View source for 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 [[Modified ups and downs notation|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.

== 1. 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. 

== 2. Interval regions ==
Each neutral interval defines a series of regions extending outwards from it, which are defined by Leriendil in terms of equal divisions of 15/14. The reasoning for the use of 15/14 is twofold: firstly, 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, and secondly, it is not itself an equal division of the octave, ensuring that no edo intervals land on region boundaries.
{| class="wikitable"
|+
!\25ed(15/14)
!Cents
!Major
!Minor
|-
|0
|0
| colspan="2" |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.

== 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. 
{| class="wikitable"
|+
!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 a category, "pentamajor", "trimajor", and "septimajor" can be substituted in for major thirds within 4.78{{c}} (1\25ed(15/14)) of the characteristic intervals 5/4, 19/15, and 9/7 respectively; if any major third validly acquires a quality, it is propagated to its complement and other interval degrees.

== 4. 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.

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").

== 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.