Xenharmonic Reference - Recent changes [en]

Chromatic semitone

Vector — Fri, 16 Oct 2026 03:15:44 GMT

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	{{Infobox interval\|2187/2048\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Chromatic semitone, augmented unison, apotome}}		{{Infobox interval\|2187/2048\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Chromatic semitone, augmented unison, apotome}}

	The 3-limit '''chromatic semitone''', also called the '''augmented unison''' ('''A1''') or the '''apotome''' (''ə-POT-ə-mee'') and represented by the ratio '''2187/2048,''' is the difference between the large and small steps of the MOS [[diatonic]] scale. It is generated by stacking 7 [[Perfect fifth\|fifths]] and octave-reducing. In [[Pythagorean tuning\|Pythagorean]] tuning (and thus purely-tuned [[just intonation]]), it is approximately {{Cents from ratio\|2187/2048\|ratio=2187/2048}} cents in size, but as an interval involved in the structure of the abstract diatonic scale it ranges from 0 to 240 cents depending on the tuning.		The 3-limit '''chromatic semitone''', also called the '''augmented unison''' ('''A1''') or the '''apotome''' (''ə-POT-ə-mee'') and represented by the ratio '''2187/2048,''' is the difference between the large and small steps of the MOS [[diatonic]] scale, therefore being a type of [[Interval arithmetic\|unison]] (in terms of interval arithmetic, at least) and a type of [[semitone]]. It is generated by stacking 7 [[Perfect fifth\|fifths]] and octave-reducing. In [[Pythagorean tuning\|Pythagorean]] tuning (and thus purely-tuned [[just intonation]]), it is approximately {{Cents from ratio\|2187/2048\|ratio=2187/2048}} cents in size, but as an interval involved in the structure of the abstract diatonic scale it ranges from 0 to 240 cents depending on the tuning. In 12edo, it is tuned to 100 cents (the same as the [[diatonic semitone]]); flatter tunings narrow it (leading to a "mellower" diatonic scale), and sharper tunings widen it.

	The chromatic semitone is significant in music notation. As it is the difference between, for example, the [[diatonic major third]] and [[diatonic minor third]], it is the interval represented by the # and b accidentals used in standard music notation. The chromatic semitone is distinct from the [[diatonic semitone]], which is the diatonic minor second and differs from the [[diatonic major second]] by a chromatic semitone.		The chromatic semitone is significant in music notation. As it is the difference between, for example, the [[diatonic major third]] and [[diatonic minor third]], it is the interval represented by the # and b accidentals used in standard music notation. The chromatic semitone is distinct from the [[diatonic semitone]], which is the diatonic minor second and differs from the [[diatonic major second]] by a chromatic semitone.

Whitewood

Hotcrystal0 — Fri, 16 Oct 2026 01:53:52 GMT

Redirected page to 7edo#Whitewood temperament

New page

#REDIRECT [[7edo#Whitewood temperament]]

7edo

Vector — Thu, 15 Oct 2026 02:08:41 GMT

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	=== JI approximation ===		=== JI approximation ===
	7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and all intervals are perfect.		7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and in Kite's ups and downs notation all intervals are perfect.
	{{Harmonics in ED\|7\|31}}		{{Harmonics in ED\|7\|31}}
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Tuning stability

Inthar — Wed, 14 Oct 2026 20:04:50 GMT

Inthar moved page Tuning stability to Tempering stability without leaving a redirect

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	'''~~Tuning~~ stability''' is, in a sense, the sensitivity of an interval to the ~~detuning~~ of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[~~Temperament\|temperamen~~]]~~<nowiki/>t~~ it is more likely to have an extreme detuning.		'''Tempering stability''' is, in a sense, the sensitivity of an interval to the tempered values of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very tempering-unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[temperament]] it is more likely to have an extreme detuning.

	As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(33) vs (35)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.		As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(33) vs (35)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.
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	The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).		The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).

	The unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.		The tempering-unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.

	The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.		The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.

Schismic

Vector — Wed, 14 Oct 2026 07:35:05 GMT

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	==== Primes 11 and 13 ====		==== Primes 11 and 13 ====
	A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.		A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 16/13 is set to the opposite neutral third to 11/9 to extend to the 13-limit.

			In cassandra, the sequence of intervals 16/13 - 5/4 - 19/15 - 9/7 - 13/10 is made equidistant.

	=== Prime 17 ===		=== Prime 17 ===

37edo

Vector — Wed, 14 Oct 2026 07:17:48 GMT

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 === JI approximation ===
-edo's most notable feature is a very good representation of the no-3 13-limit and a reasonably good representation of the no-3 19-limit. When 3 is included, it is a "dual-3" system, meaning that not only are there two tunings of 3 that can be combined to create an accurate 9/8, but those tunings of 3 are also passably accurate on their own in many contexts, such as approximating 4:5:6. The sharp fifth functions as a diatonic fifth and the primary fifth in a regular temperament interpretation of the EDO. As a dual-3 system, it features the dual-3 diatonic scales 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps, and 5L 2m 8s "trackdye".{{Harmonics in ED|37|31|0}}
+edo's most notable feature is a very good representation of the no-3 13-limit and a reasonably good representation of the no-3 19-limit. When 3 is included, it is a "dual-3" system, meaning that not only are there two tunings of 3 that can be combined to create an accurate 9/8, but those tunings of 3 are also passably accurate on their own in many contexts, such as approximating 4:5:6. The sharp fifth functions as a diatonic fifth and the primary fifth in a regular temperament interpretation of the EDO. As a dual-3 system, it features the dual-3 diatonic scales 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps, and 5L 2m 8s "trackdye".
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31edo

