Miscellaneous 7-limit temperaments: Difference between revisions

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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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Line 67:

[[Badness]] (Sintel): 0.661

== ~~Canopic a~~.k.a. ~~mirkwai~~ ==

== 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~~, 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.

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

[[Subgroup]]: 2.3.5.7

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

== ~~Schismean~~ ==

== Prograde ==

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

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 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 }}

[[Badness]] (Sintel): 2.71

== Aberdietic ==

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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: ''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

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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

== 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 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 ==

: ''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]].

[[Subgroup]]: 2.3.5.7

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== 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]].

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: ''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

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[[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 (with the chain of fifths from [[147edo]]), but it can be tuned even more accurately than that.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: {{monzo| 40 -22 -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 ==

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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 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 45: / Line 67: @@
 [[Badness]] (Sintel): 0.661
-== Canopic a.k.a. mirkwai ==
+== 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, 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.
+For the no-twos, rank-2 version, see [[canopus]].
 [[Subgroup]]: 2.3.5.7
@@ Line 167: / Line 211: @@
 [[Projection pair]]: <code>5 2401/486</code> to 2.3.7
-== Schismean ==
+== Prograde ==
-Schismean tempers out the [[3645/3584|schismean comma]] in the 7-limit. It equates [[7/5]] with a stack of three [[9/8]]'s.
+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 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 181: / Line 245: @@
 * [[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 208: / Line 272: @@
 : ''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 252: / Line 316: @@
 : ''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
@@ Line 270: / Line 334: @@
 [[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
+== 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 ==
-: ''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]].
 [[Subgroup]]: 2.3.5.7
@@ Line 342: / Line 450: @@
 == 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]].
@@ Line 440: / Line 548: @@
 : ''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
@@ Line 458: / Line 566: @@
 [[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 (with the chain of fifths from [[147edo]]), 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 ==
@@ Line 500: / Line 628: @@
 [[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 ==

Miscellaneous 7-limit temperaments: Difference between revisions

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