when you compare mode 16 and mode 15-over, you'll see that they share 13 notes, which is kinda absurd. i call the intersection scale "major miniscale" and the combined scale "major megascale", because these are both pretty great scales for basing music around one specific root chord like a major seventh chord, with nice otonal and utonal extensions.
for prime harmonics after 19, you've got 23, which is pretty much as flat as 9 and 11, so 23/18 and 23/22 are really accurate. 29 and 31 are also about equally sharp, at about three quarter commas. if you get any more out of tune than three quarter commas, the interval gets inconsistent, being closer to some other step that doesn't represent it. an example of this is 13/9 being closest to 16\31, though the difference between 9/8 (5\31) and 13/8 (22\31) is 17\31. so, anything flatter than 23\31 for 27/16 or sharper than 21\31 for 19/12 should be avoided.
i use the terms diesis, chromatone, diatone, neutratone, whole tone, and supertone on this site to refer to the small step sizes in 31 because they're the most descriptive and easy to understand, but in my own personal work I prefer using diesis, alphastep, secor, nusecond, tutone/meantone, and slendra, because they come from the temperaments associated with these steps.
| Steps in 31edo | Cents |
|---|---|
| Perfect Unison | persu |
| Super Unison | nusu |
| Augmented Unison | sito |
| Minor Second | mito |
| Neutral Second | nuto |
| Major Second | mato |
| Supermajor Second | sato |
| Subminor Third | sidi |
| Minor Third | midi |
| Neutral Third | nudi |
| Major Third | madi |
| Supermajor Third | sadi |
| Sub Fourth | sifo |
| Perfect Fourth | perfo |
| Super Fourth | nufo |
| Augmented Fourth | mafo |
| Diminished Fifth | mifi |
| Sub Fifth | nufi |
| Perfect Fifth | perfi |
| Super Fifth | safi |
| Subminor Sixth | sixa |
| Minor Sixth | mixa |
| Neutral Sixth | nuxa |
| Major Sixth | maxa |
| Supermajor Sixth | saxa |
| Subminor Seventh | sivu |
| Minor Seventh | mivu |
| Neutral Seventh | nuvu |
| Major Seventh | mavu |
| Supermajor Seventh | savu |
| Sub Octave | sico |
| Perfect Octave | perco |
if we want a shortened name for every interval in 31, i've got a system that i personally like. it's a form of shortspeak, but i think the names are more descriptive than what i've seen before. for the suffixes for unison, second, third, etc, i use su, to (tone), di, fo, fi, xa, vu, co. these are the same as the standard shortspeak. for the prefixes, i used essentially the same idea as in my solfege, just trying to preserve the prefixes when increasing or decreasing by a perfect fifth when possible, so i have 17\31 as "nufi" and 1\31 as "nusu". the prefixes from subminor to supermajor are si, mi, nu, ma, sa, with per for perfect. there are also additional ones for downdiminished, diminished, augmented, and upaugmented, those being fri (infra), di, ga, and tra (ultra). check the table to the right for the full names.
harmonic entropy is one measure of how consonant or dissonant an interval is, and if we rank every 31edo interval by it, we get 0\31 31\31 18\31 13\31 23\31 10\31 25\31 8\31 15\31 7\31 21\31 26\31 6\31 5\31 16\31 11\31 27\31 22\31 20\31 28\31 12\31 24\31 4\31 9\31 19\31 14\31 29\31 17\31 3\31 30\31 1\31 2\31. pretty cool
screamapillar is the scale P1 M2 M3 S4 P5 M6 M7 P8, basically major with a superfourth. it has a major seven tonic chord, and due to the 81/80 and 121/120 commas being tempered, the other notes all act as both overtones on the root and undertones on the major seventh, so they all sound good as extensions or pedal points.
if we want to "modulate" tonality in the same key, like go from B major to more of a B subminor, there are chords that can act as pivot chords due to having notes in both tonalities. i put a table for these below.
| Subminor | Minor | Neutral | Major | Supermajor | |
|---|---|---|---|---|---|
| Subminor to | X | 5:6:7 | Orwell Triad | h7, sM6 | 14/12/11, 7:9:11 |
| Minor to | 5:6:7 | X | 5:6:9:11 | 10/8/7, mMaj7 | mS6, Sm7 |
| Neutral to | Orwell Triad | 5:6:9:11 | X | 8:10:11:12 | Squares Triad |
| Major to | h7, sM6 | 10/8/7, mMaj7 | 8:10:11:12 | X | 10/8/7 |
| Supermajor to | 14/12/11, 7:9:11 | mS6, Sm7 | Squares Triad | 10/8/7 | X |
when designing a guitar fretboard, meantone[12] is the clear place to start, and there's a couple ways you can go about it. if you want straight, even frets, you may have notes differing by a diesis, like a D#5 on string 1 and Eb4 and Eb3 on strings 3 and 4. this isn't the case for more than a few notes, and may be desired, to allow access to a better range in certain keys. the other option is by altering the frets so that all notes are from the Eb-G# gamut, which works exactly how you'd expect, and would be a bit simpler to learn. either one works well. to add some extra color, we could split all or some chromatones in half, giving us access to neutral intervals in some keys and septimal ones in others, though the diesis gaps are much harder to play. we could also use a meantone[19] fretboard, giving us access to septimal intervals in every key, with fewer dieses.
