Ground's composition theory

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This page is written from the perspective of User:Ground and may feature proposed concepts.

This article is about composition and the practical considerations of tuning. I always have more content to add and revise. It has an emphasis on my other theories, aberrismic and straddle-prime, which may be useful to know.

It should be kept in mind that my style is generally unrestrained and chaotic, and I am in the mesotempered category. My approach to writing chords is loosely based on non-functional jazz harmony. They usually outline and support the melody and have few defined cadences involving more than two chords; the most common is probably iv → i. As a result, I don't have much to say about harmonic functions, something which is also in dire need of thorough documentation.

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. 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

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.

KC

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.

9-note diatonic analogue

ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.

When it comes to extending the diatonic scale's circle of fifths as seen in ILD, there's no alteration more important than the modes of the melodic minor scale, which I'll be referring to together as melodiatonic. It's the only non-diatonic scale that's possible to construct by making a single bubble deviation from diatonic, and it makes sense looking at the circle of fifths. Assume that some altered diatonic scale must contain only L and s steps to preserve as much step logic as possible. To do this, only ordinals following a semitone may be sharpened (C and F when the unaltered notes are all naturals), and only ordinals preceding a semitone may be flattened (B and E). Sharpening F or flattening B just changes the key signature and the result is still diatonic. Sharpening C or flattening E results in melodiatonic. I have found melodiatonic to be a remarkably reliable way to make diatonic tunings (including 12edo) sound exotic yet familiar enough to hit the same emotional beats diatonic can provide. After diatonic, it has the smallest span across the chain of fifths without duplicating a diatonic ordinal, meaning that it contains no pairs of notes that are 7 generators apart. Just like how diatonic is the framework for all quasi-diatonic scales, melodiatonic can be used as the framework for certain alterations of those quasi-diatonic scales. Unconventional uses of melodiatonic I can think of off the top of my head are parts of Undertale's Heartache in Bb Dorian b2 and the main theme of Animusic's Harmonic Voltage in C Mixolydian b6.

Quasi-Diatygic Scales

Altered Blackdye	Altered Diasem
10/9 9/8 6/5 5/4 4/3 27/20 3/2 8/5 5/3 16/9 9/5 2/1	9/8 8/7 7/6 9/7 21/16 4/3 3/2 32/21 14/9 12/7 7/4 16/9 2/1

Diatonic Tertian Triads in D Gadygic

Root	Triads
D	minor, Major
E	dim., minor
F	Major
F♯	dim.
G	minor, Major
A	minor
B♭	Major
B	dim., minor
C	Major

The gaps in the fifth chain can be filled to result in a contiguous generator chain of 9 notes rather than 7, allowing some modes to freely switch between major and minor thirds and sixths. This is probably the earliest music theory I've ever consciously developed, dating to 2014 or so (example). It has been scale #1 on the 9-note page of AllTheScales for at least as long, and I tentatively propose calling the scale diatygic using the seemingly arbitrary -ygic suffix from AllTheScales since it seems like a term I would have chosen at the time. It has stood the test of time and I find it to be the natural extension of the diatonic framework that doesn't require microtuning, so it remains my favorite 12edo-compatible scale. The scale word for the symmetrical mode is LdcdLdcdL, with d and c for diatonic and chromatic semitones. It is called Gadygic by AllTheScales. I use plenty of extended interval chains when I compose, but diatygic is intended to be a proper albitonic scale with simple self-contained rules, prescriptive rather than descriptive in the sense that it is the new template to be extended and deviated from. For some reason, the first examples of this scale I can think of (besides my own music) all come from the brony fandom. Morning in Ponyville, Awesome as I Wanna Be, and Awoken. It should leave no guesswork as to what I was doing in 2014. All of them are coincidentally in the key of D.