Vector — Wed, 14 Oct 2026 07:06:19 GMT

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	31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth\|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].		31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth\|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].
			[[File:Mohajira extensions in 31edo.png\|thumb\|31edo combines two forms of mohajira: mohajira with the ordinary mapping of 7 (left) and "migration" (i.e. septimal meantone, right). It can be demonstrated that this defines 7-limit 31 equal temperament (also found in 62edo).]]
	In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.		In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

File:Mohajira extensions in 31edo.png

Vector — Wed, 14 Oct 2026 07:01:38 GMT

Vector uploaded File:Mohajira extensions in 31edo.png

New page

wrywryw

Mohajira

Vector — Wed, 14 Oct 2026 06:57:03 GMT

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	As a tuning generated by the [[Tuning stability\|unstable]] interval 11/9, it is perhaps viable to directly map 11/9 to edos and determine the tuning of 11 from there, instead of vice versa.		As a tuning generated by the [[Tuning stability\|unstable]] interval 11/9, it is perhaps viable to directly map 11/9 to edos and determine the tuning of 11 from there, instead of vice versa.

	Generally, Mohajira is also extended to include prime 7 as the semidiminished seventh, though an alternate mapping also available in 31edo shares septimal Meantone's mapping of 7 (and contributes much to 31edo's 7-limit structure).		Generally, Mohajira is also extended to include prime 7 as the semidiminished seventh, though an alternate mapping also available in 31edo shares septimal Meantone's mapping of 7 (and contributes much to 31edo's 7-limit structure). Combining both of these mappings results in 31edo, as it forces the diatonic semitone to be comprised of three chromatic quartertones, which defines 31edo.

	== Interval table ==		== Interval table ==

Tuning stability

Vector — Wed, 14 Oct 2026 06:55:58 GMT

Created page with "'''Tuning stability''' is, in a sense, the sensitivity of an interval to the detuning of the primes relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple simil..."

New page

'''Tuning stability''' is, in a sense, the sensitivity of an interval to the detuning of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[Temperament|temperamen]]<nowiki/>t it is more likely to have an extreme detuning.

As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(3*3) vs (3*5)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.

The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).

The unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.

The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.

37edo

Ground — Tue, 13 Oct 2026 22:28:51 GMT

Show changes

Ground's composition theory

Ground — Mon, 12 Oct 2026 23:42:51 GMT

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	=== Neutral ~~diachrome~~ ===		=== Neutral diatonic extensions ===

	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.		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.

			A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.

			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.

			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.
			* Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.
			* Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.
			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.

	== Adaptive diatonic modification system ==		== Adaptive diatonic modification system ==

10-form

Vector — Mon, 12 Oct 2026 06:53:25 GMT

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	The 10-form has clean mappings of 3, 5, 7, and 13, and functions as an organization scheme for the 7-limit. There are also not fixed ranges, the boundaries may vary. The boundaries chosen here are loosely based on [[22edo]].		The 10-form has clean mappings of 3, 5, 7, and 13, and functions as an organization scheme for the 7-limit, however 11 is not cleanly mapped and may either be represented as a fourth or a tritone (the latter is preferred by Vector due to fourths being biased flat in the 10-form). There are also not fixed ranges, the boundaries may vary. The boundaries chosen here are loosely based on [[22edo]].
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Tertian structure

Vector — Mon, 12 Oct 2026 02:54:17 GMT

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	'''Tertian structure''' is the relative spacing of the [[Third\|thirds]] in a tempered tuning system, such as an equal temperament. In general, it usually refers to the spacing of the 7-limit color-quality thirds 7/6, 6/5, 5/4, and 9/7, which can be expressed as the relative size of 25/24 and 36/35. Other thirds like 19/16, 19/15, or 11/9 may be invoked to describe more complex structures.		'''Tertian structure''' is the relative spacing of the [[Third\|thirds]] in a tempered tuning system, such as an equal temperament. In general, it usually refers to the spacing of the 7-limit color-quality thirds 7/6, 6/5, 5/4, and 9/7, which can be expressed as the relative size of 25/24 and 36/35. Other thirds like 19/16, 19/15, or 11/9 may be invoked to describe more complex structures.

	Common rank-3 temperaments can be expressed as tertian structures~~, and in fact, a tertian structure can always be thought of as a rank-3 temperament~~. These include Mint, Dicot.7, Keemic, and Mandos.		A tertian structure takes the form of a [[regular temperament]] which tempers out a single comma to create a rational relationship between 25/24 and 36/35. Common rank-3 temperaments can be expressed as tertian structures. These include Mint, Dicot.7, Keemic, and Mandos.

	In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log<sub>36/35</sub>(25/24) = 1.44908497.		In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log<sub>36/35</sub>(25/24) = 1.44908497.

	All tertian structures are supported by 7edo, which collapses all distinctions between the four thirds in question down to a single interval, outlining the structure of diatonic harmony.		All tertian structures are supported by 7edo, which tempers out 25/24 and 36/35 themselves and thus collapses all distinctions between the four thirds in question down to a single interval, outlining the structure of diatonic harmony.

