Miscellaneous 7-limit temperaments: Difference between revisions

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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

== 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 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 ==

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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): 3.55

== 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]].

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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 ==

@@ Line 91: / Line 91: @@
 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 208: / Line 210: @@
 [[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 ==
@@ Line 294: / 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 332: / Line 354: @@
 [[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]].
@@ Line 406: / 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 504: / 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 522: / 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 ==

Miscellaneous 7-limit temperaments: Difference between revisions

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