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).

Temperaments discussed elsewhere are:

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

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

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* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]

* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]

* ''[[Pluto]]'' (+4000/3969) → [[~~Mirkwai clan~~ #Pluto|~~Mirkwai clan~~]]

* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]

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

* ''[[Hendecatonic (temperament)|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]]

Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].

== Chromat ==

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[~~Amity~~ family|amity extension]] with third-octave period.

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[amity family|amity extension]] with third-octave period.

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]] ([[~~POTE~~]]): ~63/50 = ~~1\3~~, ~28/27 = 60.~~528~~

[[Optimal tuning]]s:

* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}

: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}

: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}

[[Badness]]: 0.~~057499~~

[[Badness]] (Sintel): 1.46

=== 11-limit ===

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Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~28/27 = 60.~~430~~

Optimal tunings:

* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}

Badness: 0.~~050379~~

Badness (Sintel): 1.67

==== 13-limit ====

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Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~28/27 = 60.~~428~~

Optimal tunings:

* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}

Badness: 0.~~046006~~

Badness (Sintel): 1.90

===== 17-limit =====

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Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~28/27 = 60.~~438~~

Optimal tunings:

* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}

Badness: 0.~~031678~~

Badness (Sintel): 1.61

==== Catachrome ====

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Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~28/27 = 60.~~378~~

Optimal tunings:

* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}

Badness: 0.~~043844~~

Badness (Sintel): 1.81

===== 17-limit =====

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Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~28/27 = 60.~~377~~

Optimal tunings:

* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}

Badness: 0.~~030218~~

Badness (Sintel): 1.54

==== Chromic ====

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Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}

Optimal ~~tuning (POTE)~~: ~44/35 = ~~1\3~~, ~27/26 = 60.~~456~~

Optimal tunings:

* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}

Badness: 0.~~049857~~

Badness (Sintel): 2.06

===== 17-limit =====

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Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}

Optimal ~~tuning (POTE)~~: ~63/50 = ~~1\3~~, ~27/26 = 60.~~459~~

Optimal tunings:

* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}

* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}

Badness: 0.~~031043~~

Badness (Sintel): 1.58

=== Hemichromat ===

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Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}

Optimal ~~tuning (CTE)~~: ~63/50 = ~~1\3~~, ~55/54 = 30.~~2511~~

Optimal tunings:

* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}

{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}

{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}

Badness: 0.~~067173~~

Badness (Sintel): 2.22

==== 13-limit ====

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Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}

Optimal ~~tuning (CTE)~~: ~63/50 = ~~1\3~~, ~55/54 = 30.~~2527~~

Optimal tunings:

* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}

{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}

{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}

Badness: 0.~~033420~~

Badness (Sintel): 1.38

== Bisupermajor ==

~~{{See also|~~ Very high accuracy temperaments #Kwazy }}

: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}

[[~~Optimal tuning~~]] ([[~~POTE~~]]): ~1225/864 = ~~1\2~~, ~192/175 = 162.~~806~~

: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}

: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}

{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}

[[Badness]]: 0.~~065492~~

[[Badness]] (Sintel): 1.66

=== 11-limit ===

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Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}

Optimal ~~tuning (POTE)~~: ~99/70, ~11/10 = 162.~~773~~

Optimal tunings:

* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}

{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}

{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}

Badness: 0.~~032080~~

Badness (Sintel): 1.06

== ~~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

[[Optimal tuning]]s:

* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}

: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}

* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}

: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}

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

~~{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}~~

[[Badness]] (Sintel): 2.13

[[Badness]]: 0.~~084317~~

=== 11-limit ===

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Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}

Optimal ~~tuning (POTE)~~: ~99/70 = ~~1\2~~, ~81/80 = 20.~~390~~

Optimal tunings:

* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}

Badness: 0.~~030461~~

Badness (Sintel): 1.01

=== 13-limit ===

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Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}

Optimal ~~tuning (POTE)~~: ~99/70 = ~~1\2~~, ~66/65 = 20.~~427~~

Optimal tunings:

* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}

Badness: 0.~~026336~~

Badness (Sintel): 1.09

=== 17-limit ===

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Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}

Optimal ~~tuning (POTE)~~: ~17/12 = ~~1\2~~, ~66/65 = 20.~~378~~

Optimal tunings:

* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}

Badness: 0.~~022396~~

Badness (Sintel): 1.14

== 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.