	== Mint ==		== List of tertian structures ==
	Ratio (36/35:25/24): 0:1
			=== Mint ===
			Ratio (36/35:25/24): 0:1 (~inf)

	Comma: (36/35)^1 / (25/24)^0 = 36/35		Comma: (36/35)^1 / (25/24)^0 = 36/35
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	Supported edos: 5, 9, 12, 14, 16, 21, 23, 28, 33		Supported edos: 5, 9, 12, 14, 16, 21, 23, 28, 33

	Mint tempers out 36/35 entirely, equating the pental and septimal third qualities to some tuning around 300 and 400 cents~~. Unusually enough, as a temperament structure in and of itself, this is best optimized with a flat fifth~~. The most prominent mint temperaments are dominant (meantone, supported by 5edo and 12edo), mavila (supported by 9edo, 16edo, and 23edo), whitewood (supported by 14edo, 21edo, and 28edo), and dimisept (supported by 12edo and 16edo alongside 28edo). Interestingly, mint temperaments overwhelmingly have flat fifths, except for dominant. This makes sense, as mint essentially equally spaces 3/2 between 5/4 and 7/4, and if both are kept just then 3/2 is set to about 678 cents.		Mint tempers out 36/35 entirely, equating the pental and septimal third qualities to some tuning around 300 and 400 cents. The most prominent mint temperaments are dominant (meantone, supported by 5edo and 12edo), mavila (supported by 9edo, 16edo, and 23edo), whitewood (supported by 14edo, 21edo, and 28edo), and dimisept (supported by 12edo and 16edo alongside 28edo). Interestingly, mint temperaments overwhelmingly have flat fifths, except for dominant. This makes sense, as mint essentially equally spaces 3/2 between 5/4 and 7/4, and if both are kept just then 3/2 is set to about 678 cents.

	== Dicot ==		=== Dicot ===
	Ratio (36/35:25/24): 1:0		Ratio (36/35:25/24): 1:0 (0)

	Comma: (25/24)^1 / (36/35)^0 = 25/24		Comma: (25/24)^1 / (36/35)^0 = 25/24
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	Dicot is a much rarer archetype even in the small edos due to the fact that it can be described as a rank-2 temperament in a prime subgroup (although as a tertian structure 7 is given as an independent generator); additionally, its comma is the largest of any tertian structure discussed here. The temperaments supporting the main form of dicot, called dichotic, are 10, 13, 17, and 20. Unlike mint temperaments which prefer a flat fifth, dicot temperaments prefer a sharp fifth. Again, this makes sense if you examine the 5/4 eigenmonzo tuning, which jumps 3/2 all the way up to 773 cents; accurate tunings of dicot fall between the just tuning of 3/2 and that sharp tuning.		Dicot is a much rarer archetype even in the small edos due to the fact that it can be described as a rank-2 temperament in a prime subgroup (although as a tertian structure 7 is given as an independent generator); additionally, its comma is the largest of any tertian structure discussed here. The temperaments supporting the main form of dicot, called dichotic, are 10, 13, 17, and 20. Unlike mint temperaments which prefer a flat fifth, dicot temperaments prefer a sharp fifth. Again, this makes sense if you examine the 5/4 eigenmonzo tuning, which jumps 3/2 all the way up to 773 cents; accurate tunings of dicot fall between the just tuning of 3/2 and that sharp tuning.

	== Keemic ==		=== Keemic ===
	Ratio (36/35:25/24): 1:1		Ratio (36/35:25/24): 1:1 (1)

	Comma: (25/24)^1 / (36/35)^1 = 875/864		Comma: (25/24)^1 / (36/35)^1 = 875/864
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	Keemic is characterized by the narrowing of 25/24 and/or the widening of 36/35; eigenmonzo 54/49 (the distance between 7/6 and 9/7) and 3/2 leads to a 5/4 tuning of 379c; eigenmonzo 25/24 and 3/2 leads to a 9/7 tuning of 457c.		Keemic is characterized by the narrowing of 25/24 and/or the widening of 36/35; eigenmonzo 54/49 (the distance between 7/6 and 9/7) and 3/2 leads to a 5/4 tuning of 379c; eigenmonzo 25/24 and 3/2 leads to a 9/7 tuning of 457c.

	== Mandos ==		=== Mandos ===
	Ratio (36/35:25/24): 1:2		Ratio (36/35:25/24): 1:2 (2)

	Comma: (36/35)^2 / (25/24)^1 = 31104/30625		Comma: (36/35)^2 / (25/24)^1 = 31104/30625
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	''TODO: rank-3 name for amity? also cover "oddball" structures like sixix, 3-2-3, 1-4-1, 1-5-1, etc''		''TODO: rank-3 name for amity? also cover "oddball" structures like sixix, 3-2-3, 1-4-1, 1-5-1, etc''

			== Other mediant structures ==
			''todo: chthonic structure. 10/9 8/7 7/6 6/5. semaphore mint orwellismic?''

Keemic

Vector — Mon, 12 Oct 2026 02:37:27 GMT

Redirected page to Tertian structure

New page

#REDIRECT [[Tertian structure]]

Ground's composition theory

Ground — Sun, 11 Oct 2026 23:05:29 GMT

← Older revision		Revision as of 23:05, 11 October 2026
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	I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.		I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

	== 9-note ~~diatonic analogue~~ ==		== Diatonic analogues ==

			=== 9-note "diatygic" ===

	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.		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.
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	\|1200		\|1200
	\|}		\|}

			=== Neutral diachrome ===

			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.

