Miscellaneous 7-limit temperaments - Revision history

FloraC: Text replacement - "rank-3 temperament" to "rank-3 temperament"

2026-06-08T08:36:19Z

Text replacement - "rank-3 temperament" to "rank-3 temperament"

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	{{Technical data page}}		{{Technical data page}}
	Below are listed some [[7-limit]] [[rank-3 ~~temperament\|rank-3]] [[regular temperament\|~~temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit\|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:		Below are listed some [[7-limit]] [[rank-3 temperament]]s that do not belong to some other temperament collection, the majority of which are [[restriction]]s to the 7-limit of temperaments that emerge more fully in higher [[prime limit\|limits]] or [[subgroup]]s; they are sorted by [[TE logflat badness]]. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:
	* Individual [[temperament families and clans]]		* Individual [[temperament families and clans]]
	* [[Very low accuracy temperaments]]		* [[Very low accuracy temperaments]]

FloraC: Move mirkwai here as we're dissolving the mirkwai family

2026-06-06T12:06:19Z

Move mirkwai here as we're dissolving the mirkwai family

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	[[Badness]] (Sintel): 0.661		[[Badness]] (Sintel): 0.661

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

			Canopic, a.k.a. mirkwai, tempers out the [[canopic comma]] in the 7-limit.

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 16875/16807

			{{Mapping\|legend=1\| 1 0 -5 -4 \| 0 1 3 3 \| 0 0 5 4 }}
			: Mapping generators: ~2, ~3, ~10/7

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0944{{c}}
			: [[error map]]: {{val\| -0.000 -0.172 -0.493 +0.900 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0945{{c}}
			: error map: {{val\| 0.000 -0.172 -0.493 +0.900 }}

			[[Minimax tuning]]:
			* [[7-odd-limit]]
			: {{monzo list\| 1 0 0 0 \| 0 4/7 -4/7 5/7 \| 0 -3/7 3/7 5/7 \| 0 0 0 1 }}
			: [[Eigenmonzo basis\|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
			* [[9-odd-limit]]
			: {{monzo list\| 1 0 0 0 \| 0 8/11 -4/11 5/11 \| 0 -6/11 3/11 10/11 \| 0 0 0 1 }}
			: eigenmonzo (unchanged-interval) basis: 2.9/5.7

			{{Optimal ET sequence\|legend=1\| 31, 41, 72, 152, 224 }}

			[[Badness]] (Sintel): 1.51

			[[Projection pair]]s: <code>5 84375/16807 7 16875/2401</code> to 2.3.7/5

	== Greenwoodmic ==		== Greenwoodmic ==

FloraC: /* Greenwoodmic */ note det chord and 11-limit extension

2026-04-09T19:31:05Z

Greenwoodmic: note det chord and 11-limit extension

← Older revision		Revision as of 19:31, 9 April 2026
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	== Greenwoodmic ==		== Greenwoodmic ==
	Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval\|primes]] [[3/1\|3]] and [[5/1\|5]] should be tuned flat, and [[7/1\|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.		Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval\|primes]] [[3/1\|3]] and [[5/1\|5]] should be tuned flat, and [[7/1\|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.

			In contrast to [[sensamagic]], where two [[9/7]]'s stack to [[5/3]], here two 9/7's stack to [[8/5]]. As such, greenwoodmic induces [[essentially tempered chord]]s in the [[9-odd-limit]]. An obvious 11-limit extension then equates [[5/4]] with [[11/9]] and equates 9/7 with [[14/11]], tempering out [[45/44]] as well as [[99/98]] using the identity 405/392 = (45/44)⋅(99/98).

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

FloraC: /* Avicennmic */ note about some possible extensions

2026-04-09T18:48:05Z

Avicennmic: note about some possible extensions

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	== Avicennmic ==		== Avicennmic ==
	Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval\|primes]] [[3/1\|3]], [[5/1\|5]] and [[7/1\|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic.		Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval\|primes]] [[3/1\|3]], [[5/1\|5]] and [[7/1\|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic.

			One possible extension of avicennmic to the 11-limit is via [[45/44]] and [[385/384]], using the identity 525/512 = (45/44)⋅(385/384), but the result is somewhat less accurate. Instead, it is more natural to extend it to the [[2.3.5.7.13 subgroup]] by tempering out [[65/64]] and [[105/104]], using the identity 525/512 = (65/64)⋅(105/104).

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

FloraC: Move decovulture here

2026-02-06T12:40:54Z

Move decovulture here

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	[[Badness]] (Sintel): 3.40		[[Badness]] (Sintel): 3.40

			== Decovulture ==
			: ''For extensions, see [[Olympic clan #Baffin]].''

