Ground's composition theory
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, aberrismic and 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.
| 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.
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
| Gens | 949¢ | 349¢ |
|---|---|---|
| -12 | 612 | |
| -11 | 361 | 961 |
| -10 | 110 | |
| -9 | 1059 | 459 |
| -8 | 808 | |
| -7 | 557 | 1157 |
| -6 | 306 | |
| -5 | 55 | 655 |
| -4 | 1004 | |
| -3 | 753 | 153 |
| -2 | 502 | |
| -1 | 251 | 851 |
| 0 | 0 | |
| 1 | 949 | 349 |
| 2 | 698 (98) | |
| 3 | 447 | 1047 |
| 4 | 196 | |
| 5 | 1145 | 545 |
| 6 | 894 (294) | |
| 7 | 643 | 43 |
| 8 | 392 | |
| 9 | 141 | 741 |
| 10 | 1090 | |
| 11 | 839 | 239 |
| 12 | 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. 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.
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.