	== Adaptive diatonic modification system ==		== Adaptive diatonic modification system ==

17edo

Inthar — Sun, 11 Oct 2026 19:44:25 GMT

Scales

← Older revision		Revision as of 19:44, 11 October 2026
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	* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)		* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)
	* [[Mosh]] (3L4s, 2323232)		* [[Mosh]] (3L4s, 2323232)
	* Lissotonic<sup>1</sup> (3223322)		* Lissotonic<sup>1</sup> (3223322): modes of the ascending mode of Maqam Rast
	* Trachytonic<sup>1</sup> (3313322)		* Trachytonic<sup>1</sup> or screamapillar<sup>2</sup> (3313322)

	<sup>1</sup> CellularAutomaton's names		<sup>1</sup> CellularAutomaton's names

			<sup>2</sup> Xen Wiki's name

	{{Navbox EDO}}		{{Navbox EDO}}

	{{cat\|Edos}}		{{cat\|Edos}}

Straddle primes

Hotcrystal0 — Sun, 11 Oct 2026 12:51:33 GMT

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	Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:		Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:
	* 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.		* [[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.
	* 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.		* [[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.

	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:		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:

Straddle primes

Ground — Sat, 10 Oct 2026 17:32:09 GMT

SRTT

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	'''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.<3.>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]].		'''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.<3.>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]].

			=== Cluster temperaments ===

			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.

			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.
			* -3 generators: 7/4 a b
			* 2 generatora: 7/4 a/b
			At 5edo, b is assumed to be about 1, so a is less than 1 and accounts for the flat tuning of 7/4.
			* In Buzzard, the accurate 7/4 is at -3 generators. This is where a and b are about reciprocal, cancelling their errors via multiplication.
			* In Hemiseven, the accurate 7/4 is at 2 generators. This is where a and b are about equal, cancelling their errors via division.
			* 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.
			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.

	== Straddle-2 ==		== Straddle-2 ==

User:Lumithesilly

Lumithesilly — Fri, 09 Oct 2026 18:50:55 GMT

← Older revision		Revision as of 18:50, 9 October 2026
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	hii, i'm lumi		hii, i'm lumi

			i'm part of luxia , a group of nerds sharing one physical body , and a handful of us like hanging out in the xen community

	i make music and post it on [https://www.youtube.com/@ImLumiV my youtube channel]		i make music and post it on [https://www.youtube.com/@ImLumiV my youtube channel]

	i want to help contribute to the wiki but i'm not really sure what to do		i want to help contribute to the wiki but i'm not really sure what to do

Ground's composition theory

Ground — Fri, 09 Oct 2026 18:34:07 GMT

Edo multiplication

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	* 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.		* 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.
	With all this in mind, 24x6 is clearly the better choice to start with, although it's close.		With all this in mind, 24x6 is clearly the better choice to start with, although it's close.

			=== Channel management ===

			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.

	== Anchor concordances ==		== Anchor concordances ==
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	\|?		\|?
	\|"Septal"		\|"Septal"
	\|"~~Eridian~~"		\|"Eridial"
	\|"Novemdecal"		\|"Novemdecal"
	\|Pental		\|Pental
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	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.		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.

	'''~~Eridian~~''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named for Eris, ~~who~~ 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 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.}}		'''Eridial''' diasem is a term I coined for diasem with shrub- or sometimes neo- major/minor thirds. It is named after "eridian", 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.}}

	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.		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.
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	\|}		\|}
	</div>		</div>
	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 '''~~hiberidian~~''' 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.		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.

	I have a lot more to write about how blackdye and diasem work harmonically.		I have a lot more to write about how blackdye and diasem work harmonically.

21edo

Inthar — Thu, 08 Oct 2026 11:37:01 GMT

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	'''21edo''' is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.		'''21edo''' is an equal division of the octave into 21 steps of 1200c/21 ~= 57.1c each.

	21edo is unusual from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; the reverse is true of certain triad inversions, since 0-514-914 is close to being +1+1 [[DR]] (approximately 23:31:39).		21edo is unusual from a regular temperament perspective due to its mixture of very accurate (e.g. 23/16, 16/15) and very inaccurate (3/2, 6/5, 7/6) approximations. On the one hand, one could say 21edo has a 5 since its 15/8 is accurate due to error cancellation of the sharp 5 with the flat 3. In root-position major triads, though, the 400c major third sounds much less isodifferential (less close to having a +1+1 [[Delta-rational chord\|delta signature]]) than even in 12edo due to the 3/2 being much flatter, and root-position major and minor triads sound somewhat neogothic as a result and also because of the neominor third; the reverse is true of certain triad inversions, since 0-514-914 is close to being +1+1 [[DR]] (approximately 23:31:39).

