Pythagorean family - Revision history

FloraC: This is ambiguous too

2023-06-17T07:10:10Z

This is ambiguous too

← Older revision		Revision as of 07:10, 17 June 2023
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	#REDIRECT [[~~Compton family~~]]		#REDIRECT [[Pythagorean]]

Lhearne: moved content to 'Compton family' and changed to redirect page

2021-04-21T15:42:13Z

moved content to 'Compton family' and changed to redirect page

Show changes

FloraC: The 5-limit version should be called compton

2021-04-21T11:43:27Z

The 5-limit version should be called compton

← Older revision		Revision as of 11:43, 21 April 2021
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			: ''Not to be confused with [[Pythagorean tuning]].''

	The '''Pythagorean family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.		The '''Pythagorean family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

	= ~~Pythagorean~~ =		= Compton =
	~~: ''Not~~ to be ~~confused with~~ [[Pythagorean tuning]].''
			In terms of the normal list, compton adds 413343/409600 = {{monzo\| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds [[225/224]]. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.

			In either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

			In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this [[72edo]] can be recommended as a tuning.

	Comma list: 531441/524288		Comma list: 531441/524288
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	{{Val list\|legend=1\| 12, 72, 84, 156, 240, 396b }}		{{Val list\|legend=1\| 12, 72, 84, 156, 240, 396b }}

	~~= Compton =~~
	In terms of the normal list, compton adds 413343/409600 = {{monzo\| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds [[225/224]]. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.

	In either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

	~~In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this [[72edo]] can be recommended as a tuning.~~

	== 7-limit (aka Waage) ==		== 7-limit (aka Waage) ==
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	Badness: 0.022235		Badness: 0.022235

	== 13-limit ==		=== 13-limit ===
	Comma list: 225/224, 351/350, 364/363, 441/440		Comma list: 225/224, 351/350, 364/363, 441/440

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	Badness: 0.021852		Badness: 0.021852

	== 17-limit ==		==== 17-limit ====
	Comma list: 221/220, 225/224, 289/288, 351/350, 441/440		Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

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	Badness: 0.017131		Badness: 0.017131

	== Comptone ==		=== Comptone ===
	Comma list: 225/224, 325/324, 441/440, 1001/1000		Comma list: 225/224, 325/324, 441/440, 1001/1000

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	Badness: 0.025144		Badness: 0.025144

	=== 17-limit ===		==== 17-limit ====
	Comma list: 225/224, 273/272, 289/288, 325/324, 441/440		Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

Xenllium: /* Compton */

2021-03-19T11:54:39Z

Compton

← Older revision		Revision as of 11:54, 19 March 2021
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	In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this [[72edo]] can be recommended as a tuning.		In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this [[72edo]] can be recommended as a tuning.

			== 7-limit (aka Waage) ==
	Comma list: 225/224, 250047/250000		Comma list: 225/224, 250047/250000

	[[POTE generator]]: ~5/4 = 383.~~775 or ~81/80 = 16.225~~		[[POTE generator]]: ~5/4 = 383.7752

	Mapping: [{{val\| 12 19 0 -22 }}, {{val\| 0 0 1 2 }}]		Mapping: [{{val\| 12 19 0 -22 }}, {{val\| 0 0 1 2 }}]

	{{Val list\|legend=1\| 12, 60, 72, 228, 300c, 372bc, 444bc }}		{{Val list\|legend=1\| 12, 60, 72, 228, 300c, 372bc, 444bc }}

			Badness: 0.035686

	== 11-limit ==		== 11-limit ==
	Comma list: 225/224, 441/440, 4375/4356		Comma list: 225/224, 441/440, 4375/4356

	[[POTE generator]]: ~5/4 = 383.~~266 or ~81/80 = 16.734~~		[[POTE generator]]: ~5/4 = 383.2660

