Highly composite equal division: Difference between revisions

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

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 4.05 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.

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 4.05 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.

=== Extensions ===

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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 [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|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): {{EDOs|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 ===

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 The factor of being relative to the number's size is important with superabundant edos too. 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.
-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 4.05 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.
+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 4.05 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.
 === Extensions ===
@@ Line 167: / Line 167: @@
 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 [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|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): {{EDOs|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 ===