I derived diatygic from my own observation of my favorite music, intuition, and the methods above, including that 9 notes is the closest thing to a MOS between 7 and 12 notes. For one thing, MOS scales with generators approaching half an octave will have sizes that are successive odd numbers; the diatonic range jumps from 7 to 12, but the adjacent antidiatonic range follows 7 with 9. Stacking 9 fifths results in a scale with groups of 3 semitones in a row, which is substantially easier to navigate than the 10-note "supermode" in Temperley 2011, with a group of 5 semitones. I've found it more useful to use various modes of diatygic than to add additional notes, since needing more than two non-diatonic notes at a time is very rare. This is so that the scale can contain the diatonic modes Aeolian to Ionian as subsets, whereas diatygic only manages Aeolian to Mixolydian. I've never used the major seventh degree much myself, despite its high apparent importance in Western music. Even back when I used major modes more than minor, I preferred Mixolydian to Ionian. It's about as important to me as the minor second degree, which is easy to achieve through using modes other than the symmetrical one.

This extension of diatonic also works for quasi-diatonic scales. Pental blackdye is two chains of 5 of 3/2 separated by 10/9. Extending both of those chains by 2 is analogous to what I'm doing to diatonic, but separating them by 81/80 instead makes 5L2m7s. Separating by 10/9 results in a quaternary scale that is very nearly ternary; temper out the Schisma and you get 3L6m5s with a non-standard step order to what I've established in my aberrismic theory. The altered pental blackdye is almost the 3sA Aeolian mode with 5/4 and 5/3 added. The retroversion of this can be obtained by adding 6/5 and 8/5 to 3sG Mixolydian, but it's inferior because it lacks the perfect fifth over 6/5. The one additional change is 81/80 to 10/9 to fix the wolf fifth from 9/8 to 5/3. All 12 notes are required to make a chain of pure 3/2 sometimes altered by 81/80 just as blackdye does, except 4/3-16/9 and 27/20-9/5 are redundant. The extra note is kept to keep the scale to a minimum of four unique step sizes, similar to what is done with blackdye. A similar process may be done to septal diasem.

One of my favorite parts of the scale is a major third on D, a minor third on E, and a major third on F, using notes from D Gadygic.

Interval Matrix of Gadygic, DR 3:4:5 Meantone

	0	1	2	3	4	5	6	7	8	(9)
0.	0	194.6	308.2	389.1	502.7	697.3	810.9	891.8	1005.4	1200
194.6	0	113.6	194.6	308.2	502.7	616.3	697.3	810.9	1005.4	1200
308.2	0	80.9	194.6	389.1	502.7	583.7	697.3	891.8	1086.4	1200
389.1	0	113.6	308.2	421.8	502.7	616.3	810.9	1005.4	1119.1	1200
502.7	0	194.6	308.2	389.1	502.7	697.3	891.8	1005.4	1086.4	1200
697.3	0	113.6	194.6	308.2	502.7	697.3	810.9	891.8	1005.4	1200
810.9	0	80.9	194.6	389.1	583.7	697.3	778.2	891.8	1086.4	1200
891.8	0	113.6	308.2	502.7	616.3	697.3	810.9	1005.4	1119.1	1200
1005.4	0	194.6	389.1	502.7	583.7	697.3	891.8	1005.4	1086.4	1200

Adaptive diatonic modification system

This section is incomplete.

In order to make modified diatonic scales easier to use and quickly understand the intervals I use in my head, I have a proposal for an interval scheme based on splitting diatonic or quasi-diatonic intervals, somewhat similar to ADIN.

First specify which is the default diatonic or quasi-diatonic quality. This is usually pental (4:5:6), pyth, or septal (6:7:9).

Neutral seconds are about 3/7 to 4/7 of the minor third.

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:

1. Which edos do I like enough due to their combination of structure and approximation of a few target intervals, usually including LCJI?
2. 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?

A selection of large edos I use

Edo	Justification
27x3 = 81	81 is an 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.
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.
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.
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.
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.

Edo multiplication

I use midi channels interpreted by custom tuning scripts to alter notes by fractions of a step, allowing much larger edos than is otherwise practical in a piano roll without sacrificing the freedom and simplicity of the base edo to be multiplied. This creates an interesting problem for tuning selection because both the base and any multiples are necessary to consider. I prefer x6 multiplication because it's the superset of x2 and x3, after which it becomes very complicated to juggle all the different channels.

