Adaptive diatonic interval names

Adaptive diatonic interval names are Vector's attempt to characterize the labelling of certain EDOs' degrees of thirds in a manner inconsistent with conventional diatonic notation in a formal, systematic way.

The fundamental premise of ADIN goes like this.

For an EDO (let's say, 58edo), identify the "central intervals" for each category. These are the diatonic neutral intervals, which may be between edosteps, and the tritone, which is always the semioctave.

What interval qualities will label is distances from these "central intervals". For the most complex case (intervals with major and minor quality), follow the following procedure.

• Find a) the smallest interval greater than 25c above the neutral interval. Take the interval BEFORE this and label it "submajor" (if its offset is still positive, otherwise, interpret it as submajor but do not actually assign it that name).
• Find a) the closest interval to 85c above the neutral interval or b) the smallest interval greater than 75c above the neutral interval, whichever is higher. Label this interval "supermajor".
  • If supermajor and submajor coincide, label the interval "major" and then label the next interval up "supermajor".
  • If there are no unlabelled steps between submajor and supermajor, label whichever is closer to 50c above the neutral interval "major". If this is supermajor, label the next interval up "supermajor"; if this is submajor, label the next interval down "submajor" if its offset is still positive.
• There should now be unlabelled steps between submajor and supermajor. These correspond to different varieties of major. Follow the table:

Number of steps	Qualities
1	major
2	pentamajor, neomajor
3	pentamajor, novamajor, neomajor
4	pentamajor, novamajor, trimajor, neomajor
5	pentamajor, novamajor, trimajor, neomajor, shrubmajor
6	magimajor, pentamajor, novamajor, trimajor, neomajor, shrubmajor

• Find the closest otherwise unlabelled interval to 103c above the neutral interval which is higher than 75c; call it "ultramajor". (Note: you may be ending up with this interval being many semitones above the neutral, that's okay because extraneous names will be clipped off once we actually apply everything to interval classes.)
• There may now be one or more unlabelled steps between supermajor and ultramajor: this scheme currently supports one interval between these two points, which is labelled "sensamajor".
• There may also be one or more unlabelled steps below submajor. Follow the table:

Number of steps	Qualities
1	tendoneutral
2	tendoneutral, supraneutral

The neutral interval, if present in the edo, is called neutral.

• If the step above ultramajor lies below 120 cents above neutral, label it tendo.

For the minor side of any interval class, swap "major" and "minor", "sub" and "super/supra" (super applies only to major and unqualified intervals, other types use supra), "ultra" and "infra", and "arto" and "tendo" (in all senses).

The perfect fourth is, as a rule, the same degree as the diatonic minor third (whatever that happens to be) - for example, if the diatonic minor third is subminor, the perfect fourth is also interpreted as subminor, and "perfect fourth" replaces "subfourth".

As a result, in standard diatonic edos, "minor fourth" contracts to "fourth", and "major fifth" contracts to "fifth" For example, "subminor fourth" -> "subfourth". Instead of "minor fifth", there is "diminished fifth" (and instead of "major fourth", there is "augmented fourth").

This behavior is reversed in antidiatonic edos, so that the perfect fourth is major and the perfect fifth is minor. No contraction is done in equiheptatonic edos (where the perfect fourth is neither major nor minor) or oneirotonic edos (where labels as thirds/sixths take priority over the assignment for the perfect fourth).

Similar logic applies for the perfect fifth and the diatonic major third.

The semioctave may be explicitly labelled.

The unison/octave are treated as a diatonic major interval when going sharper than it, and a diatonic minor interval when going flatter than it. (Again, reversed when antidiatonic).

Priority decreases by one as you go further from the central interval and increases by one as you go closer to it. Where the central interval is halfway between an edostep, the two closest intervals have a priority of -0.5. The interval name with the highest priority is used.

If there is only one type of major or minor (not including submajor or supraminor), the qualifier on major or minor (i.e. penta- or magi-) is removed.

Finally, if all this results in a distinction between "perfect fourth" and "fourth" (which happens in 22edo), its abbreviated quality is reinstated (usually minor). Same for the fifth.

The full notation of 68edo is as follows:

1	superunison	-5	18	pentaminor third	-2
2	ultraunison / inframinor second	-6	19	supraminor third	-1
3	subminor second	-5	20	neutral third	0
4	neominor second	-4	21	submajor third	-1
5	novaminor second	-3	22	pentamajor third	-2
6	pentaminor second	-2	23	novamajor third	-3
7	supraminor second	-1	24	neomajor third	-4
8	neutral second	0	25	supermajor third	-5
9	submajor second	-1	26	ultramajor third, infrafourth	-6
10	pentamajor second	-2	27	subfourth	-5
11	novamajor second	-3	28	perfect fourth	-4
12	neomajor second	-4	29	novafourth	-3
13	supermajor second	-5	30	pentafourth	-2
14	ultramajor second, inframinor third	-6	31	suprafourth	-1
15	subminor third	-5	32	neutral fourth	0
16	neominor third	-4	33	subaugmented fourth	-1
17	novaminor third	-3	34	pentaaugmented fourth / pentadiminished fifth	-2

The system also holds up in antidiatonic cases, such as 25edo as notated with its flat fifth:

1	superunison	-2
2	subminor second	-2
3	minor second	-1
4	neutral second	0
5	major second	-1
6	minor third	-1
7	neutral third	0
8	major third	-1
9	diminished fourth	-1
10	neutral fourth	0
11	perfect fourth	-1
12	supermajor fourth	-2

And 23edo:

1	superunison	-1.5
2	subminor second	-1.5
3	minor second	-0.5
4	major second	-0.5
5	supermajor second / subminor third	-1.5
6	minor third	-0.5
7	major third	-0.5
8	supermajor third / subdiminished fourth	-1.5
9	diminished fourth	-0.5
10	perfect fourth	-0.5
11	superfourth	-1.5

Note that superfourth instead of suprafourth is used because the fourth is technically major.

In equiheptatonic cases, such as 28edo:

1	diesis	-1
2	subminor second	-2
3	minor second	-1
4	neutral second	0
5	major second	-1
6	subminor third, supermajor second	-2
7	minor third	-1
8	neutral third	0
9	major third	-1
10	supermajor third, subminor fourth	-2
11	minor fourth	-1
12	perfect fourth	0
13	major fourth	-1
14	supermajor fourth, subminor fifth	-2

And in equipentatonic cases, such as 15edo:

1	minor second	-0.5
2	major second	-0.5
3	supermajor second, subminor third	-1.5
4	minor third	-0.5
5	major third	-0.5
6	supermajor third, perfect fourth	-1.5
7	minor fourth	-0.5

It can even handle EDOs with only oneirotonic fifths, because the numbering from neutral intervals and the priority system balances out the negative limmas in those systems - though fifths and fourths kinda swap places. Here's 18edo:

1	minor second	-0.5
2	major second	-0.5
3	supermajor second	-1.5
4	subminor third	-1.5
5	minor third	-0.5
6	major third	-0.5
7	supermajor third / subminor fifth	-1.5
8	minor fifth	-1.5
9	major fifth / minor fourth	-0.5

And just as a sanity check, here's 12edo:

1	minor second	-0.5
2	major second	-0.5
3	minor third	-0.5
4	major third	-0.5
5	perfect fourth	-0.5
6	augmented fourth / diminished fifth	-0.5

And to ensure that the naming system achieves its goal, here is 22edo:

1	subminor second / superunison
2	minor second
3	major second
4	supermajor second
5	subminor third
6	minor third
7	major third
8	supermajor third
9	perfect fourth
10	minor fourth
11	major fourth