjoyable for the same reasons, but also complements the subset edo well?<br /> {| class=&quot;wikitable&quot;<br /> |+A selection of large edos I use<br /> !Edo<br /> !Justification<br /> |-<br /> | style=&quot;text-align: right;&quot; |27x3&amp;nbsp;=&amp;nbsp;81<br /> |81 is an [[Monarch|Interarch]] edo, which I find useful. 27x2 = 54 is also uniquely bad, an example of poor LCJI approximations providing insufficient advantages to be worth using.<br /> |-<br /> | style=&quot;text-align: right;&quot; |29x3&amp;nbsp;=&amp;nbsp;87<br /> |I don't like 29 enough to use it much on its own, but 87 is my favorite highly accurate LCJI edo, the intersection of Aberschismic and a variant of [[Hemiseven]].<br /> |-<br /> | style=&quot;text-align: right;&quot; |37x2&amp;nbsp;=&amp;nbsp;74<br /> |Note that this is not 37x3 = 111, a much better LCJI edo. 111 offers little to complement the existing structure of 37edo that 74 doesn't already do.<br /> |-<br /> | style=&quot;text-align: right;&quot; |43x2&amp;nbsp;=&amp;nbsp;86<br /> |I found 86 to be the best Interarch edo for my edo multiplication approach. It clearly straddles every prime up to 19 with the possible exception of 13, but it's large enough that their accuracy is not a major issue.<br /> |-<br /> | style=&quot;text-align: right;&quot; |67x2ed4&amp;nbsp;=&amp;nbsp;67<br /> |Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.<br /> |}<br /> <br /> == Edo multiplication ==<br /> <br /> I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the '''base edo''' to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.<br /> <br /> Example tier list of 37edo multiples:<br /> * Base ([[37edo|37]]): S (One of a kind.)<br /> * x2 ([[74edo|74]]): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)<br /> * x3 ([[111edo|111]]): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)<br /> * x6 ([[222edo|222]]): D (Has extra approximations and generators, but too cumbersome to do much.)<br /> This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:<br /> * Base: remainder<br /> * x2: 12%<br /> * x3: 8%<br /> * x6: 4%<br /> <br /> This should show just how important the base edo is. No matter how many intervals you have, large edos or edo multiplication will never be as efficient to write in as an elegant mid-sized edo that does what you want the most. Everything is filtered through the base edo's structure. Because of this, there's no such thing as the perfect edo. At this point, I use whatever I feel like at the time and multiply it by 6 to fill any gaps that might come up.<br /> <br /> When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.<br /> <br /> Edos that are multiples of 6 usually have more than one reasonable way to split them. I will use 144edo to explain my thought process in deciding whether to use 24x6 or 36x4.<br /> * The most important consideration is which base edo the composer enjoys the most. Both options have features that are important to me, but I've written a good amount in 24 because I believe its lack of a good 5 is less important, so it's been proven by time.<br /> * In terms of the number of useful scales available, 36 is superior. To some extent, this comes with having more notes in general.<br /> * x6 has more channels overall, but contains both the x2 and x3 subsets. x4 requires a larger base edo and has only the option to use x2 and x4 notes. Because of the distribution of the number of channels I typically use, I expect x6 to have more practical options.<br /> * x6 is more of a standard for me, so I'm set up better for that.<br /> * I mentioned how I like to make clear switches between x2 and x3 to make it easier to read my piano roll notes. x6 is better for this because x4 has only one distinct subset besides the base edo, x2.<br /> * 36x2 makes accessing 72edo especially convenient because fewer channels is easier. It wins for 11-limit stuff.<br /> * Two important subsets of 144 are 16 and 18. 24x6 has them as x2 and x3, but 36x4 has them as x4 and x1. This is a tie.<br /> * I use 5-form structures a lot, and some of my favorite edos are 43 and 67. Add 5 to bring their 3/2 closer to just, resulting in 48 and 72, 24x2 and x3. This is an elegant way to approximate both 43 and 67.<br /> * Using a smaller edo gets more range in the piano roll, which is useful for quick creative writing.<br /> * 36 has clearer concordances because it's so good at 2.3.7. It's good to have some anchor concordance in the base edo, but 24 also has 3/2. Having 2.3.7 outside the base edo naturally guides composition towards using other channels and makes it hard to forget about them. There's a nice symmetry with 24x6 putting 7/6 and 6/5 on the same channel as well as 9/7 and 5/4.<br /> With all this in mind, 24x6 is clearly the better choice to start with, although it's close.<br /> <br /> === Channel management ===<br /> <br /> Blackdye and diasem structures have a generator of a perfect fifth and some offset for certain notes. Because all of the offsets are in the same direction, using an offset that is part of a superset of the generator doubles the number of distinct channels required. This most notably results in scales where the generator is from the x2 edo and the offset is from the x3 or x6 edo only requiring 4 channels.<br /> <br /> == Anchor concordances ==<br /> <br /> We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an '''anchor concordance''' because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.<br /> <br /> For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.<br /> <br /> I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.<br /> <br /> In 37edo, both 4:5:&gt;6 and 4:5:&lt;&lt;6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides &lt;&lt;6:8:&lt;&lt;&lt;&lt;9 = 6:&gt;&gt;8:&lt;&lt;9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:<br /> * Using the very flat &lt;&lt;3 results in &gt;&gt;16/15, quite audibly wider while keeping some semblance of 4:5:6.<br /> * &lt;&lt;6/5 is found in the [[Pentagoth]] triad, which uses the exaggerated quality of the minor third and the stability of the major third.<br /> <br /> Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:<br /> * ~69edo: DR 4:5:6, 10:12:15<br /> * ~31edo: 5/4 anchor concordance<br /> * ~74edo: DR 3:4:5<br /> * ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)<br /> <br /> == Quartertone composition ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;<br /> |+Island Rastmic with Straddle Primes<br /> !Gens<br /> !949¢<br /> !349¢<br /> ! colspan=&quot;2&quot; |&lt;nowiki&gt;Proposed Names (semi- | hemi-)&lt;/nowiki&gt;<br /> |-<br /> | -12<br /> | colspan=&quot;2&quot; |612<br /> | colspan=&quot;2&quot; |Diminished Fifth<br /> |-<br /> | -11<br /> |361<br /> | class=&quot;thl&quot; |961<br /> |&quot;Semififth&quot;<br /> |&quot;Hemidim.&quot; Seventh<br /> |-<br /> | -10<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |110<br /> | colspan=&quot;2&quot; |Minor Second<br /> |-<br /> | -9<br /> |1059<br /> |459<br /> |&quot;Semithirteenth&quot;<br /> |&quot;Hemidim.&quot; Fourth<br /> |-<br /> | -8<br /> | colspan=&quot;2&quot; |808<br /> | colspan=&quot;2&quot; |Minor Sixth<br /> |-<br /> | -7<br /> | class=&quot;thl&quot; |557<br /> |1157<br /> |&quot;Semiseventh&quot;<br /> |&quot;Hemidim.&quot; Octave<br /> |-<br /> | -6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |306<br /> | colspan=&quot;2&quot; |Minor Third<br /> |-<br /> | -5<br /> |55<br /> |655<br /> |&quot;Semisecond&quot;<br /> |&quot;Hemidim.&quot; Fifth<br /> |-<br /> | -4<br /> | colspan=&quot;2&quot; |1004<br /> | colspan=&quot;2&quot; |Minor Seventh<br /> |-<br /> | -3<br /> |753<br /> |153<br /> |Semitenth<br /> |Neutral Second<br /> |-<br /> | -2<br /> | colspan=&quot;2&quot; |502<br /> | colspan=&quot;2&quot; |Perfect Fourth<br /> |-<br /> | -1<br /> |251<br /> |851<br /> |Semifourth<br /> |Neutral Sixth<br /> |-<br /> |0<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |0<br /> | colspan=&quot;2&quot; |Unison<br /> |-<br /> |1<br /> |949<br /> |349<br /> |Semitwelfth<br /> |Neutral Third<br /> |-<br /> |2<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |698 ''(98)''<br /> | colspan=&quot;2&quot; |Perfect Fifth<br /> |-<br /> |3<br /> |447<br /> |1047<br /> |Semisixth<br /> |Neutral Seventh<br /> |-<br /> |4<br /> | colspan=&quot;2&quot; |196<br /> | colspan=&quot;2&quot; |Major Second<br /> |-<br /> |5<br /> |1145<br /> | class=&quot;thl&quot; |545<br /> |&quot;Semifourteenth&quot;<br /> |&quot;Hemiaug.&quot; Fourth<br /> |-<br /> |6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |894 ''(294)''<br /> | colspan=&quot;2&quot; |Major Sixth<br /> |-<br /> |7<br /> |643<br /> |43<br /> |&quot;Semininth&quot;<br /> |&quot;Hemiaug.&quot; Unison<br /> |-<br /> |8<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |392<br /> | colspan=&quot;2&quot; |Major Third<br /> |-<br /> |9<br /> |141<br /> |741<br /> |&quot;Semithird&quot;<br /> |&quot;Hemiaug.&quot; Fifth<br /> |-<br /> |10<br /> | colspan=&quot;2&quot; |1090<br /> | colspan=&quot;2&quot; |Major Seventh<br /> |-<br /> |11<br /> | class=&quot;thl&quot; |839<br /> |239<br /> |&quot;Semieleventh&quot;<br /> |&quot;Hemiaug.&quot; Second<br /> |-<br /> |12<br /> | colspan=&quot;2&quot; |588<br /> | colspan=&quot;2&quot; |Augmented Fourth<br /> |}<br /> &lt;/div&gt;<br /> This topic isn't directly related to the other sections, but I don't have anywhere else to put it. I've long been interested in what I'm calling '''quartertone composition''', which is composition based on the 24-form. It offers both familiarity to 12edo and all the intervals that are the most alien. The most important generator chains are the ones that split 4/3 in half (semiquartal, aesthetically my favorite, see [[Intergan]]) and split 3/2 in half (mosh or dicoid). Together, they can be analyzed as a diatonic generator chain that can be deviated from by either a semifourth or hemififth. Quartertone tunings may have one chain or both.<br /> <br /> I fond it very useful to examine tertian functions by stacking different combinations inframinor/minor/neutral/major/ultramajor thirds.<br /> <br /> TODO: List<br /> <br /> === Scale theory ===<br /> <br /> One of the best ways I've found to introduce quartertone intervals into the diatonic framework is by altering either B or F (assuming all natural notes) by a quartertone. This results in a scale with only one fewer perfect fifth, since the other fifth containing the altered note is diminished. This gives four options:<br /> * A Bd C D E F G A (-&gt; mosh)<br /> * A Bt C D E F G A (-&gt; sq)<br /> * A B C D E Fd G A (-&gt; sq)<br /> * A B C D E Ft G A (-&gt; mosh)<br /> These introduce &quot;dimimajor&quot;, &quot;diminimor&quot;, and neutral chords for the mosh-leaning scales and &quot;small/large subdim&quot; and inframinor/ultramajor chords for the semiquartal-leaning scales.<br /> <br /> === Scale tempering ===<br /> <br /> Having two separate generator chains is ideal, but it can be inconvenient. This can be fixed by using a half-octave period instead of an octave, where the semitwelfth and hemififth differ by half an octave. {{adv|I would argue that the most important quartertone temperament overall is '''Island Rastmic''', a half-octave 24&amp;34 2.3.11.13/5 subgroup temperament that splits 4/3 into two 15/13s and 3/2 into two 11/9s. Like in Intergan, all the quartertone intervals don't leave much room to introduce comma steps if prime 5 is desired, so a mild Meantone tempering is the best option. In addition, I find this tuning range around Mohajira to have the most pleasant-sounding neutral triads. 17/12 may be equated to half an octave and 19/17 to 9/8 as in Intergan. The final result is 2.3.5.11.13.17.19 24&amp;62.}}<br /> <br /> In the accompanying generator table, diatonic intervals are aligned, while adjacent quartertone intervals always differ by 600¢. Primes are highlighted. {{adv|The top half of the table shows an optional 7 (the simplest mapping is in the hemififth chain due to the semitwelfth already being so close to 7/4, but it's in the wrong direction) and an alternate sharp 11, 17, and 19 all found in Intergan, but none of these were included in 2.3.5.11.13.17.19 24&amp;62. The flat 17 and 19 are shown in parenthesis because they differ from diatonic intervals by half an octave.}} My proposed interval names are extrapolated from existing names in order to be as unambiguous as possible.<br /> <br /> This way of displaying intervals in a half-octave temperament, having two full-octave generators that differ by half an octave, is mostly similar to the standard way. The columns don't correspond to which of the two periods the intervals are in, but it's simple enough to guess because the smaller one is first period and the larger one is second period. The major difference is that intervals an even number of generators from unison only show one interval in one period. For temperament reasons, I included some of the ones not shown in parenthesis. This matters a lot because it ignores an important Diaschismic equivalence: sqrt(2) / (9/8) ≈ 5/4.<br /> <br /> This may result in a new way of generating scales. An Aeolian diatonic scale ranges from -4 to 2 generators. This temperament divides the generator in half, so take all intervals in the table from -8 to 4 generators, ignoring anything in parenthesis. The resulting scale has 19 notes: 55 153 196 251 306 349 447 502 557 655 698 753 808 851 949 1004 1047 1157 1200. No more notes can be added without introducing some extremely small steps to the scale, which would be tempered out in 24edo. The scale has four unique step sizes: 43 55 98 110. There is only one step of 110¢, which can be removed by replacing 1047 or 1157 (the most extreme two intervals unique to the 349¢ column) with 1059 or 1145 (the next two intervals unique 949¢ column). This results in the two chiralities of 5L9m5s. Sharpening the two generators to 950¢ and 350¢ results in 5L14s, and further sharpening them flips m and s to 5L5m9s.<br /> <br /> === Melodic intervals ===<br /> <br /> The diatonic half of quartertone composition does not need an explanation. Semiquartal and dicoid have their own compositional practices, much of which can be inherited from diasem and blackdye respectively due to being degenerate cases. What's left to explain is how this all fits together.<br /> * The quartertone has function similar to a wide aberrisma and a narrow semitone, since semiquartal is just diasem with the two equated. It's useful for altering intervals like an aberrisma and has a distinctive &quot;metallic&quot; sound as a semitone.<br /> * The neutral second is best known for occurring in dicoid and antidiatonic, which it can take most of its function from. It has a distinctive &quot;sour&quot; sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.<br /> * The semifourth is most useful as an inframinor third, but it also works as an ultramajor second. It pairs best with wider minor thirds which make it sound more like a second in comparison. It's useful for shrinking the semitone between the second and third without shrinking the third.<br /> <br /> == Tertian categorization ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Thirds categorized as a fraction of a diatonic fifth.<br /> !Fraction f<br /> !*7/12 =<br /> !4\7 (¢)<br /> !7\12 (¢)<br /> !3/2 (¢)<br /> !3\5 (¢)<br /> !Lower bound of:<br /> |-<br /> !1/3<br /> |7/36<br /> |228.6<br /> |233.3<br /> |234.0<br /> |240.0<br /> |&quot;Horiminor&quot;<br /> |-<br /> !12/35<br /> |1/5<br /> |235.1<br /> |240.0<br /> |240.7<br /> |246.9<br /> |Inframinor<br /> |-<br /> !18/49<br /> |3/14<br /> |251.9<br /> |257.1<br /> |257.9<br /> |264.5<br /> |Subminor<br /> |-<br /> !11/28<br /> |11/48<br /> |269.4<br /> |275.0<br /> |275.8<br /> |282.9<br /> |Neominor<br /> |-<br /> !59/140<br /> |59/240<br /> |289.0<br /> |295.0<br /> |295.8<br /> |303.4<br /> |Grave Minor<br /> |-<br /> !61/140<br /> |61/240<br /> |298.8<br /> |305.0<br /> |305.9<br /> |313.7<br /> |Classic Minor<br /> |-<br /> !23/49<br /> |23/84<br /> |321.9<br /> |328.6<br /> |329.5<br /> |338.0<br /> |Supraminor<br /> |-<br /> !24/49<br /> |2/7<br /> |335.9<br /> |342.9<br /> |343.8<br /> |352.7<br /> |Neutral<br /> |-<br /> !25/49<br /> |25/84<br /> |349.9<br /> |357.1<br /> |358.1<br /> |367.3<br /> |Submajor<br /> |-<br /> !26/49<br /> |13/42<br /> |363.8<br /> |371.4<br /> |372.5<br /> |382.0<br /> |Classic Major<br /> |-<br /> !79/140<br /> |79/240<br /> |386.9<br /> |395.0<br /> |396.1<br /> |406.3<br /> |Acute Major<br /> |-<br /> !81/140<br /> |81/240<br /> |396.7<br /> |405.0<br /> |406.1<br /> |416.6<br /> |Neomajor<br /> |-<br /> !17/28<br /> |17/48<br /> |416.3<br /> |425.0<br /> |426.2<br /> |437.1<br /> |Supermajor<br /> |-<br /> !31/49<br /> |31/84<br /> |433.8<br /> |442.9<br /> |444.1<br /> |455.5<br /> |Ultramajor<br /> |-<br /> !23/35<br /> |23/60<br /> |450.6<br /> |460.0<br /> |461.3<br /> |473.1<br /> |&quot;Horimajor&quot;<br /> |-<br /> !2/3<br /> |7/18<br /> |457.1<br /> |466.7<br /> |468.0<br /> |480.0<br /> |[end of range]<br /> |}<br /> &lt;/div&gt;<br /> The accompanying table shows my own way of categorizing triads and tetrads bounded by ~3/2, with the third being a logarithmic fraction of the fifth. {{adv|Note that the thirds are presented as a logarithmic fraction of the fifth, which scales the major and minor thirds by the same ratio to fit the fifth. Delta-rational logarithmically scales the lower third in the triad slightly more than the upper third.}} This is not an exhaustive list of qualities, but rather groups of my major use cases.<br /> <br /> * I've proposed the terms &quot;horiminor&quot; and &quot;horimajor&quot; after the same root in the word &quot;horizon&quot;. These are the chords that are technically possible to hear as containing thirds, but it requires special care in the melody to make it convincing, such as splitting the horiminor third. I find this class especially useful in 10edo, which is often considered to only have neutral chords.<br /> * Inframinor and ultramajor chords share the metallic quality with the quartertone, as mentioned in the previous section. They are similar in function to the subminor and supermajor chords based on 6:7:9, but have a harsher and exaggerated sound, usually interpreted as based on 10:13:15.<br /> * Subminor and supermajor chords are in the vicinity of 6:7:9 and 14:18:21. Subminor especially has a pure sound like 4:5:6, but these chords may sound too different from Western tuning to be desirable in many cases.<br /> * Neominor and neomajor are used in this case to refer to any thirds not in the concordance wells of 6:7:9 or 4:5:6. This gives them a sound that I would describe as impure but even. They're the most versatile for transcribing modern 12edo music while remaining somewhat xenharmonic. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.<br /> * Grave minor and acute major chords are approximately the ones found in 12edo, around 16:19:24. I consider these to be a separate category less often than any other, but this presence of this otonal chord justifies them. A grave minor third is also characteristic of [[Hibernal]].<br /> * Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15, so they may also be called pental. They are considered to be the default in xenharmonic circles, and have been used in Western music for hundreds of years at least, largely falling out of consideration once 12edo became the standard. I think it's always good to have these to some extent because they provide a consonance that can anchor everything else.<br /> * Supraminor and submajor chords lean towards, but I find them to work better as a consonance because they still resemble minor and major functions. They're usually interpreted as based on 14:17:21.<br /> * Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered largely concordant, between 24edo and 31edo. This is where two slightly sharp 11/9s stack to make a slightly flat 3/2. In this range, they sound like ambiguous minor/major chords, which may be useful for modulation or when unsure of which quality is better for a given chord. More broadly, they function as an extremely versatile mild dissonance. People tend to see them as a cross between major and minor, but I actually see them as a cross between major, minor, diminished, and augmented. They're weird as a consonance in a quasi-diatonic context, not because of the complexity of 18:22:27, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.<br /> ** I call this technique a &quot;neutral substitution&quot;. It involves replacing a tertian chord with a neutral one in order to modulate. The easiest way to do this is line up a neutral chord with the root or fifth of where a diminished chord would be, then use whichever of the two is changed to alter the key.<br /> <br /> === Grave fifth chords and mainline fifths ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Successive fifths from 22/15 to 40/27,&lt;br&gt;thirds between logarithmic 12/35 and 23/35 of the fifth<br /> ! style=&quot;border-right: double;&quot; |D<br /> ! colspan=&quot;5&quot; style=&quot;border-right: double;&quot; |N<br /> !&lt;3/7f<br /> !&lt;1/2f<br /> !&gt;1/2f<br /> !&gt;4/7f<br /> ! style=&quot;border-right: double;&quot; |Fifth<br /> ! colspan=&quot;4&quot; |Fifth Complement<br /> |-<br /> ! style=&quot;border-right: double;&quot; |15<br /> |<br /> |18<br /> |<br /> | style=&quot;border-right: double;&quot; |19<br /> | style=&quot;border-right: double;&quot; |22<br /> |<br /> |315.6<br /> |<br /> | style=&quot;border-right: double;&quot; |409.2<br /> | style=&quot;border-right: double;&quot; |663.0<br /> |<br /> |347.4<br /> |<br /> |253.8<br /> |-<br /> ! style=&quot;border-right: double;&quot; |17<br /> |<br /> | class=&quot;thl&quot; |20<br /> |21<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |25<br /> |<br /> | class=&quot;thl&quot; |281.4<br /> |365.8<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |667.7<br /> |<br /> | class=&quot;thl&quot; |386.3<br /> |301.8<br /> |<br /> |-<br /> ! style=&quot;border-right: double;&quot; |19<br /> |22<br /> |23<br /> |<br /> | style=&quot;border-right: double;&quot; |24<br /> | style=&quot;border-right: double;&quot; |28<br /> |253.8<br /> |330.8<br /> |<br /> | style=&quot;border-right: double;&quot; |404.4<br /> | style=&quot;border-right: double;&quot; |671.3<br /> |417.5<br /> |340.6<br /> |<br /> |266.9<br /> |-<br /> ! style=&quot;border-right: double;&quot; |21<br /> |24<br /> | class=&quot;thl&quot; |25<br /> | class=&quot;thl&quot; |26<br /> | style=&quot;border-right: double;&quot; |27<br /> | style=&quot;border-right: double;&quot; |31<br /> |231.2<br /> | class=&quot;thl&quot; |301.8<br /> | class=&quot;thl&quot; |369.7<br /> | style=&quot;border-right: double;&quot; |435.1<br /> | style=&quot;border-right: double;&quot; |674.3<br /> |443.1<br /> | class=&quot;thl&quot; |372.4<br /> | class=&quot;thl&quot; |304.5<br /> |239.2<br /> |-<br /> ! style=&quot;border-right: double;&quot; |23<br /> |<br /> |27<br /> |28<br /> | style=&quot;border-right: double;&quot; |29<br /> | style=&quot;border-right: double;&quot; |34<br /> |<br /> |277.6<br /> |340.6<br /> | style=&quot;border-right: double;&quot; |401.3<br /> | style=&quot;border-right: double;&quot; |676.7<br /> |<br /> |399.1<br /> |336.1<br /> |275.4<br /> |-<br /> ! style=&quot;border-right: double;&quot; |25<br /> |29<br /> |30<br /> |31<br /> | style=&quot;border-right: double;&quot; |32<br /> | style=&quot;border-right: double;&quot; |37<br /> |256.9<br /> |315.6<br /> |372.4<br /> | style=&quot;border-right: double;&quot; |427.4<br /> | style=&quot;border-right: double;&quot; |678.7<br /> |421.8<br /> |363.1<br /> |306.3<br /> |251.3<br /> |-<br /> ! style=&quot;border-right: double;&quot; |27<br /> |31<br /> | class=&quot;thl&quot; |32<br /> |33<br /> | style=&quot;border-right: double;&quot; |34<br /> | style=&quot;border-right: double;&quot; |40<br /> |239.2<br /> | class=&quot;thl&quot; |294.1<br /> |347.4<br /> | style=&quot;border-right: double;&quot; |399.1<br /> | style=&quot;border-right: double;&quot; |680.4<br /> |441.3<br /> | class=&quot;thl&quot; |386.3<br /> |333.0<br /> |281.4<br /> |}<br /> &lt;/div&gt;<br /> I use a similar system for antidiatonic tertian harmony, but there's no set JI approximation for a flat fifth, so I use multiple to get the simplest ratios possible. [[Pentagoth]] equates many similar intervals in this table.<br /> <br /> I refer to the range from about 23\40 to 25\42 as '''mainline fifths'''. The full diatonic range includes other fifths that generate scales I have found over time to resemble the usual understanding of diatonic less than they resemble 7 or 5edo, and resemble 3/2 less than sqrt(20/9) and sqrt(16/7). The 25\42 boundary is clearer to me because I prefer very hard scales to very soft ones and thus I have more experience. But conveniently, these are the points where an alternate diatonic fifth can be generated at a low complexity. Analogously, I could refer to the range from about 11\20 to 17\30 as '''mainline antidiatonic fifths''' based on my personal experience.<br /> <br /> I've found 25 and 39edo to be the two regions of the armotonic tuning range that sound better than others. {{adv|29edo sounds more pure than both, and is the basis for an excellent 29&amp;67 straddle temperament I've called Gravel, but it involves the 15:19:22 chord with a fifth at my limit for resembling 3/2 which does not generate armotonic. 2.15.11.19 9&amp;56 also generates this, and the subgroup can be extended to 2.3.5.11.19 while hardly changing the edo list. 11:13:16 and 13:16:19 are proper subfifth triads and are not counted in the range. They are also nearly retroversions of each other.}}<br /> <br /> 25edo represents the two otonal chords on the table with 5/4, with flatter fifths like 34 and 43edo being better for 17:20:25 and sharper fifths like 41 and 57edo being better for 27:32:40. Both of these are found in Pentagoth, so I propose calling this range '''pentagothic'''. 43edo is especially notable for how well it also approximates 17:21:25.<br /> <br /> 39edo represents 21:26:31, which is notably a +1+1 chord, and 21:25:31, very close to its retroversion. I propose calling this range '''mavilic''' after Erv Wilson's original Meta-Mavila, the &quot;proper&quot; tuning range for Mavila. {{adv|Mavila possibly has more variants of its name than any other temperament. I'm genuinely surprised that Mavilic isn't a temperament name.}} [[Pentagoth#Otomavila|Otomavila]] is my temperament using these chords.<br /> <br /> For the most part, antidiatonic tertian harmony works like diatonic tertian harmony, but with the reversed qualities of the antidiatonic scale. The pentagothic neutral triad generates Cohemimabila temperament and has a similar function to the neutral triad explained above.<br /> <br /> === Quasi-diatonic scales ===<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Blackdye and diasem JI tunings, sorted from flat to sharp minor third, denominators up to 20<br /> !Tertian Triad<br /> !16:21:24<br /> !20:23:30<br /> !10:13:15<br /> !6:7:9&lt;br&gt;(14:18:21)<br /> !18:23:27<br /> !16:19:24<br /> !4:5:6&lt;br&gt;(10:12:15)<br /> !14:17:21<br /> !18:22:27<br /> |-<br /> |Subgroup<br /> |2.3.7<br /> |2.3.23/5<br /> |2.3.13/5<br /> |2.3.7<br /> |2.3.23<br /> |2.3.19<br /> |2.3.5<br /> |2.3.17/7<br /> |2.3.11<br /> |-<br /> |Name<br /> |&quot;Septal&quot;<br /> |?<br /> |?<br /> |&quot;Septal&quot;<br /> |&quot;Eridial&quot;<br /> |&quot;Novemdecal&quot;<br /> |Pental<br /> |?<br /> |&quot;Undecal&quot;<br /> |-<br /> |Aberrisma<br /> |28/27<br /> |640/621<br /> |416/405<br /> |64/63<br /> |736/729<br /> |513/512<br /> |81/80<br /> |459/448<br /> |33/32<br /> |-<br /> |Quasi-&lt;br&gt;Aeolian<br /> |9/8&lt;br&gt;8/7&lt;br&gt;9/7&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;12/7&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;23/20&lt;br&gt;207/160&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;23/15&lt;br&gt;69/40&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;15/13&lt;br&gt;135/104&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;20/13&lt;br&gt;45/26&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;7/6&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;14/9&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |513/512&lt;br&gt;9/8&lt;br&gt;19/16&lt;br&gt;4/3&lt;br&gt;171/128&lt;br&gt;3/2&lt;br&gt;19/12&lt;br&gt;16/9&lt;br&gt;57/32&lt;br&gt;2/1<br /> |81/80&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |459/448&lt;br&gt;9/8&lt;br&gt;17/14&lt;br&gt;4/3&lt;br&gt;153/112&lt;br&gt;3/2&lt;br&gt;34/21&lt;br&gt;16/9&lt;br&gt;51/28&lt;br&gt;2/1<br /> |33/32&lt;br&gt;9/8&lt;br&gt;11/9&lt;br&gt;4/3&lt;br&gt;11/8&lt;br&gt;3/2&lt;br&gt;33/20&lt;br&gt;16/9&lt;br&gt;11/6&lt;br&gt;2/1<br /> |-<br /> |Similar&lt;br&gt;Unlisted&lt;br&gt;Aberrismas<br /> | colspan=&quot;2&quot; |17/13: 1088/1053&lt;br&gt;(swap m/s)<br /> |17/11: 1408/1377&lt;br&gt;19/11: 304/297<br /> |<br /> |25: 2048/2025&lt;br&gt;11/7: 896/891&lt;br&gt;13/11: 352/351&lt;br&gt;17/5: 136/135&lt;br&gt;23/13: 208/207<br /> |25/7: 225/224&lt;br&gt;17: 4131/4096&lt;br&gt;19/5: 1216/1215&lt;br&gt;(diasem)<br /> |<br /> |23/7: 189/184&lt;br&gt;23/19: 621/608<br /> |13: 1053/1024<br /> |}<br /> The diatonic scale and its tertian harmony are the framework of all Western music. It makes sense that using the lowest prime as the period and second lowest prime as the generator would produce a versatile scale, and the fact that it clusters around 12edo rather than 3edo in the case of prime 5 produces more useful step sizes. Modifying this into an aberrismic scale containing the simplest tertian triad 4:5:6 and its retroversion in as many places as possible produces blackdye, and doing the same with 6:7:9 produces diasem. These are the main scales I use to represent my take on Western music: melodically interesting, harmonically pure, and not far from diatonic. However, there are other tunings of them that are useful in different places. These are generalized forms and their aberrismas may be tempered out for a true diatonic scale. 64/63 and 81/80 are common, but 736/729 and especially 513/512 are smaller and make more sense to temper out. If not, inflating them is recommended to keep the melodic significance of the aberrisma.<br /> <br /> '''Eridial''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named after &quot;eridian&quot;, the demonym for Eris, with the -al suffix of subgroups. Eris is probably one of the most goth deities of Ancient Greece and is also associated with the number 23 by Discordians. It targets 18:23:27, a little-known and surprisingly concordant neogothic tertian triad. Many microtonalists contend that 12edo pop music, while historically tracing back to Meantone, tends to sound better in Pythagorean tuning. I can really hear it after comparing different quasi-diatonic options for retuning my melodies. I've taken a particular liking to gentle/neogothic diatonic scales in the past for this reason, but I believe eridian diasem is a further refinement. As with diasem in general, I prefer this with the 3 tuned slightly flat to inflate the aberrisma, although it is especially important in this case because 736/729 has a size of only 17¢. When inflated in tunings such as 43edo, the aberrisma is subtle enough to not disturb the Pythagorean sound much, but impactful in cases like the subminor seventh, which is tuned close to 7/4. Its melody is not particularly xenharmonic, but it's one of my default scales when available because it works for almost everything. It works well as an extension to [[Septimal meantone]]. {{adv|The difference between eridian diasem's analogous 2.3.25 and 2.3.17/5 thirds (e.g. 75/64 and 20/17) is 256/255, also the difference between 16/15 and 17/16, supporting it as one of the most important 17-limit commas.}}<br /> <br /> The default set of qualities to consider is septal and pental due to their simplicity, possibly with a Pythagorean diatonic scale in the middle as found in Aberschismic edos. But I've found that I particularly like shifting both septal and pental towards neutral, to eridian and exaggerated pental. This is probably because it puts both of them about an equal distance from diatonic when using a prime 3 within a few cents of just. Eridian provides a more versatile sound and pental sounds more interesting while still working about the same. In some edos that do this such as 48 and 86, the pental tuning is Negri, meaning that semiquartal is also available.<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Cursed Eridian Scales<br /> !Standard&lt;br&gt;(23/18)(27/23)<br /> !Wolf → Perfect&lt;br&gt;(23/18)(32/27)<br /> !the other one&lt;br&gt;(81/64)(27/23)<br /> |-<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |398688256/387420489&lt;br&gt;67712/59049&lt;br&gt;32/27&lt;br&gt;243/184&lt;br&gt;2166784/1594323&lt;br&gt;368/243&lt;br&gt;36/23&lt;br&gt;59049/33856&lt;br&gt;11776/6561&lt;br&gt;2/1<br /> |4782969/4333568&lt;br&gt;27/23&lt;br&gt;129140163/99672064&lt;br&gt;2944/2187&lt;br&gt;2187/1472&lt;br&gt;128/81&lt;br&gt;59049/33856&lt;br&gt;8667136/4782969&lt;br&gt;2/1<br /> |}<br /> &lt;/div&gt;<br /> 27/23 is close enough to 32/27 that it can be replaced and result in another 2.3.23 blackdye scale with 32/27*23/18 = 368/243 triads. The fifth is the same as the existing Eridian wolf fifth, which is barely diatonic in JI. I might call this '''hiberidial''' because it applies the principle of [[Hibernal]] to the eridian subgroup. 82edo is the near-perfect 2.3.23 tuning, but 20 and 37edo have a close approximation of the scale despite not being good for the subgroup. Switching thirds to 32/27 or 81/64 to remain in the subgroup with a wolf fifth as the main one can be applied to other scales in the table.<br /> <br /> I have a lot more to write about how blackdye and diasem work harmonically.<br /> <br /> == Aberrismic details ==<br /> <br /> Note that this section is incomplete.<br /> <br /> As described in the [[Monarch]] article, I like to tune pental blackdye (3sA Aeolian: tempered 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1) with a sharp 3 and thus inflated 81/80. When extreme, more interesting intervals show up along with new problems. The first is what is seen in 11-limit Superpyth and Porcupine (55/54), turning 27/20 into 11/8, although it could be interpreted as any other superfourth. I like this interval a lot because the 1/1-9/8-6/5-27/20 tetrachord is a natural-sounding stretched version of the diatonic minor tetrachord. In fact, it may sound better melodically to use the superfourth rather than the perfect fourth to avoid the very deflated 10/9 between 6/5 and 4/3. It would be beneficial to have a triad on 27/20 to replace the one on 4/3, and this makes use of the 81/80 in the next octave, the least harmonically useful note in the scale. 32/27 is available as the more conventional quasi-diatonic third, but also 320/243, one of the least recognizeable intervals that all correspond to septal intervals in Aberschismic temperament (5120/5103). 320/243 is 476.539¢ in JI and 729/640 is 225.416¢, but these approach each other with a sharper 3. Most notably in this tuning range, tempering out 2187/2240 equates them to 9/7 and 7/6, different from the same ratios that would normally be generated by Archy. Interpreting 320/243 as 14/11 allows for 8:11:14. The exact tempering is irrelevant as long as they're some sort of supermajor and subminor thirds.<br /> <br /> The size of the aberrisma presents other issues.<br /> <br /> == Song structure ==<br /> <br /> This is my system for categorizing the songs I write by structure. Something similar can be applied to other people's music, but the standards will be different due to the unusual complexity of my music.<br /> <br /> * '''straightforward''': The simplest type, being in a similar style throughout. Always works.<br /> * '''opposing sections''': Two straightforward parts that are very different.<br /> * '''medium''': Somewhat straightforward, somewhat disjointed.<br /> * '''episodic''': Containing many different sections with themes and a progression throughout the song. The most desirable when done well and in moderation.<br /> * '''disjointed''': Containing many different sections with no clear pattern. The least desirable.<br /> <br /> I've noticed that I write more &quot;disjointed&quot; songs than I would like, and fewer &quot;straightforward&quot; and &quot;episodic&quot;. This system is intended to keep track of what I'm doing and help me work toward a better result.<br /> <br /> {| class=&quot;wikitable sortable&quot;<br /> |+Song Structure Types of The Gralbums<br /> !Track<br /> !Title<br /> !Type<br /> |-<br /> |g1t1<br /> |The Lake Reflects a Black Sky<br /> |'''disjointed'''<br /> |-<br /> |g1t2<br /> |The Art of It All<br /> |'''episodic'''<br /> |-<br /> |g1t3<br /> |Rail Twinge<br /> |'''straightforward'''<br /> |-<br /> |g1t4<br /> |Back Stalk<br /> |'''medium'''<br /> |-<br /> |g1t5<br /> |Superior Intermedial<br /> |'''episodic'''<br /> |-<br /> |g1t6<br /> |Snowboard Index<br /> |'''straightforward'''<br /> |-<br /> |g1t7<br /> |Winter's Mortal Hope<br /> |'''medium'''<br /> |-<br /> |g1t8<br /> |Transpiration<br /> |'''opposing sections'''<br /> |-<br /> |g1t9<br /> |Revelation of Your Forever<br /> |'''medium'''<br /> |-<br /> |g2t1<br /> |The Life Unreachable<br /> |'''disjointed'''<br /> |-<br /> |g2t2<br /> |Monolithium<br /> |'''opposing sections'''<br /> |-<br /> |g2t3<br /> |Nocturne Paranoia<br /> |'''disjointed'''<br /> |-<br /> |g2t4<br /> |Glimmer Extrication<br /> |'''episodic'''<br /> |-<br /> |g2t5<br /> |Not This Time<br /> |'''straightforward'''<br /> |-<br /> |g2t6<br /> |Sakura Blade Minivan<br /> |'''medium'''<br /> |-<br /> |g2t7<br /> |Resolute Prelude<br /> |'''straightforward'''<br /> |-<br /> |g2t8<br /> |Life and Limb<br /> |'''straightforward'''<br /> |-<br /> |g2t9<br /> |Residual Soliloquy<br /> |'''disjointed'''<br /> |}</div>

