Xenharmonic Reference - User contributions [en]

Ground's composition theory

2026-07-29T21:48:28Z

Inthar: /* Structural tuning selection */ typo

{{proposed}}

I'm [[User:Ground]]. This document is going to be very long. I have more content to add and a plan to revise the existing content eventually.

== Interval logic deviation theory ==

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, with an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]]. 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.

=== Local tonality ===

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Diatonic example of probabilities (up to word length 3)
!Sequence
!Next step probability
|-
|s
|L 1/1
|-
|L
|L 5/7, s 2/7
|-
|sL
|L 1/1
|-
|Ls
|L 1/1
|-
|sLL
|s 1/2, L 1/2
|-
|LsL
|L 1/1
|-
|LLs
|L 1/1
|-
|LLL
|s 2/3, L 1/3
|}
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.

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 "bubble deviations" 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.

Melodic interval sequences, or "words" 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.

=== Axes of deviation ===

"Bubble deviations" 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 "probable" (less unexpected) than something like ssL.

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.

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.

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 <<3 and >3 in 37edo straddles 74edo meantone and results in trackdye.

{{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).}}

=== Vague interval logic ===

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.

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 >4/3 and sssL becomes >3/2.

=== Aberrismic theory ===

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.

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.

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.

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.

I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

== Structural tuning selection ==

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.

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.

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:
# Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?
# 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?
{| class="wikitable"
|+A selection of large edos I use
!Edo
!Justification
|-
| style="text-align: right;" |27x3 = 81
|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.
|-
| style="text-align: right;" |29x3 = 87
|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.
|-
| style="text-align: right;" |37x2 = 74
|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.
|-
| style="text-align: right;" |43x2 = 86
|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.
|-
| style="text-align: right;" |67x2ed4 = 67
|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.
|}

== Quartertone composition ==

{| class="wikitable" style="float:right; margin-left: 12px; text-align: center;"
|+Island Rastmic with Straddle Primes
!Gens
!949¢
!349¢
! colspan="2" |<nowiki>Proposed Names (semi- | hemi-)</nowiki>
|-
| -12
| colspan="2" |612
| colspan="2" |Diminished Fifth
|-
| -11
|361
| class="thl" |961
|"Semififth"
|"Hemidim." Seventh
|-
| -10
| colspan="2" class="thl" |110
| colspan="2" |Minor Second
|-
| -9
|1059
|459
|"Semithirteenth"
|"Hemidim." Fourth
|-
| -8
| colspan="2" |808
| colspan="2" |Minor Sixth
|-
| -7
| class="thl" |557
|1157
|"Semiseventh"
|"Hemidim." Octave
|-
| -6
| colspan="2" class="thl" |306
| colspan="2" |Minor Third
|-
| -5
|55
|655
|"Semisecond"
|"Hemidim." Fifth
|-
| -4
| colspan="2" |1004
| colspan="2" |Minor Seventh
|-
| -3
|753
|153
|Semitenth
|Neutral Second
|-
| -2
| colspan="2" |502
| colspan="2" |Perfect Fourth
|-
| -1
|251
|851
|Semifourth
|Neutral Sixth
|-
|0
| colspan="2" class="thl" |0
| colspan="2" |Unison
|-
|1
|949
|349
|Semitwelfth
|Neutral Third
|-
|2
| colspan="2" class="thl" |698 ''(98)''
| colspan="2" |Perfect Fifth
|-
|3
|447
|1047
|Semisixth
|Neutral Seventh
|-
|4
| colspan="2" |196
| colspan="2" |Major Second
|-
|5
|1145
| class="thl" |545
|"Semifourteenth"
|"Hemiaug." Fourth
|-
|6
| colspan="2" class="thl" |894 ''(294)''
| colspan="2" |Major Sixth
|-
|7
|643
|43
|"Semininth"
|"Hemiaug." Unison
|-
|8
| colspan="2" class="thl" |392
| colspan="2" |Major Third
|-
|9
|141
|741
|"Semithird"
|"Hemiaug." Fifth
|-
|10
| colspan="2" |1090
| colspan="2" |Major Seventh
|-
|11
| class="thl" |839
|239
|"Semieleventh"
|"Hemiaug." Second
|-
|12
| colspan="2" |588
| colspan="2" |Augmented Fourth
|}
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.

=== Scale theory ===

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&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&62.}}

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&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.

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.

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.

=== Melodic intervals ===

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.
* 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 "metallic" sound as a semitone.
* 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 "sour" sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.
* 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.

== Tertian triad categories ==

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Thirds categorized as a fraction of a diatonic fifth.
!Fraction
!*7/12 =
!4\7 (¢)
!7\12 (¢)
!3/2 (¢)
!3\5 (¢)
!Lower bound of:
|-
!1/3
|7/36
|228.6
|233.3
|234.0
|240.0
|"Horiminor"
|-
!12/35
|1/5
|235.1
|240.0
|240.7
|246.9
|Inframinor
|-
!18/49
|3/14
|251.9
|257.1
|257.9
|264.5
|Subminor
|-
!11/28
|11/48
|269.4
|275.0
|275.8
|282.9
|Neominor
|-
!3/7
|1/4
|293.9
|300.0
|300.8
|308.6
|Classic Minor
|-
!23/49
|23/84
|321.9
|328.6
|329.5
|338.0
|Supraminor
|-
!24/49
|2/7
|335.9
|342.9
|343.8
|352.7
|Neutral
|-
!25/49
|25/84
|349.9
|357.1
|358.1
|367.3
|Submajor
|-
!26/49
|13/42
|363.8
|371.4
|372.5
|382.0
|Classic Major
|-
!4/7
|1/3
|391.8
|400.0
|401.1
|411.4
|Neomajor
|-
!17/28
|17/48
|416.3
|425.0
|426.2
|437.1
|Supermajor
|-
!31/49
|31/84
|433.8
|442.9
|444.1
|455.5
|Ultramajor
|-
!23/35
|23/60
|450.6
|460.0
|461.3
|473.1
|"Horimajor"
|-
!2/3
|7/18
|457.1
|466.7
|468.0
|480.0
|[end of range]
|}
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.

* I've proposed the terms "horiminor" and "horimajor" after the same root in the word "horizon". 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.
* 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.
* 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.
* 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. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.
* Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15. 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.
* 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.
* 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 just because they're not a simple otonal chord, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.

== Quasi-diatonic theory ==

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Blackdye and diasem JI tunings, sorted from flat to sharp minor third
!Tertian Triad
!6:7:9 (14:18:21)
!18:23:27
!16:19:24
!4:5:6 (10:12:15)
!14:17:21
|-
|Subgroup
|2.3.7
|2.3.23
|2.3.19
|2.3.5
|2.3.17/7
|-
|Name
|Septal
|"Eridian"
|?
|Pental
|?
|-
|Aberrisma
|64/63
|736/729
|513/512
|81/80
|459/448
|-
|Quasi- Aeolian
|9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
|9/8 27/23 243/184 4/3 3/2 36/23 81/46 16/9 2/1
|513/512 9/8 19/16 4/3 171/128 3/2 19/12 16/9 57/32 2/1
|81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1
|459/448 9/8 17/14 4/3 153/112 3/2 34/21 16/9 51/28 2/1
|-
|Similar Aberrismas
|
|25: 2048/2025 13/11: 352/351 17/5: 136/135
|17: 4131/4096
|
|11: 33/32 13: 1053/1024
|}
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 [[Aberrisma|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. {{adv|The obvious temperament in this range is something I'm calling '''Eridian Meantone''', a 2.3.5.23 temperament which tempers out 16767/16384 (equating the diminished fifth to a flat 23/16) instead of Septimal Meantone's 225/224, although it's reasonable to temper out both. 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.}}

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

Ground's composition theory

2026-07-29T00:43:05Z

Inthar: /* Quasi-diatonic theory */

{{proposed}}

I'm [[User:Ground]]. This document is going to be very long. I have more content to add and a plan to revise the existing content eventually.

== Interval logic deviation theory ==

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, with an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]]. 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.

=== Local tonality ===

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Diatonic example of probabilities (up to word length 3)
!Sequence
!Next step probability
|-
|s
|L 1/1
|-
|L
|L 5/7, s 2/7
|-
|sL
|L 1/1
|-
|Ls
|L 1/1
|-
|sLL
|s 1/2, L 1/2
|-
|LsL
|L 1/1
|-
|LLs
|L 1/1
|-
|LLL
|s 2/3, L 1/3
|}
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.

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 "bubble deviations" 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.

Melodic interval sequences, or "words" 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.

=== Axes of deviation ===

"Bubble deviations" 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 "probable" (less unexpected) than something like ssL.

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.

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.

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 <<3 and >3 in 37edo straddles 74edo meantone and results in trackdye.

{{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).}}

=== Vague interval logic ===

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.

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 >4/3 and sssL becomes >3/2.

=== Aberrismic theory ===

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.

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.

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.

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.

I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

== Structural tuning selection ==

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.

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.