	Notable scales:		Notable scales:

17edo

Inthar — Thu, 08 Oct 2026 11:34:43 GMT

Scales

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	* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)		* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)
	* [[Mosh]] (3L4s, 2323232)		* [[Mosh]] (3L4s, 2323232)
	* Lissotonic (3223322)		* Lissotonic<sup>1</sup> (3223322)
	* Trachytonic (3313322)		* Trachytonic<sup>1</sup> (3313322)

			<sup>1</sup> CellularAutomaton's names

	{{Navbox EDO}}		{{Navbox EDO}}

	{{cat\|Edos}}		{{cat\|Edos}}

List of commas

Vector — Thu, 08 Oct 2026 00:35:13 GMT

@@ Line 497: / Line 497: @@
 |625/624
 |2.7c
+|
 |}
@@ Line 563: / Line 556: @@
 |3042/3025
 |9.7c
+|
 |}

17edo

Unque — Thu, 08 Oct 2026 00:32:55 GMT

← Older revision		Revision as of 00:32, 8 October 2026
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	=== Intervals ===		=== Intervals ===
	17edo can be notated with [[diatonic notation]] with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too.		17edo can be notated with [[diatonic notation]] with #/b = 2\17. Since the sharp can be split in two, half-sharps and half-flats for 1\17 alterations can be used too. These can be written in Unicode-supported text as t/d, or with ^/v; occasionally ‡ is used for a semisharp.
	{\| class="wikitable"		{\| class="wikitable"
	!Edostep		!Edostep
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	\|70.6		\|70.6
	\|(Far)minor second		\|(Far)minor second
	\|Bb, At		\|B♭, At
	\|25/24		\|25/24
	\|-		\|-
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	\|141.2		\|141.2
	\|Neutral second		\|Neutral second
	\|A#, Bd		\|A♯, C♭ Bd
	\|13/12		\|13/12
	\|-		\|-
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	\|211.8		\|211.8
	\|(Far)major second		\|(Far)major second
	\|B		\|B, Cd
	\|9/8		\|9/8
	\|-		\|-
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	\|282.4		\|282.4
	\|(Far)minor third		\|(Far)minor third
	\|C		\|C, Bt
	\|33/28, 32/27		\|33/28, 32/27
	\|-		\|-
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	\|352.9		\|352.9
	\|Neutral third		\|Neutral third
	\|Db, Ct		\|B♯, D♭, Ct
	\|16/13		\|16/13
	\|-		\|-
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	\|423.5		\|423.5
	\|(Far)major third		\|(Far)major third
	\|C#, Dd		\|C♯, Dd
	\|14/11		\|14/11
	\|-		\|-
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	\|564.7		\|564.7
	\|Semiaugmented/neutral fourth		\|Semiaugmented/neutral fourth
	\|Eb, Dt		\|E♭, Dt
	\|25/18, 18/13		\|25/18, 18/13
	\|-		\|-
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	\|635.3		\|635.3
	\|Semidiminished/neutral fifth		\|Semidiminished/neutral fifth
	\|D#, Ed		\|D♯, F♭, Ed
	\|36/25, 13/9		\|36/25, 13/9
	\|-		\|-
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	\|705.9		\|705.9
	\|Perfect fifth		\|Perfect fifth
	\|E		\|E, Fd
	\|3/2		\|3/2
	\|-		\|-
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	\|776.5		\|776.5
	\|(Far)minor sixth		\|(Far)minor sixth
	\|F		\|F, Et
	\|11/7		\|11/7
	\|-		\|-
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	\|847.1		\|847.1
	\|Neutral sixth		\|Neutral sixth
	\|Gb, Ft		\|E♯, G♭, F
	\|13/8		\|13/8
	\|-		\|-
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	\|917.6		\|917.6
	\|(Far)major sixth		\|(Far)major sixth
	\|F#, Gd		\|F♯, Gd
	\|56/33, 27/16		\|56/33, 27/16
	\|-		\|-
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	\|1058.9		\|1058.9
	\|Neutral seventh		\|Neutral seventh
	\|Ab, Gt		\|A♭, Gt
	\|24/13		\|24/13
	\|-		\|-
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	\|1129.4		\|1129.4
	\|(Far)major seventh		\|(Far)major seventh
	\|G#, Ad		\|G♯, Ad
	\|48/25		\|48/25
	\|-		\|-
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	== Scales ==		== Scales ==
	Some scales:		Some scales:
	* MOS diatonic (5L2s, 3331331)		* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)
	* Mosh~~/"malacotonic"~~ (3L4s, 2323232)		* [[Mosh]] (3L4s, 2323232)
	* Lissotonic (3223322)		* Lissotonic (3223322)
	* Trachytonic (3313322)		* Trachytonic (3313322)

List of commas

Vector — Tue, 06 Oct 2026 06:40:19 GMT

Show changes

List of commas

Vector — Thu, 01 Oct 2026 03:24:40 GMT

Created page with "This is a list of small intervals separating just intonation intervals in the 63-odd-limit. == 5-limit == === 31-odd-limit (commas < 50c) === {| class="wikitable" !Name !S-expression !Simplest interval pair !Ratio !Cents !Associated temperaments and other notes |- |syntonic comma |S9 |9/8 ~ 10/9 |81/80 |21.5c |Meantone. Formal comma for 5-limit. |- |augmented diesis | |25/24 ~ 16/15 |128/125 |41.1c |Augmented (temperament) |- |porcupine comma | |27/25 ~ 10/..."

New page

This is a list of small intervals separating just intonation intervals in the 63-odd-limit.