Ground https://xenreference.com/wiki/index.php?title=Mosh&diff=8454 2026-10-25T09:27:30Z

<p>Ground: </p> <hr /> <div>The '''mosh''' scale, or '''3L 4s''', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range. Soft tunings of mosh also generate '''dicoid''', or '''7L 3s'''.<br /> <br /> == General scale theory ==<br /> Mosh has seven modes named by Andrew Heathwaite.<br /> {| class=&quot;wikitable&quot;<br /> |+Modes of mosh<br /> !Mode Name<br /> !Pattern<br /> !2nd<br /> !3rd<br /> !4th<br /> !5th<br /> !6th<br /> !7th<br /> |-<br /> |Dril<br /> |LsLsLss<br /> |M<br /> |P<br /> |M<br /> |M<br /> |A<br /> |M<br /> |-<br /> |Gil<br /> |LsLssLs<br /> |M<br /> |P<br /> |M<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Kleeth<br /> |LssLsLs<br /> |M<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Bish<br /> |sLsLsLs<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Fish<br /> |sLsLssL<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |m<br /> |-<br /> |Jwl<br /> |sLssLsL<br /> |m<br /> |P<br /> |m<br /> |m<br /> |P<br /> |m<br /> |-<br /> |Led<br /> |ssLsLsL<br /> |m<br /> |d<br /> |m<br /> |m<br /> |P<br /> |m<br /> |}<br /> Like all [[MOS]] scales, tunings of mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.<br /> <br /> The qualities of scale degrees in soft and hard mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:<br /> {| class=&quot;wikitable&quot;<br /> |+Mosh vs. Diatonic<br /> !Mosh Degree (1-indexed)<br /> !Mosh Quality<br /> !Soft<br /> !Hard<br /> |-<br /> | rowspan=&quot;2&quot; |2<br /> |minor<br /> |dia n2<br /> |m2<br /> |-<br /> |major<br /> |dia M2<br /> |A2<br /> |-<br /> | rowspan=&quot;2&quot; |3<br /> |diminished<br /> |dia m3<br /> |d3<br /> |-<br /> |perfect<br /> |dia n3<br /> |dia M3<br /> |-<br /> | rowspan=&quot;2&quot; |4<br /> |minor<br /> |dia P4<br /> |dia d4<br /> |-<br /> |major<br /> |dia A4<br /> |dia AA4<br /> |-<br /> | rowspan=&quot;2&quot; |5<br /> |minor<br /> |dia d5<br /> |dia dd5<br /> |-<br /> |major<br /> |dia P5<br /> |dia A5<br /> |-<br /> | rowspan=&quot;2&quot; |6<br /> |perfect<br /> |dia n6<br /> |dia m6<br /> |-<br /> |augmented<br /> |dia M6<br /> |dia A6<br /> |-<br /> | rowspan=&quot;2&quot; |7<br /> |minor<br /> |dia m7<br /> |dia d7<br /> |-<br /> |major<br /> |dia n7<br /> |dia M7<br /> |}<br /> <br /> === Tetrachordal mosh ===<br /> The step-reordering LssLLss of mosh is significant. Certain modes of LssLLss are used in Middle Eastern music when 24edo tuning is used (though this is an oversimplification, as contextual alterations are used in a given maqam): LssLLss is (basically) Maqam Rast and ssLLssL is Maqam Bayati. Inthar calls this scale '''tetrachordal mosh''' (as five of the modes contains two identical fourth-spanning tetrachords, either conjunct or disjunct), while CellularAutomaton calls this scale '''lissotonic'''.<br /> <br /> === Pentachordal dicoid ===<br /> Degenerate tunings of [[blackdye]] (sLmLsLsLmL) with L = m yield a scale sLLLsLsLLL, which is not the dicoid MOS but a step-reordering thereof. We may call this scale '''pentachordal dicoid''', as it consists of two identical fourth-spanning [[tetrachord|genera]] (pentachords) with 3 L steps and 1 s step, and a &quot;disjunction&quot; consisting of 1 L step and 1 s step which spans a diatonic major second.<br /> <br /> Pentachordal dicoid retains the interleaved-pentic generator structure of blackdye and the same constraint that the larger 6-step interval (4L + 2s) be a diatonic fifth. {{Adv|Like standard dicoid, pentachordal dicoid has two 5-note diatonic-fifth chains; however, they are separated by the L step, the neutral second, not by the neutral third (2L + s) as in standard dicoid.}}<br /> <br /> Pentachordal dicoid modes with disjunct tetrachords:<br /> * LLLs Ls LLLs<br /> * LLsL sL LLsL<br /> * LsLL Ls LsLL<br /> * sLLL sL sLLL<br /> Pentachordal dicoid modes with conjunct tetrachords:<br /> * LLLs LLLs Ls<br /> * LLsL LLsL sL<br /> * LsLL LsLL Ls (or Ls LLLs LLLs)<br /> * sLLL sLLL sL (or sL LLsL LLsL)<br /> * sL sLLL sLLL<br /> * Ls LsLL LsLL<br /> <br /> == Chords and tendency tones ==<br /> {{Wip}}<br /> <br /> === Soft mosh ===<br /> In soft tunings of mosh, two generators make a diatonic fifth, which is a reasonable approximation of 3/2. Thus the generator sequence creates useful tertian chords; on most degrees, these chords will be &quot;neutral,&quot; splitting that fifth in half. The simplest otonal approximation is 14:17:21, although 18:22:27 is more accurate. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9) ≈ 11:13:16. For the sake of clarity, these chords will be referred to with the names of their characteristic modes. [[User:Ground]] proposes &quot;dimiminor&quot; (between diminished and minor) for Led and &quot;dimimajor&quot; (between diminished and major) for Jwl, which are intended to be easier to remember.<br /> <br /> Unlike in diatonic, soft mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.<br /> <br /> Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the &quot;diminished third&quot; interval of the mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.<br /> <br /> From all this, we can see a presently emergent paradigm for functional harmony in soft mosh, a &quot;neutral tonality&quot; which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.<br /> {| class=&quot;wikitable&quot;<br /> |+Chords of Kleeth<br /> !Degrees<br /> !Quality<br /> !Functional description<br /> |-<br /> |1 - 3 - 5<br /> |Neutral<br /> |Tonic triad.<br /> |-<br /> |'''2''' - '''4''' - 6<br /> |Led/Dimiminor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |-<br /> |3 - 5 - '''7'''<br /> |Neutral<br /> |Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.<br /> |-<br /> |'''4''' - 6 - 1<br /> |Neutral<br /> |Expansion of subdominant area. Creates motion from tonic to elsewhere.<br /> |-<br /> |5 - '''7''' - '''2'''<br /> |Neutral<br /> |Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).<br /> |-<br /> |6 - 1 - 3<br /> |Neutral<br /> |Primary subdominant chord, &quot;opposite&quot; of III. Creates motion from tonic to elsewhere.<br /> |-<br /> |7 - '''2''' - '''4'''<br /> |Jwl/Dimimajor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |}<br /> Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.<br /> <br /> The functional paradigm for tonal music in mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.<br /> <br /> === Dicoid ===<br /> Dicoid is the superset of soft mosh. It may be most easily interpreted as an [[Aberrisma|aberrismized]] mosh, or in other words, a degenerate tuning of (a step-reordering of) [[blackdye]] where L=m. (Note: Such degenerate blackdye tunings actually result in the pentachordal dicoid scale.) It can also be seen as a chromatic scale of mosh.<br /> <br /> === Hard mosh ===<br /> In hard tunings of mosh, two small generators make an oneirotonic fifth and the s steps become semitone-sized. [[User:Ground]] proposes that 2.3.5.7.13.17.23 10 &amp; 33, an extension of Demibuzzard[https://en.xen.wiki/w/Buzzardsmic_clan#Demibuzzard] is the strongest temperament in this range. It has a neutral third of 16/13~21/17~26/21, a subfourth of 17/13~21/16, and a superfifth of ~23/15. The neutral triad is 17:21:26~21:26:32, which becomes 21:26:34~13:16:21 in first inversion and 13:17:21~16:21:26 in second inversion.<br /> <br /> {{Navbox MOS}}</div>

Ground https://xenreference.com/wiki/index.php?title=Mosh&diff=8409 2026-10-24T19:06:45Z

<p>Ground: </p> <hr /> <div>The '''mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range. Soft tunings of mosh also generate '''dicoid''', or ''7L 3s''.<br /> <br /> == General scale theory ==<br /> Mosh has seven modes named by Andrew Heathwaite.<br /> {| class=&quot;wikitable&quot;<br /> |+Modes of mosh<br /> !Mode Name<br /> !Pattern<br /> !2nd<br /> !3rd<br /> !4th<br /> !5th<br /> !6th<br /> !7th<br /> |-<br /> |Dril<br /> |LsLsLss<br /> |M<br /> |P<br /> |M<br /> |M<br /> |A<br /> |M<br /> |-<br /> |Gil<br /> |LsLssLs<br /> |M<br /> |P<br /> |M<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Kleeth<br /> |LssLsLs<br /> |M<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Bish<br /> |sLsLsLs<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Fish<br /> |sLsLssL<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |m<br /> |-<br /> |Jwl<br /> |sLssLsL<br /> |m<br /> |P<br /> |m<br /> |m<br /> |P<br /> |m<br /> |-<br /> |Led<br /> |ssLsLsL<br /> |m<br /> |d<br /> |m<br /> |m<br /> |P<br /> |m<br /> |}<br /> Like all [[MOS]] scales, tunings of mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.<br /> <br /> The qualities of scale degrees in soft and hard mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:<br /> {| class=&quot;wikitable&quot;<br /> |+Mosh vs. Diatonic<br /> !Mosh Degree<br /> !Mosh Quality<br /> !Soft<br /> !Hard<br /> |-<br /> | rowspan=&quot;2&quot; |2<br /> |minor<br /> |dia n2<br /> |m2<br /> |-<br /> |major<br /> |dia M2<br /> |A2<br /> |-<br /> | rowspan=&quot;2&quot; |3<br /> |diminished<br /> |dia m3<br /> |d3<br /> |-<br /> |perfect<br /> |dia n3<br /> |dia M3<br /> |-<br /> | rowspan=&quot;2&quot; |4<br /> |minor<br /> |dia P4<br /> |dia d4<br /> |-<br /> |major<br /> |dia A4<br /> |dia AA4<br /> |-<br /> | rowspan=&quot;2&quot; |5<br /> |minor<br /> |dia d5<br /> |dia dd5<br /> |-<br /> |major<br /> |dia P5<br /> |dia A5<br /> |-<br /> | rowspan=&quot;2&quot; |6<br /> |perfect<br /> |dia n6<br /> |dia m6<br /> |-<br /> |augmented<br /> |dia M6<br /> |dia A6<br /> |-<br /> | rowspan=&quot;2&quot; |7<br /> |minor<br /> |dia m7<br /> |dia d7<br /> |-<br /> |major<br /> |dia n7<br /> |dia M7<br /> |}<br /> <br /> == Chords and tendency tones ==<br /> {{Wip}}<br /> <br /> === Soft mosh ===<br /> In soft tunings of mosh, two generators make a diatonic fifth, which is a reasonable approximation of 3/2. Thus the generator sequence creates useful tertian chords; on most degrees, these chords will be &quot;neutral,&quot; splitting that fifth in half. The simplest otonal approximation is 14:17:21, although 18:22:27 is more accurate. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9) ≈ 11:13:16. For the sake of clarity, these chords will be referred to with the names of their characteristic modes. [[User:Ground]] proposes &quot;dimiminor&quot; (between diminished and minor) for Led and &quot;dimimajor&quot; (between diminished and major) for Jwl, which are intended to be easier to remember.<br /> <br /> Unlike in diatonic, soft mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.<br /> <br /> Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the &quot;diminished third&quot; interval of the mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.<br /> <br /> From all this, we can see a presently emergent paradigm for functional harmony in soft mosh, a &quot;neutral tonality&quot; which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.<br /> {| class=&quot;wikitable&quot;<br /> |+Chords of Kleeth<br /> !Degrees<br /> !Quality<br /> !Functional description<br /> |-<br /> |1 - 3 - 5<br /> |Neutral<br /> |Tonic triad.<br /> |-<br /> |'''2''' - '''4''' - 6<br /> |Led/Dimiminor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |-<br /> |3 - 5 - '''7'''<br /> |Neutral<br /> |Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.<br /> |-<br /> |'''4''' - 6 - 1<br /> |Neutral<br /> |Expansion of subdominant area. Creates motion from tonic to elsewhere.<br /> |-<br /> |5 - '''7''' - '''2'''<br /> |Neutral<br /> |Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).<br /> |-<br /> |6 - 1 - 3<br /> |Neutral<br /> |Primary subdominant chord, &quot;opposite&quot; of III. Creates motion from tonic to elsewhere.<br /> |-<br /> |7 - '''2''' - '''4'''<br /> |Jwl/Dimimajor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |}<br /> Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.<br /> <br /> The functional paradigm for tonal music in mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.<br /> <br /> === Dicoid ===<br /> Dicoid is the superset of soft mosh. It may be most easily interpreted as an [[Aberrisma|aberrismized]] mosh, or in other words, a degenerate tuning of [[blackdye]] where L=m.<br /> <br /> === Hard mosh ===<br /> In hard tunings of mosh, two small generators make an oneirotonic fifth and the s steps become semitone-sized. [[User:Ground]] proposes that 2.3.7.13.17.23 10 &amp; 23 is the strongest temperament in this range, with a neutral third of 16/13~21/17~26/21 and a subfourth of 17/13~21/16. The neutral triad is 17:21:26~21:26:32, which becomes 21:26:34~13:16:21 in first inversion and 13:17:21~16:21:26 in second inversion.<br /> <br /> {{Navbox MOS}}</div>

Ground https://xenreference.com/wiki/index.php?title=Mosh&diff=8408 2026-10-24T19:04:15Z

<p>Ground: </p> <hr /> <div>The '''mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range. Soft tunings of mosh also generate '''dicoid''', or ''7L 3s''.<br /> <br /> == General scale theory ==<br /> Mosh has seven modes named by Andrew Heathwaite.<br /> {| class=&quot;wikitable&quot;<br /> |+Modes of mosh<br /> !Mode Name<br /> !Pattern<br /> !2nd<br /> !3rd<br /> !4th<br /> !5th<br /> !6th<br /> !7th<br /> |-<br /> |Dril<br /> |LsLsLss<br /> |M<br /> |P<br /> |M<br /> |M<br /> |A<br /> |M<br /> |-<br /> |Gil<br /> |LsLssLs<br /> |M<br /> |P<br /> |M<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Kleeth<br /> |LssLsLs<br /> |M<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Bish<br /> |sLsLsLs<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Fish<br /> |sLsLssL<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |m<br /> |-<br /> |Jwl<br /> |sLssLsL<br /> |m<br /> |P<br /> |m<br /> |m<br /> |P<br /> |m<br /> |-<br /> |Led<br /> |ssLsLsL<br /> |m<br /> |d<br /> |m<br /> |m<br /> |P<br /> |m<br /> |}<br /> Like all [[MOS]] scales, tunings of mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.<br /> <br /> The qualities of scale degrees in soft and hard mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:<br /> {| class=&quot;wikitable&quot;<br /> |+Mosh vs. Diatonic<br /> !Mosh Degree<br /> !Mosh Quality<br /> !Soft<br /> !Hard<br /> |-<br /> | rowspan=&quot;2&quot; |2<br /> |minor<br /> |dia n2<br /> |m2<br /> |-<br /> |major<br /> |dia M2<br /> |A2<br /> |-<br /> | rowspan=&quot;2&quot; |3<br /> |diminished<br /> |dia m3<br /> |d3<br /> |-<br /> |perfect<br /> |dia n3<br /> |dia M3<br /> |-<br /> | rowspan=&quot;2&quot; |4<br /> |minor<br /> |dia P4<br /> |dia d4<br /> |-<br /> |major<br /> |dia A4<br /> |dia AA4<br /> |-<br /> | rowspan=&quot;2&quot; |5<br /> |minor<br /> |dia d5<br /> |dia dd5<br /> |-<br /> |major<br /> |dia P5<br /> |dia A5<br /> |-<br /> | rowspan=&quot;2&quot; |6<br /> |perfect<br /> |dia n6<br /> |dia m6<br /> |-<br /> |augmented<br /> |dia M6<br /> |dia A6<br /> |-<br /> | rowspan=&quot;2&quot; |7<br /> |minor<br /> |dia m7<br /> |dia d7<br /> |-<br /> |major<br /> |dia n7<br /> |dia M7<br /> |}<br /> <br /> == Chords and tendency tones ==<br /> {{Wip}}<br /> <br /> === Soft mosh ===<br /> In soft tunings of mosh, two generators make a diatonic fifth, which is a reasonable approximation of 3/2. Thus the generator sequence creates useful tertian chords; on most degrees, these chords will be &quot;neutral,&quot; splitting that fifth in half. The simplest otonal approximation is 14:17:21, although 18:22:27 is more accurate. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9) ≈ 11:13:16. For the sake of clarity, these chords will be referred to with the names of their characteristic modes. [[User:Ground]] proposes &quot;dimiminor&quot; (between diminished and minor) for Led and &quot;dimimajor&quot; (between diminished and major) for Jwl, which are intended to be easier to remember.<br /> <br /> Unlike in diatonic, soft mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.<br /> <br /> Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the &quot;diminished third&quot; interval of the mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.<br /> <br /> From all this, we can see a presently emergent paradigm for functional harmony in soft mosh, a &quot;neutral tonality&quot; which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.<br /> {| class=&quot;wikitable&quot;<br /> |+Chords of Kleeth<br /> !Degrees<br /> !Quality<br /> !Functional description<br /> |-<br /> |1 - 3 - 5<br /> |Neutral<br /> |Tonic triad.<br /> |-<br /> |'''2''' - '''4''' - 6<br /> |Led/Dimiminor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |-<br /> |3 - 5 - '''7'''<br /> |Neutral<br /> |Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.<br /> |-<br /> |'''4''' - 6 - 1<br /> |Neutral<br /> |Expansion of subdominant area. Creates motion from tonic to elsewhere.<br /> |-<br /> |5 - '''7''' - '''2'''<br /> |Neutral<br /> |Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).<br /> |-<br /> |6 - 1 - 3<br /> |Neutral<br /> |Primary subdominant chord, &quot;opposite&quot; of III. Creates motion from tonic to elsewhere.<br /> |-<br /> |7 - '''2''' - '''4'''<br /> |Jwl/Dimimajor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |}<br /> Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.<br /> <br /> The functional paradigm for tonal music in mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.<br /> <br /> === Dicoid ===<br /> Dicoid is the superset of soft mosh. It may be most easily interpreted as an [[Aberrisma|aberrismized]] mosh, or in other words, a degenerate tuning of [[blackdye]] where L=m.<br /> <br /> === Hard mosh ===<br /> In hard tunings of mosh, two small generators make an oneirotonic fifth and the s steps become semitone-sized. [[User:Ground]] proposes that 2.3.7.13.17.23 10 &amp; 23 is the strongest temperament in this range, with a neutral third of 16/13~21/17~26/21 and a subfourth of 17/13~21/16. The neutral triad is ~21:26:32, which simplifies to ~13:16:21 in first inversion and ~16:21:26 in second inversion.<br /> <br /> {{Navbox MOS}}</div>

Ground https://xenreference.com/wiki/index.php?title=Mosh&diff=8407 2026-10-24T19:02:41Z

