Adaptive diatonic interval names

The system of adaptive diatonic interval names (ADIN), developed by Vector and Leriendil, is a way to (mostly) uniquely label the intervals in an EDO based on size and relation to that EDO's patent fifth. It is diatonic because it attempts to behave predictably relative to MOSdiatonic staff notation, and it is adaptive because the differing qualities of diatonic intervals in different tunings are reflected in the interval names (that is to say, it "adapts" to different diatonic tunings). Finally, it is an interval naming system, not a notation system, because it provides no way to write notes and labels intervals based on "what they are", not "what they do". (The creator of the ADIN system endorses ups and downs notation for the latter.)

It is an attempt at formalizing the systems of interval qualities used by various xenharmonic resources on the internet.

Premise

ADIN names qualities, and then applies those names to intervals based on their distance from the nearest (possibly imaginary) diatonic neutral interval. The diatonic neutral intervals are as follows:

• Semidiminished unison (-3.5 fifths)
• Neutral second (-1.5 fifths)
• Neutral third (+0.5 fifths)
• Semiaugmented fourth (+2.5 fifths)
• Semidiminished fifth (-2.5 fifths)
• Neutral sixth (-0.5 fifths)
• Neutral seventh (+1.5 fifths)
• Semiaugmented octave (+3.5 fifths)

Intervals are named on a per-octave basis (that is, by octave-reducing, naming the interval, and adding back octaves according to conventional interval arithmetic), so the semidiminished unison and semiaugmented octave (which are lesser than and greater than the unison and octave respectively) do not actually appear in any interval names. Instead, they are chosen to ensure that the boundary between "unison" and "second" always falls precisely halfway between the perfect unison and the minor second.

These intervals may not exist in an edo (for instance, if it maps the fifth to an odd number of steps). This is okay, as they are being used as points of reference to compare to, not as actual necessary steps in the edo.

Interval regions

Each neutral interval defines a series of regions (or "qualities") extending outwards from it, which are defined in terms of equal divisions of 15/14. The use of 15/14 was proposed by Lériendil for threefold reasons:

• Firstly, 15/14 is a mapping of the apotome in aberschismic tunings: that is, it is the interval between 7/6 and 5/4 and between 6/5 and 9/7, and therefore the interval between the midpoint of 7/6 and 6/5, and the midpoint of 6/5 and 9/7;
• Secondly, it is close to 120 cents, which is the maximum amount of separation an interval can have from a diatonic neutral (assuming the fifth does indeed generate a diatonic scale), ensuring all intervals can be named;
• Finally, it is not itself an equal division of the octave, ensuring that no EDO intervals (aside from the true neutrals) land on region boundaries.

\25ed(15/14)	Cents	Major	Minor
0	0	neutral
0-2	0-9.6	tendoneutral	artoneutral
2-5	9.6-23.9	submajor	supraminor
5-10	23.9-47.8	nearmajor	nearminor
10-15	47.8-71.7	farmajor	farminor
15-20	71.7-95.6	supermajor	subminor
20+	95.6+	ultramajor	inframinor

For instance, assuming a fifth is tuned to JI, the categories of thirds are found at <255c (inframinor), 256-279c (subminor), 280-303c (farminor), 304-327c (nearminor), 327-341c (supraminor), 342-360c (neutral, arto/tendo-), 361-375c (submajor), 376-398c (nearmajor), 399-422c (farmajor), 423-446c (supermajor), and >446c (ultramajor).

With these, the complete sets of intervals of each edo may be given a name. When an interval is an equal distance from two neutrals, thirds are always given precedence over fourths (so that an interval equidistant between the neutral third and neutral fourth is always a kind of third), and over seconds, which take precedence over unisons (except for the perfect unison and octave). The same rules apply to the complementary region of the octave. Fourths always take precedence below the tritone, and fifths always take precedence above it.

The exception is when the diatonic intervals coincide, in which case the conflated interval belongs to the category corresponding to its simplest diatonic interpretation (i.e. 240c is a second, not a third, and 480c is a fourth, not a third or (diminished) fifth). The same applies to oneirotonic and antidiatonic structures.

If there is only one kind of major or minor, drop all prefixes on major and minor. For example, if the only interval qualities found are "farminor", "neutral", and "farmajor", then rename "farminor" to "minor" and "farmajor" to "major". As a result, skip step 3.

Disambiguation

In large edos, multiple intervals may be assigned the same name at the current point. This is where the disambiguation scheme comes into play. Based on the number of intervals in each category, a fixed set of names is assigned in order of size.

Quality	2	3
inframinor	arto, inframinor
subminor	sensaminor, gothminor	sensaminor, septiminor, gothminor
farminor	neominor, novaminor	neominor, triminor, novaminor
nearminor	valaminor, magiminor	valaminor, pentaminor, magiminor
supraminor	daemominor, aurominor
artoneutral	subneutral, artoneutral
tendoneutral	tendoneutral, supraneutral
submajor	auromajor, daemomajor
nearmajor	magimajor, valamajor	magimajor, pentamajor, valamajor
farmajor	novamajor, neomajor	novamajor, trimajor, neomajor
supermajor	gothmajor*, sensamajor	gothmajor*, septimajor, sensamajor
ultramajor	ultramajor, tendo

In cases where there are two intervals belonging to the nearminor/major, farminor/major, and subminor/supermajor qualities, "pentamajor", "trimajor", and "septimajor" are substituted in for major thirds within 4.78¢ (1\25ed(15/14)) of the characteristic just intervals 5/4, 19/15, and 9/7 respectively^**. If any major third acquires one of these subqualities, it is then propagated to its complement and other interval degrees.

