Highly composite EDO

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Highly composite EDO is a tuning with highly composite number of pitches in an octave. They are the opposite of Prime EDOs.

Highly composite numbers in EDOs

The defining feature of a SA-EDO is that the EDO has the largest number of notes within sub-EDOs that it contains, in proportion to the EDO's number. This means that:

  • The EDO contains the largest count of notes in symmetrical chords, and correspondingly, in uniform octave-repeating scales, relative to its size.
  • The EDO has the largest amount of modes of limited transposition relative to its size.
  • The EDO has the largest amount of rank-2 temperaments whose period is a fraction of the octave, relative to its size.
  • By the virtue of point 1, the EDO has the largest amount of familiar scales relative to its size

The factor of being relative to the number's size is important. For example, 36 has a sum of divisors of 91, while 130 has a sum of divisors of 252. If all the sub-scales were stretched end-to-end, this means 36 has 91 notes, and 130 has 252. However, 91/36 = 2.527, while 252/130 = 1.938, meaning that 36 carries a more impressive task in replicating sub-EDOs and symmetrical chords relative to it's size.

Superabundant EDOs and the difference with highly composite

The difference between SA and HC is the following: the highly composite numbers count the amount of divisors, that is sub-EDOs, going to a record., while superabundant EDOs count the amount of note in those divisors if they were stretched end-to-end.

The first 19 superabundant and highly composite numbers are the same.

An example when they are not the same: 50400 is the 27th highly composite number, that is not on the superabundant list. The count of divisors of 50400 is 108, which means it supports 106 symmetrical scales that aren't the chromatic and whole-octave scale. However, by the note count in all those scales, which is 102311 not counting 1 and 50400, the EDO lags slightly behind 27720edo, with it's coefficient being 4.05 and 50400edo's coefficient being 4.03. This means that while 27720 is less composite than 50400, it carries a more impressive task in providing notes to compose with, if the composer is interested in smaller EDOs as subscales.

And indeed it's somewhat obvious - 27720 is divisible by 11, therefore contains 11edo, while 50400 recycles EDOs from 1 to 10 multiple times.

First highly composite EDOs

First highly composite EDOs: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160.

Superabundant EDOs that are also highly composite, excluding the first 19: 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 367567200, 698377680, 735134400.

The sequence is finite and has 430 terms starting with 10080.

12edo is the predominantly used tuning in the world today, and in addition it is the only known so far highly composite EDO that's also a zeta edo, and the only one with a step size above just noticeable difference. Others have not been found yet, and given the lack of such EDOs until hundreds of thousands it's likely if another one is found, it would not be of any harmonic use since it's amount of steps would be astronomical.

Variations upon the theme

Extensions

It is possible to define N-generalized highly composite numbers as a set of numbers, for which sum of powers of divisors relative to the number is greater than all the ones before it. This means that 0-generalized highly composite numbers are plain highly composite numbers, 1-generalized are superabundant numbers, etc.

In terms of composition, this means that the EDO has the largest number of N-note cobinations in its divisors relative to its size than all other EDOs before it. 0-generalized, highly composite numbers, have the largest amount of divisor scales in EDO. 1-generalized have the largest amount of notes in those scales, and etc.

Largely composite numbers

Largely composite numbers are a variant of a sequence of numbers where the divisor count is nondecreasing, as opposed to strictly increasing.

First few are (OEIS: A067128): 1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240.

While it contains the well known highly composite numbers, it also contains some historically notable temperaments such as 72edo and 96edo. 72edo has been used in Byzantine chanting, has been theoreticized by Alois Haba and Ivan Wyschnegradsky, and has been used by jazz musician Joe Maneri. 96edo has been used by Julian Carrillo.

Prominence

Prominence (term proposed by Eliora) is the product of the number's count of divisors (highly composite) or its abundance index (superabundant) with its amount of distinct prime factors. Prominence serves as a good measure of different varieties of subsets that an EDO can provide, since a more prominent number is capable of approximating more distinct temperaments efficiently for its size.

Example: 1848edo is more prominent than 3456edo despite both having 32 divisors, due to 1848 having 2, 3, 5, 7, and 11 as prime factors, while 3456 having only 2 and 3.

Just as there are generalized N-highly composite numbers, one can generalize N,M-highly prominent numbers, since this time there is another dimension.

External links

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