the "quasi-equal" scales in 31, i.e. those that are maximally even, are tritonic[2], würschmidt[3], myna[4], slendric/mothra[5], tutone[6], neutral/mohajira[7], greeley/nusecond[8], orwell[9], miracle[10], joan[11], meantone[12], a-team[13], squares[14], and valentine[15]. for every number after it's the same pattern in reverse, starting at valentine[16].
the scale "harrison major" is a septimal major scale with a regular major sixth, with supermajor triads on the root and fifth, a subminor triad on the third, a major triad on the fourth, and a minor triad on the second. the variety is nice, and the VSm7-IS resolution is good, but another interesting thing about the scale is that it has a superfifth on the sixth, so the VI minor chord, which would usually be a substitute tonic, is a VIm(t5), which sounds "tonic-y" but clearly lets you know that the chord progression isn't stopping there, which is a cool option for deceptive cadences.
there are some cool pentatonic scales in 31. meantone[5] is the standard pentatonic we know and love, just a bit harder than in 12. orwell[5] can be thought of as a chord, but it's also a quite decent scale, with a leading tone, rooted orwell tetrad, and rooted joan triad. squares[5] includes a resolution from a neutral sixth between the super seventh and sub sixth to the perfect fifth, which is one of the most satisfying resolutions when done with full chords, so it's cool to see here. slendric[5] is approximately equal, but the slight differences give you one key with a 4:6:7 root chord, which is nice. a-team[5] sounds like standard pentatonic but with three minor thirds instead of two, which is cool. there's also an orwell triad, 13:17:19, and 19/17/13. joan[5] has a 4:7:11:15, which is absurd, so writing music solely within it is actually feasible. the supermajor and subminor pentatonics are modmoses of slendric[5] so melodically they sound similar, but their harmonies include more standard septimal chords. lastly, mode 5 is an interesting minor chord scale, though it can be hard to avoid tonicizing the minor sixth.
chords like the "terrain triad", "orgone triad", and "mothwellsmic dominant" are references to temperaments in which these chords are relevant. 5:7:9 and 8:11:14 are the essential chords of terrain and orgone respectively, so with no other name, that's what i call these triads in 31. the mothwellsmic dominant is a reference to the mothwellsma, the comma between, for instance, 49/36 and 11/8, so when you have the chord 1/1-9/7-3/2-7/4, 49/36 shows up between the third and seventh, and is represented by the superfourth.
there are some pretty cool tone clusters in 31, which are just notes very close to each other in a chord. we use the six types of seconds for these, with the diatones working well, because they represent 14/13, 15/14, 16/15, and 17/16, so a five note cluster is 13:14:15:16:17, and the 14:15:16 works alright. 8:9:10 and 10:11:12 make use of commas like that too, and we can combine them into a 8:9:10:11:12 harmonic chord. two stacked supertones work as well, and three gets to a perfect fifth, for an interesting type of sus chord. there are obviously other options as well.
other chords making use of smaller intervals are split chords, of which a few stand out. 20:24:25:30 is a regular minor/major split chord and 28:33:36:42 is a subminor/supermajor one. the yb and gr ones are much simpler, as 12:14:15:18 and 18/15/14/12, and they sound pretty cool.
hexanies are a type of JI scale where you choose a set of four harmonics and the resulting scale includes every chord between the four that's three notes or less. we actually only need six notes for this. if we choose 1 3 5 and 7, we build our hexany by multiplying them in pairs, so we get 3 5 7 15 21 35. adding in the octaves and making 3 our root note, we get P1 s3 M3 s5 M6 s7 P8. we have 4:5:6 and 10/8/7 on the s3, 6:7:8 and 7/6/5 on the M3, 5:6:7 and 6/5/4 on the M6, and 7:8:10 and 8/7/6 on the s7. lot of variety for such a simple scale. the wider class of scales that this belongs to is combination product sets, hexanies are just a particularly popular example.
the chords Ih7 and IVsM6 work as an interesting two chord vamp, with IVsM6 working as a dominant for I, and Ih7 acting as a secondary Vh7 for the IV. these exist together in YB melodic major.