Alternative system

Primarily in the case of tuning systems other than EDOs, or large EDOs ( where more than 3 intervals exist within the space of a single quality band, another fallback system can be used to assign subqualities to specific intervals.

"Trimajor" is defined as a radius of 1\25ed(15/14) around 19/15^**, the same way as it is above. "Septimajor" then directly occupies the band 1\5ed(15/14) sharp of trimajor, while "pentamajor" occupies the band 9\50ed(15/14) flat of trimajor. The sharp edge of pentamajor is then taken to be the edge between auromajor and daemomajor. Subneutral and supraneutral intervals are not distinguished in this system.

In cent values, with a justly tuned 3/2, the subqualities sharpward of the neutral third are then bounded as follows:

• 350.978 <- tendoneutral -> 360.533 <- auromajor -> 368.634 <- daemomajor -> 374.866
• 374.866 <- magimajor -> 382.967 <- pentamajor -> 392.522 <- valamajor -> 398.755
• 398.755 <- novamajor -> 404.467 <- trimajor -> 414.022 <- neomajor -> 422.643
• 422.643 <- shrubmajor^* -> 428.355 <- septimajor -> 437.911 <- sensamajor -> 446.532

In that case, two intervals falling within the same subquality can then be disambiguated as "small" and "large", or three as "small", "mid", and "large".

^* "Shrub-" can be replaced with "goth-".

^** A variation would be for 5/4, 19/15, and 9/7 to be substituted here with sqrt(25/24), sqrt(722/675), and sqrt(54/49) above the neutral third, snapping all subqualities to the same positions relative the neutral third.

Final steps

There are some additional replacements to be done:

1) Examine the diatonic fourth and whether it is major or minor. Remove the corresponding quality from all fourth names (for example, if the diatonic fourth is a farminor fourth, replace all instances of "minor fourth" with simply "fourth". Rename the opposing quality from "major" to "augmented", or "minor" to "diminished". If the fourth is any kind of neutral, no change is necessary to any interval names.

2) Label the diatonic fourth "perfect fourth" regardless of its quality.

3) Repeat for the unison, fifth, and octave.

3a) The result may create ambiguities with terms like "far octave" in some edos (the smallest edo to feature this problem being 26edo, between 25\26 and 27\26). In that case, restore "major" to octaves, fifteenths, etc above their perfect counterparts and which have ambiguous labels, and "minor" to fifteenths and above.

4) If quality is not necessary to distinguish intervals at all, remove it entirely (i.e. if there are only neutral intervals, do not specify "neutral").

Qualities in small diatonic EDOs

Below lists the palettes of neutral and major qualities (noting that minor qualities always exist as the complements of major qualities) that can be found in diatonic EDOs below about 60, that is, the EDOs that do not require the disambiguation step. A few EDOs have two diatonic fifths, one which is divisible in two and one which is not. Both fifths are kept track of, but non-patent fifths are in parentheses.

Ultramajor qualities are treated separately, since they are ambiguous in degree. However, for EDOs with flat fifths (19edo or flatter) and which divide the perfect fourth in two, subminor and supermajor qualities are in fact interordinal (e.g. supermajor thirds are the same as sub(minor) fourths). These EDOs will be marked with an asterisk. Some EDOs with sharp fifths have ultramajor (and inframinor) intervals which are, however, not interordinal; these will be marked with a superscript plus sign.

Without a neutral third

EDOs without a neutral third have:

• with a step size 21.25-27.3¢ -> submajor, nearmajor, farmajor, supermajor
  • diatonic fifths: 46, 47*, 49, 50, 53, 56⁺ (52b⁺, 54b)

• with a step size 27.3-28.65¢ -> submajor, nearmajor, farmajor
  • diatonic fifths: 42⁺, 43

• with a step size 28.65-31.85¢ -> submajor, nearmajor, supermajor
  • diatonic fifths: 39, 40

• with a step size 31.85-38.2¢ -> submajor, farmajor, supermajor
  • diatonic fifths: 32, 33*, 36

• with a step size 38.2-47.8¢ -> submajor, farmajor
  • diatonic fifths: 26, 29

• with a step size 47.8-63.7¢ -> nearmajor, supermajor
  • diatonic fifths: 19*, 22

• with a step size > 63.7¢ -> major
  • diatonic fifths: 12

With a neutral third

EDOs with a neutral third have:

• with a step size 19.1-23.9c -> neutral, submajor, nearmajor, farmajor, supermajor
  • diatonic fifths: 51, 52*, 54, 55, 58, 61⁺, 62 (57b⁺, 59b)

• with a step size 23.9-31.85c -> neutral, nearmajor, farmajor, supermajor
  • diatonic fifths: 38*, 41, 44, 45, 48 (47b⁺)

• with a step size 31.85-35.85c -> neutral, nearmajor, farmajor
  • diatonic fifths: 34, 37⁺

• with a step size 35.85-47.8c -> neutral, nearmajor, supermajor
  • diatonic fifths: 27, 31

• with a step size 47.8-71.65c -> neutral, (far)major
  • diatonic fifths: 17, 24

Notes

The first EDO this system fails to name the intervals for is currently 159edo, as it has four intervals within each supermajor range.

Extensions

Oneirotonic

Add an extra ordinal for "tritone" rather than just treating it as a special case for even edos. The chroma is the moschroma of oneirotonic.

Antidiatonic

The chroma is the moschroma of antidiatonic. Note that the pythagorean semidiminished unison is still the center of the unison range, despite being larger than 0c.