16th-octave temperaments - Revision history

Tristanbay: /* 7-limit */minor formatting change

2025-07-18T06:58:46Z

7-limit: minor formatting change

← Older revision		Revision as of 06:58, 18 July 2025
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.63		* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.630
	* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.817		* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.817

Tristanbay: /* Hilarity */Added optimal CTE and CWE tunings

2025-07-18T06:58:06Z

Hilarity: Added optimal CTE and CWE tunings

← Older revision		Revision as of 06:58, 18 July 2025
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	== Hilarity ==		== Hilarity ==
	Hilarity tempers [[40/27]] to 9\16.		Hilarity, the [[80edo\|80]] & [[128edo\|128]] temperament, tempers [[40/27]] to 9\16.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	: mapping generators: ~10460353203/10240000000, ~3		: mapping generators: ~10460353203/10240000000, ~3

			[[Optimal tuning]]s:
			* [[CTE]]: ~10460353203/1024000000 = 1\16, ~3/2 = 703.422
			* [[CWE]]: ~10460353203/1024000000 = 1\16, ~3/2 = 703.621

	{{Optimal ET sequence\|legend=1\| 16cc, 32c, 48, 80, 128, 208 }}		{{Optimal ET sequence\|legend=1\| 16cc, 32c, 48, 80, 128, 208 }}
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			[[Optimal tuning]]s:
			* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.63
			* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.817

	{{Optimal ET sequence\|legend=1\| 32cdd, 48d, 80, 128, 208, 496bd }}		{{Optimal ET sequence\|legend=1\| 32cdd, 48d, 80, 128, 208, 496bd }}
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			[[Optimal tuning]]s:
			* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.593
			* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.882

	{{Optimal ET sequence\|legend=1\| 48d, 80, 128, 208, 288be, 496bdee }}		{{Optimal ET sequence\|legend=1\| 48d, 80, 128, 208, 288be, 496bdee }}
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			[[Optimal tuning]]s:
			* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.653
			* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.902

	{{Optimal ET sequence\|legend=1\| 80, 128, 208, 496bdeef }}		{{Optimal ET sequence\|legend=1\| 80, 128, 208, 496bdeef }}

Tristanbay: /* Hilarity */Added optimal et sequences and badnesses (Smith logflat)

2025-07-18T06:45:49Z

Hilarity: Added optimal et sequences and badnesses (Smith logflat)

← Older revision		Revision as of 06:45, 18 July 2025
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	: mapping generators: ~10460353203/10240000000, ~3		: mapping generators: ~10460353203/10240000000, ~3

			{{Optimal ET sequence\|legend=1\| 16cc, 32c, 48, 80, 128, 208 }}

			[[Badness]]: 4.22

	=== 7-limit ===		=== 7-limit ===
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			{{Optimal ET sequence\|legend=1\| 32cdd, 48d, 80, 128, 208, 496bd }}

			[[Badness]]: 0.302

	=== 11-limit ===		=== 11-limit ===
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			{{Optimal ET sequence\|legend=1\| 48d, 80, 128, 208, 288be, 496bdee }}

			[[Badness]]: 0.0886

	=== 13-limit ===		=== 13-limit ===
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	: mapping generators: ~392/375, ~3		: mapping generators: ~392/375, ~3

			{{Optimal ET sequence\|legend=1\| 80, 128, 208, 496bdeef }}

			[[Badness]]: 0.0611

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Tristanbay: /* Ntiscifer */Added hilarity temperament

2025-07-10T17:32:15Z

Ntiscifer: Added hilarity temperament

← Older revision		Revision as of 17:32, 10 July 2025
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	[[Badness]]: 3.05		[[Badness]]: 3.05

			== Hilarity ==
			Hilarity tempers [[40/27]] to 9\16.

			[[Subgroup]]: 2.3.5

			[[Comma list]]: {{monzo\| 39 -48 16 }}

			{{Mapping\|legend=1\| 16 0 -39 \| 0 1 3 }}

			: mapping generators: ~10460353203/10240000000, ~3

			=== 7-limit ===
			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 179200/177147, 390625/388962

			{{Mapping\|legend=1\| 16 0 -39 -82 \| 0 1 3 5 }}

			: mapping generators: ~392/375, ~3

			=== 11-limit ===
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 896/891, 2200/2187, 125000/124509

			{{Mapping\|legend=1\| 16 0 -39 -82 30 \| 0 1 3 5 1 }}

			: mapping generators: ~392/375, ~3

			=== 13-limit ===
			[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: 325/324, 352/351, 896/891, 540000/539539

			{{Mapping\|legend=1\| 16 0 -39 -82 30 110 \| 0 1 3 5 1 -2 }}

			: mapping generators: ~392/375, ~3

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

ArrowHead294 at 03:57, 14 February 2025

2025-02-14T03:57:52Z

← Older revision		Revision as of 03:57, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|16}}		{{Infobox fractional-octave\|16}}
	16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.		16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.