<p>Ground: </p> <hr /> <div>The '''mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range. Soft tunings of mosh also generate '''dicoid''', or ''7L 3s''.<br /> <br /> == General scale theory ==<br /> Mosh has seven modes named by Andrew Heathwaite.<br /> {| class=&quot;wikitable&quot;<br /> |+Modes of mosh<br /> !Mode Name<br /> !Pattern<br /> !2nd<br /> !3rd<br /> !4th<br /> !5th<br /> !6th<br /> !7th<br /> |-<br /> |Dril<br /> |LsLsLss<br /> |M<br /> |P<br /> |M<br /> |M<br /> |A<br /> |M<br /> |-<br /> |Gil<br /> |LsLssLs<br /> |M<br /> |P<br /> |M<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Kleeth<br /> |LssLsLs<br /> |M<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Bish<br /> |sLsLsLs<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Fish<br /> |sLsLssL<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |m<br /> |-<br /> |Jwl<br /> |sLssLsL<br /> |m<br /> |P<br /> |m<br /> |m<br /> |P<br /> |m<br /> |-<br /> |Led<br /> |ssLsLsL<br /> |m<br /> |d<br /> |m<br /> |m<br /> |P<br /> |m<br /> |}<br /> Like all [[MOS]] scales, tunings of mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.<br /> <br /> The qualities of scale degrees in soft and hard mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:<br /> {| class=&quot;wikitable&quot;<br /> |+Mosh vs. Diatonic<br /> !Mosh Degree<br /> !Mosh Quality<br /> !Soft<br /> !Hard<br /> |-<br /> | rowspan=&quot;2&quot; |2<br /> |minor<br /> |dia n2<br /> |m2<br /> |-<br /> |major<br /> |dia M2<br /> |A2<br /> |-<br /> | rowspan=&quot;2&quot; |3<br /> |diminished<br /> |dia m3<br /> |d3<br /> |-<br /> |perfect<br /> |dia n3<br /> |dia M3<br /> |-<br /> | rowspan=&quot;2&quot; |4<br /> |minor<br /> |dia P4<br /> |dia d4<br /> |-<br /> |major<br /> |dia A4<br /> |dia AA4<br /> |-<br /> | rowspan=&quot;2&quot; |5<br /> |minor<br /> |dia d5<br /> |dia dd5<br /> |-<br /> |major<br /> |dia P5<br /> |dia A5<br /> |-<br /> | rowspan=&quot;2&quot; |6<br /> |perfect<br /> |dia n6<br /> |dia m6<br /> |-<br /> |augmented<br /> |dia M6<br /> |dia A6<br /> |-<br /> | rowspan=&quot;2&quot; |7<br /> |minor<br /> |dia m7<br /> |dia d7<br /> |-<br /> |major<br /> |dia n7<br /> |dia M7<br /> |}<br /> <br /> == Chords and tendency tones ==<br /> {{Wip}}<br /> <br /> === Soft mosh ===<br /> In soft tunings of mosh, two generators make a diatonic fifth, which is a reasonable approximation of 3/2. Thus the generator sequence creates useful tertian chords; on most degrees, these chords will be &quot;neutral,&quot; splitting that fifth in half. The simplest otonal approximation is 14:17:21, although 18:22:27 is more accurate. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9) ≈ 11:13:16. For the sake of clarity, these chords will be referred to with the names of their characteristic modes. [[User:Ground]] proposes &quot;dimiminor&quot; (between diminished and minor) for Led and &quot;dimimajor&quot; (between diminished and major) for Jwl, which are intended to be easier to remember.<br /> <br /> Unlike in diatonic, soft mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.<br /> <br /> Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the &quot;diminished third&quot; interval of the mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.<br /> <br /> From all this, we can see a presently emergent paradigm for functional harmony in soft mosh, a &quot;neutral tonality&quot; which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.<br /> {| class=&quot;wikitable&quot;<br /> |+Chords of Kleeth<br /> !Degrees<br /> !Quality<br /> !Functional description<br /> |-<br /> |1 - 3 - 5<br /> |Neutral<br /> |Tonic triad.<br /> |-<br /> |'''2''' - '''4''' - 6<br /> |Led/Dimiminor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |-<br /> |3 - 5 - '''7'''<br /> |Neutral<br /> |Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.<br /> |-<br /> |'''4''' - 6 - 1<br /> |Neutral<br /> |Expansion of subdominant area. Creates motion from tonic to elsewhere.<br /> |-<br /> |5 - '''7''' - '''2'''<br /> |Neutral<br /> |Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).<br /> |-<br /> |6 - 1 - 3<br /> |Neutral<br /> |Primary subdominant chord, &quot;opposite&quot; of III. Creates motion from tonic to elsewhere.<br /> |-<br /> |7 - '''2''' - '''4'''<br /> |Jwl/Dimimajor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |}<br /> Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.<br /> <br /> The functional paradigm for tonal music in mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.<br /> <br /> === Dicoid ===<br /> Dicoid is the superset of soft mosh. It may be most easily interpreted as an [[Aberrisma|aberrismized]] mosh, or in other words, a degenerate tuning of [[blackdye]] where L=m.<br /> <br /> === Hard mosh ===<br /> In hard tunings of mosh, two small generators make an oneirotonic fifth and the s steps become semitone-sized. [[User:Ground]] proposes that 2.3.7.13.17.23 10 &amp; 23 is the strongest temperament in this range, with a neutral third of 16/13~21/26 and a subfourth of 17/13~21/16. The neutral triad is ~21:26:32, which simplifies to ~13:16:21 in first inversion and ~16:21:26 in second inversion.<br /> <br /> {{Navbox MOS}}</div>

Ground https://xenreference.com/wiki/index.php?title=Mosh&diff=8406 2026-10-24T18:56:44Z

<p>Ground: </p> <hr /> <div>The '''mosh''' scale, or ''3L 4s'', serves as a common framework for many temperaments and structures generated by a stack of diatonic thirds in the neutral to major range, or diatonic sixths in the minor to neutral range. Soft tunings of mosh also generate '''dicoid''', or ''7L 3s''.<br /> <br /> == General scale theory ==<br /> Mosh has seven modes named by Andrew Heathwaite.<br /> {| class=&quot;wikitable&quot;<br /> |+Modes of mosh<br /> !Mode Name<br /> !Pattern<br /> !2nd<br /> !3rd<br /> !4th<br /> !5th<br /> !6th<br /> !7th<br /> |-<br /> |Dril<br /> |LsLsLss<br /> |M<br /> |P<br /> |M<br /> |M<br /> |A<br /> |M<br /> |-<br /> |Gil<br /> |LsLssLs<br /> |M<br /> |P<br /> |M<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Kleeth<br /> |LssLsLs<br /> |M<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Bish<br /> |sLsLsLs<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |M<br /> |-<br /> |Fish<br /> |sLsLssL<br /> |m<br /> |P<br /> |m<br /> |M<br /> |P<br /> |m<br /> |-<br /> |Jwl<br /> |sLssLsL<br /> |m<br /> |P<br /> |m<br /> |m<br /> |P<br /> |m<br /> |-<br /> |Led<br /> |ssLsLsL<br /> |m<br /> |d<br /> |m<br /> |m<br /> |P<br /> |m<br /> |}<br /> Like all [[MOS]] scales, tunings of mosh can be placed into two categories: soft and hard. Soft tunings have a generator between 2\7 (343c) and 3\10 (360c), while hard tunings have a generator between 3\10 (360c) and 1\3 (400c). Notice that soft tunings are structured such that two generators will always make a [[Diatonic#MOS diatonic|diatonic]] fifth, whereas hard tunings create an [[MOS#Table of MOS scales|antipentic]] generator; this creates a clear divide in how these two types of tunings are handled.<br /> <br /> The qualities of scale degrees in soft and hard mosh can be compared to diatonic qualities with neutral, augmented, and diminished intervals:<br /> {| class=&quot;wikitable&quot;<br /> |+Mosh vs. Diatonic<br /> !Mosh Degree<br /> !Mosh Quality<br /> !Soft<br /> !Hard<br /> |-<br /> | rowspan=&quot;2&quot; |2<br /> |minor<br /> |dia n2<br /> |m2<br /> |-<br /> |major<br /> |dia M2<br /> |A2<br /> |-<br /> | rowspan=&quot;2&quot; |3<br /> |diminished<br /> |dia m3<br /> |d3<br /> |-<br /> |perfect<br /> |dia n3<br /> |dia M3<br /> |-<br /> | rowspan=&quot;2&quot; |4<br /> |minor<br /> |dia P4<br /> |dia d4<br /> |-<br /> |major<br /> |dia A4<br /> |dia AA4<br /> |-<br /> | rowspan=&quot;2&quot; |5<br /> |minor<br /> |dia d5<br /> |dia dd5<br /> |-<br /> |major<br /> |dia P5<br /> |dia A5<br /> |-<br /> | rowspan=&quot;2&quot; |6<br /> |perfect<br /> |dia n6<br /> |dia m6<br /> |-<br /> |augmented<br /> |dia M6<br /> |dia A6<br /> |-<br /> | rowspan=&quot;2&quot; |7<br /> |minor<br /> |dia m7<br /> |dia d7<br /> |-<br /> |major<br /> |dia n7<br /> |dia M7<br /> |}<br /> <br /> == Chords and tendency tones ==<br /> {{Wip}}<br /> <br /> === Soft mosh ===<br /> In soft tunings of mosh, two generators make a diatonic fifth, which is a reasonable approximation of 3/2. Thus the generator sequence creates useful tertian chords; on most degrees, these chords will be &quot;neutral,&quot; splitting that fifth in half. The simplest otonal approximation is 14:17:21, although 18:22:27 is more accurate. The minor fifth degree, on the other hand, roughly approximates 13/9, which makes the characteristic triad of the Jwl mode roughly approximate 9:11:13, and the characteristic triad of Led roughly approximates its utonal counterpart, 1/(13:11:9) ≈ 11:13:16. For the sake of clarity, these chords will be referred to with the names of their characteristic modes. [[User:Ground]] proposes &quot;dimiminor&quot; (between diminished and minor) for Led and &quot;dimimajor&quot; (between diminished and major) for Jwl, which are intended to be easier to remember.<br /> <br /> Unlike in diatonic, soft mosh does not possess clear leading tones, with the smaller of its two step sizes representing a size category between the whole tone and semitone of diatonic; this mosh minor second step is not tense, but still creates tendency tones as its motion is sweet and concordant compared to the more sturdy and sure sound of the major second. Because each mode of mosh contains a minor second interval either above or below its tonic degree (with Bish uniquely having both), there is a clear path to resolution in all modes, and a means of creating secondary leading functions in melodies extended to chromatic harmony.<br /> <br /> Whereas the location of the tritone in diatonic is often the clearest way to orient oneself in a mode, the &quot;diminished third&quot; interval of the mosh scale is similarly a very clear way to orient oneself in a mode. Specifically, this tension tends to resolve inwards by minor second intervals to reach a unison; this causes the degrees which form this diminished third to become important tendency tones.<br /> <br /> From all this, we can see a presently emergent paradigm for functional harmony in soft mosh, a &quot;neutral tonality&quot; which can be described cleanly with the ''Kleeth'' mode as its default tonal center. Below are the chords of the Kleeth mode; tendency tones are shown in '''bold'''.<br /> {| class=&quot;wikitable&quot;<br /> |+Chords of Kleeth<br /> !Degrees<br /> !Quality<br /> !Functional description<br /> |-<br /> |1 - 3 - 5<br /> |Neutral<br /> |Tonic triad.<br /> |-<br /> |'''2''' - '''4''' - 6<br /> |Led/Dimiminor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |-<br /> |3 - 5 - '''7'''<br /> |Neutral<br /> |Primary dominant chord, containing the tendency tone 7 and moving about the generator circle.<br /> |-<br /> |'''4''' - 6 - 1<br /> |Neutral<br /> |Expansion of subdominant area. Creates motion from tonic to elsewhere.<br /> |-<br /> |5 - '''7''' - '''2'''<br /> |Neutral<br /> |Expansion of dominant or predominant area, containing degree 2 (creating pull towards 3).<br /> |-<br /> |6 - 1 - 3<br /> |Neutral<br /> |Primary subdominant chord, &quot;opposite&quot; of III. Creates motion from tonic to elsewhere.<br /> |-<br /> |7 - '''2''' - '''4'''<br /> |Jwl/Dimimajor<br /> |Useful predominant function. Creates motion towards III by way of tendency tones 2 and 4.<br /> |}<br /> Note that in the above analysis, subdominant (motion away from tonic) and predominant (motion towards dominant) are treated as different tonal areas, unlike in diatonic-based harmony where these functions use the same chords and thus become functionally identical.<br /> <br /> The functional paradigm for tonal music in mosh is therefore some form of T - (Sd) - (Pd) - D - T, with the clearest representation of these functions being I - (VI) - (vii) - III - I; this motion is roughly analogous to the I - (IV) - V - I skeleton which is characteristic of diatonic music.<br /> <br /> === Dicoid ===<br /> Dicoid is the superset of soft mosh. It may be most easily interpreted as an [[Aberrisma|aberrismized]] mosh, or in other words, a degenerate tuning of [[blackdye]] where L=m.<br /> <br /> === Hard mosh ===<br /> In hard tunings of mosh, two small generators make an oneirotonic fifth and the s steps become semitone-sized. [[User:Ground]] proposes that 2.3.7.13.17.23 10 &amp; 23 is the strongest temperament in this range, with a neutral third of ~16/13 and a subfourth of ~17/13.<br /> <br /> {{Navbox MOS}}</div>

Ground https://xenreference.com/wiki/index.php?title=Dicoid&diff=8384 2026-10-24T18:22:54Z

