Highly composite EDO: Difference between revisions

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== Superabundant numbers in EDOs ==

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

The defining feature 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 ~~amount~~ of symmetrical chords, and correspondingly, uniform octave-repeating scales, relative to its size.

* 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 [[Wikipedia:Mode of limited transposition|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, [[36edo|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 vs. highly composite ==

The highly composite numbers count the amount of divisors, that is sub-EDOs, going to a record. The superabundant numbers count the sum of divisors being the largest in proportion going to a record, and therefore the amount of notes in all those sub-EDOs.

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

An example when they are not the same: [[50400edo|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 4.05195 and 50400edo's coefficient being 4.03.

== First superabundant EDOs ==

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 == Superabundant numbers in EDOs ==
-The defining feature is that the EDO has the largest number of sub-EDOs that it contains, in proportion to the EDO's number. This means that:
+The defining feature 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 amount of symmetrical chords, and correspondingly, uniform octave-repeating scales, relative to its size.
+* 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 [[Wikipedia:Mode of limited transposition|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, [[36edo|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 vs. highly composite ==
+The highly composite numbers count the amount of divisors, that is sub-EDOs, going to a record. The superabundant numbers count the sum of divisors being the largest in proportion going to a record, and therefore the amount of notes in all those sub-EDOs.
+The first 19 superabundant and highly composite numbers are the same.
+An example when they are not the same: [[50400edo|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 4.05195 and 50400edo's coefficient being 4.03.
+== First superabundant EDOs ==