Miscellaneous 7-limit temperaments - Revision history

Overthink: /* Aristoschismic */ important detail

2026-09-01T22:56:55Z

Aristoschismic: important detail

← Older revision		Revision as of 22:56, 1 September 2026
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	== Aristoschismic ==		== Aristoschismic ==
	Aristoschismic tempers out the [[aristoschisma]], which is the difference between the [[septischisma]] and the [[schisma]]. It sets the [[Pythagorean comma]] to the mean of the [[syntonic comma]] and the [[septimal comma]], so like [[orthoschismic]] and [[septischismic]], it functions as a [[detemperament]] of [[garibaldi]]. [[441edo]] shows an excellent example of an edo tuning, but it can be tuned even more accurately than that.		Aristoschismic tempers out the [[aristoschisma]], which is the difference between the [[septischisma]] and the [[schisma]]. It sets the [[Pythagorean comma]] to the mean of the [[syntonic comma]] and the [[septimal comma]], so like [[orthoschismic]] and [[septischismic]], it functions as a [[detemperament]] of [[garibaldi]]. [[441edo]] shows an excellent example of an edo tuning (with the chain of fifths from [[147edo]]), but it can be tuned even more accurately than that.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: + aristoschismic

2026-08-30T07:37:32Z

+ aristoschismic

← Older revision		Revision as of 07:37, 30 August 2026
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	[[Badness]] (Sintel): 8.77		[[Badness]] (Sintel): 8.77

			== Aristoschismic ==
			Aristoschismic tempers out the [[aristoschisma]], which is the difference between the [[septischisma]] and the [[schisma]]. It sets the [[Pythagorean comma]] to the mean of the [[syntonic comma]] and the [[septimal comma]], so like [[orthoschismic]] and [[septischismic]], it functions as a [[detemperament]] of [[garibaldi]]. [[441edo]] shows an excellent example of an edo tuning, but it can be tuned even more accurately than that.

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: {{monzo\| 40 -22 -1 -1 }}

			{{Mapping\|legend=1\| 1 0 0 40 \| 0 1 0 -22 \| 0 0 1 -1 }}
			: Mapping generators: ~2, ~3, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9738{{c}}, ~3/2 = 702.0173{{c}}, ~5/4 = 386.3696{{c}}
			: [[Error map]]: {{val\| -0.026 +0.036 +0.004 +0.005 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0343{{c}}, ~5/4 = 386.3610{{c}}
			: Error map: {{val\| 0.000 +0.079 +0.047 +0.058 }}

			{{Optimal ET sequence\|legend=1\| 41, 53, 94, …, 294, 306, 335, 347, 388, 400, 441, 935, 1376, 3193, 4087, 4528, 5463, 5904, 7280, 12743bcd, 13184bcd, 18647bbccdd }}

			[[Badness]] (Sintel): 11.2

	== Septimagic ==		== Septimagic ==

FloraC: + intro to wizmic & others (hopefully to expand later)

2026-07-30T10:53:53Z

+ intro to wizmic & others (hopefully to expand later)

← Older revision		Revision as of 10:53, 30 July 2026
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	== Wizmic ==		== Wizmic ==
			Wizmic tempers out the [[wizma]], the sum of the two smallest [[superparticular ratio]]s in the 7-limit. It splits the [[septimal comma]] into two equal parts, which can serve as the last generator of this temperament, besides the [[octave]] and the [[3/2\|perfect fifth]]. Adding a perfect fifth to this gives a [[245/162\|sensamagic fifth]], which is the generator shown in the data below. Two sensamagic fifths minus an octave give [[~]][[8/7]], and five minus two octaves give a sub-octave short of the octave by a [[syntonic comma]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	== Breeze ==		== Breeze ==
			Breeze tempers out the [[breeze comma]] in the 7-limit.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	== Scheme ==		== Scheme ==
			Scheme tempers out the [[scheme comma]] in the 7-limit.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: + wizmic

2026-07-30T09:46:07Z

+ wizmic

← Older revision		Revision as of 09:46, 30 July 2026
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	See also [[Miscellaneous 5-limit temperaments]].		See also [[Miscellaneous 5-limit temperaments]].