<p>Ground: Redirected page to Mosh</p> <hr /> <div>#REDIRECT [[Mosh]]</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8383 2026-10-24T18:22:08Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition. 12edo is the perfect tuning for people who don't want to have to think about tuning theory, but there are some close options.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == Ground's list ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> {{adv|My favorite edos are not my best edos. 25edo is one of my favorites, but I can't reasonably recommend it to most people. This is an optimization of several specific tuning properties I'm interested in, being 7+18, 9+16, and 10+15 (summing them combines the properties they have in common). If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following. This is the philosophical distinction that I have to make in order to stop thinking so much about tuning theory.}}<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> * Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> * Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. I had to invent [[Straddle primes|straddle theory]] and the trackdye scale structure (aberrated compressed diatonic) just to be satisfied with it. I classify it as straddle-3, and the only edos I've found to be even more confusing are straddle-3-5 (42, 57).<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> These are the main types of MOS-based (rank-2) composition theory that I recommend for the edos on this list:<br /> * [[Diatonic]]: Already known from 12edo, affects everything<br /> * [[Dicoid]]: 17, 24, 27, 31, 37, 41<br /> * [[Slendric]]: 26, 31, 41<br /> * [[Semiquartal]]: 19, 24, 29<br /> * [[Porcupine]]: 22, 29, 37<br /> I add rank-3 [[Aberrisma|aberrismic theory]] for all edos above 19. Aberrismic is a broad category where other scale theories combine. For example, blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on. 37 is the only edo on this list that I think requires an understanding of straddle theory to be used effectively.<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet:<br /> * [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.<br /> <br /> == Inthar's list ==<br /> {{UTBox|KC|Inthar|000000}}<br /> <br /> === Edos ===<br /> Note: I have a pro-13edo bias.<br /> <br /> * JI approximation: 41edo is the first structurally accurate 7-limit edo in that it observes both 81/80 and 64/63. Smaller edos sacrifice approximation of either prime 5 or 7, or it tempers at least one of 81/80 and 64/63.<br /> * Straddle-fifth edos: I recommend 11edo, 13edo, and 18edo for melodic familiarity and harmonic unfamiliarity. 18edo in particular provides a [[straddle primes|straddle-fifth]] 12edo approximation and [[smitonic|4L3s]] and [[oneirotonic|5L3s]] MOS scales, both melodically familiar but warped. However, if you just want to specialize in one of 4L3s and 5L3s, you can't go wrong with 11edo (for 4L3s) or 13edo (for 5L3s).<br /> * Multiples of 5 and 7: I recommend 15edo if crude 7-limit approximations and melodically challenging 160c and 240c &quot;whole tones&quot; are your thing. 21edo is the same thing with conventional-sounding major and minor thirds like 15edo, but less challenging from a melodic perspective.<br /> <br /> === JI scales ===<br /> On the JI side, I recommend<br /> * modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation (like maqam)<br /> * [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo, allowing for a kind of functional harmony in 2.3.7 just intonation.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8381 2026-10-24T18:14:53Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition. 12edo is the perfect tuning for people who don't want to have to think about tuning theory, but there are some close options.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == Ground's list ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> {{adv|My favorite edos are not my best edos. 25edo is one of my favorites, but I can't reasonably recommend it to most people. This is an optimization of several specific tuning properties I'm interested in, being 7+18, 9+16, and 10+15 (summing them combines the properties they have in common). If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following. This is the philosophical distinction that I have to make in order to stop thinking so much about tuning theory.}}<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> * Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> * Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. I had to invent [[Straddle primes|straddle theory]] and the trackdye scale structure (aberrated compressed diatonic) just to be satisfied with it. I classify it as straddle-3, and the only edos I've found to be even more confusing are straddle-3-5 (42, 57).<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> These are the main types of MOS-based (rank-2) composition theory that I recommend for the edos on this list:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> I add rank-3 [[Aberrisma|aberrismic theory]] for all edos above 19. Aberrismic is a broad category where other scale theories combine. For example, blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on. 37 is the only edo on this list that I think requires an understanding of straddle theory to be used effectively.<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet:<br /> * [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.<br /> <br /> == Inthar's list ==<br /> {{UTBox|KC|Inthar|000000}}<br /> <br /> === Edos ===<br /> I have a pro-13edo bias.<br /> <br /> On the edo side, I recommend 11edo, 13edo, and 18edo for melodic familiarity and harmonic unfamiliarity. 18edo in particular provides a [[straddle primes|straddle-fifth]] 12edo approximation and [[smitonic|4L3s]] and [[oneirotonic|5L3s]] MOS scales, both melodically familiar but warped. However, if you just want to specialize in one of 4L3s and 5L3s, you can't go wrong with 11edo (for 4L3s) or 13edo (for 5L3s).<br /> <br /> I recommend 15edo if crude 7-limit approximations and melodically challenging 160c and 240c &quot;whole tones&quot; are your thing.<br /> <br /> === JI scales ===<br /> On the JI side, I recommend<br /> * modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation (like maqam)<br /> * [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo, allowing for a kind of functional harmony in 2.3.7 just intonation.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8374 2026-10-24T17:41:51Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == Ground's list ==<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> I draw a distinction between my favorite edos and the best edos. 25edo is one of my favorites because it has a lot of tuning properties I'm interested in, but I can't reasonably recommend it to most people. If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following.<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet:<br /> * [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.<br /> <br /> == Inthar's list ==<br /> {{UTBox|KC|Inthar|000000}}<br /> <br /> On the edo side, I recommend 11edo, 13edo, and 18edo for the melodic familiarity and harmonic unfamiliarity.<br /> <br /> On the JI side, I recommend<br /> * modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation<br /> * [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8373 2026-10-24T17:40:01Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> I draw a distinction between my favorite edos and the best edos. 25edo is one of my favorites because it has a lot of tuning properties I'm interested in, but I can't reasonably recommend it to most people. If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following.<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet:<br /> * [[5edo]] (2), [[7edo]] (1): Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * [[6edo]] (2): A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * [[8edo]] (3), [[9edo]] (3), [[10edo]] (3): 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * [[11edo]] (1), [[13edo]] (1): Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * [[14edo]] (2), [[15edo]] (2): Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * [[16edo]] (1), [[18edo]] (1): Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.<br /> <br /> === Inthar ===<br /> {{UTBox|KC|Inthar|000000}}<br /> <br /> On the edo side, I recommend 11edo, 13edo, and 18edo for the melodic familiarity and harmonic unfamiliarity.<br /> <br /> On the JI side, I recommend<br /> * modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation<br /> * [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8372 2026-10-24T17:38:45Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> I draw a distinction between my favorite edos and the best edos. 25edo is one of my favorites because it has a lot of tuning properties I'm interested in, but I can't reasonably recommend it to most people. If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following.<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet: [[5edo]] (2), [[6edo]] (2), [[7edo]] (1), [[8edo]] (3), [[9edo]] (3), [[10edo]] (3), [[11edo]] (1), [[13edo]] (1), [[14edo]] (2), [[15edo]] (2), [[16edo]] (1), [[18edo]] (1).<br /> * 5 and 7edo: Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * 6edo: A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * 8, 9 and 10edo: 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * 11 and 13edo: Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * 14 and 15edo: Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * 16 and 18edo: Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.<br /> <br /> <br /> === Inthar ===<br /> {{UTBox|KC|Inthar|000000}}<br /> <br /> On the edo side, I recommend 11edo, 13edo, and 18edo for the melodic familiarity and harmonic unfamiliarity.<br /> <br /> On the JI side, I recommend<br /> * modes of 8:9:10:11:12:13:14:16 or 8:9:10:11:12:13:15:16 if you want a simple harmonic series scale and you're okay with a modal approach with limited modulation<br /> * [[diasem]] if you want a scale rich in both fifths and 7-limit JI concordances not found in 12edo</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8371 2026-10-24T17:36:40Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> I draw a distinction between my favorite edos and the best edos. 25edo is one of my favorites because it has a lot of tuning properties I'm interested in, but I can't reasonably recommend it to most people. If I want to write an antidiatonic piece, I use 25edo. If I just want to write a piece, I use 37edo, because that's my favorite of the following.<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.<br /> <br /> === Small non-diatonic edos ===<br /> <br /> I would not use these to write a piece anymore, but I recommend them to beginners at least because they force experimentation with more extreme xenharmony. My expectations for the stability and versatility of these are much lower.<br /> <br /> This is every edo between 5 and 19 that I haven't listed yet: [[5edo]] (2), [[6edo]] (2), [[7edo]] (1), [[8edo]] (3), [[9edo]] (3), [[10edo]] (3), [[11edo]] (1), [[13edo]] (1), [[14edo]] (2), [[15edo]] (2), [[16edo]] (1), [[18edo]] (1).<br /> * 5 and 7edo: Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * 6edo: A 12edo subset, not technically xenharmonic, but still unusual and underrepresented.<br /> * 8, 9 and 10edo: 9 and 10 are systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but 8 and 9 nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * 11 and 13edo: Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * 14 and 15edo: Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * 16 and 18edo: Structurally similar to 8, 9, 11, and 13, but with more notes and more options to melodically approximate 12edo.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8369 2026-10-24T17:06:33Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. Both of these categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> I also include a list of small non-diatonic edos for beginner-friendly experimentation with more extreme xenharmony: [[5edo]] (2), [[7edo]] (1), [[9edo]] (3), [[10edo]] (3), [[11edo]] (1), [[13edo]] (1), [[14edo]] (2), [[15edo]] (2).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 5 and 7edo: Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * 9 and 10edo: Systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have recognizable minor and major chords, but has nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * 11 and 13edo: Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones). My first ever xenharmonic song was in 11edo because I could retune the 12 available notes to it.<br /> * 14 and 15edo: Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8368 2026-10-24T17:01:09Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. All of the first two categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> Special: [[5edo]] (2), [[7edo]] (1), [[9edo]] (3), [[10edo]] (3), [[11edo]] (1), [[13edo]] (1), [[14edo]] (2), [[15edo]] (2).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 5 and 7edo: Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * 9 and 10edo: Systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible. 9edo is the smallest to have minor and major chords, but has nothing even resembling a whole tone. 10edo has no minor third besides the dubious semifourth found in 5edo, which splits into two wide semitones.<br /> * 11 and 13edo: Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones).<br /> * 14 and 15edo: Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8367 2026-10-24T16:56:58Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> I specialize in edos, tuning systems with a pure octave split into equally sized steps. I categorize the difficulty of them based on their size (more notes is more difficult to handle) and size-independent difficulty, which is based on how easily structures and desirable intervals translate from 12edo and the number of concepts required to understand them (1 is easy and 3 is hard).<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. All of the first two categories are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P 0), [[19edo]] (M 1), [[22edo]] (A 2), [[24edo]] (M/P 2), [[31edo]] (M 1), [[41edo]] (P 1).<br /> <br /> Second best: [[17edo]] (A/P 1), [[26edo]] (M 2), [[27edo]] (A 2), [[29edo]] (P 1), [[37edo]] (A 3).<br /> <br /> Special: [[5edo]] (2), [[7edo]] (1), [[9edo]] (3), [[10edo]] (3), [[11edo]] (1), [[13edo]] (1), [[14edo]] (2), [[15edo]] (2).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 5 and 7edo: Toy-like in their simplicity, an equalized pentatonic and diatonic scale. 5edo's lack of tertian chords makes it more difficult.<br /> * 9 and 10edo: Systems comparable to 12edo in their own right. Constructing 12edo-like melodies is virtually impossible.<br /> * 11 and 13edo: Wildly xenharmonic harmony but familiar melodic intervals (whole tones and semitones).<br /> * 14 and 15edo: Supersets of 7 and 5edo with a crude approximation of 2...7.<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 24edo: Contains all of 12edo as a subset, making familiar sounds trivial to access while adding the most xenharmonic intervals possible. Despite this, the difficulty is higher because the additional intervals don't fit into a 12edo-like framework at all. The easiest to play on 12edo instruments, which I have not factored into the difficulty score.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8366 2026-10-24T16:24:29Z

<p>Ground: </p> <hr /> <div>'''This article contains lists of various users' recommendations for general-purpose tuning systems and beginner-friendly explanations.''' It is recommended to pick a tuning early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. All of them are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P), [[19edo]] (M), [[22edo]] (A), [[24edo]] (M/P), [[31edo]] (M), [[41edo]] (P).<br /> <br /> Second best: [[17edo]] (A/P), [[26edo]] (M), [[27edo]] (A), [[29edo]] (P), [[37edo]] (A).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Starter_tunings&diff=8365 2026-10-24T16:20:55Z

<p>Ground: Ground moved page Starter tunings to Best tuning systems</p> <hr /> <div>#REDIRECT [[Best tuning systems]]</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8364 2026-10-24T16:20:55Z

<p>Ground: Ground moved page Starter tunings to Best tuning systems</p> <hr /> <div>This article contains lists of various users' optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. All of them are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P), [[19edo]] (M), [[22edo]] (A), [[24edo]] (M/P), [[31edo]] (M), [[41edo]] (P).<br /> <br /> Second best: [[17edo]] (A/P), [[26edo]] (M), [[27edo]] (A), [[29edo]] (P), [[37edo]] (A).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8363 2026-10-24T16:17:39Z

<p>Ground: </p> <hr /> <div>This article contains lists of various users' optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> To provide a simple overview, consonance of a musical interval is based on approximating simple frequency ratios, where you multiply the frequency of one note by the ratio to get the frequency of the other note. &quot;Subgroups&quot; are used to categorize intervals, where for example 2.3.5 describes intervals that can be made by multiplying and dividing the prime numbers 2, 3, and 5. Arbitrary intervals may be achieved with a complex enough ratio, but this article focuses on the simplest ones as they are the most clearly audible. The perfect fifth is 3/2.<br /> * 2.3.5: Known as &quot;pental&quot;, widely considered the default rational interpretation of Western harmony. The simple thirds are around 6/5 and 5/4.<br /> * 2.3.7: Known as &quot;septal&quot;, more xenharmonic with a narrower semitone. The simple thirds are around 7/6 and 9/7.<br /> * 2.3.5.7 (2...7): Known as &quot;septimal&quot;, the combination of the two, describing some Western forms of music like blues, jazz, and barbershop. A common subgroup for basic xenharmonic analysis.<br /> * 2.3.5.7.11.13 (2...13): Known as &quot;tridecimal&quot;, an extension that contains intervals further from diatonic. Its novel third-like intervals are the 11/9 and 16/13 neutral thirds approximated in some Middle-Eastern music, 15/13 semifourth (second-third), and 13/10 semisixth (third-fourth).<br /> <br /> The diatonic perfect fifth is the most important interval in 12edo theory after the octave, as it generates the diatonic scale. Scales in the middle of the diatonic range can be split into three camps:<br /> * Meantone (M): The diatonic scale is 2.3.5 with a flat 3, used in Baroque music. Intervals involving 7 are small alterations of diatonic.<br /> * Archy (A): The diatonic scale is 2.3.7 with a sharp 3, equally useful but with little historical precedent. Intervals involving 5 are small alterations of diatonic.<br /> * Pythagorean (P): The diatonic scale is 2.3 with an especially accurate 3, the most common interpretation of popular 12edo music. In a sufficiently large edo, accurately tuned intervals involving both 5 and 7 are alterations of diatonic.<br /> <br /> == User lists ==<br /> === Ground ===<br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile edo tunings, split into two tiers. All of them are diatonic to have a baseline level of harmonic stability:<br /> <br /> Absolute best: [[12edo]] (M/P), [[19edo]] (M), [[22edo]] (A), [[24edo]] (M/P), [[31edo]] (M), [[41edo]] (P).<br /> <br /> Second best: [[17edo]] (A/P), [[26edo]] (M), [[27edo]] (A), [[29edo]] (P), [[37edo]] (A).<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> {{adv|The list contains all diatonic edos up to 41, with the exception of 32, 33, 34, 36, 38, and 39. These can be understood in pairs of a sharp 3 and about equally flat 3: [32, 33] have an especially inaccurate 3, and the best 5 is tuned in the opposite direction, giving pental chords an unstable sound. [34, 36] are great for 2.3.5 and 2.3.7 repectively, but are too large not to have the full 2.3.5.7. [39, 38] are fine, but nothing special for their size.}}<br /> <br /> Tuning descriptions (incomplete):<br /> * 19edo: Good 2.3.5 with debatable 7. The best when it comes to simple practical composition. Clearly audibly xenharmonic, but not extreme. Like a smaller 31edo.<br /> * 22edo: Good 2...11. About as xenharmonic as 19edo in a different way.<br /> * 26edo: Debatable 2...13.<br /> * 27edo: Good 2...7.13, all tuned sharp.<br /> * 29edo: Excellent 2.3 with critically poor 5...13, meaning that ratios like 7/5 cancel out the errors and become accurate. Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Excellent 2...7 with debatable 11. Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: Excellent 2.5...13, which makes it a huge outlier in the list. It has a highly efficient and interesting array of intervals, but a barely usable diatonic fifth, so the other intervals have to carry the harmonic stability. Requires an understanding of [[Straddle primes|straddling]] the fifth to be used on the same level as the others.<br /> * 41edo: Excellent 2...11. Like 31edo, but more complex and accurate.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24, 29<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> {{adv|I was not expecting dicoid to be so prevalent, but this list implies that it's the most broadly applicable MOS theory after Diatonic. This is because it splits the diatonic fifth in half while semiquartal (specifically the 5L9s superset) splits the diatonic fourth in half, but their nearby LCJI intervals are not equal. 6/5 and 5/4 are about twice as far apart as 8/7 and 7/6, meaning that a semiquartal edo has to be about twice the size to represent all of these intervals accurately without equating those in each pair.}}<br /> <br /> '''Dicoid''' is LLLsLLsLLs, generated by a neutral third in the range (2\7, 3\10), half of a diatonic fifth. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings near the middle of the range melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8358 2026-10-23T22:33:30Z

<p>Ground: </p> <hr /> <div>This article contains lists of various users' optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> == User lists ==<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile microtonal tunings, split into two tiers:<br /> <br /> Absolute best: [[19edo]], [[22edo]], [[24edo]], [[31edo]], [[41edo]].<br /> <br /> Second best: [[17edo]], [[26edo]], [[27edo]], [[29edo]], [[37edo]].<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> * 19edo: The best when it comes to simple practical composition. Similar to 31edo, but smaller.<br /> * 29edo: Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: A highly efficient and interesting array of intervals, but no clear perfect fifth. Requires an understanding of [[Straddle primes|straddling]] to be used on the same level as the others.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> I was not expecting dicoid to be so prevalent, but apparently it can't be avoided. '''Dicoid''' is LLLsLLsLLs, generated by a neutral third [2\7, 3\10], half of a diatonic fifth.. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings in the ~16/13 category melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8357 2026-10-23T22:31:54Z

<p>Ground: </p> <hr /> <div>This article is intended to list the optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile microtonal tunings, split into two tiers:<br /> <br /> Absolute best: [[19edo]], [[22edo]], [[24edo]], [[31edo]], [[41edo]].<br /> <br /> Second best: [[17edo]], [[26edo]], [[27edo]], [[29edo]], [[37edo]].<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> * 19edo: The best when it comes to simple practical composition. Similar to 31edo, but smaller.<br /> * 29edo: Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: A highly efficient and interesting array of intervals, but no clear perfect fifth. Requires an understanding of [[Straddle primes|straddling]] to be used on the same level as the others.<br /> <br /> Main types of scale-based composition theory required for the list, according to me:<br /> * Diatonic: Already known from 12edo, affects everything<br /> * Dicoid: 17, 24, 27, 31, 37, 41<br /> * Slendric: 26, 31, 41<br /> * Semiquartal: 19, 24<br /> * Porcupine: 22, 29, 37<br /> * [[Aberrisma|Aberrismic]]: 22+<br /> <br /> I was not expecting dicoid to be so prevalent, but apparently it can't be avoided. '''Dicoid''' is LLLsLLsLLs, generated by a neutral third [2\7, 3\10], half of a diatonic fifth.. Its subset '''mosh''' is LsLsLss is more melodically useful, but has a wider generator range. The non-dicoid part is unusual and not represented here. I suggest writing composition theory based on two categories of the tuning range, just like we do for diatonic xen theory: ~11/9 and ~16/13, with 17edo about in the middle. Tunings in the ~16/13 category melodically resemble 12edo depictions of Middle Eastern scales, which new xen composers should basically know what to do with.<br /> <br /> Aberrismic is a broad category because blackdye theory can be synthesized from diatonic + dicoid, diasem from diatonic + semiquartal, and so on.</div>

Ground https://xenreference.com/wiki/index.php?title=Tuning_recommendations_for_beginners&diff=8356 2026-10-23T22:20:32Z

<p>Ground: Created page with &quot;This article is intended to list the optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition. {{UTBox|g_|Ground|7766ff}} This is my list of the most versatile microtonal tunings, split into two tiers: Absolute best: 19edo, 22edo, 24edo, 31edo, 41edo. Second best: 17edo, 26edo, 27edo, 29edo, 37edo. I intend to...&quot;</p> <hr /> <div>This article is intended to list the optimal tunings for a new xenharmonist to start with. It is recommended to pick one early and focus on it so that tuning theory does not become more important than composition.<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> <br /> This is my list of the most versatile microtonal tunings, split into two tiers:<br /> <br /> Absolute best: [[19edo]], [[22edo]], [[24edo]], [[31edo]], [[41edo]].<br /> <br /> Second best: [[17edo]], [[26edo]], [[27edo]], [[29edo]], [[37edo]].<br /> <br /> I intend to add sound examples to make the selection process easier.<br /> <br /> * 19edo: The best when it comes to simple practical composition. Similar to 31edo, but smaller.<br /> * 29edo: Translates popular 12edo music well because it has a perfect fifth very close to 3/2. Most composers will be fine using a different tuning though.<br /> * 31edo: Something of a standard. Easy to use and full of consonant intervals, but on the larger side.<br /> * 37edo: A highly efficient and interesting array of intervals, but no clear perfect fifth. Requires an understanding of [[Straddle primes|straddling]] to be used on the same level as the others.</div>

Ground https://xenreference.com/wiki/index.php?title=Template:UTBox&diff=8355 2026-10-23T21:58:13Z

<p>Ground: </p> <hr /> <div>&lt;includeonly&gt;{{Mbox|message={{UserTag|{{{1}}}|{{{2}}}|{{{3}}}|'''This section is written from the perspective of [[User:{{{2}}}]]''' and may feature [[:Category:Pages with proposed concepts|proposed concepts]].}}}}&lt;/includeonly&gt;&lt;noinclude&gt;<br /> UTBox is a messagebox form of [[Template:UserTag|UserTag]] that applies to entire sections of articles about a particular person's theory. <br /> <br /> &lt;pre&gt;{{UTBox|g_|Ground|7766ff}}&lt;/pre&gt;<br /> <br /> # User tag, ideally one or two letters<br /> # Username for the link to the user page<br /> # Hex color without #<br /> <br /> {{UTBox|g_|Ground|7766ff}}<br /> [[Category:Message boxes]]<br /> &lt;/noinclude&gt;</div>

Ground https://xenreference.com/wiki/index.php?title=Ground%27s_composition_theory&diff=8346 2026-10-22T03:50:25Z

