Highly composite equal division
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.
Generalizations of the concept of "anti-prime" edos include superabundant edos, largely composite numbers, etc.
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, the predominantly used tuning in the world today, is the only known so far highly composite edo that is also a zeta edo and the only one with a step size above just-noticeable difference, except for the trival ones. Others have not been found yet, and given the lack of such edos until hundreds of thousands it is likely if another one is found, it would not be of any harmonic use since its 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.
|2||3||3.4190226||Completely misses the octave.|
|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.|
|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.|
|180||308||307.71203||Corresponds to 308edo, but with quite a stretch.|
|360||-||615.42406||Falls halfway between 615 and 616edo. Also, one step is quite close to the schisma.|
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 can be intuitively inferred from the fact that 27720 is divisible by 11, therefore contains 11edo and thus introduces a new factor, while 50400 recycles edos from 1 to 10 multiple times.
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.
Highly factorable numbers
Highly factorable numbers are similar to highly composite numbers, but the difference is that they have the largest number of factorizations compared to the numbers before them, which means different ways of dividing edo into subsets as opposed to counting subsets themselves.
For example, 24edo has ways of division as 3 x 8, 4 x 6, 2 x 12, 2 x 2 x 6, 2 x 3 x 4, and 2 x 2 x 2 x 3, which is a total of 6 ways of factorizing, and is greater than any number before it.
First few are (OEIS:A033833): 1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160. The sequence has a few notable members like 72edo and 96edo, alongside the 23-limit giant 4320edo and a few already known highly composite numbers.
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.
Just as there are generalized N-highly composite numbers, one can generalize N,M-highly prominent numbers, since this time there is another dimension.