Example tier list of 37edo multiples:

• Base (37): S (One of a kind.)
• x2 (74): B (Nothing special besides the two meantone diasems and various Diashismic straddle temperaments. A pretty straightforward extension of 37's interval set.)
• x3 (111): A (Very good at approximating the 17-limit and containing a few temperaments I like. More of a self-contained microtemperament, very different from 74.)
• x6 (222): D (Has extra approximations and generators, but too cumbersome to do much.)

This is a top-tier edo multiplication setup when you weight these based on their expected frequency, which is estimated based on existing projects:

• Base: remainder
• x2: 12%
• x3: 8%
• x6: 4%

When using edo multiples, it's important to keep better track of the intervals in use and minimize the effect of the added complexity to compose efficiently. I do this by switching between discrete scales in the multiples (rather than flowing freely between structures and selecting all notes by ear as in the base edo) and by primarily using notes of the base edo as the tonic to minimize the versions of each scale to keep track of when I modulate. No matter what, the things I write will always be filtered through the base edo. It's natural that the extra precision and features of multiples won't be necessary most of the time, and the freedom of a reasonably sized base edo will outweigh them.

Anchor concordances

We conventionally make the distinction between consonance, which is dependent on musical context, and concordance, which is not. This gives the false impression that concordance does not vary significantly between people, and based on register and timbre. Dyadic harmonic entropy is the most well-known measure of concordance, and it works pretty well for me, but not everyone. Triadic harmonic entropy fits less well because I've personally noticed that having at least one interval in a chord tuned close to LCJI sounds more concordant than balancing the errors of multiple LCJI intervals. I refer to this special interval as an anchor concordance because it stabilizes the whole chord. This way of optimization is opposed to delta-rational, which I also find useful.

For example, the patent 3/2 in 25 and 28edo is far off, but their representations of 4:5:6 mask the beating with an accurate 5/4. I prefer to have anchor concordances be 3/2, thirds, or sometimes sixths because they show up the most often in tertian chords, and major thirds in particular. My theory is that this is because the major third range has stronger concordance wells than minor thirds, which have something close to a plateau on the harmonic entropy curve. My favorites to have are 5/4, 9/7, and 13/10.

I consider 2/1, 3/2, and 5/4 to be the most important anchor concordances. If any two of them are approximated within a few cents, the tuning system will be capable of comfortably stable harmony. This is why I believe 12edo works so well, and why 25, 28, and 37edo are more capable than people expect. The original purpose of straddle primes was to account for the extreme variability of the reasonable tunings of 3/2 in 37edo and stacking both to approximate the pythagorean chain.

In 37edo, both 4:5:>6 and 4:5:<<6 (a very close approximation of 1/(40:32:27)) serve as reasonable consonances. I've never been able to observe different functions between them. The same applies to the different tunings of 6:8:9, besides <<6:8:<<<<9 = 6:>>8:<<9 which just doesn't sound good. This is most obviously useful when an accurate 3 isn't available, but it's also beneficial for altering intervals to follow a melody with optimal step sizes for my compositional goal:

• Using the very flat <<3 results in >>16/15, quite audibly wider while keeping some semblance of 4:5:6.
• <<6/5 is found in the Pentagoth triad, which uses the exaggerated quality of the minor third and the stability of the major third.