	Mapping: [{{val\|12 19 0 -22 -42 }}, {{val\| 0 0 1 2 3 }}]		Mapping: [{{val\|12 19 0 -22 -42 }}, {{val\| 0 0 1 2 3 }}]
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	{{Val list\|legend=1\| 12, 60e, 72 }}		{{Val list\|legend=1\| 12, 60e, 72 }}

	~~=== 13-limit ===~~		Badness: 0.022235
	~~Comma list~~: ~~225/224, 441/440, 351/350, 364/363~~

	POTE generator: ~5/4 = 383.~~963 or ~81/80 = 16.037~~		== 13-limit ==
			Comma list: 225/224, 351/350, 364/363, 441/440

			POTE generator: ~5/4 = 383.9628

	Mapping: [{{val\| 12 19 0 -22 -42 -67 }}, {{val\| 0 0 1 2 3 4 }}]		Mapping: [{{val\| 12 19 0 -22 -42 -67 }}, {{val\| 0 0 1 2 3 4 }}]

	{{Val list\|legend=1\| 72, 228f, 300cf }}		{{Val list\|legend=1\| 12f, 72, 84, 156, 228f, 300cf }}

			Badness: 0.021852

			== 17-limit ==
			Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

			POTE generator: ~5/4 = 383.7500

			Mapping: [{{val\| 12 19 0 -22 -42 -67 49 }}, {{val\| 0 0 1 2 3 4 0 }}]

			{{Val list\|legend=1\| 12f, 72, 84, 156g, 228fg }}

	Badness: 0.~~0219~~		Badness: 0.017131

	=== Comptone ===		== Comptone ==
	Comma list: 225/224, 441/440~~, 325/324~~, 1001/1000		Comma list: 225/224, 325/324, 441/440, 1001/1000

	POTE generator: ~5/4 = 382.~~612 or ~81/80 = 17.388~~		POTE generator: ~5/4 = 382.6116

	Mapping: [{{val\| 12 19 0 -22 -42 100 }}, {{val\| 0 0 1 2 3 -2 }}]		Mapping: [{{val\| 12 19 0 -22 -42 100 }}, {{val\| 0 0 1 2 3 -2 }}]
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	{{Val list\|legend=1\| 12, 60e, 72, 204cdef, 276cdef }}		{{Val list\|legend=1\| 12, 60e, 72, 204cdef, 276cdef }}

	Badness: 0.~~0251~~		Badness: 0.025144

			=== 17-limit ===
			Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

			POTE generator: ~5/4 = 382.5968

			Mapping: [{{val\| 12 19 0 -22 -42 100 49 }}, {{val\| 0 0 1 2 3 -2 0 }}]

			{{Val list\|legend=1\| 12, 60e, 72, 132deg, 204cdefg }}

			Badness: 0.016361

	= Catler =		= Catler =

FloraC: Bold lemma; +dis-confusion reminder

2021-03-13T05:00:39Z

Bold lemma; +dis-confusion reminder

← Older revision		Revision as of 05:00, 13 March 2021
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	The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.		The '''Pythagorean family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

	= Pythagorean =		= Pythagorean =
			: ''Not to be confused with [[Pythagorean tuning]].''

	Comma list: 531441/524288		Comma list: 531441/524288

FloraC: Cleanup

2021-03-11T08:52:26Z

Cleanup

← Older revision		Revision as of 08:52, 11 March 2021
Line 1:		Line 1:
	The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.		The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

	= Pythagorean =		= Pythagorean =
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	= Compton =		= Compton =
	In terms of the normal list, compton adds 413343/409600 = {{monzo\| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.		In terms of the normal list, compton adds 413343/409600 = {{monzo\| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds [[225/224]]. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.

	In ~~the~~ either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.		In either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

	In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this [[72edo]] can be recommended as a tuning.		In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this [[72edo]] can be recommended as a tuning.