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:
# Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?
# 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?
{| class="wikitable"
|+A selection of large edos I use
!Edo
!Jusfitication
|-
| style="text-align: right;" |27x3 = 81
|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.
|-
| style="text-align: right;" |29x3 = 87
|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.
|-
| style="text-align: right;" |37x2 = 74
|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.
|-
| style="text-align: right;" |43x2 = 86
|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.
|-
| style="text-align: right;" |67x2ed4 = 67
|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.
|}

== Quartertone composition ==

{| class="wikitable" style="margin-left: 12px; text-align: center;"
|+Island Rastmic with Straddle Primes
!Gens
!949¢
!349¢
! colspan="2" |<nowiki>Proposed Names (semi- | hemi-)</nowiki>
|-
| -12
| colspan="2" |612
| colspan="2" |Diminished Fifth
|-
| -11
|361
| class="thl" |961
|"Semififth"
|"Hemidim." Seventh
|-
| -10
| colspan="2" class="thl" |110
| colspan="2" |Minor Second
|-
| -9
|1059
|459
|"Semithirteenth"
|"Hemidim." Fourth
|-
| -8
| colspan="2" |808
| colspan="2" |Minor Sixth
|-
| -7
| class="thl" |557
|1157
|"Semiseventh"
|"Hemidim." Octave
|-
| -6
| colspan="2" class="thl" |306
| colspan="2" |Minor Third
|-
| -5
|55
|655
|"Semisecond"
|"Hemidim." Fifth
|-
| -4
| colspan="2" |1004
| colspan="2" |Minor Seventh
|-
| -3
|753
|153
|Semitenth
|Neutral Second
|-
| -2
| colspan="2" |502
| colspan="2" |Perfect Fourth
|-
| -1
|251
|851
|Semifourth
|Neutral Sixth
|-
|0
| colspan="2" class="thl" |0
| colspan="2" |Unison
|-
|1
|949
|349
|Semitwelfth
|Neutral Third
|-
|2
| colspan="2" class="thl" |698 ''(98)''
| colspan="2" |Perfect Fifth
|-
|3
|447
|1047
|Semisixth
|Neutral Seventh
|-
|4
| colspan="2" |196
| colspan="2" |Major Second
|-
|5
|1145
| class="thl" |545
|"Semifourteenth"
|"Hemiaug." Fourth
|-
|6
| colspan="2" class="thl" |894 ''(294)''
| colspan="2" |Major Sixth
|-
|7
|643
|43
|"Semininth"
|"Hemiaug." Unison
|-
|8
| colspan="2" class="thl" |392
| colspan="2" |Major Third
|-
|9
|141
|741
|"Semithird"
|"Hemiaug." Fifth
|-
|10
| colspan="2" |1090
| colspan="2" |Major Seventh
|-
|11
| class="thl" |839
|239
|"Semieleventh"
|"Hemiaug." Second
|-
|12
| colspan="2" |588
| colspan="2" |Augmented Fourth
|}
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.

=== Scale theory ===

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&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&62.}}

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&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.

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.

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.

=== Intervals and chords ===

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.
* 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 "metallic" sound as a semitone.
* 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 "sour" sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.
* 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.
* Inframinor and ultramajor chords share the metallic quality with the quartertone. 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.
* Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered generally concordant, between 24edo and 31edo. 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 stretched diminished chords.

== Quasi-diatonic theory ==

{| class="wikitable" style="margin-left: 12px;"
|+Blackdye and diasem JI tunings, sorted from flat to sharp minor third
!Tertian Triad
!6:7:9 (14:18:21)
!18:23:27
!16:19:24
!4:5:6 (10:12:15)
!14:17:21
|-
|Subgroup
|2.3.7
|2.3.23
|2.3.19
|2.3.5
|2.3.17/7
|-
|Name
|Septal
|"Eridian"
|?
|Pental
|?
|-
|Aberrisma
|64/63
|736/729
|513/512
|81/80
|459/448
|-
|Quasi- Aeolian
|9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
|9/8 27/23 243/184 4/3 3/2 36/23 81/46 16/9 2/1
|513/512 9/8 19/16 4/3 171/128 3/2 19/12 16/9 57/32 2/1
|81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1
|459/448 9/8 17/14 4/3 153/112 3/2 34/21 16/9 51/28 2/1
|-
|Similar Aberrismas
|
|25: 2048/2025 13/11: 352/351 17/5: 136/135
|17: 4131/4096
|
|11: 33/32 13: 1053/1024
|}
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 [[Aberrisma|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. {{adv|The obvious temperament in this range is something I'm calling '''Eridian Meantone''', a 2.3.5.23 temperament which tempers out 16767/16384 (equating the diminished fifth to a flat 23/16) instead of Septimal Meantone's 225/224, although it's reasonable to temper out both. 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.}}

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

Ground's composition theory

2026-07-29T00:42:47Z

Inthar: /* Quartertone composition */

{{proposed}}

I'm [[User:Ground]]. This document is going to be very long. I have more content to add and a plan to revise the existing content eventually.

== Interval logic deviation theory ==

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, with an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]]. 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.

=== Local tonality ===

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Diatonic example of probabilities (up to word length 3)
!Sequence
!Next step probability
|-
|s
|L 1/1
|-
|L
|L 5/7, s 2/7
|-
|sL
|L 1/1
|-
|Ls
|L 1/1
|-
|sLL
|s 1/2, L 1/2
|-
|LsL
|L 1/1
|-
|LLs
|L 1/1
|-
|LLL
|s 2/3, L 1/3
|}
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.

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 "bubble deviations" 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.

Melodic interval sequences, or "words" 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.

=== Axes of deviation ===

"Bubble deviations" 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 "probable" (less unexpected) than something like ssL.

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.

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.

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 <<3 and >3 in 37edo straddles 74edo meantone and results in trackdye.

{{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).}}

=== Vague interval logic ===

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.

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 >4/3 and sssL becomes >3/2.

=== Aberrismic theory ===

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.

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.

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.

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.

I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

== Structural tuning selection ==

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.

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.

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:
# Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?
# 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?
{| class="wikitable"
|+A selection of large edos I use
!Edo
!Jusfitication
|-
| style="text-align: right;" |27x3 = 81
|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.
|-
| style="text-align: right;" |29x3 = 87
|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.
|-
| style="text-align: right;" |37x2 = 74
|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.
|-
| style="text-align: right;" |43x2 = 86
|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.
|-
| style="text-align: right;" |67x2ed4 = 67
|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.
|}

== Quartertone composition ==

{| class="wikitable" style="margin-left: 12px; text-align: center;"
|+Island Rastmic with Straddle Primes
!Gens
!949¢
!349¢
! colspan="2" |<nowiki>Proposed Names (semi- | hemi-)</nowiki>
|-
| -12
| colspan="2" |612
| colspan="2" |Diminished Fifth
|-
| -11
|361
| class="thl" |961
|"Semififth"
|"Hemidim." Seventh
|-
| -10
| colspan="2" class="thl" |110
| colspan="2" |Minor Second
|-
| -9
|1059
|459
|"Semithirteenth"
|"Hemidim." Fourth
|-
| -8
| colspan="2" |808
| colspan="2" |Minor Sixth
|-
| -7
| class="thl" |557
|1157
|"Semiseventh"
|"Hemidim." Octave
|-
| -6
| colspan="2" class="thl" |306
| colspan="2" |Minor Third
|-
| -5
|55
|655
|"Semisecond"
|"Hemidim." Fifth
|-
| -4
| colspan="2" |1004
| colspan="2" |Minor Seventh
|-
| -3
|753
|153
|Semitenth
|Neutral Second
|-
| -2
| colspan="2" |502
| colspan="2" |Perfect Fourth
|-
| -1
|251
|851
|Semifourth
|Neutral Sixth
|-
|0
| colspan="2" class="thl" |0
| colspan="2" |Unison
|-
|1
|949
|349
|Semitwelfth
|Neutral Third
|-
|2
| colspan="2" class="thl" |698 ''(98)''
| colspan="2" |Perfect Fifth
|-
|3
|447
|1047
|Semisixth
|Neutral Seventh
|-
|4
| colspan="2" |196
| colspan="2" |Major Second
|-
|5
|1145
| class="thl" |545
|"Semifourteenth"
|"Hemiaug." Fourth
|-
|6
| colspan="2" class="thl" |894 ''(294)''
| colspan="2" |Major Sixth
|-
|7
|643
|43
|"Semininth"
|"Hemiaug." Unison
|-
|8
| colspan="2" class="thl" |392
| colspan="2" |Major Third
|-
|9
|141
|741
|"Semithird"
|"Hemiaug." Fifth
|-
|10
| colspan="2" |1090
| colspan="2" |Major Seventh
|-
|11
| class="thl" |839
|239
|"Semieleventh"
|"Hemiaug." Second
|-
|12
| colspan="2" |588
| colspan="2" |Augmented Fourth
|}
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.

=== Scale theory ===

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&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&62.}}

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&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.

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.

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.

=== Intervals and chords ===

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.
* 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 "metallic" sound as a semitone.
* 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 "sour" sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.
* 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.
* Inframinor and ultramajor chords share the metallic quality with the quartertone. 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.
* Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered generally concordant, between 24edo and 31edo. 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 stretched diminished chords.

== Quasi-diatonic theory ==

{| class="wikitable" style="float:right; margin-left: 12px;"
|+Blackdye and diasem JI tunings, sorted from flat to sharp minor third
!Tertian Triad
!6:7:9 (14:18:21)
!18:23:27
!16:19:24
!4:5:6 (10:12:15)
!14:17:21
|-
|Subgroup
|2.3.7
|2.3.23
|2.3.19
|2.3.5
|2.3.17/7
|-
|Name
|Septal
|"Eridian"
|?
|Pental
|?
|-
|Aberrisma
|64/63
|736/729
|513/512
|81/80
|459/448
|-
|Quasi- Aeolian
|9/8 7/6 21/16 4/3 3/2 14/9 7/4 16/9 2/1
|9/8 27/23 243/184 4/3 3/2 36/23 81/46 16/9 2/1
|513/512 9/8 19/16 4/3 171/128 3/2 19/12 16/9 57/32 2/1
|81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1
|459/448 9/8 17/14 4/3 153/112 3/2 34/21 16/9 51/28 2/1
|-
|Similar Aberrismas
|
|25: 2048/2025 13/11: 352/351 17/5: 136/135
|17: 4131/4096
|
|11: 33/32 13: 1053/1024
|}
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 [[Aberrisma|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. {{adv|The obvious temperament in this range is something I'm calling '''Eridian Meantone''', a 2.3.5.23 temperament which tempers out 16767/16384 (equating the diminished fifth to a flat 23/16) instead of Septimal Meantone's 225/224, although it's reasonable to temper out both. 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.}}

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

Stretching and compression

2026-07-27T08:11:38Z

Inthar: /* Chords */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

=== Tens- and tractaberration ===
In [[aberrismic theory]], the ''tens-aberrated'' and ''tract-aberrated'' versions of a [[MOS]] scale aLbm are given by the [[MOS substitution]] operations [a+b-1]s(aLbm) and [a+b+1]s(aLbm) respectively. For example, sLsmsLsLsLsmsLs is a tractaberrated diatonic scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in [[TAMNAMS]] notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Ground's composition theory

2026-07-27T05:04:34Z

Inthar: /* Axes of deviation */

{{proposed}}

I'm [[User:Ground]]. This document is going to be very long. I have more content to add and a plan to revise the existing content eventually.