			== Wizmic ==
			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 420175/419904

			{{Mapping\|legend=1\| 1 0 3 0 \| 0 1 4 0 \| 0 0 5 -2 }}
			: Mapping generators: ~2, ~3, ~245/162

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0161{{c}}, ~3/2 = 701.9929{{c}}, ~245/162 = 715.6723{{c}}
			: [[Error map]]: {{val\| +0.016 +0.054 -0.029 -0.106 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9981{{c}}, ~245/162 = 715.6543{{c}}
			: Error map: {{val\| 0.000 +0.043 -0.050 -0.134 }}

			{{Optimal ET sequence\|legend=1\| 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4777d, 4918d, 5089d, 5701d }}

			[[Badness]] (Sintel): 0.381

	== Breeze ==		== Breeze ==

FloraC: + scheme. Officialize canopic as the sole name for {16875/16807}. Schismean -> aberdietic and review its optimal ET sequence

2026-07-29T10:16:46Z

+ scheme. Officialize canopic as the sole name for {16875/16807}. Schismean -> aberdietic and review its optimal ET sequence

← Older revision		Revision as of 10:16, 29 July 2026
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	[[Badness]] (Sintel): 0.661		[[Badness]] (Sintel): 0.661

	== ~~Canopic a~~.k.a. ~~mirkwai~~ ==		== Scheme ==
			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 14348907/14336000

			{{Mapping\|legend=1\| 1 0 0 -14 \| 0 1 0 15 \| 0 0 1 -3 }}
			: Mapping generators: ~2, ~3, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0267{{c}}, ~3/2 = 701.8565{{c}}, ~5/4 = 386.2912{{c}}
			: [[Error map]]: {{val\| +0.027 -0.072 +0.031 +0.015 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8494{{c}}, ~5/4 = 386.3135{{c}}
			: Error map: {{val\| 0.000 -0.106 -0.000 -0.025 }}

			{{Optimal ET sequence\|legend=1\| 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd }}

			[[Badness]] (Sintel): 1.47

			== Canopic ==
	: ''For extensions, see [[Swetismic temperaments #Indra]].''		: ''For extensions, see [[Swetismic temperaments #Indra]].''

	Canopic~~, a.k.a.~~ mirkwai, tempers out the [[canopic comma]] in the 7-limit.		Canopic (formerly ''mirkwai'') tempers out the [[canopic comma]] in the 7-limit.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	[[Projection pair]]: <code>5 2401/486</code> to 2.3.7		[[Projection pair]]: <code>5 2401/486</code> to 2.3.7

	== ~~Schismean~~ ==		== Aberdietic ==
	~~Schismean~~ tempers out the [[3645/3584\|~~schismean comma~~]] in the 7-limit. It equates [[7/5]] with a stack of three [[9/8]]'s.		Aberdietic tempers out the [[3645/3584\|aberdiesis]] in the 7-limit. It equates [[7/5]] with a stack of three [[9/8]]'s.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 697.459{{c}}, ~5/4 = 383.104{{c}}		* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 697.459{{c}}, ~5/4 = 383.104{{c}}

	{{Optimal ET sequence\|legend=1\| 5c, 7d, 12, 19, 31, 81 }}		{{Optimal ET sequence\|legend=1\| 5c, 7d, 12, 19, 31, 81, 98c, 117c, 129c, 148bc, 160bc, 179bcd }}

	[[Badness]] (Sintel): 2.96		[[Badness]] (Sintel): 2.96

FloraC: + triwellismic

2026-07-26T05:34:38Z

+ triwellismic

← Older revision		Revision as of 05:34, 26 July 2026
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	[[Badness]] (Sintel): 3.40		[[Badness]] (Sintel): 3.40

			== Triwellismic ==
			Triwellismic tempers out the [[triwellisma]]. Besides the [[octave]] and [[3/2\|perfect fifth]], it is generated by a [[15/14]] semitone, with six of them making a retroptolemaic fifth, [[243/160]], and seven making a septimal augmented fifth, [[729/448]]. An alternative generator is the [[10/7]] tritone, with two minus an octave giving the [[jubilisma]] and three jubilismas being equated with [[16/15]].