== 5-limit ==

=== 31-odd-limit (commas < 50c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|syntonic comma
|S9
|9/8 ~ 10/9
|81/80
|21.5c
|[[Meantone]]. Formal comma for [[5-limit]].
|-
|augmented diesis
|
|25/24 ~ 16/15
|128/125
|41.1c
|[[Augmented (temperament)]]
|-
|porcupine comma
|
|27/25 ~ 10/9
|250/243
|49.2c
|The 5-limit restriction of [[Porcupine]]
|}

=== 63-odd-limit (commas < 25c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|diaschisma
|
|45/32 ~ 64/45
|2048/2025
|19.6c
|[[Diaschismic]]
|}

== 2.3.7 ==

=== 31-odd-limit (commas < 50c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|archytas comma
|S8
|9/8 ~ 8/7
|64/63
|27.3c
|[[Archy]] (Superpyth). Formal comma for [[7-limit]].
|-
|interseptimal comma
|S7
|8/7 ~ 7/6
|49/48
|35.7c
|[[Semaphore]]
|}

=== 63-odd-limit (commas < 25c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|gamelisma
|S7/S8
|64/49 ~ 21/16
|1029/1024
|8.4c
|[[Slendric]]
|}

== 7-limit ==

=== 31-odd-limit (commas < 50c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|marvel comma
|S15
|16/15 ~ 15/14
|225/224
|7.7c
|[[Marvel]], such as septimal meantone
|-
|avicennma
|
|21/16 ~ 32/25
|525/512
|43.4c
|
|-
|starling comma
|
|25/21 ~ 6/5
|126/125
|13.8c
|
|-
|jubilisma
|S5/S7
|10/7 ~ 7/5
|50/49
|35c
|Shows up in temperaments like [[pajara]] and [[26edo]]
|-
|mint comma
|S6
|7/6 ~ 6/5
|36/35
|48.8c
|Shows up in [[dominant]] and [[dimisept]]. Idiosyncratic 7-limit formal comma used by
Ben Johnston notation.
|}

=== 63-odd-limit (commas < 25c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|breedsma
|S49
|50/49 ~ 49/48
|2401/2400
|0.7c
|{{UserTag|SS|Vector|cebaff|stupid-sounding comma name tbh}}
|-
|orwellisma
|S6/S7
|49/48 ~ 36/35
|1728/1715
|13.1c
|Shows up in [[orwell]]
|-
|sensamagic comma
|S7/S9
|9/7 ~ 35/27
|245/243
|14.2c
|Important to [[tritave]]-based harmony.
|-
|keema
|
|36/35 ~ 25/24
|875/864
|21.9c
|Evenly spaces the basic 7-limit thirds. Found in [[magic]] and [[porcupine]].
|-
|nuwell comma
|
|54/49 ~ 49/45
|2430/2401
|20.8c
|Found in [[orwell]] and [[nusecond]].
|-
|octagar comma
|S19 * S20^2
|63/50 ~ 80/63
|4000/3969
|13.5c
|
|}

== 2.3.5.11 ==

=== 31-odd-limit (commas < 50c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|cake comma
|S9*S10
|11/9 ~ 5/4
|45/44
|38.9c
|Related to the main 11-limit formal comma (not included here) by the syntonic comma.
|-
|ptolemisma
|S10
|11/10 ~ 10/9
|100/99
|17.4c
|Shows up in porcupine and flattone.
|-
|telepathma
|S10*S11
|12/11 ~ 10/9
|55/54
|31.8c
|Shows up in pajara (with the 11-limit extension for the sharp tuning) and in porcupine.{{UserTag|SS|Vector|cebaff|A432 music is detuned by this amount (but is not in and of itself xenharmonic). (Surprisingly, this is not related to the origin of the comma name.)}}
|-
|biyatisma
|S11
|12/11 ~ 11/10
|121/120
|14.4c
|
|-
|rastma
|S9/S11
|11/9 ~ 27/22
|243/242
|7.1c
|Shows up in 11-limit [[neutral temperaments]].
|}

== 11-limit ==

=== 31-odd-limit (commas < 40c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|werckisma
|S21
|22/21 ~ 21/20
|441/440
|3.9c
|
|-
|mothwellsma
|
|9/7 ~ 14/11
|99/98
|17.6c
|Involved in the flat pajara extension.
|-
|konbini comma
|S12 * S13 * S14
|7/5 ~ 11/8
|56/55
|31.2c
|
|-
|valinorsma
|S8/S10
|25/22 ~ 8/7
|176/175
|9.9c
|
|-
|pakkanen comma
|
|28/25 ~ 25/22
|625/616
|25.1c
|
|}

=== 63-odd-limit (commas < 15c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|keenanisma
|S33*S34*S35
|12/11 ~ 35/32
|385/384
|4.5c
|
|-
|fwiwisma
|
|45/44 ~ 64/63
|2835/2816
|11.6c
|
|-
|lehmerisma
|S22/S24, S55
|56/55 ~ 55/54
|3025/3024
|0.6c
|
|-
|moctdelisma
|
|55/49 ~ 28/25
|1375/1372
|3.8c
|
|-
|swetisma
|S12/S14
|49/40 ~ 27/22
|540/539
|3.2c
|Equates the 7-limit and 11-limit neutral thirds
|-
|pentacircle comma
|
|33/28 ~ 32/27
|896/891
|9.7c
|
|-
|''unnamed''
|
|49/44 ~ 55/49
|(2420/2401)
|13.6c
|
|}

== 2.3.7.13 ==

=== 31-odd-limit (commas < 50c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|squbema
|S27
|28/27 ~ 27/26
|729/728
|2.4c
|
|-
|buzurgisma
|S13
|14/13 ~ 13/12
|169/168
|17.6c
|
|}

=== 63-odd-limit (commas < 25c) ===
{| class="wikitable"
!Name
!S-expression
!Simplest interval pair
!Ratio
!Cents
!Associated temperaments and other notes
|-
|harmoneutral comma
|
|16/13 ~ 39/32
|512/507
|17c
|Shows up in 13-limit neutral temperaments
|}