Compare the different optimizations of 2.3.5 Meantone which I have noticed sound better than those immediately around them:

• ~69edo: DR 4:5:6, 10:12:15
• ~31edo: 5/4 anchor concordance
• ~74edo: DR 3:4:5
• ~67edo: 1/6-comma (equal error on 9 and 5, around where 3 becomes truly stackable by my standards)

Quartertone composition

Island Rastmic with Straddle Primes

Gens	949¢	349¢	Proposed Names (semi- | hemi-)
-12	612		Diminished Fifth
-11	361	961	"Semififth"	"Hemidim." Seventh
-10	110		Minor Second
-9	1059	459	"Semithirteenth"	"Hemidim." Fourth
-8	808		Minor Sixth
-7	557	1157	"Semiseventh"	"Hemidim." Octave
-6	306		Minor Third
-5	55	655	"Semisecond"	"Hemidim." Fifth
-4	1004		Minor Seventh
-3	753	153	Semitenth	Neutral Second
-2	502		Perfect Fourth
-1	251	851	Semifourth	Neutral Sixth
0	0		Unison
1	949	349	Semitwelfth	Neutral Third
2	698 (98)		Perfect Fifth
3	447	1047	Semisixth	Neutral Seventh
4	196		Major Second
5	1145	545	"Semifourteenth"	"Hemiaug." Fourth
6	894 (294)		Major Sixth
7	643	43	"Semininth"	"Hemiaug." Unison
8	392		Major Third
9	141	741	"Semithird"	"Hemiaug." Fifth
10	1090		Major Seventh
11	839	239	"Semieleventh"	"Hemiaug." Second
12	588		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. 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. 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 categorization

Thirds categorized as a fraction of a diatonic fifth.

Fraction f	*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
59/140	59/240	289.0	295.0	295.8	303.4	Grave Minor
61/140	61/240	298.8	305.0	305.9	313.7	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
79/140	79/240	386.9	395.0	396.1	406.3	Acute Major
81/140	81/240	396.7	405.0	406.1	416.6	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. 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 while remaining somewhat xenharmonic. My favorite chords in this region are based on 18:23:27, although this often called shrub- rather than neo-.
• Grave minor and acute major chords are approximately the ones found in 12edo, around 16:19:24. I consider these to be a separate category less often than any other, but this presence of this otonal chord justifies them. A grave minor third is also characteristic of Hibernal.
• Classic minor and major chords are in the vicinity of 4:5:6 and 10:12:15, so they may also be called pental. They are considered to be the default in xenharmonic circles, and have been used in Western music for hundreds of years at least, largely falling out of consideration once 12edo became the standard. I think it's always good to have these to some extent because they provide a consonance that can anchor everything else.
• 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 because of the complexity of 18:22:27, but because this mix of qualities makes them more complicated. They resemble diminished seventh chords in the way they connect anything to anything.
  • I call this technique a "neutral substitution". It involves replacing a tertian chord with a neutral one in order to modulate. The easiest way to do this is line up a neutral chord with the root or fifth of where a diminished chord would be, then use whichever of the two is changed to alter the key.

Grave fifth chords and mainline fifths

Successive fifths from 22/15 to 40/27,
thirds between logarithmic 12/35 and 23/35 of the fifth

D	N					<3/7f	<1/2f	>1/2f	>4/7f	Fifth	Fifth Complement
15		18		19	22		315.6		409.2	663.0		347.4		253.8
17		20	21		25		281.4	365.8		667.7		386.3	301.8
19	22	23		24	28	253.8	330.8		404.4	671.3	417.5	340.6		266.9
21	24	25	26	27	31	231.2	301.8	369.7	435.1	674.3	443.1	372.4	304.5	239.2
23		27	28	29	34		277.6	340.6	401.3	676.7		399.1	336.1	275.4
25	29	30	31	32	37	256.9	315.6	372.4	427.4	678.7	421.8	363.1	306.3	251.3
27	31	32	33	34	40	239.2	294.1	347.4	399.1	680.4	441.3	386.3	333.0	281.4

I use a similar system for antidiatonic tertian harmony, but there's no set JI approximation for a flat fifth, so I use multiple to get the simplest ratios possible. Pentagoth equates many similar intervals in this table.

I refer to the range from about 23\40 to 25\42 as mainline fifths. The full diatonic range includes other fifths that generate scales I have found over time to resemble the usual understanding of diatonic less than they resemble 7 or 5edo, and resemble 3/2 less than sqrt(20/9) and sqrt(16/7). The 25\42 boundary is clearer to me because I prefer very hard scales to very soft ones and thus I have more experience. But conveniently, these are the points where an alternate diatonic fifth can be generated at a low complexity. Analogously, I could refer to the range from about 11\20 to 17\30 as mainline antidiatonic fifths based on my personal experience.

