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.
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, 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
| 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.
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?
| 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. |
Quartertone composition
| 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 triad categories
| 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 |
| 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 | [Pental] 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 | [Pental] 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.
- Pental 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 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.
Quasi-diatonic theory
| 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" | "Eridian" | ? | 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.
I have a lot more to write about how blackdye and diasem work harmonically.
