Miscellaneous 7-limit temperaments: Difference between revisions

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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]]

* [[Very low accuracy temperaments]]

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See also [[Miscellaneous 5-limit temperaments]].

== 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

[[Comma list]]: 420175/419904

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

[[Subgroup]]: 2.3.5.7

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[[Badness]] (Sintel): 0.661

== Scheme ==

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 14348907/14336000

: 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]].''

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807

: 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

[[Badness]] (Sintel): 1.51

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

== Greenwoodmic ==

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[[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

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* [[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, 98c, 117c, 129c, 148bc, 160bc, 179bcd }}

[[Badness]] (Sintel): 2.96

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: ''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

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[[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 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 ==

: ''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

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[[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 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 ==

@@ Line 1: / Line 1: @@
 {{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]]
 * [[Very low accuracy temperaments]]
@@ Line 6: / Line 6: @@
 See also [[Miscellaneous 5-limit temperaments]].
+== 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
+[[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 tempers out the [[breeze comma]] in the 7-limit.
 [[Subgroup]]: 2.3.5.7
@@ Line 44: / Line 66: @@
 [[Badness]] (Sintel): 0.661
+== Scheme ==
+Scheme tempers out the [[scheme comma]] in the 7-limit.
+[[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]].''
+Canopic (formerly ''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 ==
@@ Line 135: / Line 209: @@
 [[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
@@ Line 149: / Line 223: @@
 * [[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
@@ Line 176: / Line 250: @@
 : ''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
@@ Line 238: / Line 312: @@
 [[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 ==
 : ''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
@@ Line 468: / Line 564: @@
 [[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 ==