<p>Ground: /* Quartertone composition */</p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> This article is about composition and the practical considerations of tuning. I always have more content to add and revise. It has an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]], which may be useful to know.<br /> <br /> It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the [[Levels of tempering|mesotempered category]]. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.<br /> <br /> == Interval logic deviation theory ==<br /> <br /> Much of my music has had a distinctively shifting tonality since 2018 or earlier, which started in 12edo. This article is an attempt to explain how it works. I'm introducing a placeholder term for it, '''interval logic deviation''' theory (ILD), which can be replaced if this turns out to be something already described.<br /> <br /> === Local tonality ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic example of probabilities (up to word length 3)<br /> !Sequence<br /> !Next step probability<br /> |-<br /> |s<br /> |L 1/1<br /> |-<br /> |L<br /> |L 5/7, s 2/7<br /> |-<br /> |sL<br /> |L 1/1<br /> |-<br /> |Ls<br /> |L 1/1<br /> |-<br /> |sLL<br /> |s 1/2, L 1/2<br /> |-<br /> |LsL<br /> |L 1/1<br /> |-<br /> |LLs<br /> |L 1/1<br /> |-<br /> |LLL<br /> |s 2/3, L 1/3<br /> |}<br /> &lt;/div&gt;<br /> I have a simultaneous regard and disregard for standard Western tonality. This is because I view it as an important but strictly local property, meaning it fundamentally only applies on the scope of a single path from tension to release, however long that is. Thus, modulations are only generally uncommon because phrases usually resolve to the same key center they started from, but changing tonality is just as much of a choice as not changing it. Modulation flows just like any other melodic or harmonic movement.<br /> <br /> This flow is facilitated by ILD, in which scales aren't a fixed set of notes, but a template for interval logic to be rearranged and deviated from. As such, their main features are probabilities in an interval sequence and &quot;bubble deviations&quot; from that interval sequence. ILD is best for music with a strong melodic focus, such as mine, where the melody informs the harmony instead of the reverse. Other concepts may be used instead with the same general goal.<br /> <br /> Melodic interval sequences, or &quot;words&quot; of step sizes, are the most minimal expression of tonal tension and release. For example, if a diatonic melody were to play C then B, the listener is likely to expect A to be next and feel a small resolution upon hearing it. This is the descending sL word. Melodies are full of small sequences like this based on the scale that they are in. It's possible to use sequences with notes outside the scale while still feeling like they belong, and how much they belong can be predicted. The longer the word and the greater probability of occurring indicates that it's more likely to sound like it belongs in the scale.<br /> <br /> === Axes of deviation ===<br /> <br /> &quot;Bubble deviations&quot; in ILD are named after the bubble sort algorithm, which repeatedly swaps adjacent items in an array. Steps in a base scale can be swapped in the same way to modify the scale with no requirement for a clear structure like a generator chain or lattice splotch, which is my term for a collection of generator chains in aberrismic theory. Suppose you want the scale word sLs in diatonic. This would require only one bubble deviation from the expected step order, turning sLLs to sLsL. While the probability of encountering it in the base diatonic scale is zero, it sounds more &quot;probable&quot; (less unexpected) than something like ssL.<br /> <br /> This explains why I used Dimininished[8] in 12edo more often than Augmented[6], because its stepwise interval logic has less deviation from diatonic. Diminished[8] is made of a repeating sequence of 1\12 and 2\12, common in diatonic, whereas Augmented[6]'s steps of 1\12 and 3\12 do not occur in diatonic at all. As a result, melodies in Diminished sound less exotic.<br /> <br /> Bubble deviations are only one axis of deviation. There is another axis which I've found to be exclusively useful in tuning systems with aberrismic-sized steps or smaller: the axis of microtonal deviation from expected intervals. This involves changing the pitch of an expected interval only slightly, so it is heard as a variation of the expected interval rather than a different interval entirely. This axis interacts with the base scale by introducing or modifying an aberrismic offset, for example diatonic being diasem or blackdye with the offset removed, 2.3.7 diasem having a larger offset than 2.3.23, or 2.3.5 blackdye having a smaller offset than 2.3.17/7. The intervals affected by the offset, usually thirds and sixths, differ microtonally when the offset is changed.<br /> <br /> Straddling intervals that are stacked the most (usually 3/2) introduce a third axis that can be simplified into a combination of the other two. It's possible to have bubble deviation from a scale that isn't even in the tuning system being used, like how alternating &lt;&lt;3 and &gt;3 in 37edo straddles 74edo meantone and results in trackdye.<br /> <br /> {{UserTag|KC|Inthar|000000|[[Stretching and compression]] constitute yet another axis of deviation separate from straddling. Diatonic-based example: This is important in 4L3s and 5L3s which are warped-diatonic MOSes. Note that dual-3 diatonic 5L1m1s is a subset of interleaved diatonic 7s(5L2m), tens-interleaved diatonic 6s(5L2m) ''and'' tract-interleaved diatonic 8s(5L2m).}}<br /> <br /> === Vague interval logic ===<br /> <br /> Scales are useful for ILD, but not technically necessary. One may pick the desired step sizes and find just a few arrangements leading to useful intervals, like the perfect fifth. This should be especially useful when trying to avoid making quasi-diatonic music in any tuning system.<br /> <br /> Take 3\24 s and 5\24 L for example. The words LL and sssL make 10\24 and 14\24 respectively, so the two can together be used to infer vague probabilities. Edos that straddle important intervals are also useful for this because they increase the chances of landing on those intervals. If s and L were replaced with 11\86 and 18\86, LL becomes &gt;4/3 and sssL becomes &gt;3/2.<br /> <br /> === Aberrismic theory ===<br /> <br /> The above concepts also apply to ternary scales, where they may be even more useful. Conventional aberrismic scales feature an alternating generator sequence that creates a 2-dimensional lattice, which can be used similarly to a generator sequence, but is less intuitive due to its complexity. Thus, ILD proves to be a practical alternative when modulating. This is how I write 2.3.5 and 2.3.7 music in tunings that aren't Meantone or Archy.<br /> <br /> Quasi-diatonic aberrismic scales are the ideal case due to the general importance of stacking prime 3 and internalized diatonic logic of Western music. They typically have four main step sizes: '''aberrisma''' (s), '''semitone''' (m), and the two whole tones '''solitone''' (L) and '''magnitone''' (L+s). Other scales or ILD can introduce more sizes such as the '''magnisemitone''' (m+s). The decision of whether to use a solitone or magnitone depends on which intervals in a chord are preferred and which sounds more melodically to the composer in a given situation. The aberrisma is a new class so its theory is less well defined, but it works best as a passing tone. It provides the ability to change the length of scale runs without repeating any notes.<br /> <br /> I find it most generally useful to define the size range of an aberrisma as being between 81/80 and half of 16/15, but my personal usage has ranged from about 12¢ (the size of 81/80 in 5-limit CWE Negri and around one step of the largest full edos I use) to about 100¢ (81/80 in some tunings of Blackwood with a flat 5). I provide 40¢ as the general ideal, but the actual ideal varies. Faster music prefers a larger aberrisma because it is more audible. Scales with a large magnitone can handle a larger aberrisma without it sounding entirely like a semitone, such as 37edo pental blackdye. '''Subaberrismas''', aberrismas small enough that they are not reliably recognizable as a melodic step, have a unique sound that may be desirable in certain situations.<br /> <br /> Because they are often passing tones when used melodically, it can be relatively easy to swap an aberrisma for one of a more preferable size. For example, I may swap 1\43 in 43edo diasem for the semiquartal 2\43 to increase its audibility.<br /> <br /> I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.<br /> <br /> == Diatonic analogues ==<br /> <br /> === 9-note &quot;diatygic&quot; ===<br /> <br /> ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.<br /> <br /> When it comes to extending the diatonic scale's circle of fifths as seen in ILD, there's no alteration more important than the modes of the melodic minor scale, which I'll be referring to together as '''melodiatonic'''. It's the only non-diatonic scale that's possible to construct by making a single bubble deviation from diatonic, and it makes sense looking at the circle of fifths. Assume that some altered diatonic scale must contain only L and s steps to preserve as much step logic as possible. To do this, only ordinals following a semitone may be sharpened (C and F when the unaltered notes are all naturals), and only ordinals preceding a semitone may be flattened (B and E). Sharpening F or flattening B just changes the key signature and the result is still diatonic. Sharpening C or flattening E results in melodiatonic. I have found melodiatonic to be a remarkably reliable way to make diatonic tunings (including 12edo) sound exotic yet familiar enough to hit the same emotional beats diatonic can provide. After diatonic, it has the smallest span across the chain of fifths without duplicating a diatonic ordinal, meaning that it contains no pairs of notes that are 7 generators apart. Just like how diatonic is the framework for all quasi-diatonic scales, melodiatonic can be used as the framework for certain alterations of those quasi-diatonic scales. {{adv|Unconventional uses of melodiatonic I can think of off the top of my head are parts of Undertale's Heartache in Bb Dorian b2 and the main theme of Animusic's Harmonic Voltage in C Mixolydian b6.}}<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Quasi-Diatygic Scales<br /> !Altered&lt;br&gt;Blackdye<br /> !Altered&lt;br&gt;Diasem<br /> |-<br /> |10/9&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;5/4&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;5/3&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |9/8&lt;br&gt;8/7&lt;br&gt;7/6&lt;br&gt;9/7&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;14/9&lt;br&gt;12/7&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |}<br /> {| class=&quot;wikitable&quot;<br /> |+Diatonic Tertian Triads in D Gadygic<br /> !Root<br /> !Triads<br /> |-<br /> |D<br /> |minor, Major<br /> |-<br /> |E<br /> |dim., minor<br /> |-<br /> |F<br /> |Major<br /> |-<br /> |F♯<br /> |dim.<br /> |-<br /> |G<br /> |minor, Major<br /> |-<br /> |A<br /> |minor<br /> |-<br /> |B♭<br /> |Major<br /> |-<br /> |B<br /> |dim., minor<br /> |-<br /> |C<br /> |Major<br /> |}<br /> &lt;/div&gt;<br /> The gaps in the fifth chain can be filled to result in a contiguous generator chain of 9 notes rather than 7, allowing some modes to freely switch between major and minor thirds and sixths. This is probably the earliest music theory I've ever consciously developed, dating to 2014 or so ([https://soundcloud.com/groundfault/doodle-62 example]). It has been scale #1 on the [https://allthescales.org/scales.php?n=9 9-note page of AllTheScales] for at least as long, and I tentatively propose calling the scale '''diatygic''' using the seemingly arbitrary -ygic suffix from AllTheScales since it seems like a term I would have chosen at the time. It has stood the test of time and I find it to be the natural extension of the diatonic framework that doesn't require microtuning, so it remains my favorite 12edo-compatible scale. The scale word for the symmetrical mode is LdcdLdcdL, with d and c for diatonic and chromatic semitones. It is called '''Gadygic''' by AllTheScales. I use plenty of extended interval chains when I compose, but diatygic is intended to be a proper albitonic scale with simple self-contained rules, prescriptive rather than descriptive in the sense that it is the new template to be extended and deviated from. {{adv|For some reason, the first examples of this scale I can think of (besides my own music) all come from the brony fandom. Morning in Ponyville, Awesome as I Wanna Be, and Awoken. It should leave no guesswork as to what I was doing in 2014. All of them are coincidentally in the key of D.}}<br /> <br /> I derived diatygic from my own observation of my favorite music, intuition, and the methods above, including that 9 notes is the closest thing to a MOS between 7 and 12 notes. For one thing, MOS scales with generators approaching half an octave will have sizes that are successive odd numbers; the diatonic range jumps from 7 to 12, but the adjacent antidiatonic range follows 7 with 9. Stacking 9 fifths results in a scale with groups of 3 semitones in a row, which is substantially easier to navigate than the 10-note &quot;supermode&quot; in [https://mtosmt.org/issues/mto.11.17.4/mto.11.17.4.temperley.html Temperley 2011], with a group of 5 semitones. I've found it more useful to use various modes of diatygic than to add additional notes, since needing more than two non-diatonic notes at a time is very rare. This is so that the scale can contain the diatonic modes Aeolian to Ionian as subsets, whereas diatygic only manages Aeolian to Mixolydian. I've never used the major seventh degree much myself, despite its high apparent importance in Western music. Even back when I used major modes more than minor, I preferred Mixolydian to Ionian. It's about as important to me as the minor second degree, which is easy to achieve through using modes other than the symmetrical one.<br /> <br /> This extension of diatonic also works for quasi-diatonic scales. {{adv|Pental blackdye is two chains of 5 of 3/2 separated by 10/9. Extending both of those chains by 2 is analogous to what I'm doing to diatonic, but separating them by 81/80 instead makes 5L2m7s. Separating by 10/9 results in a quaternary scale that is very nearly ternary; temper out the Schisma and you get 3L6m5s with a non-standard step order to what I've established in my aberrismic theory.}} The altered pental blackdye is almost the 3sA Aeolian mode with 5/4 and 5/3 added. The retroversion of this can be obtained by adding 6/5 and 8/5 to 3sG Mixolydian, but it's inferior because it lacks the perfect fifth over 6/5. The one additional change is 81/80 to 10/9 to fix the wolf fifth from 9/8 to 5/3. All 12 notes are required to make a chain of pure 3/2 sometimes altered by 81/80 just as blackdye does, except 4/3-16/9 and 27/20-9/5 are redundant. The extra note is kept to keep the scale to a minimum of four unique step sizes, similar to what is done with blackdye. A similar process may be done to septal diasem.<br /> <br /> One of my favorite parts of the scale is a major third on D, a minor third on E, and a major third on F, using notes from D Gadygic.<br /> {| class=&quot;wikitable&quot;<br /> |+Interval Matrix of Gadygic, DR 3:4:5 Meantone<br /> !<br /> !0<br /> !1<br /> !2<br /> !3<br /> !4<br /> !5<br /> !6<br /> !7<br /> !8<br /> !(9)<br /> |-<br /> !0.<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !194.6<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1200<br /> |-<br /> !308.2<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !389.1<br /> |0<br /> |113.6<br /> |308.2<br /> |421.8<br /> |502.7<br /> |616.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !502.7<br /> |0<br /> |194.6<br /> |308.2<br /> |389.1<br /> |502.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |-<br /> !697.3<br /> |0<br /> |113.6<br /> |194.6<br /> |308.2<br /> |502.7<br /> |697.3<br /> |810.9<br /> |891.8<br /> |1005.4<br /> |1200<br /> |-<br /> !810.9<br /> |0<br /> |80.9<br /> |194.6<br /> |389.1<br /> |583.7<br /> |697.3<br /> |778.2<br /> |891.8<br /> |1086.4<br /> |1200<br /> |-<br /> !891.8<br /> |0<br /> |113.6<br /> |308.2<br /> |502.7<br /> |616.3<br /> |697.3<br /> |810.9<br /> |1005.4<br /> |1119.1<br /> |1200<br /> |-<br /> !1005.4<br /> |0<br /> |194.6<br /> |389.1<br /> |502.7<br /> |583.7<br /> |697.3<br /> |891.8<br /> |1005.4<br /> |1086.4<br /> |1200<br /> |}<br /> <br /> === Neutral diatonic extensions ===<br /> <br /> Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.<br /> <br /> A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.<br /> <br /> I believe xenharmony involves a balance of being unusual without being jarring. Neutral seconds around 150¢ are excellent for the former, but are more difficult to fit into a diatonic framework. When given the choice, I prefer to have neutral seconds around 135¢ to fit in place of a semitone and 165¢ to fit in place of a whole tone. Temperaments in small edos are Flattone for the former and Porcupine for the latter.<br /> <br /> When converting a simple subgroup to contain these intervals, it makes sense to use 2.3.5 because it's closer to neutral whereas 2.3.7 is closer to interordinal.<br /> * Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.<br /> * Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.<br /> This is an excellent use case for straddle-3 temperaments. 37edo has mild Mavila and extreme Archy fifths, but their distance from 3/2 is somewhat masked in 5-limit chords due to the 5/4 anchor concordance.<br /> <br /> == Adaptive diatonic modification system ==<br /> <br /> This section is incomplete.<br /> <br /> In order to make modified diatonic scales easier to use and quickly understand the intervals I use in my head, I have a proposal for an interval scheme based on splitting diatonic or quasi-diatonic intervals, somewhat similar to [[Adaptive diatonic interval names|ADIN]].<br /> <br /> First specify which is the default diatonic or quasi-diatonic quality. This is usually pental (4:5:6), pyth, or septal (6:7:9).<br /> <br /> Neutral seconds are about 3/7 to 4/7 of the minor third.<br /> <br /> == Structural tuning selection ==<br /> <br /> Edos are the most common type of tuning system, and the one I use exclusively. They're a simple and discrete set, which makes them very easy to compare to one another and list in the titles of music made with them. This creates a common assumption that they sound more different from each other than they actually do, which I suffer from myself sometimes. Small edos really do have distinct personalities, but as they get larger and more flexible, those differences gradually disappear. I try to guess the tuning system of a piece whenever it isn't stated, and I'm very often wrong.<br /> <br /> I propose that the most important distinction between tuning systems of sufficient flexibility is not how they sound, but how their unique combinations of structures guide the composer. I like to believe that I have sole control over what I write, but my choice of tuning system prioritizes certain scales, chords, and intervals over others by making them easier to use in practice.<br /> <br /> I use large edos in an unusual way. I like to have multiple octaves of range directly in the piano roll, so I access large edos by multiplying smaller ones using midi channels and my custom tuning scripts. This creates a more complex process of tuning selection, involving the combination of two considerations:<br /> # Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?