Xenllium at 12:56, 30 December 2024

2024-12-30T12:56:32Z

← Older revision		Revision as of 12:56, 30 December 2024
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	16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.		16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.

	~~A temperament~~ discussed elsewhere is [[hexadecoid]]~~, a weak extension of octopus / octoid to the 19-limit with slightly different mappings~~.		Temperaments discussed elsewhere include [[hexadecoid]] and [[Jubilismic clan\|sedecic]].

	== Sulfur ==		== Sulfur ==
	Subgroup: 2.3.5		Subgroup: 2.3.5

	Comma list: {{monzo\| 115 96 -16 }}		[[Comma list]]: {{monzo\| -115 96 -16 }}

	{{Mapping\|legend=1\| 16 0 -115 \| 0 1 6 }}		{{Mapping\|legend=1\| 16 0 -115 \| 0 1 6 }}

	: ~~mapping~~ generators: ~214748364800000/205891132094649 = ~~1\16~~, ~3		: Mapping generators: ~214748364800000/205891132094649 = 1＼16, ~3

	[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.~~980~~		[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.8895

	[[Support]]ing [[ET]]s: {{EDOs\|48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472}}		[[Support]]ing [[ET]]s: {{EDOs\|48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472}}

			=== 7-limit ===
			Subgroup: 2.3.5.7

			Comma list: 14348907/14336000, 2147483648/2144153025

			{{Mapping\|legend=1\| 16 0 -115 121 \| 0 1 6 -3 }}

			: Mapping generators: ~256/245 = 1＼16, ~3

			[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9129

			[[Optimal ET sequence]]: {{EDOs\|48, 128c, 176, 224, 400, 624 }}

			[[Badness]]: 0.166051

			=== 11-limit ===
			Subgroup: 2.3.5.7.11

			Comma list: 9801/9800, 46656/46585, 131072/130977

			Mapping: {{Mapping\| 16 0 -115 121 30 \| 0 1 6 -3 1 }}

			: Mapping generators: ~256/245 = 1＼16, ~3

			Optimal tuning (CTE): ~3/2 = 701.9070

			Optimal ET sequence: {{EDOs\|48, 128c, 176, 224, 400, 624 }}

			Badness: 0.041764

			=== 13-limit ===
			Subgroup: 2.3.5.7.11.13

			Comma list: 1716/1715, 2080/2079, 4096/4095, 39366/39325

			Mapping: {{Mapping\| 16 0 -115 121 30 186 \| 0 1 6 -3 1 -5 }}

			: Mapping generators: ~117/112 = 1＼16, ~3

			Optimal tuning (CTE): ~3/2 = 701.9047

			Optimal ET sequence: {{EDOs\|48, 128cf, 176, 224, 400, 624, 848 }}

			Badness: 0.025967

			=== 17-limit ===
			Subgroup: 2.3.5.7.11.13.17

			Comma list: 936/935, 1701/1700, 1716/1715, 4096/4095, 11016/11011

			Mapping: {{Mapping\| 16 0 -115 121 30 186 319 \| 0 1 6 -3 1 -5 -10 }}

			: Mapping generators: ~117/112 = 1＼16, ~3

			Optimal tuning (CTE): ~3/2 = 701.9418

			Optimal ET sequence: {{EDOs\|176, 224, 400, 624 }}

			Badness: 0.023704

	=== 19-limit ===		=== 19-limit ===
			Subgroup: 2.3.5.7.11.13.17.19

	~~Subgroup~~: ~~2.3.5.7.11.13.17.19~~		Comma list: 936/935, 1521/1520, 1701/1700, 1716/1715, 4096/4095, 11016/11011