== Introduction and motivation ==

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, with an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]]. 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.

== Local tonality ==

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.

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 "bubble deviations" 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.

Melodic interval sequences, or "words" 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.
{| class="wikitable"
|+Diatonic example of probabilities (up to word length 3)
!Sequence
!Next step probability
|-
|s
|L 1/1
|-
|L
|L 5/7, s 2/7
|-
|sL
|L 1/1
|-
|Ls
|L 1/1
|-
|sLL
|s 1/2, L 1/2
|-
|LsL
|L 1/1
|-
|LLs
|L 1/1
|-
|LLL
|s 2/3, L 1/3
|}

== Axes of deviation ==

"Bubble deviations" 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 "probable" (less unexpected) than something like ssL.

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.

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.

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 <<3 and >3 in 37edo straddles 74edo meantone and results in trackdye.

{{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).}}

== Vague interval logic ==

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.

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 >4/3 and sssL becomes >3/2.

== Aberrismic theory ==

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.

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.

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.

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.

I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

== Quartertone composition ==

{| class="wikitable" style="float:right; margin-left: 12px; text-align: center;"
|+Island Rastmic with Straddle Primes
!Gens
!949¢
!349¢
! colspan="2" |<nowiki>Proposed Names (semi- | hemi-)</nowiki>
|-
| -12
| colspan="2" |612
| colspan="2" |Diminished Fifth
|-
| -11
|361
| class="thl" |961
|Semififth
|Hemidim Seventh
|-
| -10
| colspan="2" class="thl" |110
| colspan="2" |Minor Second
|-
| -9
|1059
|459
|Semithirteenth
|Hemidim Fourth
|-
| -8
| colspan="2" |808
| colspan="2" |Minor Sixth
|-
| -7
| class="thl" |557
|1157
|Semiseventh
|Hemidim Octave
|-
| -6
| colspan="2" class="thl" |306
| colspan="2" |Minor Third
|-
| -5
|55
|655
|Semisecond
|Hemidim Fifth
|-
| -4
| colspan="2" |1004
| colspan="2" |Minor Seventh
|-
| -3
|753
|153
|Semitenth
|Neutral Second
|-
| -2
| colspan="2" |502
| colspan="2" |Perfect Fourth
|-
| -1
|251
|851
|Semifourth
|Neutral Sixth
|-
|0
| colspan="2" class="thl" |0
| colspan="2" |Unison
|-
|1
|949
|349
|Semitwelfth
|Neutral Third
|-
|2
| colspan="2" class="thl" |698 ''(98)''
| colspan="2" |Perfect Fifth
|-
|3
|447
|1047
|Semisixth
|Neutral Seventh
|-
|4
| colspan="2" |196
| colspan="2" |Major Second
|-
|5
|1145
| class="thl" |545
|Semifourteenth
|Hemiaug Fourth
|-
|6
| colspan="2" class="thl" |894 ''(294)''
| colspan="2" |Major Sixth
|-
|7
|643
|43
|Semininth
|Hemiaug Unison
|-
|8
| colspan="2" class="thl" |392
| colspan="2" |Major Third
|-
|9
|141
|741
|Semithird
|Hemiaug Fifth
|-
|10
| colspan="2" |1090
| colspan="2" |Major Seventh
|-
|11
| class="thl" |839
|239
|Semieleventh
|Hemiaug Second
|-
|12
| colspan="2" |588
| colspan="2" |Augmented Fourth
|}
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.

=== Scale theory ===

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&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&62.}}

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&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.

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.

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.

=== Intervals and chords ===

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.
* 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 "metallic" sound as a semitone.
* 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 "sour" sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.
* 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.
* Inframinor and ultramajor chords share the metallic quality with the quartertone. 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.
* Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered generally concordant, between 24edo and 31edo. 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 stretched diminished chords.

Aberrisma

2026-07-27T01:14:40Z

Inthar: /* Glossary */

{{Proposed}}

An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Its name originates from the comparison of its sound to a chromatic aberration, with the -isma suffix often used for commas. Though they are typically between 20 and 55 cents (around 81/80 to half of 16/15) and often inflated commas, Larger and smaller sizes can function as aberrismas depending on the scale tuning. 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prejubilic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem).

Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

This is subject to change as aberrismic theory notation is updated in the future.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).
* [[Erac|Eracs]] may be used to represent the inflating of commas typically found in aberrismic scales, although they represent a specific unit of error.

== Aberrismic theory and RTT ==
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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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.
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than the typical limit for an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Stretching and compression

2026-07-27T01:14:21Z

Inthar: /* Tens- and tractaberration */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

=== Tens- and tractaberration ===
In [[aberrismic theory]], the ''tens-aberrated'' and ''tract-aberrated'' versions of a [[MOS]] scale aLbm are given by the [[MOS substitution]] operations [a+b-1]s(aLbm) and [a+b+1]s(aLbm) respectively. For example, sLsmsLsLsLsmsLs is a tractaberrated diatonic scale made from diatonic (a MOS substitution scale of type 8s(5L2m)), also called trackdye.

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:13:56Z

Inthar: /* Tens- and tractaberration */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

=== Tens- and tractaberration ===
In [[aberrismic theory]], the ''tens-aberrated'' and ''tract-aberrated'' versions of a [[MOS]] scale aLbm are given by the [[MOS substitution]] operations [a+b-1]s(aLbm) and [a+b+1]s(aLbm) respectively. For example, sLsmsLsLsLsmsLs is a tractaberrated diatonic scale made from diatonic (a MOS substitution scale of type 8s(5L2m)).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:13:07Z

Inthar: /* Tens- and tractaberration */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

=== Tens- and tractaberration ===
In [[aberrismic theory]], the ''tens-aberrated'' and ''tract-aberrated'' versions of a [[MOS]] scale aLbm are given by the [[MOS substitution]] operations [a+b-1]s(aLbm) and [a+b+1]s(aLbm) respectively.

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:12:19Z

Inthar: /* Scales */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

=== Tens- and tractaberration ===
In [[aberrismic theory]], the tens-aberrated and tract-aberrated versions of a [[MOS]] scale aLbm are given by the [[MOS substitution]] patterns [a+b-1]s(aLbm) and [a+b+1]s(aLbm) respectively.

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:09:32Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these may be prefixed to chords or scales to indicate stretching and compression. For example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:08:26Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression; for example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c), whereas [[11edo]]'s 0-4-7\11 (0c-436c-763c) is a tensmajor chord.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:07:17Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression; for example, [[13edo]]'s 0-4-7\13 (0c-369c-646c) may be called a tractmajor chord since it is a compressed version of the [[12edo]] major chord 0-4-7\12 (0c-400c-700c).

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:06:26Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression, for example 0-4-7\13 (0c-369c-646c) is a tractmajor chord since it is a compressed version of 0-4-7\12 (0c-400c-700c).

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:03:35Z

Inthar: /* Linear stretching and compression */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves subtracting or adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:02:39Z

Inthar: /* Linear stretching and compression */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

The psychoacoustic significance of linearly stretching is that it preserves a chord's [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:01:37Z

Inthar: /* Linear stretching and compression */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a linearly stretched 4:5:6.

Linearly stretching a chord preserves its [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:01:25Z

Inthar: /* Linear stretching and compression */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 (= 39:49:59) is a lunear stretched 4:5:6.

Linearly stretching a chord preserves its [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T01:01:10Z

Inthar: /* Linear stretching and compression */

{{stub}}
'''Stretching and compression''' are (usually logarithmic or approximately logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==
Linear stretching and compression involves adding a uniform linear increment to a chord written in frequency ratios: for example, 4.1:5.1:6.1 (= 41:51:61) is a linearly compressed 4:5:6, and 3.9:4.9:5.9 is a lunear stretched 4:5:6.

Linearly stretching a chord preserves its [[delta signature]], unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. Note that all the chords in the examples above are isodifferential (+1+1).

A small logarithmic stretch nevertheless ''approximates'' a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, the 19edo major triad 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

Stretching and compression

2026-07-27T00:58:47Z

Inthar: /* Linear stretching and compression */

Stretching and compression

2026-07-27T00:57:54Z

Inthar:

Stretching and compression

2026-07-27T00:57:26Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''; these morphemes may be prefixed to chords or scales to indicate stretching and compression.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==

Tract

2026-07-27T00:55:16Z

Inthar: Redirected page to Stretching and compression

#redirect [[Stretching and compression]]

Tens

2026-07-27T00:55:05Z

Inthar: Redirected page to Stretching and compression

#redirect [[Stretching and compression]]

Ground's composition theory

2026-07-25T23:59:27Z

Inthar: /* Aberrismic theory */

{{proposed}}

I'm [[User:Ground]]. This document is going to be very long. I have more content to add and a plan to revise the existing content eventually.