<br /> # Which of those can be multiplied by 2 or 3 to achieve a superset that's enjoyable for the same reasons, but also complements the subset edo well?<br /> {| class=&quot;wikitable&quot;<br /> |+A selection of large edos I use<br /> !Edo<br /> !Justification<br /> |-<br /> | style=&quot;text-align: right;&quot; |27x3&amp;nbsp;=&amp;nbsp;81<br /> |81 is an [[Monarch|Interarch]] edo, which I find useful. 27x2 = 54 is also uniquely bad, an example of poor LCJI approximations providing insufficient advantages to be worth using.<br /> |-<br /> | style=&quot;text-align: right;&quot; |29x3&amp;nbsp;=&amp;nbsp;87<br /> |I don't like 29 enough to use it much on its own, but 87 is my favorite highly accurate LCJI edo, the intersection of Aberschismic and a variant of [[Hemiseven]].<br /> |-<br /> | style=&quot;text-align: right;&quot; |37x2&amp;nbsp;=&amp;nbsp;74<br /> |Note that this is not 37x3 = 111, a much better LCJI edo. 111 offers little to complement the existing structure of 37edo that 74 doesn't already do.<br /> |-<br /> | style=&quot;text-align: right;&quot; |43x2&amp;nbsp;=&amp;nbsp;86<br /> |I found 86 to be the best Interarch edo for my edo multiplication approach. It clearly straddles every prime up to 19 with the possible exception of 13, but it's large enough that their accuracy is not a major issue.<br /> |-<br /> | style=&quot;text-align: right;&quot; |67x2ed4&amp;nbsp;=&amp;nbsp;67<br /> |Doubling ed4s is very annoying, but it's the only way for me to practically reach large prime edos. 67 is the only one significant enough to be worth the trouble.<br /> |}<br /> <br /> == Edo multiplication ==<br /> <br /> I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the '''base edo''' to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.<br /> <br /> Example tier list of 37edo multiples:<br /> * Base ([[37edo|37]]): S (One of a kind.)<br /> * x2 ([[74edo|74]]): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)<br /> * x3 ([[111edo|111]]): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)<br /> * x6 ([[222edo|222]]): D (Has extra approximations and generators, but too cumbersome to do much.)<br /> This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:<br /> * Base: remainder<br /> * x2: 12%<br /> * x3: 8%<br /> * x6: 4%<br /> <br /> This should show just how important the base edo is. No matter how many intervals you have, large edos or edo multiplication will never be as efficient to write in as an elegant mid-sized edo that does what you want the most. Everything is filtered through the base edo's structure. Because of this, there's no such thing as the perfect edo. At this point, I use whatever I feel like at the time and multiply it by 6 to fill any gaps that might come up.<br /> <br /> When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.<br /> <br /> Edos that are multiples of 6 usually have more than one reasonable way to split them. I will use 144edo to explain my thought process in deciding whether to use 24x6 or 36x4.<br /> * The most important consideration is which base edo the composer enjoys the most. Both options have features that are important to me, but I've written a good amount in 24 because I believe its lack of a good 5 is less important, so it's been proven by time.<br /> * In terms of the number of useful scales available, 36 is superior. To some extent, this comes with having more notes in general.<br /> * x6 has more channels overall, but contains both the x2 and x3 subsets. x4 requires a larger base edo and has only the option to use x2 and x4 notes. Because of the distribution of the number of channels I typically use, I expect x6 to have more practical options.<br /> * x6 is more of a standard for me, so I'm set up better for that.<br /> * I mentioned how I like to make clear switches between x2 and x3 to make it easier to read my piano roll notes. x6 is better for this because x4 has only one distinct subset besides the base edo, x2.<br /> * 36x2 makes accessing 72edo especially convenient because fewer channels is easier. It wins for 11-limit stuff.<br /> * Two important subsets of 144 are 16 and 18. 24x6 has them as x2 and x3, but 36x4 has them as x4 and x1. This is a tie.<br /> * I use 5-form structures a lot, and some of my favorite edos are 43 and 67. Add 5 to bring their 3/2 closer to just, resulting in 48 and 72, 24x2 and x3. This is an elegant way to approximate both 43 and 67.<br /> * Using a smaller edo gets more range in the piano roll, which is useful for quick creative writing.<br /> * 36 has clearer concordances because it's so good at 2.3.7. It's good to have some anchor concordance in the base edo, but 24 also has 3/2. Having 2.3.7 outside the base edo naturally guides composition towards using other channels and makes it hard to forget about them. There's a nice symmetry with 24x6 putting 7/6 and 6/5 on the same channel as well as 9/7 and 5/4.<br /> With all this in mind, 24x6 is clearly the better choice to start with, although it's close.<br /> <br /> === Channel management ===<br /> <br /> Blackdye and diasem structures have a generator of a perfect fifth and some offset for certain notes. Because all of the offsets are in the same direction, using an offset that is part of a superset of the generator doubles the number of distinct channels required. This most notably results in scales where the generator is from the x2 edo and the offset is from the x3 or x6 edo only requiring 4 channels.<br /> <br /> == Anchor concordances ==<br /> <br /> We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an '''anchor concordance''' because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.<br /> <br /> For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.<br /> <br /> I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.<br /> <br /> In 37edo, both 4:5:&gt;6 and 4:5:&lt;&lt;6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides &lt;&lt;6:8:&lt;&lt;&lt;&lt;9 = 6:&gt;&gt;8:&lt;&lt;9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:<br /> * Using the very flat &lt;&lt;3 results in &gt;&gt;16/15, quite audibly wider while keeping some semblance of 4:5:6.<br /> * &lt;&lt;6/5 is found in the [[Pentagoth]] triad, which uses the exaggerated quality of the minor third and the stability of the major third.<br /> <br /> Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:<br /> * ~69edo: DR 4:5:6, 10:12:15<br /> * ~31edo: 5/4 anchor concordance<br /> * ~74edo: DR 3:4:5<br /> * ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)<br /> <br /> == Quartertone composition ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot;<br /> |+Island Rastmic with Straddle Primes<br /> !Gens<br /> !949¢<br /> !349¢<br /> ! colspan=&quot;2&quot; |&lt;nowiki&gt;Proposed Names (semi- | hemi-)&lt;/nowiki&gt;<br /> |-<br /> | -12<br /> | colspan=&quot;2&quot; |612<br /> | colspan=&quot;2&quot; |Diminished Fifth<br /> |-<br /> | -11<br /> |361<br /> | class=&quot;thl&quot; |961<br /> |&quot;Semififth&quot;<br /> |&quot;Hemidim.&quot; Seventh<br /> |-<br /> | -10<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |110<br /> | colspan=&quot;2&quot; |Minor Second<br /> |-<br /> | -9<br /> |1059<br /> |459<br /> |&quot;Semithirteenth&quot;<br /> |&quot;Hemidim.&quot; Fourth<br /> |-<br /> | -8<br /> | colspan=&quot;2&quot; |808<br /> | colspan=&quot;2&quot; |Minor Sixth<br /> |-<br /> | -7<br /> | class=&quot;thl&quot; |557<br /> |1157<br /> |&quot;Semiseventh&quot;<br /> |&quot;Hemidim.&quot; Octave<br /> |-<br /> | -6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |306<br /> | colspan=&quot;2&quot; |Minor Third<br /> |-<br /> | -5<br /> |55<br /> |655<br /> |&quot;Semisecond&quot;<br /> |&quot;Hemidim.&quot; Fifth<br /> |-<br /> | -4<br /> | colspan=&quot;2&quot; |1004<br /> | colspan=&quot;2&quot; |Minor Seventh<br /> |-<br /> | -3<br /> |753<br /> |153<br /> |Semitenth<br /> |Neutral Second<br /> |-<br /> | -2<br /> | colspan=&quot;2&quot; |502<br /> | colspan=&quot;2&quot; |Perfect Fourth<br /> |-<br /> | -1<br /> |251<br /> |851<br /> |Semifourth<br /> |Neutral Sixth<br /> |-<br /> |0<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |0<br /> | colspan=&quot;2&quot; |Unison<br /> |-<br /> |1<br /> |949<br /> |349<br /> |Semitwelfth<br /> |Neutral Third<br /> |-<br /> |2<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |698 ''(98)''<br /> | colspan=&quot;2&quot; |Perfect Fifth<br /> |-<br /> |3<br /> |447<br /> |1047<br /> |Semisixth<br /> |Neutral Seventh<br /> |-<br /> |4<br /> | colspan=&quot;2&quot; |196<br /> | colspan=&quot;2&quot; |Major Second<br /> |-<br /> |5<br /> |1145<br /> | class=&quot;thl&quot; |545<br /> |&quot;Semifourteenth&quot;<br /> |&quot;Hemiaug.&quot; Fourth<br /> |-<br /> |6<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |894 ''(294)''<br /> | colspan=&quot;2&quot; |Major Sixth<br /> |-<br /> |7<br /> |643<br /> |43<br /> |&quot;Semininth&quot;<br /> |&quot;Hemiaug.&quot; Unison<br /> |-<br /> |8<br /> | colspan=&quot;2&quot; class=&quot;thl&quot; |392<br /> | colspan=&quot;2&quot; |Major Third<br /> |-<br /> |9<br /> |141<br /> |741<br /> |&quot;Semithird&quot;<br /> |&quot;Hemiaug.&quot; Fifth<br /> |-<br /> |10<br /> | colspan=&quot;2&quot; |1090<br /> | colspan=&quot;2&quot; |Major Seventh<br /> |-<br /> |11<br /> | class=&quot;thl&quot; |839<br /> |239<br /> |&quot;Semieleventh&quot;<br /> |&quot;Hemiaug.&quot; Second<br /> |-<br /> |12<br /> | colspan=&quot;2&quot; |588<br /> | colspan=&quot;2&quot; |Augmented Fourth<br /> |}<br /> &lt;/div&gt;<br /> This topic isn't directly related to the other sections, but I don't have anywhere else to put it. I've long been interested in what I'm calling '''quartertone composition''', which is composition based on the 24-form. It offers both familiarity to 12edo and all the intervals that are the most alien. The most important generator chains are the ones that split 4/3 in half (semiquartal, aesthetically my favorite, see [[Intergan]]) and split 3/2 in half (mosh or dicoid). Together, they can be analyzed as a diatonic generator chain that can be deviated from by either a semifourth or hemififth. Quartertone tunings may have one chain or both.<br /> <br /> I fond it very useful to examine tertian functions by stacking different combinations inframinor/minor/neutral/major/ultramajor thirds.<br /> <br /> TODO: List<br /> <br /> === Scale theory ===<br /> <br /> One of the best ways I've found to introduce quartertone intervals into the diatonic framework is by altering either B or F (assuming all natural notes) by a quartertone. This results in a scale with only one fewer perfect fifth, since the other fifth containing the altered note is diminished. This gives four options:<br /> * A Bd C D E F G A (-&gt; mosh)<br /> * A Bt C D E F G A (-&gt; sq)<br /> * A B C D E Fd G A (-&gt; sq)<br /> * A B C D E Ft G A (-&gt; mosh)<br /> These introduce &quot;dimimajor&quot;, &quot;diminimor&quot;, and neutral chords for the mosh-leaning scales and &quot;small/large subdim&quot; and inframinor/ultramajor chords for the semiquartal-leaning scales.<br /> <br /> === Scale tempering ===<br /> <br /> Having two separate generator chains is ideal, but it can be inconvenient. This can be fixed by using a half-octave period instead of an octave, where the semitwelfth and hemififth differ by half an octave. {{adv|I would argue that the most important quartertone temperament overall is '''Island Rastmic''', a half-octave 24&amp;34 2.3.11.13/5 subgroup temperament that splits 4/3 into two 15/13s and 3/2 into two 11/9s. Like in Intergan, all the quartertone intervals don't leave much room to introduce comma steps if prime 5 is desired, so a mild Meantone tempering is the best option. In addition, I find this tuning range around Mohajira to have the most pleasant-sounding neutral triads. 17/12 may be equated to half an octave and 19/17 to 9/8 as in Intergan. The final result is 2.3.5.11.13.17.19 24&amp;62.}}<br /> <br /> In the accompanying generator table, diatonic intervals are aligned, while adjacent quartertone intervals always differ by 600¢. Primes are highlighted. {{adv|The top half of the table shows an optional 7 (the simplest mapping is in the hemififth chain due to the semitwelfth already being so close to 7/4, but it's in the wrong direction) and an alternate sharp 11, 17, and 19 all found in Intergan, but none of these were included in 2.3.5.11.13.17.19 24&amp;62. The flat 17 and 19 are shown in parenthesis because they differ from diatonic intervals by half an octave.}} My proposed interval names are extrapolated from existing names in order to be as unambiguous as possible.<br /> <br /> This way of displaying intervals in a half-octave temperament, having two full-octave generators that differ by half an octave, is mostly similar to the standard way. The columns don't correspond to which of the two periods the intervals are in, but it's simple enough to guess because the smaller one is first period and the larger one is second period. The major difference is that intervals an even number of generators from unison only show one interval in one period. For temperament reasons, I included some of the ones not shown in parenthesis. This matters a lot because it ignores an important Diaschismic equivalence: sqrt(2) / (9/8) ≈ 5/4.<br /> <br /> This may result in a new way of generating scales. An Aeolian diatonic scale ranges from -4 to 2 generators. This temperament divides the generator in half, so take all intervals in the table from -8 to 4 generators, ignoring anything in parenthesis. The resulting scale has 19 notes: 55 153 196 251 306 349 447 502 557 655 698 753 808 851 949 1004 1047 1157 1200. No more notes can be added without introducing some extremely small steps to the scale, which would be tempered out in 24edo. The scale has four unique step sizes: 43 55 98 110. There is only one step of 110¢, which can be removed by replacing 1047 or 1157 (the most extreme two intervals unique to the 349¢ column) with 1059 or 1145 (the next two intervals unique 949¢ column). This results in the two chiralities of 5L9m5s. Sharpening the two generators to 950¢ and 350¢ results in 5L14s, and further sharpening them flips m and s to 5L5m9s.<br /> <br /> === Melodic intervals ===<br /> <br /> The diatonic half of quartertone composition does not need an explanation. Semiquartal and dicoid have their own compositional practices, much of which can be inherited from diasem and blackdye respectively due to being degenerate cases. What's left to explain is how this all fits together.<br /> * The quartertone has function similar to a wide aberrisma and a narrow semitone, since semiquartal is just diasem with the two equated. It's useful for altering intervals like an aberrisma and has a distinctive &quot;metallic&quot; sound as a semitone.<br /> * The neutral second is best known for occurring in dicoid and antidiatonic, which it can take most of its function from. It has a distinctive &quot;sour&quot; sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.<br /> * The semifourth is most useful as an inframinor third, but it also works as an ultramajor second. It pairs best with wider minor thirds which make it sound more like a second in comparison. It's useful for shrinking the semitone between the second and third without shrinking the third.<br /> <br /> == Tertian categorization ==<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Thirds categorized as a fraction of a diatonic fifth.<br /> !Fraction f<br /> !*7/12 =<br /> !4\7 (¢)<br /> !7\12 (¢)<br /> !3/2 (¢)<br /> !3\5 (¢)<br /> !Lower bound of:<br /> |-<br /> !1/3<br /> |7/36<br /> |228.6<br /> |233.3<br /> |234.0<br /> |240.0<br /> |&quot;Horiminor&quot;<br /> |-<br /> !12/35<br /> |1/5<br /> |235.1<br /> |240.0<br /> |240.7<br /> |246.9<br /> |Inframinor<br /> |-<br /> !18/49<br /> |3/14<br /> |251.9<br /> |257.1<br /> |257.9<br /> |264.5<br /> |Subminor<br /> |-<br /> !11/28<br /> |11/48<br /> |269.4<br /> |275.0<br /> |275.8<br /> |282.9<br /> |Neominor<br /> |-<br /> !59/140<br /> |59/240<br /> |289.0<br /> |295.0<br /> |295.8<br /> |303.4<br /> |Grave Minor<br /> |-<br /> !61/140<br /> |61/240<br /> |298.8<br /> |305.0<br /> |305.9<br /> |313.7<br /> |Classic Minor<br /> |-<br /> !23/49<br /> |23/84<br /> |321.9<br /> |328.6<br /> |329.5<br /> |338.0<br /> |Supraminor<br /> |-<br /> !24/49<br /> |2/7<br /> |335.9<br /> |342.9<br /> |343.8<br /> |352.7<br /> |Neutral<br /> |-<br /> !25/49<br /> |25/84<br /> |349.9<br /> |357.1<br /> |358.1<br /> |367.3<br /> |Submajor<br /> |-<br /> !26/49<br /> |13/42<br /> |363.8<br /> |371.4<br /> |372.5<br /> |382.0<br /> |Classic Major<br /> |-<br /> !79/140<br /> |79/240<br /> |386.9<br /> |395.0<br /> |396.1<br /> |406.3<br /> |Acute Major<br /> |-<br /> !81/140<br /> |81/240<br /> |396.7<br /> |405.0<br /> |406.1<br /> |416.6<br /> |Neomajor<br /> |-<br /> !17/28<br /> |17/48<br /> |416.3<br /> |425.0<br /> |426.2<br /> |437.1<br /> |Supermajor<br /> |-<br /> !31/49<br /> |31/84<br /> |433.8<br /> |442.9<br /> |444.1<br /> |455.5<br /> |Ultramajor<br /> |-<br /> !23/35<br /> |23/60<br /> |450.6<br /> |460.0<br /> |461.3<br /> |473.1<br /> |&quot;Horimajor&quot;<br /> |-<br /> !2/3<br /> |7/18<br /> |457.1<br /> |466.7<br /> |468.0<br /> |480.0<br /> |[end of range]<br /> |}<br /> &lt;/div&gt;<br /> The accompanying table shows my own way of categorizing triads and tetrads bounded by ~3/2, with the third being a logarithmic fraction of the fifth. {{adv|Note that the thirds are presented as a logarithmic fraction of the fifth, which scales the major and minor thirds by the same ratio to fit the fifth. Delta-rational logarithmically scales the lower third in the triad slightly more than the upper third.}} This is not an exhaustive list of qualities, but rather groups of my major use cases.<br /> <br /> * I've proposed the terms &quot;horiminor&quot; and &quot;horimajor&quot; after the same root in the word &quot;horizon&quot;. These are the chords that are technically possible to hear as containing thirds, but it requires special care in the melody to make it convincing, such as splitting the horiminor third. I find this class especially useful in 10edo, which is often considered to only have neutral chords.<br /> * Inframinor and ultramajor chords share the metallic quality with the quartertone, as mentioned in the previous section. They are similar in function to the subminor and supermajor chords based on 6:7:9, but have a harsher and exaggerated sound, usually interpreted as based on 10:13:15.<br /> * Subminor and supermajor chords are in the vicinity of 6:7:9 and 14:18:21. Subminor especially has a pure sound like 4:5:6, but these chords may sound too different from Western tuning to be desirable in many cases.<br /> * Neominor and neomajor are used in this case to refer to any thirds not in the concordance wells of 6:7:9 or 4:5:6. This gives them a sound that I would describe as impure but even. They're the most versatile for transcribing modern 12edo music while remaining somewhat xenharmonic. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.<br /> * Grave minor and acute major chords are approximately the ones found in 12edo, around 16:19:24. I consider these to be a separate category less often than any other, but this presence of this otonal chord justifies them. A grave minor third is also characteristic of [[Hibernal]].<br /> * Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15, so they may also be called pental. They are considered to be the default in xenharmonic circles, and have been used in Western music for hundreds of years at least, largely falling out of consideration once 12edo became the standard. I think it's always good to have these to some extent because they provide a consonance that can anchor everything else.