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).
• 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).

For tritones, start at the semi-octave. If the semioctave exists in the scale, label it "neutral tritone", otherwise label the two closest intervals "artoneutral tritone" for the smaller and "tendoneutral tritone" for the larger. Label the sharpest tritone interval flatter than the neutral or artoneutral tritone "narrow tritone" (and label its complement "wide tritone"). If the interval closest to 565 cents has not already been labelled as a tritone, label it a "subtritone". If there are intervals between the subtritone and narrow tritone, label "grave tritone" if only 1 interval and "grave tritone" (for the smaller) and "small tritone" (for the larger) if there are two. Label the tritone interval one step smaller than the subtritone the infratritone. Similarly, do the same thing for fourths and fifths with the best 4/3 and 3/2 (and checking for n +- 35c rather than specifically 565c), but you won't need the artoneutral/tendoneutral case.

For unisons, the term "diesis" is used in place of "major unison" and "counterdiesis" in place of "minor octave"; "comma" is "submajor unison" and "countercomma" is "supraminor octave".

Priority: "perfect fourth", "perfect fifth", "tritone", "unison", and "octave" have the same priority as a "novamajor third", "neomajor third", or "major third", whichever is the first one listed that exists in the labeling; priority decreases by one as you go further from the central interval (or perfect interval in the case of fourths and fifths), 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 "narrow tritone", "subtritone", or "small tritone" (in decreasing order of precedence) do not appear in the list of highest-priority names, and "artoneutral tritone" exists, then relabel artoneutral to whichever is available.

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.

The full notation of 68edo is as follows:

1	comma	-4	18	pentaminor third	-2
2	diesis	-5	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	subfourth	-5
10	pentamajor second	-2	27	narrow fourth	-4
11	novamajor second	-3	28	perfect fourth	-3
12	neomajor second	-4	29	wide fourth	-4
13	supermajor second	-5	30	superfourth	-5
14	ultramajor second, inframinor third	-6	31	ultrafourth/infratritone	-6
15	subminor third	-5	32	subtritone	-5
16	neominor third	-4	33	narrow tritone	-4
17	novaminor third	-3	34	neutral tritone	-3

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

1	diesis	-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	supermajor third	-2
10	narrow fourth	-2
11	perfect fourth	-1
12	narrow tritone	-1.5

In equiheptatonic cases, such as 28edo:

1	diesis	-2
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	-2
11	narrow fourth	-2
12	perfect fourth	-1
13	narrow tritone, wide fourth	-2
14	neutral tritone	-1

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	perfect fourth	-0.5
7	narrow tritone	-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 these systems. 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	perfect fourth	-0.5
8	wide fourth, narrow tritone	-1.5
9	neutral tritone	-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	neutral tritone	-0.5

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

1	subminor second / diesis
2	minor second
3	major second
4	supermajor second
5	subminor third
6	minor third
7	major third
8	supermajor third, narrow fourth
9	perfect fourth
10	wide fourth
11	neutral tritone