Highly composite equal division - Revision history

Zhenlige: /* Highly composite edf–edo correspondence */

2025-12-12T17:23:38Z

Highly composite edf–edo correspondence

← Older revision		Revision as of 17:23, 12 December 2025
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	\| -		\| -
	\| 20.514135		\| 20.514135
	\| Completely misses the octave.		\| Completely misses the octave. Close to [[41ed4]] (one string on a [[Kite Guitar]]).
	\|-		\|-
	\| 24		\| 24

ArrowHead294: /* Highly composite edo */

2025-06-03T13:42:47Z

Highly composite edo

← Older revision		Revision as of 13:42, 3 June 2025
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	== Highly composite edo ==		== Highly composite edo ==
	[[12edo]], the predominantly used tuning in the world today, is currently the only known highly composite edo that ~~is also a~~ zeta ~~edo~~ 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.		[[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 ===		=== The first 41 highly composite edos ===

ArrowHead294: /* Highly composite edo */

2025-04-14T10:18:32Z

Highly composite edo

← Older revision		Revision as of 10:18, 14 April 2025
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	== Highly composite edo ==		== 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 highly composite edo that is also a zeta edo 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 ===		=== The first 41 highly composite edos ===

FloraC: /* Generalization */ +todo

2025-04-01T16:08:23Z

Generalization: +todo

← Older revision		Revision as of 16:08, 1 April 2025
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	== Generalization ==		== Generalization ==
			{{Todo\|inline=1\|split page\|comment=Move the definitions to [[Highly composite number]] and discuss the properties without regards to equal divisions. }}

	=== Extensions ===		=== 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.		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.

FloraC: /* Highly composite edf */ in fact, both are random. Edfs tend to correspond to an edo simply becuz there are fewer edfs.

2025-04-01T15:45:11Z

Highly composite edf: in fact, both are random. Edfs tend to correspond to an edo simply becuz there are fewer edfs.

← Older revision		Revision as of 15:45, 1 April 2025
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	== Highly composite edf ==		== 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.		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.		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		\|+Table of first highly melodic edf–edo correspondences
	! Edf		! Edf

FloraC: Style and +category

2025-04-01T15:40:11Z

Style and +category

Show changes

BudjarnLambeth: /* Highly composite EDF */ add link to orphan page

2024-10-20T07:50:51Z

Highly composite EDF: add link to orphan page

← Older revision		Revision as of 07:50, 20 October 2024
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	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.		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.

			[[Schismagonal]] temperament is useable in many highly composite EDFs.

	=== Highly composite EDF-EDO correspondence ===		=== Highly composite EDF-EDO correspondence ===

BudjarnLambeth: /* The first 41 highly composite edos */

2024-03-09T21:06:11Z

The first 41 highly composite edos

← Older revision		Revision as of 21:06, 9 March 2024
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	=== The first 41 highly composite edos ===		=== The first 41 highly composite edos ===
	The first 19 highly composite edos are also the first 19 superabundant edos.		The first 19 highly composite edos are also the first 19 superabundant edos:

	{{EDOs\| 1, 2, 4, 6, 12, 24, 36, 48, 60, 120 }}, ~~<br>~~		{{EDOs\|1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 11088 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160}}...
	~~{{EDOs\|~~ 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560 }}, ~~<br>~~
	10080, 15120, 20160, 25200, ~~[[27720edo\|~~27720]], 45360, 50400, 55440, 83160, ~~110880, <br>~~
	166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160.

	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.
			Superabundant edos that are also highly composite, excluding the first 19:

			{{EDOs\|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.

	== Highly composite EDF ==		== Highly composite EDF ==

BudjarnLambeth at 21:04, 9 March 2024

2024-03-09T21:04:24Z

← Older revision		Revision as of 21:04, 9 March 2024
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	[[Category:Equal-step tuning]]		[[Category:Equal-step tuning]]

	~~{{todo\|review\|cleanup\|comment=the way the page is laid out seems a bit chaotic and scattered, see if that can be rectified by switching up the heading structure}}~~

BudjarnLambeth: /* External links */

2024-03-09T21:04:14Z

External links

← Older revision		Revision as of 21:04, 9 March 2024
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	== External links ==		== External links ==
			* https://oeis.org/A002182 - highly composite numbers
	* https://oeis.org/A004394 - superabundant numbers		* https://oeis.org/A004394 - superabundant numbers
	* https://oeis.org/A002182 - highly composite numbers
	* https://oeis.org/A067128 - largely composite numbers		* https://oeis.org/A067128 - largely composite numbers