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, with the exception of the fifth and fourth, which use their perfect varieties, 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 or b) the closest interval to 35c above the neutral interval, whichever is higher. 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).
Fourths/fifths and tritones utilize the exact same interval qualities, except that the "neutral" is removed from the neutral qualities. Note that the terms "penta-" and "tri-" now no longer directly correspond to 5- or 3-limit JI intervals in the case of intervals in this region.
Note that here, "infra" is preferred in lieu of "supraminor", and likewise "ultra" in place of "submajor" (unless "sub-" and "super-" are not already taken, in which case, those are used).
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: "fourth", "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, 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.
An unqualified fourth or fifth is labelled "perfect"; an unqualified tritone is labelled "neutral" (importantly, subtritone and supratritone remain unchanged).
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 | minor fourth | -5 |
| 10 | pentamajor second | -2 | 27 | subfourth | -4 |
| 11 | novamajor second | -3 | 28 | perfect fourth | -3 |
| 12 | neomajor second | -4 | 29 | superfourth | -4 |
| 13 | supermajor second | -5 | 30 | pentamajor fourth | -5 |
| 14 | ultramajor second, inframinor third | -6 | 31 | novaminor tritone, novamajor fourth | -6 |
| 15 | subminor third | -5 | 32 | pentaminor tritone | -5 |
| 16 | neominor third | -4 | 33 | subtritone | -4 |
| 17 | novaminor third | -3 | 34 | 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 | subfourth | -2 |
| 11 | perfect fourth | -1 |
| 12 | subtritone | -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 | minor fourth | -2 |
| 12 | perfect fourth | -1 |
| 13 | minor tritone, major 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 | minor 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 | major fourth, minor 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 |