Equal-step tuning - Revision history

BudjarnLambeth: /* Lookalike equal divisions */ add link

2026-10-03T21:07:38Z

Lookalike equal divisions: add link

← Older revision		Revision as of 21:07, 3 October 2026
Line 37:		Line 37:

	(Unlimited resolution version: [[:File:equal.svg\|equal.svg]])		(Unlimited resolution version: [[:File:equal.svg\|equal.svg]])

			(Version that goes to 100ed2: [[User:BudjarnLambeth/ET alignment chart]])

	For the mathematically inclined, this kind of diagram is closely related to the [[Riemann zeta function]].		For the mathematically inclined, this kind of diagram is closely related to the [[Riemann zeta function]].

BudjarnLambeth: /* List of notable AS */ actually as33/32 isn't notabke anyway, removed it

2026-09-30T10:54:14Z

List of notable AS: actually as33/32 isn't notabke anyway, removed it

← Older revision		Revision as of 10:54, 30 September 2026
Line 91:		Line 91:
	* [[1ed9/8\|AS9/8]]		* [[1ed9/8\|AS9/8]]
	* [[21/20\|AS21/20]]		* [[21/20\|AS21/20]]
	* [[33/32\|AS33/32]]

	==== List of notable APS ====		==== List of notable APS ====

BudjarnLambeth: /* List of notable AS */ fix link

2026-09-30T10:53:03Z

List of notable AS: fix link

← Older revision		Revision as of 10:53, 30 September 2026
Line 91:		Line 91:
	* [[1ed9/8\|AS9/8]]		* [[1ed9/8\|AS9/8]]
	* [[21/20\|AS21/20]]		* [[21/20\|AS21/20]]
	* [[~~1ed33~~/32\|AS33/32]]		* [[33/32\|AS33/32]]

	==== List of notable APS ====		==== List of notable APS ====

Fredg999: Text replacement - "the Riemann zeta function" to "the Riemann zeta function"

2026-07-16T18:54:41Z

Text replacement - "the Riemann zeta function" to "the Riemann zeta function"

← Older revision		Revision as of 18:54, 16 July 2026
Line 38:		Line 38:
	(Unlimited resolution version: [[:File:equal.svg\|equal.svg]])		(Unlimited resolution version: [[:File:equal.svg\|equal.svg]])

	For the mathematically inclined, this kind of diagram is closely related to [[~~The Riemann zeta function and tuning\|the~~ Riemann zeta function]].		For the mathematically inclined, this kind of diagram is closely related to the [[Riemann zeta function]].

	== Catalog of equal-step tunings ==		== Catalog of equal-step tunings ==

FloraC: Split

2025-11-16T11:58:18Z

Split

Show changes

Xenllium at 03:01, 16 November 2025

2025-11-16T03:01:59Z

← Older revision		Revision as of 03:01, 16 November 2025
Line 79:		Line 79:
	* [[Ed8]] (… of the 8th harmonic)		* [[Ed8]] (… of the 8th harmonic)
	* [[Ed12]] (… of the 12th harmonic)		* [[Ed12]] (… of the 12th harmonic)


	=== Equal multiplications ===		=== Equal multiplications ===
Line 348:		Line 349:
	\|15		\|15
	\|28		\|28
			\|6, 7
	\|7, 8		\|7, 8
	~~\|8, 9~~
	\|13, 15		\|13, 15
	\|}		\|}

Contribution: /* Alpha-beta-gamma family of equal divisions */

2025-10-28T02:47:39Z

Alpha-beta-gamma family of equal divisions

← Older revision		Revision as of 02:47, 28 October 2025
Line 358:		Line 358:
	* On Gamma scales, <math>S_n</math> is mapped on the <math>(2n - 1)^\text{th}</math> degree step, and <math>B_n</math> is mapped on the <math>(2n + 1)^\text{th}</math> degree step.		* On Gamma scales, <math>S_n</math> is mapped on the <math>(2n - 1)^\text{th}</math> degree step, and <math>B_n</math> is mapped on the <math>(2n + 1)^\text{th}</math> degree step.

	Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types ~~again flatten~~ the smaller and ~~sharpen~~ the larger.		Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types also sharpen the smaller and flatten the larger.

	The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation. <math>n</math> also brings the interval pairs increasingly near to pure.		The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation. <math>n</math> also brings the interval pairs increasingly near to pure.