== Introduction and motivation ==

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, with an emphasis on my other theories, [[Aberrisma|aberrismic]] and [[Straddle_primes|straddle-prime]]. 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.

== Local tonality ==

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.

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 "bubble deviations" 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.

Melodic interval sequences, or "words" 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.
{| class="wikitable"
|+Diatonic example of probabilities (up to word length 3)
!Sequence
!Next step probability
|-
|s
|L 1/1
|-
|L
|L 5/7, s 2/7
|-
|sL
|L 1/1
|-
|Ls
|L 1/1
|-
|sLL
|s 1/2, L 1/2
|-
|LsL
|L 1/1
|-
|LLs
|L 1/1
|-
|LLL
|s 2/3, L 1/3
|}

== Axes of deviation ==

"Bubble deviations" 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 "probable" (less unexpected) than something like ssL.

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.

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.

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 <<3 and >3 in 37edo straddles 74edo meantone and results in trackdye.

{{UserTag|KC|Inthar|000000|Tens (stretching) and tract (compression) constitutes 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).}}

== Vague interval logic ==

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.

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 >4/3 and sssL becomes >3/2.

== Aberrismic theory ==

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.

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.

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.

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.

I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

== Quartertone composition ==

{| class="wikitable" style="float:right; margin-left: 12px; text-align: center;"
|+Island Rastmic with Straddle Primes
!Gens
!949¢
!349¢
|-
| -12
| colspan="2" |612
|-
| -11
|361
| class="thl" |961
|-
| -10
| colspan="2" class="thl" |110
|-
| -9
|1059
|459
|-
| -8
| colspan="2" |808
|-
| -7
| class="thl" |557
|1157
|-
| -6
| colspan="2" class="thl" |306
|-
| -5
|55
|655
|-
| -4
| colspan="2" |1004
|-
| -3
|753
|153
|-
| -2
| colspan="2" |502
|-
| -1
|251
|851
|-
|0
| colspan="2" class="thl" |0
|-
|1
|949
|349
|-
|2
| colspan="2" class="thl" |698 ''(98)''
|-
|3
|447
|1047
|-
|4
| colspan="2" |196
|-
|5
|1145
| class="thl" |545
|-
|6
| colspan="2" class="thl" |894 ''(294)''
|-
|7
|643
|43
|-
|8
| colspan="2" class="thl" |392
|-
|9
|141
|741
|-
|10
| colspan="2" |1090
|-
|11
| class="thl" |839
|239
|-
|12
| colspan="2" |588
|}
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.

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&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&62.}}

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&62. The flat 17 and 19 are shown in parenthesis because they differ from diatonic intervals by half an octave.}}

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.
* 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 "metallic" sound as a semitone.
* 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 "sour" sound that often stands out too much outside the scales that use it structurally, but splitting minor thirds in half is my favorite technique.
* 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.
* Inframinor and ultramajor chords share the metallic quality with the quartertone. 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.
* Neutral chords are the most interesting. I find them to have a pretty narrow tuning range to be considered generally concordant, between 24edo and 31edo. 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 stretched diminished chords.

9edo

2026-07-24T11:05:40Z

Inthar: /* Music in Xedo */

{{stub}}
UNDER CONSTRUCTION

'''9edo''', or 9 equal divisions of the octave (sometimes called '''9-TET''' or '''9-tone equal temperament'''), is the [[Equal temperament|equal tuning]] featuring steps of (1200/9) = 133.333… [[Cent|cents]] exactly, 9 of which stack to the perfect octave [[2/1]].

9edo is probably best known for its very flat, muddy-sounding [[fifth]] interval and [[antidiatonic]] (2L 5s) scale, a version of the diatonic scale with inverted harmonic properties (such as major and minor intervals being flipped) when you use the circle of fifths. It does not represent small [[harmonic series|harmonics]] that well, but it has extremely accurate renditions of the just intonation intervals [[27/25]] and [[7/6]], which forms the basis of an ultra-precise [[regular temperament]] called [[ennealimmal]].

== General theory ==

=== Derivation ===
[Derivation by equal division of various key intervals or of interval qualities]

=== JI approximation ===
[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]{{Harmonics in ED|9|prime}}

=== Edostep interpretations ===
[Cover these in a list]

=== Intervals and notation ===
{| class="wikitable"
|+
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="2" |[[JI]] approximation
! colspan="3" |Notation
|-
!7-limit [[Ennealimmal]]-based (accurate)
!Coarse
!Melodic antidiatonic
!Harmonic antidiatonic
![[36edo]] notation ([[ups and downs notation|ups and downs]])
|-
|0
|0
|[[1/1]]
|
|D
|D
|D
|-
|1
|133.33
|[[27/25]]
|[[16/15]]
|E
|E
|^Eb
|-
|2
|266.67
|[[7/6]]
|[[8/7]]
|E#, Fb
|Eb, F#
|vF
|-
|3
|400
|[[63/50]]
|[[5/4]]
|F
|F
|F#
|-
|5
|533.33
|[[49/36]]
|[[4/3]], [[11/8]]
|G
|G
|^G
|-
|6
|666.67
|[[72/49]]
|[[3/2]], [[16/11]]
|A
|A
|vA
|-
|6
|800
|[[100/63]]
|[[8/5]]
|B
|B
|Bb
|-
|7
|933.33
|[[12/7]]
|[[7/4]]
|B#, Cb
|Bb, C#
|^B
|-
|6
|1066.67
|[[50/27]]
|[[15/8]]
|C
|C
|vC#
|-
|9
|1200
|[[2/1]]
|
|D
|D
|D
|}

== Tempering properties ==

=== Tempered commas ===
[List commas with S-expressions and examples of what they equate]

=== Arithmetic progressions ===

=== Notable structural chains ===
[Generator chains]

== Compositional theory ==

=== Tertian structure ===
[Describe]
{| class="wikitable"
|+Thirds in Xedo
!Quality ([[ADIN]])
|'''Mosdiatonic Quality'''
|Quality
|-
!Cents
|'''XXX'''
|XXX
|-
!Just interpretation
|'''X/X'''
|X/X
|-
!Steps
|'''X'''
|X
|}
Diatonic thirds are bolded.

=== Scales ===
[if edo is composite, link to subset edos and discuss scales of subset edos there]

[List scales and scale descriptions including structure (generators if applicable), notable intervals available, associated temperaments, relationships to other scales, and logic for derivation.]

[Explain more complex scale theory topics here]

==== Tables of scales ====

=== Harmony ===
[explain notable JI and DR chords. TRY TO PROVIDE COHERENT CHORD SYSTEMS RATHER THAN JUST LISTING RANDOM CHORDS WITH NO RELATION. Explain tunings of familiar chords]

[Explain systems of harmony here, this is how you put the chords together to make music]

[Sections such as "functional harmony", "modal harmony", etc - varies based on the edo and the personal composition style]

[Use individual voices maybe?]

==== Tables of chords ====

== Instruments ==
[Put instruments and isomorphic layouts here]

== Supersets and subsets ==

== Comparisons to other tuning systems ==

== Music in 9edo ==

== See also ==

Aberrisma

2026-07-17T02:56:04Z

Inthar: /* Other aberrismic scales */

{{Proposed}}

An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Though they are often between 20 and 55 cents, Larger and smaller sizes can function as aberrismas depending on the scale tuning. 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prejubilic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem).

Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

This is subject to change as aberrismic theory notation is updated in the future.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

2.5.7 subgroup

2026-07-17T02:55:45Z

Inthar: /* JI scales */

The '''2.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[5/4]], and [[7/4]], with the exclusion of [[3/2]] (adding which would result in the full [[7-limit]]).

Notable intervals include:
* 5/4 (the pental major third)
* 7/4 (the septimal subminor seventh)
* 7/5 (the lesser septimal tritone)
* 10/7 (the greater septimal tritone)
* 28/25 (the septimal quasi-meantone)
* 35/32 (the septimal neutral second)
* 49/40 (a neutral third)

{{Adv|The 2.5.7 subgroup includes the following odd harmonics below 256: 1, 5, 7, 25, 35, 49, 125, 175, 245.}}

An especially efficient temperament in 2.5.7 is [[Didacus]], 2.5.7[25 & 31], which is generated by a tempered 28/25 and tempers out 3136/3125, the interval between a stack of two 7/5 tritones and three 5/4 major thirds. Didacus is a 6-form cluster temperament.

[[31edo]] is a particularly accurate 2.5.7 system, but [[37edo]] is more accurate for extensions to larger subgroups such as 2.5.7.11.13.

== JI scales ==
The fundamental 2.5.7 [[aberrismic]] scale is prejubilic or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or soft citric, via adding three 50/49 steps.
* Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 2/1)

Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.