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 67108864/66976875

			{{Mapping\|legend=1\| 1 0 0 13 \| 0 2 0 -7 \| 0 0 1 -2 }}
			: mapping generators: ~2, ~8192/4725, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9033{{c}}, ~8192/4725 = 951.0102{{c}}, ~5/4 = 386.5872{{c}}
			: [[error map]]: {{val\| -0.097 +0.065 +0.080 +0.059 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0899{{c}}, ~5/4 = 386.6184{{c}}
			: error map: {{val\| 0.000 +0.225 +0.305 +0.308 }}

			{{Optimal ET sequence\|legend=1\| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}

			[[Badness]] (Sintel): 3.82

	== Trimyna ==		== Trimyna ==

Xenllium at 15:51, 31 January 2026

2026-01-31T15:51:28Z

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	[[Badness]] (Sintel): 15.7		[[Badness]] (Sintel): 15.7

			== Naiad ==
			: ''For extensions, see [[Wizardharry clan #Naiad]].''

			Naiad tempers out the [[naiadisma]] in the 7-limit. An obvious 13-limit interpretation of one generator (~98/75) is [[13/10]].

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 161414428/158203125

			{{Mapping\|legend=1\| 1 5 0 2 \| 0 -9 0 -4 \| 0 0 1 1 }}
			: Mapping generators: ~2, ~98/75, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.937{{c}}, ~98/75 = 455.269{{c}}, ~5/4 = 387.946{{c}}
			: [[error map]]: {{val\| -0.063, +0.311, +1.506, -2.207 }}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~98/75 = 455.298{{c}}, ~5/4 = 387.896{{c}}
			: error map: {{val\| 0.000, +0.362, +1.582, -2.123 }}

			{{Optimal ET sequence\|legend=1\| 8d, 21, 29, 37, 50, 58, 87, 145 }}

			[[Badness]] (Sintel): 52.5

	[[Category:Temperament collections]]		[[Category:Temperament collections]]

Xenllium at 05:37, 31 January 2026

2026-01-31T05:37:09Z

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	[[Badness]] (Sintel): 7.26		[[Badness]] (Sintel): 7.26

			== History ==
			: ''For extensions, see [[Werckismic temperaments #History]].''

			History tempers out the [[historisma]] in the 7-limit, and splits the fourth in six.

			[[Comma list]]: 257298363/256000000

			{{Mapping\|legend=1\| 1 2 0 0 \| 0 -6 0 7 \| 0 0 1 1 }}
			: Mapping generators: ~2, ~21/20, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.154{{c}}, ~21/20 = 83.091{{c}}, ~5 = 2786.669{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~21/20 = 83.067{{c}}, ~5 = 2786.515{{c}}

			{{Optimal ET sequence\|legend=1\| 14c, 15, 29, 43, 58, 72, 130, 202}}

			[[Badness]] (Sintel): 7.95

	== Sensibeta ==		== Sensibeta ==

Xenllium at 13:20, 22 January 2026

2026-01-22T13:20:12Z

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	[[Badness]] (Sintel): 6.18		[[Badness]] (Sintel): 6.18

			== Tolerant ==
			: ''For extensions, see [[Pentacircle clan #Tolerant]].''

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 179200/177147

			{{Mapping\|legend=1\| 1 0 0 -10 \| 0 1 0 11 \| 0 0 1 -2 }}
			: Mapping generators: ~2, ~3, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.539{{c}}, ~3 = 1903.226{{c}}, ~5 = 2785.816{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~3 = 1903.805{{c}}, ~5 = 2786.356{{c}}

			{{Optimal ET sequence\|legend=1\| 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd }}

			[[Badness]] (Sintel): 7.26

	== Sensibeta ==		== Sensibeta ==

Xenllium at 11:46, 22 January 2026

2026-01-22T11:46:28Z

← Older revision		Revision as of 11:46, 22 January 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

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

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 3645/3584

			{{Mapping\|legend=1\| 1 0 0 -9 \| 0 1 0 6 \| 0 0 1 1 }}
			: Mapping generators: ~2, ~3, ~5

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1201.420{{c}}, ~3/2 = 698.145{{c}}, ~5/4 = 382.612{{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 }}

			[[Badness]] (Sintel): 2.96

	== Keegic ==		== Keegic ==

FloraC: + intros

2026-01-21T12:08:12Z

+ intros

← Older revision		Revision as of 12:08, 21 January 2026
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	== Keegic ==		== Keegic ==
			Keegic tempers out the [[keega]] in the 7-limit, and finds the [[3/1\|3rd]] [[harmonic]] by a stack of three [[10/7]]'s.

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

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	== Septimagic ==		== Septimagic ==
			Septimagic tempers out the [[septimagic comma]] in the 7-limit and gives the [[2.3.7 subgroup\|2.3.7-]][[subgroup]] [[magic]] [[restriction]] an independent generator for the [[5/1\|5th]] [[harmonic]].

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