5L 2s/Temperaments - Revision history

VectorGraphics: Phase out wedgies

2025-06-12T04:02:26Z

Phase out wedgies

← Older revision		Revision as of 04:02, 12 June 2025
Line 60:		Line 60:

	Mapping generators: ~2, ~3		Mapping generators: ~2, ~3

	~~[[Wedgie]]: {{wedgie\| 1 4 4 }}~~

	[[Tuning ranges]]:		[[Tuning ranges]]:
Line 788:		Line 786:

	[[Mapping]]: [{{val\|1 1 -3 4}}, {{val\|0 1 9 -2}}]		[[Mapping]]: [{{val\|1 1 -3 4}}, {{val\|0 1 9 -2}}]

	~~[[Wedgie]]: {{wedgie\|1 9 -2 12 -6 -30}}~~

	[[Val]]s: {{Optimal ET sequence\| 5, 17, 22, 27, 49 }}		[[Val]]s: {{Optimal ET sequence\| 5, 17, 22, 27, 49 }}

ArrowHead294 at 13:28, 3 March 2025

2025-03-03T13:28:21Z

Show changes

ArrowHead294 at 19:35, 10 January 2025

2025-01-10T19:35:51Z

← Older revision		Revision as of 19:35, 10 January 2025
Line 1:		Line 1:
	{{breadcrumb}}		{{breadcrumb}}
	Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.		Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.

	== Meantone ==		== Meantone ==

ArrowHead294 at 20:38, 25 November 2024

2024-11-25T20:38:34Z

Show changes

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T19:17:57Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 19:17, 7 May 2023
Line 468:		Line 468:
	[[Gencom]] [[mapping]]: [{{val\|1 1 7 11 -10 -8 6 }}, {{val\|0 1 -8 -14 23 20 -3 }}]		[[Gencom]] [[mapping]]: [{{val\|1 1 7 11 -10 -8 6 }}, {{val\|0 1 -8 -14 23 20 -3 }}]

	{{~~Val list~~\|legend=1\| 41, 53, 94 }}		{{Optimal ET sequence\|legend=1\| 41, 53, 94 }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 0.6486 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 0.6486 cents
Line 566:		Line 566:
	[[Gencom]] [[mapping]]: [{{val\| 1 1 -6 -3 -1 }}, {{val\| 0 1 15 11 8 }}]		[[Gencom]] [[mapping]]: [{{val\| 1 1 -6 -3 -1 }}, {{val\| 0 1 15 11 8 }}]

	{{~~Val list~~\|legend=1\| 17, 46, 63 }}		{{Optimal ET sequence\|legend=1\| 17, 46, 63 }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 0.7541 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 0.7541 cents
Line 626:		Line 626:
	[[Gencom mapping]]: [{{val\| 1 1 4 }}, {{val\| 0 1 -2 }}]		[[Gencom mapping]]: [{{val\| 1 1 4 }}, {{val\| 0 1 -2 }}]

	{{~~Val list~~\|legend=1\| 5, 12, 17, 22, 27, 137bd }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 27, 137bd }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 1.856 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 1.856 cents
Line 705:		Line 705:
	Gencom mapping: [{{val\| 1 1 4 7 }}, {{val\| 0 1 -2 -6 }}]		Gencom mapping: [{{val\| 1 1 4 7 }}, {{val\| 0 1 -2 -6 }}]

	~~Vals:~~ {{~~Val list~~\| 5, 12, 17, 39c, 56d }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 39c, 56d }}

	RMS error: 1.977 cents		RMS error: 1.977 cents
Line 783:		Line 783:
	[[Wedgie]]: {{wedgie\|1 9 -2 12 -6 -30}}		[[Wedgie]]: {{wedgie\|1 9 -2 12 -6 -30}}

	[[Val]]s: {{~~Val list~~\| 5, 17, 22, 27, 49 }}		[[Val]]s: {{Optimal ET sequence\| 5, 17, 22, 27, 49 }}

	[[Badness]]: 0.032318		[[Badness]]: 0.032318
Line 901:		Line 901:
	Gencom mapping: [{{val\| 1 1 -6 4 -7 }}, {{val\| 0 1 14 -2 18 }}]		Gencom mapping: [{{val\| 1 1 -6 4 -7 }}, {{val\| 0 1 14 -2 18 }}]