=== Interval matrices ===

==== Achiral ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmsL
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!smLsLmsLL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|625/392
|25/14
|-
!mLsLmsLLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|49/25
|-
!LsLmsLLsm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|224/125
|64/35
|-
!sLmsLLsmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|8/5
|80/49
|25/14
|-
!LmsLLsmLs
|28/25
|49/40
|'''5/4'''
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!msLLsmLsL
|35/32
|125/112
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLLsmLsLm
|50/49
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LLsmLsLms
|28/25
|784/625
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Right-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!sLmLsLmsL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!LmLsLmsLs
|28/25
|49/40
|343/250
|7/5
|196/125
|343/200
|'''7/4'''
|49/25
|-
!mLsLmsLsL
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLmsLsLm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|80/49
|64/35
|-
!sLmsLsLmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|500/343
|80/49
|25/14
|-
!LmsLsLmLs
|28/25
|49/40
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!msLsLmLsL
|35/32
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLsLmLsLm
|50/49
|8/7
|400/343
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LsLmLsLms
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Left-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmLs
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!smLsLmLsL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!mLsLmLsLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|343/200
|'''7/4'''
|49/25
|-
!LsLmLsLsm
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|64/35
|-
!sLmLsLsmL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|80/49
|25/14
|-
!LmLsLsmLs
|28/25
|49/40
|343/250
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!mLsLsmLsL
|35/32
|49/40
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLsmLsLm
|28/25
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!sLsmLsLmL
|50/49
|8/7
|400/343
|125/98
|10/7
|500/343
|80/49
|25/14
|}

== Tempered scales ==
=== [[Didacus]][6] ===
<pre>
let s = 28/25
let L = 7/4
s;s;s;s;s;L;
stack()
31@
</pre>
=== Didacus[13] ===
<pre>
let L = 35/32
let s = 50/49
s;L;s;L;s;L;s;L;s;L;s;L;s;
stack()
31@
</pre>
=== Jubilismic[6] ===
<pre>
let L = 7/4
let s = 35/32
L;L;s;L;L;s;
stack()
16@
</pre>

== Temperaments ==
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
* [[Didacus]] ({{e|25}} & {{e|31}}): Best accuracy-simplicity tradeoff among 2.5.7 temperaments. Generates 5L1s generated by ~28/25.
* [[Jubilismic]] ({{e|16}} & {{e|22}}): Less accurate, identifying 7/5 and 10/7. Has [[citric]] (4L2s, LLsLLs) MOS scales (L/s = 3/2 in 16edo, L/s = 4/3 in 22edo).
* [[Mabilic]] ({{e|16}} & {{e|25}}): Generated by a flat armotonic fifth around 672 cents.
* 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by [[Augmented (temperament)|Augmented]] edos such as [[15edo]], [[21edo]], and [[27edo]].
{{Cat|JI groups}}
* [[Sidewalk]] ({{e|21}} & {{e|46}}): Generated by a neominor third, two of which make a flattened 7/5; reaches 8/7 after 5 generator steps and 8/5 after 7 generator steps. Accurately extends to primes 11, 13, 17, and 23.

Aberrisma

2026-07-16T20:04:45Z

Inthar: /* Systematic naming */

{{Proposed}}

An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Though they are often between 20 and 55 cents, Larger and smaller sizes can function as aberrismas depending on the scale tuning. 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem).

Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

This is subject to change as aberrismic theory notation is updated in the future.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Aberrisma

2026-07-15T01:46:11Z

Inthar:

{{Proposed}}

An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Though they are often between 20 and 55 cents, Larger and smaller sizes can function as aberrismas depending on the scale tuning. 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).

This is subject to change as aberrismic theory notation is updated in the future.

Proposal: Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Aberrisma

2026-07-15T01:45:37Z

Inthar:

{{Proposed}}

An '''aberrisma''' is an interval representing some comma as an additional smaller type of melodic step. Though they are conventionally 20 to 55 cents, Larger and smaller sizes can function as aberrismas depending on the scale tuning. 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).

This is subject to change as aberrismic theory notation is updated in the future.

Proposal: Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Aberrisma

2026-07-14T00:14:31Z

Inthar: /* Systematic naming */

{{Proposed}}

An '''aberrisma''' is an interval between roughly 20 and 55 cents representing some comma as an additional smaller type of melodic step (that is, a [[diesis]]). 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).

This is subject to change as aberrismic theory notation is updated in the future.

Proposal: Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the added step size.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Aberrisma

2026-07-14T00:13:38Z

Inthar: /* Systematic naming */

{{Proposed}}

An '''aberrisma''' is an interval between roughly 20 and 55 cents representing some comma as an additional smaller type of melodic step (that is, a [[diesis]]). 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).

This is subject to change as aberrismic theory notation is updated in the future.

Proposal: Any systematic name can be used for a scale as long as the scale satisfies the deletion-MOS subcondition for the third step size.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

2.5.7 subgroup

2026-07-13T17:55:11Z

Inthar: /* JI scales */

The '''2.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[5/4]], and [[7/4]], with the exclusion of [[3/2]] (adding which would result in the full [[7-limit]]).

Notable intervals include:
* 5/4 (the pental major third)
* 7/4 (the septimal subminor seventh)
* 7/5 (the lesser septimal tritone)
* 10/7 (the greater septimal tritone)
* 28/25 (the septimal quasi-meantone)
* 35/32 (the septimal neutral second)
* 49/40 (a neutral third)

{{Adv|The 2.5.7 subgroup includes the following odd harmonics below 256: 1, 5, 7, 25, 35, 49, 125, 175, 245.}}

An especially efficient temperament in 2.5.7 is [[Didacus]], 2.5.7[25 & 31], which is generated by a tempered 28/25 and tempers out 3136/3125, the interval between a stack of two 7/5 tritones and three 5/4 major thirds. Didacus is a 6-form cluster temperament.

[[31edo]] is a particularly accurate 2.5.7 system, but [[37edo]] is more accurate for extensions to larger subgroups such as 2.5.7.11.13.

== JI scales ==
The fundamental 2.5.7 [[aberrismic]] scale is prelembic or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or soft citric, via adding three 50/49 steps.
* Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 2/1)

Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.

=== Interval matrices ===

==== Achiral ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmsL
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!smLsLmsLL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|625/392
|25/14
|-
!mLsLmsLLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|49/25
|-
!LsLmsLLsm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|224/125
|64/35
|-
!sLmsLLsmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|8/5
|80/49
|25/14
|-
!LmsLLsmLs
|28/25
|49/40
|'''5/4'''
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!msLLsmLsL
|35/32
|125/112
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLLsmLsLm
|50/49
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LLsmLsLms
|28/25
|784/625
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Right-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!sLmLsLmsL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!LmLsLmsLs
|28/25
|49/40
|343/250
|7/5
|196/125
|343/200
|'''7/4'''
|49/25
|-
!mLsLmsLsL
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLmsLsLm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|80/49
|64/35
|-
!sLmsLsLmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|500/343
|80/49
|25/14
|-
!LmsLsLmLs
|28/25
|49/40
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!msLsLmLsL
|35/32
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLsLmLsLm
|50/49
|8/7
|400/343
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LsLmLsLms
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Left-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmLs
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!smLsLmLsL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!mLsLmLsLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|343/200
|'''7/4'''
|49/25
|-
!LsLmLsLsm
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|64/35
|-
!sLmLsLsmL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|80/49
|25/14
|-
!LmLsLsmLs
|28/25
|49/40
|343/250
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!mLsLsmLsL
|35/32
|49/40
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLsmLsLm
|28/25
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!sLsmLsLmL
|50/49
|8/7
|400/343
|125/98
|10/7
|500/343
|80/49
|25/14
|}

== Tempered scales ==
=== [[Didacus]][6] ===
<pre>
let s = 28/25
let L = 7/4
s;s;s;s;s;L;
stack()
31@
</pre>
=== Didacus[13] ===
<pre>
let L = 35/32
let s = 50/49
s;L;s;L;s;L;s;L;s;L;s;L;s;
stack()
31@
</pre>
=== Jubilismic[6] ===
<pre>
let L = 7/4
let s = 35/32
L;L;s;L;L;s;
stack()
16@
</pre>

== Temperaments ==
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
* [[Didacus]] ({{e|25}} & {{e|31}}): Best accuracy-simplicity tradeoff among 2.5.7 temperaments. Generates 5L1s generated by ~28/25.
* [[Jubilismic]] ({{e|16}} & {{e|22}}): Less accurate, identifying 7/5 and 10/7. Has [[citric]] (4L2s, LLsLLs) MOS scales (L/s = 3/2 in 16edo, L/s = 4/3 in 22edo).
* [[Mabilic]] ({{e|16}} & {{e|25}}): Generated by a flat armotonic fifth around 672 cents.
* 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by [[Augmented (temperament)|Augmented]] edos such as [[15edo]], [[21edo]], and [[27edo]].
{{Cat|JI groups}}
* [[Sidewalk]] ({{e|21}} & {{e|46}}): Generated by a neominor third, two of which make a flattened 7/5; reaches 8/7 after 5 generator steps and 8/5 after 7 generator steps. Accurately extends to primes 11, 13, 17, and 23.

2.5.7 subgroup

2026-07-13T17:54:11Z

Inthar: /* JI scales */

The '''2.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[5/4]], and [[7/4]], with the exclusion of [[3/2]] (adding which would result in the full [[7-limit]]).

Notable intervals include:
* 5/4 (the pental major third)
* 7/4 (the septimal subminor seventh)
* 7/5 (the lesser septimal tritone)
* 10/7 (the greater septimal tritone)
* 28/25 (the septimal quasi-meantone)
* 35/32 (the septimal neutral second)
* 49/40 (a neutral third)

{{Adv|The 2.5.7 subgroup includes the following odd harmonics below 256: 1, 5, 7, 25, 35, 49, 125, 175, 245.}}

An especially efficient temperament in 2.5.7 is [[Didacus]], 2.5.7[25 & 31], which is generated by a tempered 28/25 and tempers out 3136/3125, the interval between a stack of two 7/5 tritones and three 5/4 major thirds. Didacus is a 6-form cluster temperament.

[[31edo]] is a particularly accurate 2.5.7 system, but [[37edo]] is more accurate for extensions to larger subgroups such as 2.5.7.11.13.

== JI scales ==
The fundamental 2.5.7 [[aberrismic]] scale is prelembic or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or citric, via adding three 50/49 steps.
* Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 2/1)
It sounds like soft [[citric]] (4L2s) with aberrismas.

Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.