	Comma list: 225/224, 250047/250000		Comma list: 225/224, 250047/250000
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	Mapping: [{{val\| 12 19 0 -22 -42 -67 }}, {{val\| 0 0 1 2 3 4 }}]		Mapping: [{{val\| 12 19 0 -22 -42 -67 }}, {{val\| 0 0 1 2 3 4 }}]

	{{Val list\|legend=1\| 72, 228f, 300cf }}		{{Val list\|legend=1\| 72, 228f, 300cf }}

	Badness: 0.0219		Badness: 0.0219
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	= Catler =		= Catler =
	In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7, 7/5, and most importantly, 64/63.		In terms of the normal comma list, catler is characterized by the addition of the [[schisma]], 32805/32768, to the Pythagorean comma, though it can also be characterized as adding [[81/80]], [[128/125]] or [[648/625]]. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7, 7/5, and most importantly, 64/63.

	Comma list: 81/80, 128/125		Comma list: 81/80, 128/125
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	Badness: 0.0300		Badness: 0.0300

	=Hours=		= Hours =
	~~Commas~~: 19683/19600, 33075/32768		Comma list: 19683/19600, 33075/32768

	POTE generator: ~225/224 = 2.100		POTE generator: ~225/224 = 2.100

	~~Map~~: [~~<~~24 38 0 123 83\|~~, <~~0 0 1 -1 0\|]		Mapping: [{{val\| 24 38 0 123 83 }}, {{val\| 0 0 1 -1 0 }}]

	~~Wedgie: <~~0 24 -24 38 -38 -123\|		{{Multival\|legend=1\| 0 24 -24 38 -38 -123 }}

	~~EDOs:~~ 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd		{{Val list\|legend=1\| 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd }}

	Badness: 0.1161		Badness: 0.1161

	==11-limit==		== 11-limit ==
	~~Commas~~: 243/242, 385/384, 9801/9800		Comma list: 243/242, 385/384, 9801/9800

	POTE generator: ~225/224 = 2.161		POTE generator: ~225/224 = 2.161

	~~Map~~: [~~<~~24 38 0 123 83\|~~, <~~0 0 1 -1 0\|]		Mapping: [{{val\| 24 38 0 123 83 }}, {{val\| 0 0 1 -1 0 }}]

	~~EDOs:~~ 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde		{{Val list\|legend=1\| 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde }}

	Badness: 0.0362		Badness: 0.0362

	==13-limit==		== 13-limit ==
	~~Commas~~: 243/242, 351/350, 364/363, 385/384		Comma list: 243/242, 351/350, 364/363, 385/384

	POTE generator: ~225/224 = 3.955		POTE generator: ~225/224 = 3.955

	~~Map~~: [~~<~~24 38 0 123 83 33\|~~, <~~0 0 1 -1 0 1\|]		Mapping: [{{val\| 24 38 0 123 83 33 }}, {{val\| 0 0 1 -1 0 1 }}]

	~~EDOs:~~ 24, 48f, 72, 168df, 240df		{{Val list\|legend=1\| 24, 48f, 72, 168df, 240df }}

	Badness: 0.0269		Badness: 0.0269

FloraC: Cleanup +1

2021-03-11T08:41:39Z

Cleanup +1

← Older revision		Revision as of 08:41, 11 March 2021
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	{{Val list\|legend=1\| 12, 36, 48, 132, 180 }}		{{Val list\|legend=1\| 12, 36, 48, 132, 180 }}

	==11-limit==		== 11-limit ==
	~~Commas~~: 81/80, 99/98, 128/125		Comma list: 81/80, 99/98, 128/125

	POTE generator: ~36/35 = 22.723		POTE generator: ~64/63 = 22.723

	~~Map~~: [~~<~~12 19 28 0 -26\|~~, <~~0 0 0 1 2\|]		Mapping: [{{val\| 12 19 28 0 -26 }}, {{val\| 0 0 0 1 2 }}]

	~~EDOs:~~ 12, 48c, 108cd		{{Val list\|legend=1\| 12, 48c, 108cd }}

	Badness: 0.0582		Badness: 0.0582

	==Catlat==		== Catlat ==
	~~Commas~~: 81/80, 128/125, 540/539		Comma list: 81/80, 128/125, 540/539