Ternary scale

Inthar — Wed, 30 Sep 2026 00:37:32 GMT

MV3 ternary scales

← Older revision		Revision as of 00:37, 30 September 2026
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	Single-period MV3 ternary scale patterns come in the following types:		Single-period MV3 ternary scale patterns come in the following types:
	# '''pairwise-MOS''': scales such that making any two kinds of steps the same size always results in a MOS (an example is Zarlino, where LMsLMLs gives LssLsLs, LLsLLLs, and LsLLsLL)		# '''pairwise-MOS''': scales such that making any two kinds of steps the same size always results in a MOS (an example is Zarlino, where LMsLMLs gives LssLsLs, LLsLLLs, and LsLLsLL)
	#* '''abacaba''' scales~~<sup>[explain]</sup>~~		#* '''abacaba''' scales where '''a''', '''b''', and '''c''' are three distinct step sizes: for example, LmLsLmL
	#* '''odd-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])		#* '''odd-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])
	#* '''even-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])		#* '''even-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])
	# '''abcba''' scales~~<sup>[explain]</sup>~~: ~~the unique sporadic~~ non-pairwise-MOS MV3 ternary ~~pattern~~		# '''abcba''' scales where '''a''', '''b''', and '''c''' are three distinct step sizes: a finite family of non-pairwise-MOS MV3 ternary patterns
	# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales		# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales

Notation

Vector — Sat, 26 Sep 2026 03:16:10 GMT

← Older revision		Revision as of 03:16, 26 September 2026
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	===== Etymology =====		===== Etymology =====
	Keys in Western music are usually each associated with a mode and transposition of the diatonic scale, the latter of which the key signature specifies.[[File:5555.png\|thumb\|201x201px\|A key signature using FJS accidentals to specify the 5-limit zarlino scale in C major. D, E, A, and B are flattened by a syntonic comma.]]		Keys in Western music are usually each associated with a mode (major = Ionian, minor ~= Aeolian) and transposition of the diatonic scale, the latter of which the key signature specifies. For instance, a key signature of F# implies G major or E minor, due to G Ionian being G A B C D E F# and E Aeolian being E F# G A B C D. [[File:5555.png\|thumb\|201x201px\|A key signature using FJS accidentals to specify the 5-limit zarlino scale in C major. D, E, A, and B are flattened by a syntonic comma.]]

	==== Note symbols ====		==== Note symbols ====
	While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations.		While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations, so if you're designing a notation system around using symbol shape for inflections or other pitch-related meanings, that needs to be accounted for.

	=== Annotations ===		=== Annotations ===
	Figured bass is a prominent form of notation accompanying a standard score that indicates what notes and/or chords to play, most commonly used for analytical clarity, especially where performers are expected to improvise. It utilizes the numbers 1 through 7 to indicate the degrees of a scale. One notable constraint of figured bass is that the symbols for degrees must be singular digits or letters, disallowing constructions like "1/2" and "15"; this makes it rather painful to generalize to microtonal scales while keeping numeric indices for intuition. One system works around this by utilizing the letters of the Greek alphabet, which allow for 24 distinct degrees to be notated. Another option is to utilize alternative systems to figured bass, such as placing chord symbols above the score.		Figured bass is a prominent form of notation accompanying a standard score that indicates what notes and/or chords to play, most commonly used for analytical clarity, especially where performers are expected to improvise. It utilizes the numbers 1 through 7 to indicate the degrees of a scale. One notable constraint of figured bass is that to avoid compromising any ease of readability, the symbols for degrees must be singular digits or letters, disallowing constructions like "1/2" and "15"; this makes it rather painful to generalize to microtonal scales while keeping numeric indices for intuition. One system works around this by utilizing the letters of the Greek alphabet, which allow for 24 distinct degrees to be notated. Another option is to utilize alternative systems to figured bass, such as placing chord symbols above the score.

	=== Scale degrees vs. intervals ===		=== Scale degrees vs. intervals ===
	A scale degree refers to a note defined by an offset from the tonal center, tonic, or otherwise a fixed reference point that does not change throughout a song or section. Scale degrees are conventionally notated by specifying alterations from the degrees of the characteristic scale. Intervals are the offsets themselves, which may be applied either to the tonal center (specifying a scale degree) or to any other pitch. Intervals are notated with qualities such as major or minor, so that "m3" ("minor third") is an interval while "b3" ("flat three") is a degree. Note that outside of major and minor/sharp and flat, additional accidentals tend to share their names with qualities.		A scale degree refers to a note defined by an offset from the tonal center, tonic, or otherwise a fixed reference point that does not change throughout a song or section. Scale degrees are conventionally notated by specifying alterations from the degrees of the characteristic scale (that is, the scale degree a minor third above the tonal center in diatonic, where Ionian is the characteristic scale, is b3). Intervals are the offsets themselves, which may be applied either to the tonal center (specifying a scale degree) or to any other pitch. Intervals are notated with qualities such as major or minor, so that "m3" ("minor third") is an interval while "b3" ("flat three") is a degree. Note that outside of major and minor/sharp and flat, additional accidentals tend to share their names with qualities.

	However, note that this is somewhat disputed:		However, note that this is somewhat disputed:
Line 53:		Line 53:
	* Unque characterizes scale degrees as "points in space" and intervals as "types of motion within that space", characterizing pitch space relative to the tonal center as a form of "secondary absolute" space.		* Unque characterizes scale degrees as "points in space" and intervals as "types of motion within that space", characterizing pitch space relative to the tonal center as a form of "secondary absolute" space.
	* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.		* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.
	* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa.		* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa when no further context is given.
	* Inthar treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors.		* Inthar treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors.