=== Interval matrices ===

==== Achiral ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmsL
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!smLsLmsLL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|625/392
|25/14
|-
!mLsLmsLLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|49/25
|-
!LsLmsLLsm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|224/125
|64/35
|-
!sLmsLLsmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|8/5
|80/49
|25/14
|-
!LmsLLsmLs
|28/25
|49/40
|'''5/4'''
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!msLLsmLsL
|35/32
|125/112
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLLsmLsLm
|50/49
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LLsmLsLms
|28/25
|784/625
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Right-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!sLmLsLmsL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!LmLsLmsLs
|28/25
|49/40
|343/250
|7/5
|196/125
|343/200
|'''7/4'''
|49/25
|-
!mLsLmsLsL
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLmsLsLm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|80/49
|64/35
|-
!sLmsLsLmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|500/343
|80/49
|25/14
|-
!LmsLsLmLs
|28/25
|49/40
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!msLsLmLsL
|35/32
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLsLmLsLm
|50/49
|8/7
|400/343
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LsLmLsLms
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Left-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmLs
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!smLsLmLsL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!mLsLmLsLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|343/200
|'''7/4'''
|49/25
|-
!LsLmLsLsm
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|64/35
|-
!sLmLsLsmL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|80/49
|25/14
|-
!LmLsLsmLs
|28/25
|49/40
|343/250
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!mLsLsmLsL
|35/32
|49/40
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLsmLsLm
|28/25
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!sLsmLsLmL
|50/49
|8/7
|400/343
|125/98
|10/7
|500/343
|80/49
|25/14
|}

== Tempered scales ==
=== [[Didacus]][6] ===
<pre>
let s = 28/25
let L = 7/4
s;s;s;s;s;L;
stack()
31@
</pre>
=== Didacus[13] ===
<pre>
let L = 35/32
let s = 50/49
s;L;s;L;s;L;s;L;s;L;s;L;s;
stack()
31@
</pre>
=== Jubilismic[6] ===
<pre>
let L = 7/4
let s = 35/32
L;L;s;L;L;s;
stack()
16@
</pre>

== Temperaments ==
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
* [[Didacus]] ({{e|25}} & {{e|31}}): Best accuracy-simplicity tradeoff among 2.5.7 temperaments. Generates 5L1s generated by ~28/25.
* [[Jubilismic]] ({{e|16}} & {{e|22}}): Less accurate, identifying 7/5 and 10/7. Has [[citric]] (4L2s, LLsLLs) MOS scales (L/s = 3/2 in 16edo, L/s = 4/3 in 22edo).
* [[Mabilic]] ({{e|16}} & {{e|25}}): Generated by a flat armotonic fifth around 672 cents.
* 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by [[Augmented (temperament)|Augmented]] edos such as [[15edo]], [[21edo]], and [[27edo]].
{{Cat|JI groups}}
* [[Sidewalk]] ({{e|21}} & {{e|46}}): Generated by a neominor third, two of which make a flattened 7/5; reaches 8/7 after 5 generator steps and 8/5 after 7 generator steps. Accurately extends to primes 11, 13, 17, and 23.

2.5.7 subgroup

2026-07-13T17:53:55Z

Inthar: /* JI scales */

The '''2.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[5/4]], and [[7/4]], with the exclusion of [[3/2]] (adding which would result in the full [[7-limit]]).

Notable intervals include:
* 5/4 (the pental major third)
* 7/4 (the septimal subminor seventh)
* 7/5 (the lesser septimal tritone)
* 10/7 (the greater septimal tritone)
* 28/25 (the septimal quasi-meantone)
* 35/32 (the septimal neutral second)
* 49/40 (a neutral third)

{{Adv|The 2.5.7 subgroup includes the following odd harmonics below 256: 1, 5, 7, 25, 35, 49, 125, 175, 245.}}

An especially efficient temperament in 2.5.7 is [[Didacus]], 2.5.7[25 & 31], which is generated by a tempered 28/25 and tempers out 3136/3125, the interval between a stack of two 7/5 tritones and three 5/4 major thirds. Didacus is a 6-form cluster temperament.

[[31edo]] is a particularly accurate 2.5.7 system, but [[37edo]] is more accurate for extensions to larger subgroups such as 2.5.7.11.13.

== JI scales ==
The fundamental 2.5.7 [[aberrismic]] scale is prelembic (tentative name) or citro3s (4L2m3s, L = 28/25, m = 35/32, s = 50/49). It is the JI preimage of Lemba[6], or citric, via adding three 50/49 steps.
* Achiral: LsmLsLmsL (28/25 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Right-handed: sLmLsLmsL (50/49 8/7 5/4 7/5 10/7 8/5 7/4 25/14 2/1)
* Left-handed: LsmLsLmLs (28/25 8/7 5/4 7/5 10/7 8/5 7/4 49/25 2/1)
It sounds like soft [[citric]] (4L2s) with aberrismas.

Didacus tempering sets L = m + s. 37edo equates 49/40 to 16/13.

=== Interval matrices ===

==== Achiral ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmsL
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!smLsLmsLL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|625/392
|25/14
|-
!mLsLmsLLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|49/25
|-
!LsLmsLLsm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|224/125
|64/35
|-
!sLmsLLsmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|8/5
|80/49
|25/14
|-
!LmsLLsmLs
|28/25
|49/40
|'''5/4'''
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!msLLsmLsL
|35/32
|125/112
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLLsmLsLm
|50/49
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LLsmLsLms
|28/25
|784/625
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Right-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!sLmLsLmsL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|25/14
|-
!LmLsLmsLs
|28/25
|49/40
|343/250
|7/5
|196/125
|343/200
|'''7/4'''
|49/25
|-
!mLsLmsLsL
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLmsLsLm
|28/25
|8/7
|32/25
|7/5
|10/7
|8/5
|80/49
|64/35
|-
!sLmsLsLmL
|50/49
|8/7
|'''5/4'''
|125/98
|10/7
|500/343
|80/49
|25/14
|-
!LmsLsLmLs
|28/25
|49/40
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!msLsLmLsL
|35/32
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!sLsLmLsLm
|50/49
|8/7
|400/343
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!LsLmLsLms
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|49/25
|}

==== Left-handed ====
{| class="wikitable"
!
!1
!2
!3
!4
!5
!6
!7
!8
|-
!LsmLsLmLs
|28/25
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|'''7/4'''
|49/25
|-
!smLsLmLsL
|50/49
|125/112
|'''5/4'''
|125/98
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!mLsLmLsLs
|35/32
|49/40
|'''5/4'''
|7/5
|49/32
|343/200
|'''7/4'''
|49/25
|-
!LsLmLsLsm
|28/25
|8/7
|32/25
|7/5
|196/125
|8/5
|224/125
|64/35
|-
!sLmLsLsmL
|50/49
|8/7
|'''5/4'''
|7/5
|10/7
|8/5
|80/49
|25/14
|-
!LmLsLsmLs
|28/25
|49/40
|343/250
|7/5
|196/125
|8/5
|'''7/4'''
|49/25
|-
!mLsLsmLsL
|35/32
|49/40
|'''5/4'''
|7/5
|10/7
|'''25/16'''
|'''7/4'''
|25/14
|-
!LsLsmLsLm
|28/25
|8/7
|32/25
|64/49
|10/7
|8/5
|80/49
|64/35
|-
!sLsmLsLmL
|50/49
|8/7
|400/343
|125/98
|10/7
|500/343
|80/49
|25/14
|}

== Tempered scales ==
=== [[Didacus]][6] ===
<pre>
let s = 28/25
let L = 7/4
s;s;s;s;s;L;
stack()
31@
</pre>
=== Didacus[13] ===
<pre>
let L = 35/32
let s = 50/49
s;L;s;L;s;L;s;L;s;L;s;L;s;
stack()
31@
</pre>
=== Jubilismic[6] ===
<pre>
let L = 7/4
let s = 35/32
L;L;s;L;L;s;
stack()
16@
</pre>

== Temperaments ==
Common rank-2 temperaments in 2.5.7 (i.e. temperaments that interpret intervals as 2.5.7 JI ratios):
* [[Didacus]] ({{e|25}} & {{e|31}}): Best accuracy-simplicity tradeoff among 2.5.7 temperaments. Generates 5L1s generated by ~28/25.
* [[Jubilismic]] ({{e|16}} & {{e|22}}): Less accurate, identifying 7/5 and 10/7. Has [[citric]] (4L2s, LLsLLs) MOS scales (L/s = 3/2 in 16edo, L/s = 4/3 in 22edo).
* [[Mabilic]] ({{e|16}} & {{e|25}}): Generated by a flat armotonic fifth around 672 cents.
* 3edo.7 (6 & 15): Generates 3L3s with generator ~7/4. Supported by [[Augmented (temperament)|Augmented]] edos such as [[15edo]], [[21edo]], and [[27edo]].
{{Cat|JI groups}}
* [[Sidewalk]] ({{e|21}} & {{e|46}}): Generated by a neominor third, two of which make a flattened 7/5; reaches 8/7 after 5 generator steps and 8/5 after 7 generator steps. Accurately extends to primes 11, 13, 17, and 23.

Aberrisma

2026-07-13T17:51:01Z

Inthar: /* Other aberrismic scales */

{{Proposed}}

An '''aberrisma''' is an interval between roughly 20 and 55 cents representing some comma as an additional smaller type of melodic step (that is, a [[diesis]]). 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].

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{adv|"Almost" 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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|prelembic / citro3s (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).

