Highly melodic EDO

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Highly melodic EDOs are a set of superabundant EDOs and highly composite EDOs banded together. They are the equal division scales with a superabundant or a highly composite number of pitches in an octave. They can be seen as the opposite of Prime EDOs.

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.

Name

The name "highly melodic" derives from the fact that these EDOs replicate smaller EDOs, therefore they contain large amounts of melody, that is a recognizable and a relatively standardized sequence of notes. If an EDO contains a sub-EDO within itself, such as 420 containing 10, 12, and 7, these EDOs' music can be played in 420edo directly, and their melodic perception is replicated and is put together.

Contrast this with zeta EDOs, that approximate just intonation well, being highly harmonic EDOs.

Superabundant and 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.

The difference

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 melodic EDOs

First superabundant EDOs:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600.

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.

Extension

It is possible to define N-generalized superabundant 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 superabundant numbers are the 1-generalized SA numbers, meanwhile 0-generalized numbers are highly composite.

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.

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