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	=== Diatonic notation ===		=== Diatonic notation ===
	[[Diatonic notation\|This page]] covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma.		[[Diatonic notation\|This page]] covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma. Accidentals apply to the portion of the measure after the place where the accidental appears.

	=== KISS notation ===		=== KISS notation ===
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	==== Diamond-MOS notation ====		==== Diamond-MOS notation ====
	Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used.		Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used. The accessibility features of Diamond-MOS notation may be applied to other notation systems.

	=== Just notation systems ===		=== Just notation systems ===
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	==== Color notation ====		==== Color notation ====
	Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to ~~24edo~~.		Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to 12edo.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Ternary scale

Vector — Sat, 26 Sep 2026 03:06:29 GMT

"identifying" is too mathematical

← Older revision		Revision as of 03:06, 26 September 2026
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	Single-period MV3 ternary scale patterns come in the following types:		Single-period MV3 ternary scale patterns come in the following types:
	# '''pairwise-MOS''': scales such that ~~identifying~~ any two ~~step sizes~~ always results in a MOS		# '''pairwise-MOS''': scales such that making any two kinds of steps the same size always results in a MOS (an example is Zarlino, where LMsLMLs gives LssLsLs, LLsLLLs, and LsLLsLL)
	#* '''abacaba''' scales		#* '''abacaba''' scales<sup>[explain]</sup>
	#* '''odd-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])		#* '''odd-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is odd and gcd(a, b) = 1 (example: [[diasem]])
	#* '''even-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])		#* '''even-regular''' scales: [[MOS substitution]] scales of type a'''x'''(b'''y'''b'''z''') where a is even and gcd(a, b) = 1 (example: [[penslen]])
	# '''abcba''' scales: the unique sporadic non-pairwise-MOS MV3 ternary pattern		# '''abcba''' scales<sup>[explain]</sup>: the unique sporadic non-pairwise-MOS MV3 ternary pattern
	# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales		# '''twisted MV3''' scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales

Aberschismic

Inthar — Fri, 25 Sep 2026 07:23:04 GMT

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	'''Aberschismic''' (formerly known as '''Hemifamity'''), [[41edo\|41]] & [[46edo\|46]] & [[53edo\|53]], is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''aberschisma'''), which manifests in several ~~different~~ ways:		'''Aberschismic''' (formerly known as '''Hemifamity'''), [[41edo\|41]] & [[46edo\|46]] & [[53edo\|53]], is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''aberschisma'''), which manifests in several mathematically equivalent ways:
	# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63		# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63
	# equates the Pythagorean minor second with 21/20		# equates the Pythagorean minor second with 21/20

Marvel

Inthar — Fri, 25 Sep 2026 07:17:38 GMT

Supporting temperaments

← Older revision		Revision as of 07:17, 25 September 2026
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	== Supporting temperaments ==		== Supporting temperaments ==
	Many rank-2 temperaments supported by small-to-medium-sized ~~edos~~ support Marvel:		Many rank-2 temperaments supported by small-to-medium-sized equal temperaments support Marvel:
	* 19 & 31, Septimal [[Meantone]] (Marvel + Meantone or + [[Didacus]])		* 19 & 31, Septimal [[Meantone]] (Marvel + Meantone or + [[Didacus]])
	* 19 & 22, [[Magic]] (Marvel + [[Sensamagic]])		* 19 & 22, [[Magic]] (Marvel + [[Sensamagic]])

Schismic

FloraC — Mon, 21 Sep 2026 15:45:41 GMT

Cleanup on infobox; + category

@@ Line 1: / Line 1: @@
-{{Infobox regtemp|Mapping=1; 1 -8 -3|Subgroup=2.3.5, 2.3.5.19|Edo join 1=12|Edo join 2=29|Generators=3/2|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)|Odd limit 1=5|Complexity 1=12|Generators tuning=701.7|Subgroups=2.3.5, 2.3.5.19|Mistuning 1=0.217|Optimization method=CWE|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
+{{Infobox regtemp
 [[File:Schismic17.png|thumb|Schismic 17-note MOS. This is a sharp tuning; the Pythagorean comma has been inflated to function as an [[aberrisma]].]]
 '''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.
@@ Line 1,109: / Line 1,117: @@
 == Footnotes ==
 <sup>[a]</sup> Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).
 {{Navbox regtemp}}

Ground's composition theory

Ground — Sat, 19 Sep 2026 12:24:41 GMT

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	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.		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.

			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.
			* 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.
			* In terms of the number of useful scales available, 36 is superior. To some extent, this comes with having more notes in general.
			* 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.
			* x6 is more of a standard for me, so I'm set up better for that.
			* 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.
			* 36x2 makes accessing 72edo especially convenient because fewer channels is easier. It wins for 11-limit stuff.
			* 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.
			* 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.
			* Using a smaller edo gets more range in the piano roll, which is useful for quick creative writing.
			* 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.
			With all this in mind, 24x6 is clearly the better choice to start with, although it's close.

	== Anchor concordances ==		== Anchor concordances ==

User:Hotcrystal0/Sandbox

Hotcrystal0 — Fri, 18 Sep 2026 20:27:34 GMT

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	==Intervals==		==Intervals==