I've found 25 and 39edo to be the two regions of the armotonic tuning range that sound better than others. 29edo sounds more pure than both, and is the basis for an excellent 29&67 straddle temperament I've called Gravel, but it involves the 15:19:22 chord with a fifth at my limit for resembling 3/2 which does not generate armotonic. 2.15.11.19 9&56 also generates this, and the subgroup can be extended to 2.3.5.11.19 while hardly changing the edo list. 11:13:16 and 13:16:19 are proper subfifth triads and are not counted in the range. They are also nearly retroversions of each other.

25edo represents the two otonal chords on the table with 5/4, with flatter fifths like 34 and 43edo being better for 17:20:25 and sharper fifths like 41 and 57edo being better for 27:32:40. Both of these are found in Pentagoth, so I propose calling this range pentagothic. 43edo is especially notable for how well it also approximates 17:21:25.

39edo represents 21:26:31, which is notably a +1+1 chord, and 21:25:31, very close to its retroversion. I propose calling this range mavilic after Erv Wilson's original Meta-Mavila, the "proper" tuning range for Mavila. Mavila possibly has more variants of its name than any other temperament. I'm genuinely surprised that Mavilic isn't a temperament name. Otomavila is my temperament using these chords.

For the most part, antidiatonic tertian harmony works like diatonic tertian harmony, but with the reversed qualities of the antidiatonic scale. The pentagothic neutral triad generates Cohemimabila temperament and has a similar function to the neutral triad explained above.

Quasi-diatonic scales

Blackdye and diasem JI tunings, sorted from flat to sharp minor third, denominators up to 20

Tertian Triad	16:21:24	20:23:30	10:13:15	6:7:9 (14:18:21)	18:23:27	16:19:24	4:5:6 (10:12:15)	14:17:21	18:22:27
Subgroup	2.3.7	2.3.23/5	2.3.13/5	2.3.7	2.3.23	2.3.19	2.3.5	2.3.17/7	2.3.11
Name	"Septal"	?	?	"Septal"	"Eridian"	"Novemdecal"	Pental	?	"Undecal"
Aberrisma	28/27	640/621	416/405	64/63	736/729	513/512	81/80	459/448	33/32
Quasi- Aeolian	9/8 8/7 9/7 4/3 3/2 32/21 12/7 16/9 2/1 (swap m/s)	9/8 23/20 207/160 4/3 3/2 23/15 69/40 16/9 2/1 (swap m/s)	9/8 15/13 135/104 4/3 3/2 20/13 45/26 16/9 2/1 (swap m/s)	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	33/32 9/8 11/9 4/3 11/8 3/2 33/20 16/9 11/6 2/1
Similar Unlisted Aberrismas	17/13: 1088/1053 (swap m/s)		17/11: 1408/1377 19/11: 304/297		25: 2048/2025 11/7: 896/891 13/11: 352/351 17/5: 136/135 23/13: 208/207	25/7: 225/224 17: 4131/4096 19/5: 1216/1215 (diasem)		23/7: 189/184 23/19: 621/608	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 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. 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.

The default set of qualities to consider is septal and pental due to their simplicity, possibly with a Pythagorean diatonic scale in the middle as found in Aberschismic edos. But I've found that I particularly like shifting both septal and pental towards neutral, to eridian and exaggerated pental. This is probably because it puts both of them about an equal distance from diatonic when using a prime 3 within a few cents of just. Eridian provides a more versatile sound and pental sounds more interesting while still working about the same. In some edos that do this such as 48 and 86, the pental tuning is Negri, meaning that semiquartal is also available.