<br /> * Supraminor and submajor chords lean towards, but I find them to work better as a consonance because they still resemble minor and major functions. They're usually interpreted as based on 14:17:21.<br /> * Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered largely concordant, between 24edo and 31edo. This is where two slightly sharp 11/9s stack to make a slightly flat 3/2. In this range, they sound like ambiguous minor/major chords, which may be useful for modulation or when unsure of which quality is better for a given chord. More broadly, they function as an extremely versatile mild dissonance. People tend to see them as a cross between major and minor, but I actually see them as a cross between major, minor, diminished, and augmented. They're weird as a consonance in a quasi-diatonic context, not because of the complexity of 18:22:27, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.<br /> ** I call this technique a &quot;neutral substitution&quot;. It involves replacing a tertian chord with a neutral one in order to modulate. The easiest way to do this is line up a neutral chord with the root or fifth of where a diminished chord would be, then use whichever of the two is changed to alter the key.<br /> <br /> === Grave fifth chords and mainline fifths ===<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Successive fifths from 22/15 to 40/27,&lt;br&gt;thirds between logarithmic 12/35 and 23/35 of the fifth<br /> ! style=&quot;border-right: double;&quot; |D<br /> ! colspan=&quot;5&quot; style=&quot;border-right: double;&quot; |N<br /> !&lt;3/7f<br /> !&lt;1/2f<br /> !&gt;1/2f<br /> !&gt;4/7f<br /> ! style=&quot;border-right: double;&quot; |Fifth<br /> ! colspan=&quot;4&quot; |Fifth Complement<br /> |-<br /> ! style=&quot;border-right: double;&quot; |15<br /> |<br /> |18<br /> |<br /> | style=&quot;border-right: double;&quot; |19<br /> | style=&quot;border-right: double;&quot; |22<br /> |<br /> |315.6<br /> |<br /> | style=&quot;border-right: double;&quot; |409.2<br /> | style=&quot;border-right: double;&quot; |663.0<br /> |<br /> |347.4<br /> |<br /> |253.8<br /> |-<br /> ! style=&quot;border-right: double;&quot; |17<br /> |<br /> | class=&quot;thl&quot; |20<br /> |21<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |25<br /> |<br /> | class=&quot;thl&quot; |281.4<br /> |365.8<br /> | style=&quot;border-right: double;&quot; |<br /> | style=&quot;border-right: double;&quot; |667.7<br /> |<br /> | class=&quot;thl&quot; |386.3<br /> |301.8<br /> |<br /> |-<br /> ! style=&quot;border-right: double;&quot; |19<br /> |22<br /> |23<br /> |<br /> | style=&quot;border-right: double;&quot; |24<br /> | style=&quot;border-right: double;&quot; |28<br /> |253.8<br /> |330.8<br /> |<br /> | style=&quot;border-right: double;&quot; |404.4<br /> | style=&quot;border-right: double;&quot; |671.3<br /> |417.5<br /> |340.6<br /> |<br /> |266.9<br /> |-<br /> ! style=&quot;border-right: double;&quot; |21<br /> |24<br /> | class=&quot;thl&quot; |25<br /> | class=&quot;thl&quot; |26<br /> | style=&quot;border-right: double;&quot; |27<br /> | style=&quot;border-right: double;&quot; |31<br /> |231.2<br /> | class=&quot;thl&quot; |301.8<br /> | class=&quot;thl&quot; |369.7<br /> | style=&quot;border-right: double;&quot; |435.1<br /> | style=&quot;border-right: double;&quot; |674.3<br /> |443.1<br /> | class=&quot;thl&quot; |372.4<br /> | class=&quot;thl&quot; |304.5<br /> |239.2<br /> |-<br /> ! style=&quot;border-right: double;&quot; |23<br /> |<br /> |27<br /> |28<br /> | style=&quot;border-right: double;&quot; |29<br /> | style=&quot;border-right: double;&quot; |34<br /> |<br /> |277.6<br /> |340.6<br /> | style=&quot;border-right: double;&quot; |401.3<br /> | style=&quot;border-right: double;&quot; |676.7<br /> |<br /> |399.1<br /> |336.1<br /> |275.4<br /> |-<br /> ! style=&quot;border-right: double;&quot; |25<br /> |29<br /> |30<br /> |31<br /> | style=&quot;border-right: double;&quot; |32<br /> | style=&quot;border-right: double;&quot; |37<br /> |256.9<br /> |315.6<br /> |372.4<br /> | style=&quot;border-right: double;&quot; |427.4<br /> | style=&quot;border-right: double;&quot; |678.7<br /> |421.8<br /> |363.1<br /> |306.3<br /> |251.3<br /> |-<br /> ! style=&quot;border-right: double;&quot; |27<br /> |31<br /> | class=&quot;thl&quot; |32<br /> |33<br /> | style=&quot;border-right: double;&quot; |34<br /> | style=&quot;border-right: double;&quot; |40<br /> |239.2<br /> | class=&quot;thl&quot; |294.1<br /> |347.4<br /> | style=&quot;border-right: double;&quot; |399.1<br /> | style=&quot;border-right: double;&quot; |680.4<br /> |441.3<br /> | class=&quot;thl&quot; |386.3<br /> |333.0<br /> |281.4<br /> |}<br /> &lt;/div&gt;<br /> I use a similar system for antidiatonic tertian harmony, but there's no set JI approximation for a flat fifth, so I use multiple to get the simplest ratios possible. [[Pentagoth]] equates many similar intervals in this table.<br /> <br /> I refer to the range from about 23\40 to 25\42 as '''mainline fifths'''. The full diatonic range includes other fifths that generate scales I have found over time to resemble the usual understanding of diatonic less than they resemble 7 or 5edo, and resemble 3/2 less than sqrt(20/9) and sqrt(16/7). The 25\42 boundary is clearer to me because I prefer very hard scales to very soft ones and thus I have more experience. But conveniently, these are the points where an alternate diatonic fifth can be generated at a low complexity. Analogously, I could refer to the range from about 11\20 to 17\30 as '''mainline antidiatonic fifths''' based on my personal experience.<br /> <br /> I've found 25 and 39edo to be the two regions of the armotonic tuning range that sound better than others. {{adv|29edo sounds more pure than both, and is the basis for an excellent 29&amp;67 straddle temperament I've called Gravel, but it involves the 15:19:22 chord with a fifth at my limit for resembling 3/2 which does not generate armotonic. 2.15.11.19 9&amp;56 also generates this, and the subgroup can be extended to 2.3.5.11.19 while hardly changing the edo list. 11:13:16 and 13:16:19 are proper subfifth triads and are not counted in the range. They are also nearly retroversions of each other.}}<br /> <br /> 25edo represents the two otonal chords on the table with 5/4, with flatter fifths like 34 and 43edo being better for 17:20:25 and sharper fifths like 41 and 57edo being better for 27:32:40. Both of these are found in Pentagoth, so I propose calling this range '''pentagothic'''. 43edo is especially notable for how well it also approximates 17:21:25.<br /> <br /> 39edo represents 21:26:31, which is notably a +1+1 chord, and 21:25:31, very close to its retroversion. I propose calling this range '''mavilic''' after Erv Wilson's original Meta-Mavila, the &quot;proper&quot; tuning range for Mavila. {{adv|Mavila possibly has more variants of its name than any other temperament. I'm genuinely surprised that Mavilic isn't a temperament name.}} [[Pentagoth#Otomavila|Otomavila]] is my temperament using these chords.<br /> <br /> For the most part, antidiatonic tertian harmony works like diatonic tertian harmony, but with the reversed qualities of the antidiatonic scale. The pentagothic neutral triad generates Cohemimabila temperament and has a similar function to the neutral triad explained above.<br /> <br /> === Quasi-diatonic scales ===<br /> <br /> {| class=&quot;wikitable&quot;<br /> |+Blackdye and diasem JI tunings, sorted from flat to sharp minor third, denominators up to 20<br /> !Tertian Triad<br /> !16:21:24<br /> !20:23:30<br /> !10:13:15<br /> !6:7:9&lt;br&gt;(14:18:21)<br /> !18:23:27<br /> !16:19:24<br /> !4:5:6&lt;br&gt;(10:12:15)<br /> !14:17:21<br /> !18:22:27<br /> |-<br /> |Subgroup<br /> |2.3.7<br /> |2.3.23/5<br /> |2.3.13/5<br /> |2.3.7<br /> |2.3.23<br /> |2.3.19<br /> |2.3.5<br /> |2.3.17/7<br /> |2.3.11<br /> |-<br /> |Name<br /> |&quot;Septal&quot;<br /> |?<br /> |?<br /> |&quot;Septal&quot;<br /> |&quot;Eridial&quot;<br /> |&quot;Novemdecal&quot;<br /> |Pental<br /> |?<br /> |&quot;Undecal&quot;<br /> |-<br /> |Aberrisma<br /> |28/27<br /> |640/621<br /> |416/405<br /> |64/63<br /> |736/729<br /> |513/512<br /> |81/80<br /> |459/448<br /> |33/32<br /> |-<br /> |Quasi-&lt;br&gt;Aeolian<br /> |9/8&lt;br&gt;8/7&lt;br&gt;9/7&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;32/21&lt;br&gt;12/7&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;23/20&lt;br&gt;207/160&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;23/15&lt;br&gt;69/40&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;15/13&lt;br&gt;135/104&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;20/13&lt;br&gt;45/26&lt;br&gt;16/9&lt;br&gt;2/1&lt;br&gt;(swap m/s)<br /> |9/8&lt;br&gt;7/6&lt;br&gt;21/16&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;14/9&lt;br&gt;7/4&lt;br&gt;16/9&lt;br&gt;2/1<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |513/512&lt;br&gt;9/8&lt;br&gt;19/16&lt;br&gt;4/3&lt;br&gt;171/128&lt;br&gt;3/2&lt;br&gt;19/12&lt;br&gt;16/9&lt;br&gt;57/32&lt;br&gt;2/1<br /> |81/80&lt;br&gt;9/8&lt;br&gt;6/5&lt;br&gt;4/3&lt;br&gt;27/20&lt;br&gt;3/2&lt;br&gt;8/5&lt;br&gt;16/9&lt;br&gt;9/5&lt;br&gt;2/1<br /> |459/448&lt;br&gt;9/8&lt;br&gt;17/14&lt;br&gt;4/3&lt;br&gt;153/112&lt;br&gt;3/2&lt;br&gt;34/21&lt;br&gt;16/9&lt;br&gt;51/28&lt;br&gt;2/1<br /> |33/32&lt;br&gt;9/8&lt;br&gt;11/9&lt;br&gt;4/3&lt;br&gt;11/8&lt;br&gt;3/2&lt;br&gt;33/20&lt;br&gt;16/9&lt;br&gt;11/6&lt;br&gt;2/1<br /> |-<br /> |Similar&lt;br&gt;Unlisted&lt;br&gt;Aberrismas<br /> | colspan=&quot;2&quot; |17/13: 1088/1053&lt;br&gt;(swap m/s)<br /> |17/11: 1408/1377&lt;br&gt;19/11: 304/297<br /> |<br /> |25: 2048/2025&lt;br&gt;11/7: 896/891&lt;br&gt;13/11: 352/351&lt;br&gt;17/5: 136/135&lt;br&gt;23/13: 208/207<br /> |25/7: 225/224&lt;br&gt;17: 4131/4096&lt;br&gt;19/5: 1216/1215&lt;br&gt;(diasem)<br /> |<br /> |23/7: 189/184&lt;br&gt;23/19: 621/608<br /> |13: 1053/1024<br /> |}<br /> The diatonic scale and its tertian harmony are the framework of all Western music. It makes sense that using the lowest prime as the period and second lowest prime as the generator would produce a versatile scale, and the fact that it clusters around 12edo rather than 3edo in the case of prime 5 produces more useful step sizes. Modifying this into an aberrismic scale containing the simplest tertian triad 4:5:6 and its retroversion in as many places as possible produces blackdye, and doing the same with 6:7:9 produces diasem. These are the main scales I use to represent my take on Western music: melodically interesting, harmonically pure, and not far from diatonic. However, there are other tunings of them that are useful in different places. These are generalized forms and their aberrismas may be tempered out for a true diatonic scale. 64/63 and 81/80 are common, but 736/729 and especially 513/512 are smaller and make more sense to temper out. If not, inflating them is recommended to keep the melodic significance of the aberrisma.<br /> <br /> '''Eridial''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named after &quot;eridian&quot;, the demonym for Eris, with the -al suffix of subgroups. Eris is probably one of the most goth deities of Ancient Greece and is also associated with the number 23 by Discordians. It targets 18:23:27, a little-known and surprisingly concordant neogothic tertian triad. Many microtonalists contend that 12edo pop music, while historically tracing back to Meantone, tends to sound better in Pythagorean tuning. I can really hear it after comparing different quasi-diatonic options for retuning my melodies. I've taken a particular liking to gentle/neogothic diatonic scales in the past for this reason, but I believe eridian diasem is a further refinement. As with diasem in general, I prefer this with the 3 tuned slightly flat to inflate the aberrisma, although it is especially important in this case because 736/729 has a size of only 17¢. When inflated in tunings such as 43edo, the aberrisma is subtle enough to not disturb the Pythagorean sound much, but impactful in cases like the subminor seventh, which is tuned close to 7/4. Its melody is not particularly xenharmonic, but it's one of my default scales when available because it works for almost everything. It works well as an extension to [[Septimal meantone]]. {{adv|The difference between eridian diasem's analogous 2.3.25 and 2.3.17/5 thirds (e.g. 75/64 and 20/17) is 256/255, also the difference between 16/15 and 17/16, supporting it as one of the most important 17-limit commas.}}<br /> <br /> The default set of qualities to consider is septal and pental due to their simplicity, possibly with a Pythagorean diatonic scale in the middle as found in Aberschismic edos. But I've found that I particularly like shifting both septal and pental towards neutral, to eridian and exaggerated pental. This is probably because it puts both of them about an equal distance from diatonic when using a prime 3 within a few cents of just. Eridian provides a more versatile sound and pental sounds more interesting while still working about the same. In some edos that do this such as 48 and 86, the pental tuning is Negri, meaning that semiquartal is also available.<br /> <br /> &lt;div style=&quot;float: right; margin: 0 0 1em 1em; max-width: 100%; overflow-x: auto;&quot;&gt;<br /> {| class=&quot;wikitable&quot;<br /> |+Cursed Eridian Scales<br /> !Standard&lt;br&gt;(23/18)(27/23)<br /> !Wolf → Perfect&lt;br&gt;(23/18)(32/27)<br /> !the other one&lt;br&gt;(81/64)(27/23)<br /> |-<br /> |9/8&lt;br&gt;27/23&lt;br&gt;243/184&lt;br&gt;4/3&lt;br&gt;3/2&lt;br&gt;36/23&lt;br&gt;81/46&lt;br&gt;16/9&lt;br&gt;2/1<br /> |398688256/387420489&lt;br&gt;67712/59049&lt;br&gt;32/27&lt;br&gt;243/184&lt;br&gt;2166784/1594323&lt;br&gt;368/243&lt;br&gt;36/23&lt;br&gt;59049/33856&lt;br&gt;11776/6561&lt;br&gt;2/1<br /> |4782969/4333568&lt;br&gt;27/23&lt;br&gt;129140163/99672064&lt;br&gt;2944/2187&lt;br&gt;2187/1472&lt;br&gt;128/81&lt;br&gt;59049/33856&lt;br&gt;8667136/4782969&lt;br&gt;2/1<br /> |}<br /> &lt;/div&gt;<br /> 27/23 is close enough to 32/27 that it can be replaced and result in another 2.3.23 blackdye scale with 32/27*23/18 = 368/243 triads. The fifth is the same as the existing Eridian wolf fifth, which is barely diatonic in JI. I might call this '''hiberidial''' because it applies the principle of [[Hibernal]] to the eridian subgroup. 82edo is the near-perfect 2.3.23 tuning, but 20 and 37edo have a close approximation of the scale despite not being good for the subgroup. Switching thirds to 32/27 or 81/64 to remain in the subgroup with a wolf fifth as the main one can be applied to other scales in the table.<br /> <br /> I have a lot more to write about how blackdye and diasem work harmonically.<br /> <br /> == Aberrismic details ==<br /> <br /> Note that this section is incomplete.<br /> <br /> As described in the [[Monarch]] article, I like to tune pental blackdye (3sA Aeolian: tempered 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1) with a sharp 3 and thus inflated 81/80. When extreme, more interesting intervals show up along with new problems. The first is what is seen in 11-limit Superpyth and Porcupine (55/54), turning 27/20 into 11/8, although it could be interpreted as any other superfourth. I like this interval a lot because the 1/1-9/8-6/5-27/20 tetrachord is a natural-sounding stretched version of the diatonic minor tetrachord. In fact, it may sound better melodically to use the superfourth rather than the perfect fourth to avoid the very deflated 10/9 between 6/5 and 4/3. It would be beneficial to have a triad on 27/20 to replace the one on 4/3, and this makes use of the 81/80 in the next octave, the least harmonically useful note in the scale. 32/27 is available as the more conventional quasi-diatonic third, but also 320/243, one of the least recognizeable intervals that all correspond to septal intervals in Aberschismic temperament (5120/5103). 320/243 is 476.539¢ in JI and 729/640 is 225.416¢, but these approach each other with a sharper 3. Most notably in this tuning range, tempering out 2187/2240 equates them to 9/7 and 7/6, different from the same ratios that would normally be generated by Archy. Interpreting 320/243 as 14/11 allows for 8:11:14. The exact tempering is irrelevant as long as they're some sort of supermajor and subminor thirds.<br /> <br /> The size of the aberrisma presents other issues.<br /> <br /> == Song structure ==<br /> <br /> This is my system for categorizing the songs I write by structure. Something similar can be applied to other people's music, but the standards will be different due to the unusual complexity of my music.<br /> <br /> * '''straightforward''': The simplest type, being in a similar style throughout. Always works.<br /> * '''opposing sections''': Two straightforward parts that are very different.<br /> * '''medium''': Somewhat straightforward, somewhat disjointed.<br /> * '''episodic''': Containing many different sections with themes and a progression throughout the song. The most desirable when done well and in moderation.<br /> * '''disjointed''': Containing many different sections with no clear pattern. The least desirable.<br /> <br /> I've noticed that I write more &quot;disjointed&quot; songs than I would like, and fewer &quot;straightforward&quot; and &quot;episodic&quot;. This system is intended to keep track of what I'm doing and help me work toward a better result.<br /> <br /> {| class=&quot;wikitable sortable&quot;<br /> |+Song Structure Types of The Gralbums<br /> !Track<br /> !Title<br /> !Type<br /> |-<br /> |g1t1<br /> |The Lake Reflects a Black Sky<br /> |'''disjointed'''<br /> |-<br /> |g1t2<br /> |The Art of It All<br /> |'''episodic'''<br /> |-<br /> |g1t3<br /> |Rail Twinge<br /> |'''straightforward'''<br /> |-<br /> |g1t4<br /> |Back Stalk<br /> |'''medium'''<br /> |-<br /> |g1t5<br /> |Superior Intermedial<br /> |'''episodic'''<br /> |-<br /> |g1t6<br /> |Snowboard Index<br /> |'''straightforward'''<br /> |-<br /> |g1t7<br /> |Winter's Mortal Hope<br /> |'''medium'''<br /> |-<br /> |g1t8<br /> |Transpiration<br /> |'''opposing sections'''<br /> |-<br /> |g1t9<br /> |Revelation of Your Forever<br /> |'''medium'''<br /> |-<br /> |g2t1<br /> |The Life Unreachable<br /> |'''disjointed'''<br /> |-<br /> |g2t2<br /> |Monolithium<br /> |'''opposing sections'''<br /> |-<br /> |g2t3<br /> |Nocturne Paranoia<br /> |'''disjointed'''<br /> |-<br /> |g2t4<br /> |Glimmer Extrication<br /> |'''episodic'''<br /> |-<br /> |g2t5<br /> |Not This Time<br /> |'''straightforward'''<br /> |-<br /> |g2t6<br /> |Sakura Blade Minivan<br /> |'''medium'''<br /> |-<br /> |g2t7<br /> |Resolute Prelude<br /> |'''straightforward'''<br /> |-<br /> |g2t8<br /> |Life and Limb<br /> |'''straightforward'''<br /> |-<br /> |g2t9<br /> |Residual Soliloquy<br /> |'''disjointed'''<br /> |}</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8277 2026-10-19T23:30:11Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly.--&gt;]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 of a turn clockwise in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8276 2026-10-19T23:28:49Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly.--&gt;]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8275 2026-10-19T23:28:17Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. The self-intersection may also be resolved by adding a fourth spatial dimension, also supposedly. --&gt; The self-intersection may also be resolved by adding a fourth spatial dimension.]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8274 2026-10-19T23:26:15Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. --&gt; The self-intersection may also be resolved by adding a fourth spatial dimension.]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude. Center-crossing faces twist upside-down so the outside of the shape remains outside.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div>