	POTE generator: ~36/35 = 27.864		POTE generator: ~64/63 = 27.864

	~~Map~~: [~~<~~12 19 28 0 109\|~~, <~~0 0 0 1 -2\|]		Mapping: [{{val\| 12 19 28 0 109 }}, {{val\| 0 0 0 1 -2 }}]

	~~EDOs:~~ 36, 48c, 84c		{{Val list\|legend=1\| 36, 48c, 84c }}

	Badness: 0.0819		Badness: 0.0819

	==Catcall==		== Catcall ==
	~~Commas~~: 56/55, 81/80, 128/125		Comma list: 56/55, 81/80, 128/125

	POTE generator: ~36/35 = 32.776		POTE generator: ~36/35 = 32.776

	~~Map~~: [~~<~~12 19 28 0 8\|~~, <~~0 0 0 1 1\|]		Mapping: [{{val\| 12 19 28 0 8 }}, {{val\| 0 0 0 1 1 }}]

	~~EDOs:~~ 12, 24, 36, 72ce		{{Val list\|legend=1\| 12, 24, 36, 72ce }}

	Badness: 0.0345		Badness: 0.0345

	===13-limit===		=== 13-limit ===
	~~Commas~~: 56/55, 66/65, 81/80, 105/104		Comma list: 56/55, 66/65, 81/80, 105/104

	POTE generator: ~36/35 = 37.232		POTE generator: ~36/35 = 37.232

	~~Map~~: [~~<~~12 19 28 0 8 11\|~~, <~~0 0 0 1 1 1\|]		Mapping: [{{val\| 12 19 28 0 8 11 }}, {{val\| 0 0 0 1 1 1 }}]

	~~EDOs:~~ 12f, 24, 36f, 60cf		{{Val list\|legend=1\| 12f, 24, 36f, 60cf }}

	Badness: 0.0284		Badness: 0.0284

	==Duodecic==		=== Duodecic ===
	~~Commas~~: 56/55, 81/80, 91/90, 128/125		Comma list: 56/55, 81/80, 91/90, 128/125

	POTE generator: ~36/35 = 37.688		POTE generator: ~36/35 = 37.688

	~~Map~~: [~~<~~12 19 28 0 8 78\|~~, <~~0 0 0 1 1 -1\|]		Mapping: [{{val\| 12 19 28 0 8 78 }}, {{val\| 0 0 0 1 1 -1 }}]

	~~EDOs:~~ 12, 24, 36, 60c		{{Val list\|legend=1\| 12, 24, 36, 60c }}

	Badness: 0.0383		Badness: 0.0383

	===17-limit===		==== 17-limit ====
	~~Commas~~: 51/50, 56/55, 81/80, 91/90, 128/125		Comma list: 51/50, 56/55, 81/80, 91/90, 128/125

	POTE generator: ~36/35 = 38.097		POTE generator: ~36/35 = 38.097

	~~Map~~: [~~<~~12 19 28 0 8 78 49\|~~, <~~0 0 0 1 1 -1 0\|]		Mapping: [{{val\| 12 19 28 0 8 78 49 }}, {{val\| 0 0 0 1 1 -1 0 }}]

	~~EDOs:~~ 12, 24, 36, 60c		{{Val list\|legend=1\| 12, 24, 36, 60c }}

	Badness: 0.0275		Badness: 0.0275

	===19-limit===		==== 19-limit ====
	~~Commas~~: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95		Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95

	POTE generator: ~36/35 = 38.080		POTE generator: ~36/35 = 38.080

	~~Map~~: [~~<~~12 19 28 0 8 78 49 51\|~~, <~~0 0 0 1 1 -1 0 0\|]		Mapping: [{{val\| 12 19 28 0 8 78 49 51 }}, {{val\| 0 0 0 1 1 -1 0 0 }}]