Cursed Eridian Scales

Standard (23/18)(27/23)	Wolf → Perfect (23/18)(32/27)	the other one (81/64)(27/23)
9/8 27/23 243/184 4/3 3/2 36/23 81/46 16/9 2/1	398688256/387420489 67712/59049 32/27 243/184 2166784/1594323 368/243 36/23 59049/33856 11776/6561 2/1	4782969/4333568 27/23 129140163/99672064 2944/2187 2187/1472 128/81 59049/33856 8667136/4782969 2/1

27/23 is close enough to 32/27 that it can be replaced and result in another 2.3.23 blackdye scale with 32/27*23/18 = 368/243 triads. The fifth is the same as the existing Eridian wolf fifth, which is barely diatonic in JI. I might call this hiberidian because it applies the principle of Hibernal to the eridian subgroup. 82edo is the near-perfect 2.3.23 tuning, but 20 and 37edo have a close approximation of the scale despite not being good for the subgroup. Switching thirds to 32/27 or 81/64 to remain in the subgroup with a wolf fifth as the main one can be applied to other scales in the table.

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

Aberrismic details

Note that this section is incomplete.

As described in the Monarch article, I like to tune pental blackdye (3sA Aeolian: tempered 81/80 9/8 6/5 4/3 27/20 3/2 8/5 16/9 9/5 2/1) with a sharp 3 and thus inflated 81/80. When extreme, more interesting intervals show up along with new problems. The first is what is seen in 11-limit Superpyth and Porcupine (55/54), turning 27/20 into 11/8, although it could be interpreted as any other superfourth. I like this interval a lot because the 1/1-9/8-6/5-27/20 tetrachord is a natural-sounding stretched version of the diatonic minor tetrachord. In fact, it may sound better melodically to use the superfourth rather than the perfect fourth to avoid the very deflated 10/9 between 6/5 and 4/3. It would be beneficial to have a triad on 27/20 to replace the one on 4/3, and this makes use of the 81/80 in the next octave, the least harmonically useful note in the scale. 32/27 is available as the more conventional quasi-diatonic third, but also 320/243, one of the least recognizeable intervals that all correspond to septal intervals in Aberschismic temperament (5120/5103). 320/243 is 476.539¢ in JI and 729/640 is 225.416¢, but these approach each other with a sharper 3. Most notably in this tuning range, tempering out 2187/2240 equates them to 9/7 and 7/6, different from the same ratios that would normally be generated by Archy. Interpreting 320/243 as 14/11 allows for 8:11:14. The exact tempering is irrelevant as long as they're some sort of supermajor and subminor thirds.

The size of the aberrisma presents other issues.

Song structure

This is my system for categorizing the songs I write by structure. Something similar can be applied to other people's music, but the standards will be different due to the unusual complexity of my music.

• straightforward: The simplest type, being in a similar style throughout. Always works.
• opposing sections: Two straightforward parts that are very different.
• medium: Somewhat straightforward, somewhat disjointed.
• episodic: Containing many different sections with themes and a progression throughout the song. The most desirable when done well and in moderation.
• disjointed: Containing many different sections with no clear pattern. The least desirable.

I've noticed that I write more "disjointed" songs than I would like, and fewer "straightforward" and "episodic". This system is intended to keep track of what I'm doing and help me work toward a better result.

Song Structure Types of The Gralbums

Track	Title	Type
g1t1	The Lake Reflects a Black Sky	disjointed
g1t2	The Art of It	episodic
g1t3	Rail Twinge	straightforward
g1t4	Back Stalk	medium
g1t5	Superior Intermedial	episodic
g1t6	Snowboard Index	straightforward
g1t7	Winter's Mortal Hope	medium
g1t8	Transpiration	opposing sections
g1t9	Revelation of Your Forever	medium
g2t1	The Life Unreachable	disjointed
g2t2	Monolithium	opposing sections
g2t3	Nocturne Paranoia	disjointed
g2t4	Glimmer Extrication	episodic
g2t5	Not This Time	straightforward
g2t6	Sakura Blade Minivan	medium
g2t7	Resolute Prelude	straightforward
g2t8	Life and Limb	straightforward
g2t9	Residual Soliloquy	disjointed