Highly composite equal division

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A highly composite equal division is an equal tuning that divides a given equave into a highly composite number of pitches. The opposite of a highly composite equal division is a prime equal division.

A highly composite edo therefore contains a highly composite number of pitches per octave, such as 6edo, 12edo and 60edo.

Generalizations of the concept of "anti-prime" edos include superabundant edos, largely composite numbers, etc.

Properties

The defining characteristics of highly composite equal divisions are the following:

  • The tuning contains the largest count of notes in symmetrical chords, and correspondingly, in uniform equave-repeating scales, relative to its size.
  • The tuning has the largest amount of modes of limited transposition relative to its size.
  • The tuning has the largest amount of rank-2 temperaments whose period is a fraction of the equave, relative to its size.
  • By the virtue of point 1, the tuning 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 9 divisors, while 132 has 12 divisors. However, 9/36 = 0.25, while 12/132 = 0.0909..., meaning that 36 carries a more impressive task in replicating sub-edos and symmetrical chords relative to its size.

Highly composite edo

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.

The first 41 highly composite edos

The first 19 highly composite edos are also the first 19 superabundant 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.

Highly composite EDF

Unlike highly composite edos, whose harmonic content tends to be random and usually contorted, highly composite EDFs often correspond to a useful edo.

Highly composite EDFs have a possible usage in Georgian (Kartvelian) music. Since Kartvelian scales are created by dividing the perfect fifth into an arbitrary number of steps, and complementing that with dividing 4/3 into an arbitrary number of steps, edos which correspond to highly composite EDFs have a high density of such scales per their size.

Highly composite EDF-EDO correspondence

The following is a table of first 19 highly composite/superabundant EDFs and their corresponding EDOs.

Table of first highly melodic EDF-EDO correspondences
EDF EDO log2/log1.5*EDF

(exact EDO)

Comments
1 2 1.7095112 Trivial
2 3 3.4190226 Completely misses the octave.
4 7 6.8380452
6 10 10.257068 10edo, but with a heavy stretch
12 - 20.514135 Completely misses the octave
24 41 41.028271 24edf is equivalent to 41edo. Patent vals match through the 19-limit.
36 - 61.542406
48 82 82.056542 48edf is equivalent to 82edo.
60 103 102.57067 Surprisingly, it's a match to 103edo despite 60edf falling halfway between 102 and 103.
120 205 205.14135
180 308 307.71203 Corresponds to 308edo, but with quite a stretch.
240 410 410.28271
360 - 615.42406 Falls halfway between 615 and 616edo. Also, one step is quite close to the schisma.
720 1231 1230.8481
840 1436 1435.9895
1260 2154 2153.9842
1680 2872 2871.9789
2520 4308 4397.9685
5040 8616 8615.9369

Generalization

Superabundant equal division

Superabundant numbers are slightly different from highly composite numbers, hence superabundant equal divisions can be considered separately from highly composite numbers whenever the properties of superabundant numbers are more relevant in context. Abundancy index is the ratio of the sum of divisors of the number to the number itself, and thus superabundant numbers are numbers where these indices increase to a record, and thus are the highest in proportion to the number itself.

Although neither sequence of these numbers is a subset of the other, the first 19 superabundant and highly composite numbers are the same, and there are more common terms further in the sequences. Therefore, many highly composite equal divisions are also superabundant and vice versa, but some equal divisions are only in one of the two categories (or neither at all).

Highly composite edos have a record amount of divisors, where divisors correspond to the number of sub-edos, while superabundant edos have a record number of notes in those divisors if they were stretched end to end.

The factor of being relative to the number's size is important with superabundant edos too. 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.

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.

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 equal division has the largest number of N-note combinations in its divisors relative to its size than all other equal divisions (of the same equave) before it. 0-generalized, highly composite equal divisions, have the largest amount of divisor scales, that is sub-EDOs. 1-generalized, superabundant equal divisions, have the largest amount of notes in those scales, etc.

This also means that an N-generalized highly composite number has the largest amount of rank-N temperaments that make use of sub-EDOs as generators relative to its size. For example, 2-generalized highly composite numbers therefore contain the most rank-2 temperaments that they both carry themselves as coprime to equal division's size, and also rank-2 temperaments of their subset equal divisons respectively.

2-generalized highly composite numbers are the following (OEIS: A208767): 1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720.

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 equal division 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, 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.

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