Ground https://xenreference.com/wiki/index.php?title=Generator_gronut&diff=8273 2026-10-19T23:21:40Z

<p>Ground: </p> <hr /> <div>{{UTBox|g_|Ground|7766ff}}<br /> <br /> [[File:37edo 7-12-gronut shaded.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, flattened version.]]<br /> [[File:37edo 7-12-gronut 3D.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, 3D render.]]<br /> [[File:7-12-gronut curve.png|thumb|Lemniscate-like curve around a 7\12 gronut. &lt;!--It may be adjusted into a type of torus knot, supposedly. --&gt; The self-intersection may also be resolved by adding a fourth spatial dimension.]]<br /> [[File:37edo 7-12-gronut simple.png|thumb|21\37 and 22\37 applied to a 7\12 gronut, simple diagram.]]<br /> [[File:37edo 11-19-gronut simple.png|thumb|21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.]]<br /> [[File:11-19-gronut manifold.png|thumb|11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude.]]<br /> <br /> The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.<br /> <br /> The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. [[Aberrisma|Aberrismic theory]] is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.<br /> <br /> The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a [[Straddle primes|straddled generator]]. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.<br /> <br /> The third is to map a harmonic table onto a torus. One of these is documented in the [[15edo#The harmonic table|15edo article]], involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.<br /> <br /> My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a '''gronut''' (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.<br /> <br /> When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.</div>

Ground https://xenreference.com/wiki/index.php?title=File:11-19-gronut_manifold.png&diff=8272 2026-10-19T23:20:44Z

<p>Ground: </p> <hr /> <div>11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 turns down in longitude.</div>

Ground