Contribution: /* Alpha-beta-gamma family of equal divisions */

2025-10-28T02:21:56Z

Alpha-beta-gamma family of equal divisions

← Older revision		Revision as of 02:21, 28 October 2025
Line 354:		Line 354:

	Mappings of <math>S_n</math> and <math>B_n</math> are always mapped as follow:		Mappings of <math>S_n</math> and <math>B_n</math> are always mapped as follow:
	* On Alpha scales, <math>S_n</math> is mapped on the <math>(n - 1)</math>th degree step and <math>B_n</math> is mapped on the <math>n</math>th degree step		* On Alpha scales, <math>S_n</math> is mapped on the <math>(n - 1)^\text{th}</math> degree step, and <math>B_n</math> is mapped on the <math>n^\text{th}</math> degree step.
	* On Beta scales, <math>S_n</math> is mapped on the <math>n</math>th degree step and <math>B_n</math> is mapped on the <math>(n + 1 )</math>th degree step		* On Beta scales, <math>S_n</math> is mapped on the <math>n^\text{th}</math> degree step, and <math>B_n</math> is mapped on the <math>(n + 1)^\text{th}</math> degree step.
	* On Gamma scales <math>S_n</math> is mapped on the <math>(2n - 1)</math>th degree step and <math>B_n</math> is mapped on the <math>(2n + 1)</math>th degree step		* On Gamma scales, <math>S_n</math> is mapped on the <math>(2n - 1)^\text{th}</math> degree step, and <math>B_n</math> is mapped on the <math>(2n + 1)^\text{th}</math> degree step.

	Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger.		Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger.

Contribution: /* Alpha-beta-gamma family of equal divisions */

2025-10-28T02:17:24Z

Alpha-beta-gamma family of equal divisions

← Older revision		Revision as of 02:17, 28 October 2025
Line 352:		Line 352:
	\|13, 15		\|13, 15
	\|}		\|}

			Mappings of <math>S_n</math> and <math>B_n</math> are always mapped as follow:
			* On Alpha scales, <math>S_n</math> is mapped on the <math>(n - 1)</math>th degree step and <math>B_n</math> is mapped on the <math>n</math>th degree step
			* On Beta scales, <math>S_n</math> is mapped on the <math>n</math>th degree step and <math>B_n</math> is mapped on the <math>(n + 1 )</math>th degree step
			* On Gamma scales <math>S_n</math> is mapped on the <math>(2n - 1)</math>th degree step and <math>B_n</math> is mapped on the <math>(2n + 1)</math>th degree step

	Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger.		Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger.

Contribution: /* Alpha-beta-gamma family of equal divisions */

2025-10-28T02:05:34Z

Alpha-beta-gamma family of equal divisions

← Older revision		Revision as of 02:05, 28 October 2025
Line 275:		Line 275:
	! colspan="2" \|SSCP		! colspan="2" \|SSCP
	! colspan="3" \|Number of divisions		! colspan="3" \|Number of divisions
			! colspan="3" \|SSCP mappings
	\|-		\|-
	!Small		!Small
	!Big		!Big
			!Alpha
			!Beta
			!Gamma
	!Alpha		!Alpha
	!Beta		!Beta
Line 289:		Line 293:
	\|5		\|5
	\|8		\|8
			\|1, 2
			\|2, 3
			\|3, 5
	\|-		\|-
	\|3		\|3
Line 297:		Line 304:
	\|7		\|7
	\|12		\|12
			\|2, 3
			\|3, 4
			\|5, 7
	\|-		\|-
	\|4		\|4
Line 305:		Line 315:
	\|9		\|9
	\|16		\|16
			\|3, 4
			\|4, 5
			\|7, 9
	\|-		\|-
	\|5		\|5
Line 313:		Line 326:
	\|11		\|11
	\|20		\|20
			\|4, 5
			\|5, 6
			\|9, 11
	\|-		\|-
	\|6		\|6
Line 321:		Line 337:
	\|13		\|13
	\|24		\|24
			\|5, 6
			\|6, 7
			\|11, 13
	\|-		\|-
	\|7		\|7
Line 329:		Line 348:
	\|15		\|15
	\|28		\|28
			\|7, 8
			\|8, 9
			\|13, 15
	\|}		\|}