	~~EDOs:~~ 12, 24, 36, 60c		{{Val list\|legend=1\| 12, 24, 36, 60c }}

	Badness: 0.0209		Badness: 0.0209

	==Duodecim==		= Duodecim =
	~~Commas~~: 36/35, 50/49, 64/63		Comma list: 36/35, 50/49, 64/63

	POTE generator: ~45/44 = 34.977		POTE generator: ~45/44 = 34.977

	~~Map~~: [~~<~~12 19 28 34 0\|~~, <~~0 0 0 0 1\|]		Mapping: [{{val\| 12 19 28 34 0 }}, {{val\| 0 0 0 0 1 }}]

	~~EDOs:~~ 12, 24d		{{Val list\|legend=1\| 12, 24d }}

	=Omicronbeta ~~temperament~~=		= Omicronbeta =
	~~Commas~~: 225/224, 243/242, 441/440, 4375/4356		Comma list: 225/224, 243/242, 441/440, 4375/4356

	~~Generator~~: ~13/8 = 837.814		POTE generator: ~13/8 = 837.814

	~~Map~~: [~~<~~72 114 167 202 249 266\|~~, <~~0 0 0 0 0 1\|]		Mapping: [{{val\| 72 114 167 202 249 266 }}, {{val\| 0 0 0 0 0 1 }}]

	~~EDOs:~~ 72, 144, 216c, 288cdf, 504bcdef		{{Val list\|legend=1\| 72, 144, 216c, 288cdf, 504bcdef }}

	Badness: 0.0300		Badness: 0.0300

FloraC: Cleanup

2021-03-11T08:25:47Z

Cleanup

← Older revision		Revision as of 08:25, 11 March 2021
Line 1:		Line 1:
	~~__FORCETOC__~~		The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = {{monzo\| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.

	~~The Pythagorean family tempers out the~~ Pythagorean ~~comma, 531441/524288~~ = \|-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo\|12edo]]. While the tuning of the fifth will be that of 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.		= Pythagorean =

	~~[[POTE_tuning\|POTE generator]]~~: ~5/~~4 = 384.884 (15.116)~~		Comma list: 531441/524288

	~~Map:~~ [~~<12 19 0\|, <0 0 1\|~~]		[[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116

	~~EDOs~~: [~~[12edo~~\|12~~]], [[72edo\|72]]~~, ~~[[84edo~~\|~~84]], 156, 240, 396b~~		Mapping: [{{val\| 12 19 0 }}, {{val\| 0 0 1 }}

	=Compton ~~temperament~~=		{{Val list\|legend=1\| 12, 72, 84, 156, 240, 396b }}
	In terms of the normal list, compton adds 413343/409600 = \|-14 10 -2 1~~>~~ to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.
			= Compton =
			In terms of the normal list, compton adds 413343/409600 = {{monzo\| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds 225/224. Compton, however, does not need to be used as a 7-limit temperament; in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.

	In the either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.		In the either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. The major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.
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	In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this [[72edo]] can be recommended as a tuning.		In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440. For this [[72edo]] can be recommended as a tuning.

	~~Commas~~: 225/224, 250047/250000		Comma list: 225/224, 250047/250000

	[[~~POTE_tuning\|~~POTE generator]]: ~5/4 = 383.775 (16.225)		[[POTE generator]]: ~5/4 = 383.775 or ~81/80 = 16.225

	~~Map~~: [~~<~~12 19 0 -22\|~~, <~~0 0 1 2\|]		Mapping: [{{val\| 12 19 0 -22 }}, {{val\| 0 0 1 2 }}]

	~~EDOs:~~ 12, ~~[[60edo\|~~60]], 72, 228, 300c, 372bc, 444bc		{{Val list\|legend=1\| 12, 60, 72, 228, 300c, 372bc, 444bc }}

	==11-limit==		== 11-limit ==
	~~Commas~~: 225/224, 441/440, 4375/4356		Comma list: 225/224, 441/440, 4375/4356