This is subject to change as aberrismic theory notation is updated in the future.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 groundfault'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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 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. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Semiquartal

2026-07-12T00:49:05Z

Inthar: /* See also */

'''Semiquartal''', also known as 5L 4s, is a [[MOS]] pattern consisting of 9 steps, with pattern LLsLsLsLs. It is supported by [[14edo]], [[19edo]], and [[24edo]], among others. Like [[Mosh|mosh]], semiquartal contains many intervals almost exactly between 12edo intervals, which improves [[Xenness|xenness]]. Unlike mosh, semiquartal triads have vaguely major or minor qualities.
{| class="wikitable"
|+
!Generator
!Notes
|-
|1\5
|Collapsed.
|-
|<nowiki>||||||| 9\44</nowiki>
|Notable archy. Better as a larger scale.
|-
|<nowiki>|||||| 8\39</nowiki>
|Notable archy. Better as a larger scale.
|-
|<nowiki>||||| 7\34</nowiki>
|Good approximation of Immunity temperament.
|-
|<nowiki>|||| 6\29</nowiki>
|
|-
|<nowiki>||||| 11\53</nowiki>
|
|-
|<nowiki>||| 5\24</nowiki>
|Often acceptable approximation of [[Intergan]] by a smaller edo.
|-
|<nowiki>||||| 14\67</nowiki>
|Nearly optimal [[Intergan]].
|-
|<nowiki>|||| 9\43</nowiki>
|Often acceptable approximation of [[Intergan]] with a notable meantone.
|-
|<nowiki>||||| 13\62</nowiki>
|Notable meantone.
|-
|<nowiki>|| 4\19</nowiki>
|Smallest semiquartal edo to generate a [[diatonic fifth]]. A passable representation of meantone-range semiquartal. The [[patent val]] makes Semaphore temperament viable.
|-
|<nowiki>||| 7\33</nowiki>
|Semisixth is very close to 9/7.
|-
|<nowiki>| 3\14</nowiki>
|
|-
|<nowiki>|| 5\23</nowiki>
|Good approximation of Bug temperament.
|-
|2\9
|Equalized.
|}

== See also ==
* [[Interordinal]]
* [[Chthonic harmony]]
* [[Semifourth-generated scales]]
{{Stub}}

Hedgehog

2026-07-08T23:58:39Z

Inthar:

Conventionally, '''Hedgehog''' (2.3.5.7[22 & 36c]) tempers out the Jubilisma 50/49, splitting the octave in half into a tritone representing both 7/5 and 10/7, and the [[Sensamagic]] comma 245/243, equating a stack of two 9/7s with 5/3. If the subgroup is reduced to just 2.3.7, it tempers out the Stearnsma 118098/117649, resulting in a much more accurate temperament that can be extended in other ways.

{{UserTag|g_|Ground|7766ff|The 2.3.7 temperament is extremely useful in virtually any even edo with a decent 3, 7, and 9/7 due to its low complexity. Equating the difference between 9/7 and half an octave to 9/7 / (7/6) <nowiki>=</nowiki> 54/49 is an obvious thing to do because (9/7)^2 / (7/6) <nowiki>=</nowiki> 486/343 is 603¢. It is also obvious to equate 16/13 plus half an octave to 7/4. With this in mind, I propose two temperaments on either side of 22edo:
* The meantone version, 23-limit 36&50
* The straddle version (half-octave Porcupine), 23-limit 22&74bgh with alternate flat versions of 3, 17, and 19 at 11 generators away.
}}

{{Navbox regtemp}}
{{Cat|temperaments}}

Hedgehog

2026-07-08T23:58:05Z

Inthar:

Conventionally, '''Hedgehog''' (2.3.5.7[22 & 36c]) tempers out Jubilisma 50/49, splitting the octave in half into a tritone representing both 7/5 and 10/7, and the [[Sensamagic]] comma 245/243, equating a stack of two 9/7s with 5/3. If the subgroup is reduced to just 2.3.7, it tempers out the Stearnsma 118098/117649, resulting in a much more accurate temperament that can be extended in other ways.

{{UserTag|g_|Ground|7766ff|The 2.3.7 temperament is extremely useful in virtually any even edo with a decent 3, 7, and 9/7 due to its low complexity. Equating the difference between 9/7 and half an octave to 9/7 / (7/6) <nowiki>=</nowiki> 54/49 is an obvious thing to do because (9/7)^2 / (7/6) <nowiki>=</nowiki> 486/343 is 603¢. It is also obvious to equate 16/13 plus half an octave to 7/4. With this in mind, I propose two temperaments on either side of 22edo:
* The meantone version, 23-limit 36&50
* The straddle version (half-octave Porcupine), 23-limit 22&74bgh with alternate flat versions of 3, 17, and 19 at 11 generators away.
}}

{{Navbox regtemp}}
{{Cat|temperaments}}

Pentagoth

2026-07-07T03:04:06Z

Inthar: /* Composition theory */

'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''. The rank-3 temperament is generated by a sharp 5/4 (~390¢) and half of 7/5 (~287¢ * 2 = ~574¢), interpreted as a sharp 20/17, a roughly in-tune 13/11, and a flat 19/16. 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~103¢ semitone between the two generators. Stacking the two generators results in a flat fifth of ~677¢, used in the 17:20:25 triad.

'''Sidewalk''' is a rank-2 2.5.7.17...[21 & 25] temperament that supports Pentagoth; it is generated by a neominor third, two of which stack to a flat 7/5.

== 2.5.7: an introduction ==

{{UserTag|g_|Ground|7766ff|I've always been interested in the [[2.5.7 subgroup]] and its extensions. Prime 3 is so central to how we tend to understand harmony that removing it is always interesting, and 5 and 7 are the next two simplest, thus the best alternatives to create a new harmonic system.

Here's a list of some of the best temperaments with their mappings of 5 and 7:
* {{e|6}} & {{e|25}}: [[Didacus]] 2 5
* {{e|16}} & {{e|21}}: Llywelyn 7 -1
* {{e|16}} & {{e|25}}: [[Mabilic]] 3 -5
* {{e|21}} & {{e|25}}: Sidewalk -7 -5
* {{e|21}} & {{e|31}}: [[Miracle]] -7 -2
* {{e|15}} & {{e|16}}: Rainy 5 -3
* {{e|15}} & {{e|22}}: [[Porcupine]] -5 6
* {{e|16}} & {{e|27}}: Kumonga restriction -9 -1
All of these are pretty well-established names, except for Sidewalk, which I came up with.
}}

{{UserTag|SS|Vector|cebaff|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error. 2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above). Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval.
}}

== An asidewalk ==

{{UserTag|g_|Ground|7766ff|'''Sidewalk''' is likely the least well known of the basic 2.5.7 temperaments, given that I had a chance to coin the name for its comma, 823543/800000. It's one of the few temperaments with a generator in the neominor third region, half of 7/5 in this case.
** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.

The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.
}}

== Pentagoth temperament ==

{{UserTag|g_|Ground|7766ff|The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e|37}}, {{e|46}}, and 67.

The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half. I've been referring to "Pentagoth triads" that rely on this exact type of tempering even more often than I refer to a particular temperament.
}}

{| class="wikitable"
|+2.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right
!Gens
!-3
!-2
!-1
!0
!1
!2
!3
|-
!6
|
|
|
|
|
|49/46
|
|-
!5
|
|
|
|
|49/34
|
|26/23
|-
!4
|
|
|
|49/25
|28/23 49/40
|26/17 35/23 49/32
|44/23
|-
!3
|
|
|
|28/17 38/23
|26/25 35/34
|22/17 13/10 49/38
|13/8
|-
!2
|
|
|28/25 19/17 49/44
|32/23 7/5
|40/23 7/4 44/25
|11/10 25/23 35/32
|11/8 26/19
|-
!1
|
|38/25
|32/17 19/10 49/26
|20/17 13/11 19/16
|28/19 52/35 34/23 25/17
|13/7 35/19
|22/19
|-
!0
|
|32/25 14/11
|8/5 35/22
|1/1
|5/4 44/35
|11/7 25/16
|
|-
!-1
|19/11
|14/13 38/35
|34/25 19/14 23/17 35/26
|32/19 22/13 17/10
|20/19 52/49 17/16
|25/19
|
|-
!-2
|16/11 19/13
|64/35 20/11 46/25
|8/7 23/20 25/22
|10/7 23/16
|88/49 34/19 25/14
|
|
|-
!-3
|16/13
|20/13 17/11 76/49
|68/35 25/13
|17/14 23/19
|
|
|
|-
!-4
|23/22
|64/49 17/13 46/35
|80/49 23/14
|50/49
|
|
|
|-
!-5
|23/13
|
|68/49
|
|
|
|
|-
!-6
|
|92/49
|
|
|
|
|
|}

== Tempering process ==

{{UserTag|g_|Ground|7766ff|All temperaments are 9 & 16 & 21 unless otherwise mentioned.
* Given a 2.5.7 temperament where 7/5 is split in half, 5/4 * sqrt(7/5) makes a flat fifth like 25/17. The supraminor third 49/40 is also close to 17/14. Equating these pairs tempers out 2023/2000, which I've decided to call the Pentagoth comma due to being the first and most obvious step.
** 2023/2000 -> 2.5.7.17; CWE 388.049¢ 289.369¢, CE 390.556¢ 288.428¢.
* Then, 7/5 will be reasonably biased flat due to being (20/17)^2, pulling it closer to 32/23. Also, the same supraminor third is close to 28/23 as well. This tempers out the 2.5.7.23 comma 161/160.
** 161/160 -> 2.5.7.17.23; CWE 387.534¢ 288.767¢, CE 390.950¢ 287.244¢.
* After that, things get messier. 13/11 can be easily equated to half of 7/5 by tempering out 847/845, and there are no better options than to do the same with 19/16, tempering out 1805/1792, even though this makes it very flat and the least accurate prime in the no-3 23-limit extension. It makes up for low accuracy with extremely low complexity, and makes 19/14 the octave complement of 25/17.
** 847/845, 1805/1792 -> 2.5.7.13/11.17.19.23; CWE 386.133¢ 289.374¢, CE 390.581¢ 288.351¢.
* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.
** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.
* So what do you do to add 3 and make it full 23-limit? It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).
** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.
** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.
** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.
** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.
}}

== Sidewalk again ==

{{UserTag|g_|Ground|7766ff|As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.

Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.
}}

== More commas in 2.5.7.17 ==

{{UserTag|g_|Ground|7766ff|Vengeance is a very efficient temperament. 34edo's tuning is very good for optimising 17:20:25, but adding 7 becomes difficult because 34edo almost perfectly straddles it. On one side is 25edo, mapping 7 to -5 generators. While the original discussions of this temperament were going on, I gravitated towards 25edo even though it wasn't optimal, which I now know is because of Pentagoth. On the other side is 43edo, mapping 7 to -14 generators. It is known for Amavil and Cohemimabila (the temperament splitting its grave fifth in half), both of which I independently discovered while researching 43edo and this subgroup a while before learning of their names.

In adding 7 to Vengeance to get 17/14 and such, the best temperament might be Cohemimabila, which is represented well by edos 43, 68, and 25, but not 34. 2.5.7.17 Cohemimabila is Didacus plus 4913/4900, which is like an analogue to the Pentagoth comma. Combining 4913/4900 with Sidewalk results in Pentagoth with 343/340 or Pentagoth Sidewalk. Pentagoth Didacus splits the whole tone generator in half.

* <nowiki>Vengeance comma: 78608/78125 = 16/5 / (25/17)^3</nowiki>
* <nowiki>Sidewalk comma: 823543/800000 = 7/5 / (400/343)^2</nowiki>
* <nowiki>Pentagoth comma: 2023/2000 = 7/5 / (20/17)^2</nowiki>
* <nowiki>UNNAMED: 4913/4900 = (17/10)^2 * 17/14 / (7/2)</nowiki>
* <nowiki>UNNAMED: 343/340 = 7/5 / (8/7 * 17/14) = 20/17 / (400/343)</nowiki>
}}

== Composition theory ==

{{UserTag|g_|Ground|7766ff|Pentagoth temperament represents a generalization of the '''pentagoth triad''' which includes a pental major third and neominor third. This triad is the center of Pentagoth composition.

This section is incomplete.
}}

{{Navbox regtemp}}
{{Cat|Temperaments}}

Hibernal

2026-07-07T03:03:01Z

Inthar:

'''Hibernal''' is a set of temperaments based on stacking a 25/21 minor third and 19/15 major third. 25 and 15 are both divisible by 5, resulting in 63:75:95, which is also delta-rational +3+5. The name is the adjective form of "winter" after the 32edo song that uses it extensively, Winter's Mortal Hope.

{{UserTag|g_|Ground|7766ff|I used Hibernal in 32edo for a while because I loved the melodic properties of its blackdye, without knowing why it sounded more concordant than expected. I initially guessed it was 19/16 and 14/11, but I tried slight variations of the minor and major triads and realized that I was hearing the well of 19/15 instead.

For the tempering process, I will use [[erac]]s to show error accumulation. On a basic level, 25/21 * 19/15 is equated to >3/2, tempering out <190/189. This >3 is in the Archy range, so <<64/63 can be tempered out to add >7. At this point, tempering becomes less obvious. For 25/21 to be accurate, >>25 would be required to offset the very sharp Archy >>21. Assuming >>25 splits into (>5)^2, an accurate 19/15 would need an intolerably sharp >>19 to offset >>15. In my numerous attempts to turn Hibernal into a rank-3 2.3.5.7.19 temperament, I came to the conclusion that I have to make it straddle-5 to make any sense, with a roughly accurate 5 and an extra sharp >>5, equated to 19/15. It may look silly, 32edo actually works that way in my experience. This is my definitive 32edo temperament. I use the Oceanfront mapping of 13/10, resulting in a straddled 13 as well.}}
{| class="wikitable"
|+19-limit Hibernal Stradle-5-13, JI tuning in cents (generators 95/63 and 19/15), primes highlighted
|
| -2
| -1
|0
|1
|2
|-
| -4
|1137
|346
|756
|1165
|374
|-
| -3
|648
|1057
|267
|676
|1085
|-
| -2
|159
|569
| class="thl" |'''978'''
|187
|596
|-
| -1
|870
|80
|489
|898
| class="thl" |'''107'''
|-
|0
| class="thl" |'''382'''
|791
| class="thl" |'''0'''
| class="thl" |''409''
|818
|-
|1
|1093
| class="thl" |'''302'''
| class="thl" |'''711'''
|1120
|330
|-
|2
|604
|1013
|222
|631
|1041
|-
|3
|115
|524
|933
|143
| class="thl" |'''552'''
|-
|4
| class="thl" |'''826'''
|35
|444
| class="thl" |''854''
|63
|}

A simple rank-2 reduction uses Superpyth mappings for 25/21 and 19/15 in place of 6/5 and 5/4, seen in edos like 86, 59, 32, and 69. Other useful edos may include 44, 47, 56, 64, 71, 76, 79, 83, 91, 100, 106, 108, and 111.

{{Cat|Temperaments}}

Hibernal

2026-07-07T03:02:16Z

Inthar:

'''Hibernal''' is a set of temperaments based on stacking a 25/21 minor third and 19/15 major third. 25 and 15 are both divisible by 5, resulting in 63:75:95, which is also delta-rational +3+5. The name is the adjective form of "winter" after the 32edo song that uses it extensively, Winter's Mortal Hope.

{{UserTag|g_|Ground|7766ff|I used Hibernal in 32edo for a while because I loved the melodic properties of its blackdye, without knowing why it sounded more concordant than expected. I initially guessed it was 19/16 and 14/11, but I tried slight variations of the minor and major triads and realized that I was hearing the well of 19/15 instead.

For the tempering process, I will use eracs to show error accumulation. On a basic level, 25/21 * 19/15 is equated to >3/2, tempering out <190/189. This >3 is in the Archy range, so <<64/63 can be tempered out to add >7. At this point, tempering becomes less obvious. For 25/21 to be accurate, >>25 would be required to offset the very sharp Archy >>21. Assuming >>25 splits into (>5)^2, an accurate 19/15 would need an intolerably sharp >>19 to offset >>15. In my numerous attempts to turn Hibernal into a rank-3 2.3.5.7.19 temperament, I came to the conclusion that I have to make it straddle-5 to make any sense, with a roughly accurate 5 and an extra sharp >>5, equated to 19/15. It may look silly, 32edo actually works that way in my experience. This is my definitive 32edo temperament. I use the Oceanfront mapping of 13/10, resulting in a straddled 13 as well.}}
{| class="wikitable"
|+19-limit Hibernal Stradle-5-13, JI tuning in cents (generators 95/63 and 19/15), primes highlighted
|
| -2
| -1
|0
|1
|2
|-
| -4
|1137
|346
|756
|1165
|374
|-
| -3
|648
|1057
|267
|676
|1085
|-
| -2
|159
|569
| class="thl" |'''978'''
|187
|596
|-
| -1
|870
|80
|489
|898
| class="thl" |'''107'''
|-
|0
| class="thl" |'''382'''
|791
| class="thl" |'''0'''
| class="thl" |''409''
|818
|-
|1
|1093
| class="thl" |'''302'''
| class="thl" |'''711'''
|1120
|330
|-
|2
|604
|1013
|222
|631
|1041
|-
|3
|115
|524
|933
|143
| class="thl" |'''552'''
|-
|4
| class="thl" |'''826'''
|35
|444
| class="thl" |''854''
|63
|}

A simple rank-2 reduction uses Superpyth mappings for 25/21 and 19/15 in place of 6/5 and 5/4, seen in edos like 86, 59, 32, and 69. Other useful edos may include 44, 47, 56, 64, 71, 76, 79, 83, 91, 100, 106, 108, and 111.

{{Cat|Temperaments}}

Stretching and compression

2026-07-06T03:30:33Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (MOS diatonic LLsLLLs -> machinoid LLLLLs) or making it smaller (MOS diatonic LLsLLLs -> smitonic LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==

Stretching and compression

2026-07-06T03:30:09Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how [[oneirotonic]] is obtained from MOS diatonic (LLsLLLs -> LLsLsLLs), and similarly [[pinedye]] (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==

Stretching and compression

2026-07-06T03:29:41Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how oneirotonic is obtained from diatonic (LLsLLLs -> LLsLsLLs), and similarly pinedye (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

== Linear stretching and compression ==

Stretching and compression

2026-07-06T03:29:23Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how oneirotonic is obtained from diatonic (LLsLLLs -> LLsLsLLs), and similarly pinedye (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs).

== Chords ==
Stretched versions of chords can be indicated by using the prefix ''tens-'', and compressed versions of chords can be indicated by using the prefix ''tract-''. (Example: [[oneirotonic]] tract-diatonic chords such as the tract-major triad which is 0s-M2s-M4s in TAMNAMS notation)

Stretching and compression

2026-07-06T02:52:50Z

Inthar:

{{stub}}
'''Stretching and compression''' are (usually logarithmic) operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how oneirotonic is obtained from diatonic (LLsLLLs -> LLsLsLLs), and similarly pinedye (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs).

Stretching and compression

2026-07-06T02:52:24Z

Inthar: /* Scales */

{{stub}}
'''Stretching and compression''' are operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how oneirotonic is obtained from diatonic (LLsLLLs -> LLsLsLLs), and similarly pinedye (LLmLLLm -> LLmLLsLm). A scale pattern may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs).

Stretching and compression

2026-07-06T02:51:39Z

Inthar:

{{stub}}
'''Stretching and compression''' are operations that can be done on melodies, scales, or chords. Stretching can be called '''tens''' and compression can be called '''tract'''.

== Scales ==
A scale pattern can be compressed by inserting a(n arbitrary but usually small) step size; this is how oneirotonic is obtained from diatonic (LLsLLLs -> LLsLsLLs). Or it may be stretched by removing a step (LLsLLLs -> LLLLLs) or making it smaller (LLsLLLs -> LLsLsLs)