	~~Comma list~~: ~~936/935, 1716/1715, 2080/2079, 2376/2375, 4096/4095, 11016/11011~~		Mapping: {{Mapping\| 16 0 -115 121 30 186 319 423 \| 0 1 6 -3 1 -5 -10 -14 }}

	{{Mapping~~\|legend~~=~~1\| 16 0 -115 121 30 186 319 423 \| 0 1 6 -~~3 ~~1 -5 -10 -14 }}~~		: Mapping generators: ~117/112 = 1＼16, ~3

	~~: mapping generators~~: ~~~117~~/~~112~~ = ~~1\16, ~3~~		Optimal tuning (CTE): ~3/2 = 701.9505

	[[Optimal ~~tuning]] ([[CTE]])~~: ~~~3/2 = 701.950~~		Optimal ET sequence: {{EDOs\|176h, 224, 400, 624 }}

	~~[[Support]]ing [[ET]]s~~: ~~{{EDOs\|48ghh, 176h, 224, 400, 624, 848gh, 1024e}}~~		Badness: 0.020421

	== Ntiscifer ==		== Ntiscifer ==

BudjarnLambeth: navbox

2024-12-05T07:18:14Z

navbox

← Older revision		Revision as of 07:18, 5 December 2024
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	[[Badness]]: 3.05		[[Badness]]: 3.05

			{{Navbox fractional-octave}}

FloraC at 05:41, 4 July 2024

2024-07-04T05:41:30Z

← Older revision		Revision as of 05:41, 4 July 2024
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	{{Main\| Ntiscifer }}		{{Main\| Ntiscifer }}

	Ntiscifer is equivalent to the 16edo [[circle of fifths]] with an added dimension for [[5/4]]. In 16edo, this maps 5/4 to 375 cents, as in mavila temperament. Tunings with a separate, more accurate third include [[64edo]], [[80edo]], and [[96edo]]; 96edo is a particularly accurate tuning, though 64edo might be considered more practical.		Ntiscifer tempers out the Pythagorean double-augmented second, and is equivalent to the 16edo [[circle of fifths]] with an added dimension for [[5/4]]. In 16edo, this maps 5/4 to 375 cents, as in mavila temperament. Tunings with a separate, more accurate third include [[64edo]], [[80edo]], and [[96edo]]; 96edo is a particularly accurate tuning, though 64edo might be considered more practical.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{Mapping\|legend=1\| 16 25 0 \| 0 0 1 }}		{{Mapping\|legend=1\| 16 25 0 \| 0 0 1 }}

			: mapping generators: ~2048/2187, ~5

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

FloraC: Data copypasted from the main page

2024-07-04T05:37:41Z

Data copypasted from the main page

← Older revision		Revision as of 05:37, 4 July 2024
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	[[Support]]ing [[ET]]s: {{EDOs\|48ghh, 176h, 224, 400, 624, 848gh, 1024e}}		[[Support]]ing [[ET]]s: {{EDOs\|48ghh, 176h, 224, 400, 624, 848gh, 1024e}}

			== Ntiscifer ==
			{{Main\| Ntiscifer }}

			Ntiscifer is equivalent to the 16edo [[circle of fifths]] with an added dimension for [[5/4]]. In 16edo, this maps 5/4 to 375 cents, as in mavila temperament. Tunings with a separate, more accurate third include [[64edo]], [[80edo]], and [[96edo]]; 96edo is a particularly accurate tuning, though 64edo might be considered more practical.

			[[Subgroup]]: 2.3.5

			[[Comma list]]: [[43046721/33554432]]

			{{Mapping\|legend=1\| 16 25 0 \| 0 0 1 }}

			[[Optimal tuning]]s:
			* [[CTE]]: ~2048/2187 = 1\16, ~5/4 = 386.3137 (~135/128 = 11.3137)
			* [[CWE]]: ~2048/2187 = 1\16, ~5/4 = 373.1508 (~128/135 = 1.8492)

			{{Optimal ET sequence\|legend=1\| 16 }}

			[[Badness]]: 3.05

Godtone: add hexadecoid

2024-06-30T01:42:12Z

add hexadecoid

← Older revision		Revision as of 01:42, 30 June 2024
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	{{Fractional-octave navigation\|16}}		{{Fractional-octave navigation\|16}}
	16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.		16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.

			A temperament discussed elsewhere is [[hexadecoid]], a weak extension of octopus / octoid to the 19-limit with slightly different mappings.

	== Sulfur ==		== Sulfur ==