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 235298/234375

			{{Mapping\|legend=1\| 1 0 1 1 \| 0 1 5 6 \| 0 0 -6 -7 }}
			: Mapping generators: ~2, ~3, ~15/14

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9876{{c}}, ~3/2 = 701.3045{{c}}, ~15/14 = 120.5354{{c}}
			: [[Error map]]: {{val\| -0.012 +0.031 +0.466 -0.584 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9958{{c}}, ~15/14 = 120.5308{{c}}
			: Error map: {{val\| 0.000 +0.041 +0.480 -0.567 }}

			{{Optimal ET sequence\|legend=1\| 29, 31, 60, 68, 91, 99, 229, 289, 388, 518, 906 }}

			[[Badness]] (Sintel): 3.55

	== Decovulture ==		== Decovulture ==

FloraC: /* Decovulture */ fix link

2026-07-14T15:15:41Z

Decovulture: fix link

← Older revision		Revision as of 15:15, 14 July 2026
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	: ''For extensions, see [[Olympic clan #Baffin]].''		: ''For extensions, see [[Olympic clan #Baffin]].''

	Decovulture tempers out the [[decovulture]] ~~comma~~ and splits the [[3/1\|3rd harmonic]] into two semitwelfths, separated from ~[[7/4]] and ~[[12/7]] each by a [[diaschisma]].		Decovulture tempers out the [[decovulture comma]] and splits the [[3/1\|3rd harmonic]] into two semitwelfths, separated from ~[[7/4]] and ~[[12/7]] each by a [[diaschisma]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: /* Decovulture */ + intro

2026-07-14T15:15:18Z

Decovulture: + intro

← Older revision		Revision as of 15:15, 14 July 2026
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	== Decovulture ==		== Decovulture ==
	: ''For extensions, see [[Olympic clan #Baffin]].''		: ''For extensions, see [[Olympic clan #Baffin]].''

			Decovulture tempers out the [[decovulture]] comma and splits the [[3/1\|3rd harmonic]] into two semitwelfths, separated from ~[[7/4]] and ~[[12/7]] each by a [[diaschisma]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:07Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
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	: ''For extensions, see [[Keenanismic temperaments #Uniwiz]].''		: ''For extensions, see [[Keenanismic temperaments #Uniwiz]].''

	Uniwiz tempers out the [[uniwiz comma]] in the 7-limit, equating the [[9/8\|whole tone]] with a stack of four septimal quartertones of [[36/35]], and splits the [[~~2/1\|~~octave]] in two. This means the quartertone should be sharpened a bit, leading to the natural 11-limit extension where [[385/384]] and [[9801/9800]] are tempered out.		Uniwiz tempers out the [[uniwiz comma]] in the 7-limit, equating the [[9/8\|whole tone]] with a stack of four septimal quartertones of [[36/35]], and splits the [[octave]] in two. This means the quartertone should be sharpened a bit, leading to the natural 11-limit extension where [[385/384]] and [[9801/9800]] are tempered out.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Move cloudy here

2026-06-25T11:35:25Z

Move cloudy here

← Older revision		Revision as of 11:35, 25 June 2026
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	[[Badness]] (Sintel): 13.3		[[Badness]] (Sintel): 13.3

			== Cloudy ==
			Cloudy tempers out the [[cloudy comma]], a.k.a. the 5-7-comma, equating the octave with a stack of five [[8/7]]'s.

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 16807/16384

			{{Mapping\|legend=1\| 5 0 0 14 \| 0 1 0 0 \| 0 0 1 0 }}
			: Mapping generators: ~8/7, ~3, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~8/7 = 240.3136{{c}}, ~3/2 = 700.3807{{c}}, ~5/4 = 383.1686{{c}}
			: [[error map]]: {{val\| +1.568 -0.007 -0.010 -4.436 }}
			* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 700.2940{{c}}, ~5/4 = 383.8804{{c}}
			: error map: {{val\| 0.000 -1.661 -2.433 -8.826 }}

			{{Optimal ET sequence\|legend=1\| 5, 10, 15, 35, 40, 45, 50, 60, 110d, 170cdd, 175dd, 235cddd }}

			[[Badness]] (Sintel): 13.9

	== Linus ==		== Linus ==