Hemimage temperaments: Difference between revisions

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This is a collection of [[~~Rank~~-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include ~~commatic,~~ chromat, degrees, ~~subfourth~~, and ~~bisupermajor~~, considered below, as well as the following discussed elsewhere:

This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:

* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]

* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]

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* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]

* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]

* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]

* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]

* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]

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~~{{Multival|legend=1| 15 39 48 27 34 2 }}~~

: mapping generators: ~63/50, ~28/27

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: mapping generators: ~1225/864, ~192/175

~~{{Multival|legend=1| 16 -10 34 -53 9 107 }}~~

[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806

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Badness: 0.032080

== ~~Commatic~~ ==

== Bicommatic ==

~~The~~ commatic temperament has a period of half octave and a generator of 20.4 cents~~. It is so named because the generator is~~ a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~567/400, ~81/80

~~{{Multival|legend=1| 10 38 36 37 29 -23 }}~~

[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377

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== Degrees ==

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]] with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}} we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.

An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.

By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.

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: mapping generators: ~28/27, ~3

~~{{Multival|legend=1| 20 40 60 17 39 27 }}~~

[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)

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[[Badness]]: 0.106471

Badness (~~Dirichlet~~): 2.694

Badness (Sintel): 2.694

=== 11-limit ===

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Badness: 0.046770

Badness (~~Dirichlet~~): 1.546

Badness (Sintel): 1.546

=== 13-limit ===

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Badness: 0.032718

Badness (~~Dirichlet~~): 1.352

Badness (Sintel): 1.352

=== 17-limit ===

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Badness (~~Dirichlet~~): 1.171

Badness (Sintel): 1.171

=== 19-limit ===

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Badness (~~Dirichlet~~): 1.273

Badness (Sintel): 1.273

=== 23-limit ===

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Badness (~~Dirichlet~~): 1.209

Badness (Sintel): 1.209

=== 29-limit ===

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Badness (~~Dirichlet~~): 1.134

Badness (Sintel): 1.134

=== no-31's 37-limit ===

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Badness (~~Dirichlet~~): 1.127

Badness (Sintel): 1.127

=== no-31's 41-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870

Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}

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Badness (~~Dirichlet~~): 1.100

Badness (Sintel): 1.100

== Squarschmidt ==

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~~{{Multival|legend=1| 29 4 69 -61 28 149 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643

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Badness: 0.038186

~~== Subfourth ==~~

~~[[Subgroup]]: 2.3.5.7~~

~~[[Comma list]]: 10976/10935, 65536/64827~~

~~{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}~~

~~: mapping generators: ~2, ~21/16~~

~~{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}~~

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991~~

~~{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}~~

~~[[Badness]]: 0.140722~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

~~Comma list: 540/539, 896/891, 12005/11979~~

~~Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995~~

~~{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}~~

~~Badness: 0.045323~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 352/351, 364/363, 540/539, 676/675~~

~~Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}~~

~~Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996~~

~~{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}~~

~~Badness: 0.023800~~

[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Hemimage temperaments| ]]

[[Category:Hemimage| ]]

[[Category:Rank 2]]

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-This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, and bisupermajor, considered below, as well as the following discussed elsewhere:
+{{Technical data page}}
+This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:
 * ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
 * ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
@@ Line 12: / Line 13: @@
 * ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
 * ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
+* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
 * ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
 * ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
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 {{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
-{{Multival|legend=1| 15 39 48 27 34 2 }}
 : mapping generators: ~63/50, ~28/27
@@ Line 161: / Line 161: @@
 : mapping generators: ~1225/864, ~192/175
-{{Multival|legend=1| 16 -10 34 -53 9 107 }}
 [[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
@@ Line 183: / Line 181: @@
 Badness: 0.032080
-== Commatic ==
+== Bicommatic ==
-The commatic temperament has a period of half octave and a generator of 20.4 cents. It is so named because the generator is a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.
+Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.
 [[Subgroup]]: 2.3.5.7
@@ Line 193: / Line 191: @@
 : mapping generators: ~567/400, ~81/80
-{{Multival|legend=1| 10 38 36 37 29 -23 }}
 [[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
@@ Line 242: / Line 238: @@
 == Degrees ==
+{{ See also | 20th-octave temperaments }}
 Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
-An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]] with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}} we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
+An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
 By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.
@@ Line 257: / Line 254: @@
 : mapping generators: ~28/27, ~3
-{{Multival|legend=1| 20 40 60 17 39 27 }}
 [[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
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 [[Badness]]: 0.106471
-Badness (Dirichlet): 2.694
+Badness (Sintel): 2.694
 === 11-limit ===
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 Badness: 0.046770
-Badness (Dirichlet): 1.546
+Badness (Sintel): 1.546
 === 13-limit ===
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 Badness: 0.032718
-Badness (Dirichlet): 1.352
+Badness (Sintel): 1.352
 === 17-limit ===
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 {{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
-Badness (Dirichlet): 1.171
+Badness (Sintel): 1.171
 === 19-limit ===
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 {{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
-Badness (Dirichlet): 1.273
+Badness (Sintel): 1.273
 === 23-limit ===
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 {{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
-Badness (Dirichlet): 1.209
+Badness (Sintel): 1.209
 === 29-limit ===
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 {{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
-Badness (Dirichlet): 1.134
+Badness (Sintel): 1.134
 === no-31's 37-limit ===
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 {{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
-Badness (Dirichlet): 1.127
+Badness (Sintel): 1.127
 === no-31's 41-limit ===
 Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
-Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480
+Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}
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 {{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
-Badness (Dirichlet): 1.100
+Badness (Sintel): 1.100
 == Squarschmidt ==
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 {{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
-{{Multival|legend=1| 29 4 69 -61 28 149 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
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 Badness: 0.038186
-== Subfourth ==
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 10976/10935, 65536/64827
-{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
-: mapping generators: ~2, ~21/16
-{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991
-{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
-[[Badness]]: 0.140722
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 540/539, 896/891, 12005/11979
-Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
-{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
-Badness: 0.045323
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 364/363, 540/539, 676/675
-Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
-{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
-Badness: 0.023800
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Hemimage temperaments| ]] <!-- main article -->
 [[Category:Hemimage| ]] <!-- key article -->
 [[Category:Rank 2]]