the principle mode of the oneirotonic scale is called dylathian, and has a 8:10:13 root chord. all other scale degrees get superfifth-based chords, four get subfifth-based chords like 13:17:19 and 24:30:35, and four get superfourth-based chords like orwell triads and tetrads. a specifically good resolution is from dVIO to I(x5).
when we take a "score" of a semitone based on how close it is to the 17edo semitone, we take the reciprocal of the step as a fraction of the octave, and measure its distance to 17. so, 1\31 is 1/31 of the octave, and 31-17=14, so its score is 14. pretty bad, but also the same as 1\3. 1\2 provides equal direction to the tonic and octave, so 15 is in a way the point where an interval doesn't lead at all. so, 14 is at least usable. 2\31 is 1.5, which is very good, 3\31 is 6.67, 4\31 is 9.25, 5\31 is 10.8, and 6\31 is 11.83. for curiosity's sake, 1\12 is 5, 1\19 is 2, 1\22 is 5, 2\22 is 6, and 1\24 is 7. this is very much just a convenient but nonscientific way of quantifying directionality so don't take it too seriously.
there's a custom scale i thought of that is just meant to have a bunch of options for root chords. haven't seen it anywhere else, though i guess that could always be wrong. it's a hexatonic scale of P1 s3 M3 S4 P5 s7 P8. it has a major triad on the root with the harmonic seventh and eleventh as extensions, and if you take away the fifth, the sharp nine is a solid tension for the harmonic seven shell. the sub third makes for a good split chord on the root, and the seventh is a reasonable extension. with just the sub third, you can make a subminor seven chord, or an orwell triad using the superfourth. it's also a subset of the graham orwell modmos, so i call it graham hexatonic, for lack of a better name. with the additional three notes in graham orwell, the orwell triad becomes a pentad, you get a type of augmented triad on the root, you have 8:10:14:17 and major seven chords available, and you have notes a diatone above the fifth and root and below the root for resolution.
the names of temperaments in 31edo may seem random and arbitrary, but they come from established names for temperaments that exist outside of 31edo. here, i'll go over some interesting examples. miracle is a temperament invented by george secor in 1974, one of the earliest regular temperaments besides meantone. the name stands for "multitudes of integer ratios approximated consistently, linearly, and efficiently", as all 11-odd-limit ratios are approximated soon and accurately in the generator chain, which makes it useful outside of an edo system, or within a large one like 72edo. valentine is named for robert c valentine, a baroque era composer. tutone is named as such due to being equivalent to every other step of the meantone generator chain. interestingly, there exist other common names for this temperament, such as didacus, though the origin of this name has escaped me for now. mothra similarly has a second name in 31edo, that being slendric, which comes from the controversial similarities between slendric moses and the indonesian slendro. however, abstractly, these two temperaments have different meanings. slendric refers specifically to the 2.3.7 subgroup tuning, mapping three 8/7s to a 3/2. mothra expands to the full 7-limit by including a mapping of twelve 8/7s to 5/1, essentially just tempering the syntonic comma. in 31edo, these names can be used interchangeably, except generally for larger moses that utilize intervals of 5. mothra is the name of a fictional monster in the godzilla franchise, and godzilla is the name of another meantone temperament using a sharper 8/7 as a generator. similarly, neutral and mohajira are both names for the temperament generated by the neutral third, forming a 2.3.11 temperament, but mohajira adds in the syntonic comma, as well as a mapping of -11 neutral thirds to the (generally unused) harmonic seventh. the name mohajira is a rough translation of "migratory" from arabic, as mohajira moses share similarities with certain arabic maqamat, and the slight differences from mode to mode of mohajira[7] can be described as a slow migration. the name orwell comes from 84edo, which supports orwell, as 19\84 is the generator in that system, and 1984 is a famous book by george orwell. myna is the name of a bird, similar to starling, the name of the 126/125 comma it tempers, but it is also roughly homophonous with minor, as the primary generator is a minor third. würschmidt is named for josé würschmidt, a physicist with connections to early 22edo research. his relation to the temperament is not fully clear. squares is named for the fact that 4 generators result in a 3rd harmonic, 9 result in a seventh harmonic, and 16 result in a fifth harmonic, so the 7-limit mappings are <0, 4, 16, 9>, all square numbers. a-team is a homophone of eighteen, as 18edo is an oneirotonic system that represents the temperament. finally, tritonic is named for the tritone generator, roughly equivalent to a 7/5, known as huygens' tritone, or 10/7, known as euler's tritone. the names slender and joan appear to be completely random, so if you have any idea where they come from and how they relate to the temperament, feel free to reach out.