Miscellaneous 7-limit temperaments - Revision history

CompactStar at 18:36, 20 October 2026

2026-10-20T18:36:47Z

← Older revision		Revision as of 18:36, 20 October 2026
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	Canopic (formerly ''mirkwai'') tempers out the [[canopic comma]] in the 7-limit.		Canopic (formerly ''mirkwai'') tempers out the [[canopic comma]] in the 7-limit.

			For the no-twos, rank-2 version, see [[canopus]].

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

FloraC: + prograde

2026-10-15T09:15:57Z

+ prograde

← Older revision		Revision as of 09:15, 15 October 2026
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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

			== Prograde ==
			Prograde tempers out the [[prograde comma]] in the 7-limit, and besides the octave and perfect fifth, it is generated by a very sharp [[~]][[15/14]]. Two of these generators make ~[[6/5]] and three make ~[[9/7]]. An obvious 11-limit extension then equates the generator with both [[11/10]] and [[12/11]], tempering out [[56/55]], [[77/75]], and [[121/120]].

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 392/375

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

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1197.4315{{c}}, ~3/2 = 706.5443{{c}}, ~15/14 = 150.8299{{c}}
			: [[Error map]]: {{val\| -2.568 +2.021 +13.434 -13.364 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2305{{c}}, ~15/14 = 151.3070{{c}}
			: Error map: {{val\| 0.000 +4.275 +17.303 -10.286 }}

			{{Optimal ET sequence\|legend=1\| 8d, 9, 15, 24, 32c, 39 }}

			[[Badness]] (Sintel): 2.71

	== Aberdietic ==		== Aberdietic ==

FloraC: /* Parahemif */ - duplicate info

2026-10-03T11:05:07Z

Parahemif: - duplicate info

← Older revision		Revision as of 11:05, 3 October 2026
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	: ''For extensions, see [[Rastmic rank-3 clan #Parahemif]].''		: ''For extensions, see [[Rastmic rank-3 clan #Parahemif]].''

	Parahemif tempers out the [[parahemif comma]] in the 7-limit, ~~equating a~~ [[~~Pythagorean apotome~~]] with ~~a stack of two septimal third-tones of [[28/27]], and splits~~ the ~~fifth in two. It also equates the large septimal diesis of~~ [[49/~~48]] with the [[Pythagorean comma~~]]. ~~This means the fifth should be tuned sharp and the septimal third-tone should be flattened~~ to ~~a somewhat large quartertone which can be used as~~ the ~~undecimal quartertone of [[33/32]], leading to a natural~~ 11-limit ~~extension where [[243/242]] and [[896/891]] are tempered out~~.		Parahemif tempers out the [[parahemif comma]] in the 7-limit, expanding [[hemif]] with an independent generator for the [[5/1\|5th harmonic]]. Its extensions to the 11- and 13-limit follow the same path as hemif.

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

FloraC: + mandos

2026-09-16T07:41:18Z

+ mandos

← Older revision		Revision as of 07:41, 16 September 2026
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	[[Badness]] (Sintel): 3.55		[[Badness]] (Sintel): 3.55

			== Mandos ==
			: ''For extensions, see [[Rastmic rank-3 clan #Mandos]].''

			Mandos tempers out the [[mandos comma]] in the 7-limit, equating the [[25/24\|classical chromatic semitone]] with a stack of two septimal quartertones of [[36/35]], and splits the [[3/2\|perfect fifth]] in two. A rank-2 temperament that does that is [[myna]], which introduces little extra [[damage]] over mandos. Its natural 11-limit extension tempers out [[243/242]] and [[176/175]], from the identity 31104/30625 = (176/175)<sup>2</sup>⋅(243/242).

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 31104/30625

			{{Mapping\|legend=1\| 1 1 0 6 \| 0 2 0 5 \| 0 0 1 -2 }}
			: Mapping generators: ~2, ~175/144, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.1782{{c}}, ~175/144 = 350.6514{{c}}, ~5/4 = 390.4750{{c}}
			: [[Error map]]: {{val\| -0.822 -1.474 +2.518 +1.837 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/144 = 350.5166{{c}}, ~5/4 = 390.2032{{c}}
			: Error map: {{val\| 0.000 -0.922 +3.889 +3.351 }}

			{{Optimal ET sequence\|legend=1\| 24, 27, 31, 58, 89, 154d, 181cd, 212cd, 270ccdd, 301ccd }}

			[[Badness]] (Sintel): 3.77

	== Decovulture ==		== Decovulture ==

FloraC: /* Mirwomo */ expand a little

2026-09-16T07:26:34Z

Mirwomo: expand a little

← Older revision		Revision as of 07:26, 16 September 2026
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	: ''For extensions, see [[Rastmic rank-3 clan #Mirwomo]].''		: ''For extensions, see [[Rastmic rank-3 clan #Mirwomo]].''

	Mirwomo tempers out the [[mirwomo comma]] in the 7-limit, equating the [[Pythagorean apotome]] with a stack of two septimal quartertones of [[36/35]], and splits the fifth in two. This means the fifth should be flattened a bit and the quartertone should be sharpened, ~~leading to a~~ natural 11-limit extension ~~where~~ [[243/242]] and [[385/384]] ~~are tempered out~~.		Mirwomo tempers out the [[mirwomo comma]] in the 7-limit, equating the [[Pythagorean apotome]] with a stack of two septimal quartertones of [[36/35]], and splits the [[3/2\|perfect fifth]] in two. This means the perfect fifth should be flattened a bit and the quartertone should be sharpened. A rank-2 temperament that does that is [[miracle]], which introduces little extra [[damage]] over mirwomo. Its natural 11-limit extension tempers out [[243/242]] and [[385/384]], from the identity 33075/32768 = (243/242)⋅(385/384)<sup>2</sup>.

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

FloraC: Update links

2026-09-08T13:26:50Z

Update links

← Older revision		Revision as of 13:26, 8 September 2026
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	== Decovulture ==		== Decovulture ==
	: ''For extensions, see [[~~Olympic~~ clan #Baffin]].''		: ''For extensions, see [[Vishdelismic 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]].
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	== Quasiorwellismic ==		== Quasiorwellismic ==
	: ''For extensions, see [[~~Lehmerismic temperaments~~ #Ganesha]].''		: ''For extensions, see [[Vishdelismic clan #Ganesha]].''

	Quasiorwellismic tempers out the [[quasiorwellisma]] in the 7-limit, and finds [[7/6]] by a stack of ten [[5/4]]'s octave reduced. A natural 11-limit extension thus arises from mapping [[11/9]] to a stack of four [[5/4]]'s octave reduced, leading to [[ganesha]], which tempers out [[3025/3024]] and [[5632/5625]].		Quasiorwellismic tempers out the [[quasiorwellisma]] in the 7-limit, and finds [[7/6]] by a stack of ten [[5/4]]'s octave reduced. A natural 11-limit extension thus arises from mapping [[11/9]] to a stack of four [[5/4]]'s octave reduced, leading to [[ganesha]], which tempers out [[3025/3024]] and [[5632/5625]].

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 ==