	~~Vals:~~ {{~~Val list~~\| 5, 32, 37 }}		{{Optimal ET sequence\|legend=1\| 5, 32, 37 }}

	RMS error: 2.318 cents		RMS error: 2.318 cents

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T18:05:27Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 18:05, 7 May 2023
Line 67:		Line 67:
	* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]		* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]

	{{~~Vals~~\|legend=1\| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}		{{Optimal ET sequence\|legend=1\| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}

	[[Badness]]: 0.007381		[[Badness]]: 0.007381
Line 163:		Line 163:
	Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.		Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

	{{~~Vals~~\|legend=1\| 7, 19, 26, 45f }}		{{Optimal ET sequence\|legend=1\| 7, 19, 26, 45f }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 1.742 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 1.742 cents
Line 256:		Line 256:
	Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.		Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.

	{{~~Vals~~\|legend=1\| 12, 19, 31, 81, 112b, 143b }}		{{Optimal ET sequence\|legend=1\| 12, 19, 31, 81, 112b, 143b }}

	[[Badness]]: 0.013707		[[Badness]]: 0.013707
Line 288:		Line 288:
	Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.		Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

	{{~~Vals~~\|legend=1\| 12e, 19, 31, 81 }}		{{Optimal ET sequence\|legend=1\| 12e, 19, 31, 81 }}

	[[Badness]]: 0.021543		[[Badness]]: 0.021543
Line 320:		Line 320:
	[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.		[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.

	{{~~Vals~~\|legend=1\| 12, 19e, 31, 105, 136b, 167be, 198be }}		{{Optimal ET sequence\|legend=1\| 12, 19e, 31, 105, 136b, 167be, 198be }}

	[[Badness]]: 0.017027		[[Badness]]: 0.017027

Cmloegcmluin: unchanged interval → unchanged-interval

2023-05-05T23:42:45Z

unchanged interval → unchanged-interval

← Older revision		Revision as of 23:42, 5 May 2023
Line 150:		Line 150:
	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit]]: ~3/2 = {{Monzo\| 8/13 0 1/13 -1/13 }}		* [[7-odd-limit]]: ~3/2 = {{Monzo\| 8/13 0 1/13 -1/13 }}
	: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5		: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/5
	* [[9-odd-limit]]: ~3/2 = {{monzo\| 6/11 2/11 0 -1/11 }}		* [[9-odd-limit]]: ~3/2 = {{monzo\| 6/11 2/11 0 -1/11 }}
	: Eigenmonzos (unchanged intervals): 2, 9/7		: Eigenmonzos (unchanged-intervals): 2, 9/7

	[[Tuning ranges]]:		[[Tuning ranges]]:
Line 245:		Line 245:
	[[Minimax tuning]]:		[[Minimax tuning]]:
	* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo\| 0 0 1/4 }}		* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo\| 0 0 1/4 }}
	: [[Eigenmonzo]]s (unchanged intervals): 2, 5		: [[Eigenmonzo]]s (unchanged-intervals): 2, 5

	[[Tuning ranges]]:		[[Tuning ranges]]:
Line 279:		Line 279:
	Minimax tuning:		Minimax tuning:
	* [[11-odd-limit]]: ~3/2 = {{Monzo\| 0 0 1/4 }}		* [[11-odd-limit]]: ~3/2 = {{Monzo\| 0 0 1/4 }}
	: [[Eigenmonzo]]s (unchanged intervals): 2, 5		: [[Eigenmonzo]]s (unchanged-intervals): 2, 5

	Tuning ranges:		Tuning ranges:
Line 311:		Line 311:
	Minimax tuning:		Minimax tuning:
	* [[11-odd-limit]]: ~3/2 = {{Monzo\| 9/16 -1/8 0 0 1/16 }}		* [[11-odd-limit]]: ~3/2 = {{Monzo\| 9/16 -1/8 0 0 1/16 }}
	: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9		: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9

	Tuning ranges:		Tuning ranges:

Fredg999 category edits: Categories

2022-03-02T21:32:13Z

Fredg999 category edits: Removing from Category:Scales using Cat-a-lot

2022-02-25T14:54:44Z

Removing from Category:Scales using Cat-a-lot

← Older revision		Revision as of 14:54, 25 February 2022
Line 905:		Line 905:
	RMS error: 2.318 cents		RMS error: 2.318 cents

	~~[[Category:Scales]]~~
	[[Category:MOS Scales]]		[[Category:MOS Scales]]
	[[Category:Diatonic]]		[[Category:Diatonic]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	{{todo\|rework}}		{{todo\|rework}}

Xenllium at 08:07, 22 August 2021

2021-08-22T08:07:41Z

Show changes

← Older revision		Revision as of 19:17, 7 May 2023
Line 468:		Line 468:
	[[Gencom]] [[mapping]]: [{{val\|1 1 7 11 -10 -8 6 }}, {{val\|0 1 -8 -14 23 20 -3 }}]		[[Gencom]] [[mapping]]: [{{val\|1 1 7 11 -10 -8 6 }}, {{val\|0 1 -8 -14 23 20 -3 }}]

	{{~~Val list~~\|legend=1\| 41, 53, 94 }}		{{Optimal ET sequence\|legend=1\| 41, 53, 94 }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 0.6486 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 0.6486 cents
Line 566:		Line 566:
	[[Gencom]] [[mapping]]: [{{val\| 1 1 -6 -3 -1 }}, {{val\| 0 1 15 11 8 }}]		[[Gencom]] [[mapping]]: [{{val\| 1 1 -6 -3 -1 }}, {{val\| 0 1 15 11 8 }}]

	{{~~Val list~~\|legend=1\| 17, 46, 63 }}		{{Optimal ET sequence\|legend=1\| 17, 46, 63 }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 0.7541 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 0.7541 cents
Line 626:		Line 626:
	[[Gencom mapping]]: [{{val\| 1 1 4 }}, {{val\| 0 1 -2 }}]		[[Gencom mapping]]: [{{val\| 1 1 4 }}, {{val\| 0 1 -2 }}]

	{{~~Val list~~\|legend=1\| 5, 12, 17, 22, 27, 137bd }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 27, 137bd }}

	[[Tp_tuning#T2 tuning\|RMS error]]: 1.856 cents		[[Tp_tuning#T2 tuning\|RMS error]]: 1.856 cents
Line 705:		Line 705:
	Gencom mapping: [{{val\| 1 1 4 7 }}, {{val\| 0 1 -2 -6 }}]		Gencom mapping: [{{val\| 1 1 4 7 }}, {{val\| 0 1 -2 -6 }}]

	~~Vals:~~ {{~~Val list~~\| 5, 12, 17, 39c, 56d }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 39c, 56d }}

	RMS error: 1.977 cents		RMS error: 1.977 cents
Line 783:		Line 783:
	[[Wedgie]]: {{wedgie\|1 9 -2 12 -6 -30}}		[[Wedgie]]: {{wedgie\|1 9 -2 12 -6 -30}}

	[[Val]]s: {{~~Val list~~\| 5, 17, 22, 27, 49 }}		[[Val]]s: {{Optimal ET sequence\| 5, 17, 22, 27, 49 }}

	[[Badness]]: 0.032318		[[Badness]]: 0.032318
Line 901:		Line 901:
	Gencom mapping: [{{val\| 1 1 -6 4 -7 }}, {{val\| 0 1 14 -2 18 }}]		Gencom mapping: [{{val\| 1 1 -6 4 -7 }}, {{val\| 0 1 14 -2 18 }}]

	~~Vals:~~ {{~~Val list~~\| 5, 32, 37 }}		{{Optimal ET sequence\|legend=1\| 5, 32, 37 }}

	RMS error: 2.318 cents		RMS error: 2.318 cents

← Older revision		Revision as of 21:32, 2 March 2022
Line 905:		Line 905:
	RMS error: 2.318 cents		RMS error: 2.318 cents

	[[Category:~~MOS Scales~~]]		[[Category:Lists of temperaments]]
	[[Category:Diatonic]]		[[Category:Diatonic]]
	~~[[Category:Regular temperament theory]]~~
	{{todo\|rework}}		{{todo\|rework}}