Highly composite equal division: Difference between revisions

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== 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.

[[12edo]], the predominantly used tuning in the world today, is currently the only known non-trivial highly composite edo that holds any zeta records and the only one with a step size above the [[just-noticeable difference]]. 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 ===

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== 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-inspired 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.

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The following is a table of first 19 highly composite/superabundant edfs and their corresponding edos.

{| class="wikitable"

{| class="wikitable right-1 right-2 center-3"

|+Table of first highly melodic edf–edo correspondences

! Edf

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== Generalization ==

{{Todo|inline=1|split page|comment=Move the definitions to [[Highly composite number]] and discuss the properties without regards to equal divisions. }}

=== Extensions ===

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.

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 == 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.
+[[12edo]], the predominantly used tuning in the world today, is currently the only known non-trivial highly composite edo that holds any zeta records and the only one with a step size above the [[just-noticeable difference]]. 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 ===
@@ Line 27: / Line 27: @@
 == 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-inspired 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.
@@ Line 36: / Line 34: @@
 The following is a table of first 19 highly composite/superabundant edfs and their corresponding edos.
-{| class="wikitable"
+{| class="wikitable right-1 right-2 center-3"
 |+Table of first highly melodic edf–edo correspondences
 ! Edf
@@ Line 140: / Line 138: @@
 == Generalization ==
+{{Todo|inline=1|split page|comment=Move the definitions to [[Highly composite number]] and discuss the properties without regards to equal divisions. }}
 === Extensions ===
 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.