	[[~~POTE_tuning\|~~POTE generator]]: ~5/4 = 383.266 (16.734)		[[POTE generator]]: ~5/4 = 383.266 or ~81/80 = 16.734

	~~Map~~: [~~<~~12 19 0 -22 -42\|~~, <~~0 0 1 2 3\|]		Mapping: [{{val\|12 19 0 -22 -42 }}, {{val\| 0 0 1 2 3 }}]

	~~EDOs:~~ 12, 60e, 72		{{Val list\|legend=1\| 12, 60e, 72 }}

	==13-limit==		=== 13-limit ===
	~~Commas~~: 225/224, 441/440, 351/350, 364/363		Comma list: 225/224, 441/440, 351/350, 364/363

	POTE generator: ~5/4 = 383.963 (16.037)		POTE generator: ~5/4 = 383.963 or ~81/80 = 16.037

	~~Map~~: [~~<~~12 19 0 -22 -42 -67\|~~, <~~0 0 1 2 3 4\|]		Mapping: [{{val\| 12 19 0 -22 -42 -67 }}, {{val\| 0 0 1 2 3 4 }}]

	~~EDOs:~~ 72, 228f, 300cf		{{Val list\|legend=1\| 72, 228f, 300cf }}

	Badness: 0.0219		Badness: 0.0219

	==Comptone==		=== Comptone ===
	~~Commas~~: 225/224, 441/440, 325/324, 1001/1000		Comma list: 225/224, 441/440, 325/324, 1001/1000

	POTE generator: ~5/4 = 382.612 (17.388)		POTE generator: ~5/4 = 382.612 or ~81/80 = 17.388

	~~Map~~: [~~<~~12 19 0 -22 -42 100\|~~, <~~0 0 1 2 3 -2\|]		Mapping: [{{val\| 12 19 0 -22 -42 100 }}, {{val\| 0 0 1 2 3 -2 }}]

	~~EDOs:~~ 12, 60e, 72, 204cdef, 276cdef		{{Val list\|legend=1\| 12, 60e, 72, 204cdef, 276cdef }}

	Badness: 0.0251		Badness: 0.0251

	=Catler ~~temperament~~=		= Catler =
	In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7, 7/5, and most importantly, 64/63.		In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo]]. Catler can also be characterized as the 12&24 temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 6/5, 9/7, 7/5, and most importantly, 64/63.

	~~Commas~~: 81/80, 128/125		Comma list: 81/80, 128/125

	[[~~POTE_tuning\|~~POTE generator]]: 26.790		[[POTE generator]]: ~64/63 = 26.790

	~~Map~~: [~~<~~12 19 28 0\|~~, <~~0 0 0 1\|]		Mapping: [{{val\| 12 19 28 0 }}, {{val\| 0 0 0 1 }}]

	~~EDOs:~~ 12, ~~[[36edo\|~~36]], ~~[[48edo\|~~48]], 132, 180		{{Val list\|legend=1\| 12, 36, 48, 132, 180 }}

	==11-limit==		==11-limit==

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2020-10-08T09:48:00Z

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← Older revision		Revision as of 09:48, 8 October 2020
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	[[Category:Theory]]		[[Category:Theory]]
			[[Category:Temperament family]]
	[[Category:Pythagorean]]		[[Category:Pythagorean]]
	~~[[Category:Family]]~~
	[[Category:Rank 2]]		[[Category:Rank 2]]

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2020-05-31T09:23:27Z

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← Older revision		Revision as of 09:23, 31 May 2020
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	~~[[Category:family]]~~
	~~[[Category:list]]~~		[[Category:Theory]]
	[[Category:~~overview~~]]		[[Category:Pythagorean]]
	[[Category:~~pythagorean~~]]		[[Category:Family]]
	[[Category:~~rank_2~~]]		[[Category:Rank 2]]
	[[Category:~~theory~~]]