Highly composite EDO: Difference between revisions

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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. 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 be of any harmonic use since it's amount of steps would be astronomical.

== ~~Extension~~ ==

== Variations upon the theme ==

=== Extensions ===

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.

=== Largerly composite numbers ===

Largely composite numbers are a variant of a sequence of highly composite numbers where the divisor count is nondecreasing, as opposed to strictly increasing in the case of highly composite numbers.

First few are (OEIS: A067128): {{EDOs|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]].

== External links ==

* https://oeis.org/A004394 - superabundant numbers

* https://oeis.org/A002182 - highly composite numbers

* https://oeis.org/A067128 - largely composite numbers

[[Category:Equal divisions of the octave]]

[[Category:EDO theory pages]]

[[Category:Theory]]

@@ Line 35: / Line 35: @@
 edo 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. 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 be of any harmonic use since it's amount of steps would be astronomical.
-== Extension ==
+== Variations upon the theme ==
+=== Extensions ===
 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.
+=== Largerly composite numbers ===
+Largely composite numbers are a variant of a sequence of highly composite numbers where the divisor count is nondecreasing, as opposed to strictly increasing in the case of highly composite numbers.
+First few are (OEIS: A067128): {{EDOs|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]].
 == External links ==
 * https://oeis.org/A004394 - superabundant numbers
 * https://oeis.org/A002182 - highly composite numbers
+* https://oeis.org/A067128 - largely composite numbers
 [[Category:Equal divisions of the octave]]
 [[Category:EDO theory pages]]
 [[Category:Theory]]