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.

By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~28/27, ~3

[[Optimal tuning]]s:

* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})

[[~~Optimal tuning~~]] ([[~~POTE~~]]): ~28/27 = ~~1\20~~, ~3/2 = ~~703~~.~~015~~

: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}

* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})

: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}

{{Optimal ET sequence|legend=1| ~~20cd,~~ 60, 80, 140, 640b, 780b }}

[[Badness]]~~: 0.106471~~

[[Badness]] (Sintel): 2.69

~~Badness~~ (~~Dirichlet~~): 2.~~694~~

=== 11-limit ===

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Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}

Optimal ~~tuning~~ (~~POTE~~): ~28/27 = ~~1\20~~, ~3/2 = 703.~~231~~

Optimal tunings:

* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})

~~Badness: 0.046770~~

Badness (Sintel): 1.55

Badness (~~Dirichlet~~): 1.~~546~~

=== 13-limit ===

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Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}

Optimal ~~tuning~~ (~~POTE~~): ~28/27 = ~~1\20~~, ~3/2 = 703.~~080~~

Optimal tunings:

* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})

~~Badness: 0.032718~~

Badness (Sintel): 1.35

Badness (~~Dirichlet~~): 1.~~352~~

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 1001/1000~~, 1331/1323, 289/288~~

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

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

Optimal ~~tuning~~ (~~CTE~~): ~28/27 = ~~1\20~~, ~3/2 = 703.~~107~~

Optimal tunings:

* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})

Badness (~~Dirichlet~~): 1.~~171~~

Badness (Sintel): 1.17

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 352/351, ~~1001~~/~~1000, 1331/1323, 289/288~~, ~~400~~/~~399~~

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

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

Optimal ~~tuning~~ (~~CTE~~): ~28/27 = ~~1\20~~, ~3/2 = 703.~~107~~

Optimal tunings:

* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})

Badness (~~Dirichlet~~): 1.~~273~~

Badness (Sintel): 1.27

=== 23-limit ===

~~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.~~

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 352/351, ~~1001/1000, 1331/1323, 289~~/~~288~~, 400/399~~, 460/459~~

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

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

Optimal ~~tuning~~ (~~CTE~~): ~28/27 = ~~1\20~~, ~3/2 = 703.~~169~~

Optimal tunings:

* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})

Badness (~~Dirichlet~~): 1.~~209~~

Badness (Sintel): 1.21

=== 29-limit ===

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 ([[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 doesn't appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 352/351, ~~1001/1000, 1331/1323, 289~~/~~288~~, 400/399, ~~460/459, 726~~/~~725~~

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

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

Optimal ~~tuning~~ (~~CTE~~): ~29/28 = ~~1\20~~, ~3/2 = 703.~~171~~

Optimal tunings:

* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})

* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})

Badness (~~Dirichlet~~): 1.~~134~~

Badness (Sintel): 1.13

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

~~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.~~

=== 2.3.5.7.11.13.17.19.23.29.37 subgroup ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 325/324, 352/351, ~~1001/1000, 1331/1323, 289~~/~~288~~, 400/399, ~~460/459, 726~~/~~725~~, 481/480

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480

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

Optimal ~~tuning (CTE)~~: ~29/28 = ~~1\20~~, ~3/2 = 703.~~222~~

Optimal tunings:

* WE: ~29/28 = 60.0001{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})

{{~~Optimal ET sequence|legend~~=~~1| 20cdeijl, 60el, 80, 140~~ }}

* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})

~~Badness (Dirichlet)~~: ~~1.127~~

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

~~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.~~

~~By looking at the mapping, we observe an 80-note [[MOS]] scale is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament.~~

~~We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators~~, ~~yielding:~~

37:~~44:54:56:58:60:65:69:74:82:85~~

Badness (Sintel): 1.13

~~Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above.~~

The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in Degrees. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].

=== 2.3.5.7.11.13.17.19.23.29.37.41 subgroup ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

Comma list: 325/324, 352/351, ~~1001/1000, 1331/1323, 289~~/~~288~~, 400/399, ~~460~~/~~459~~, ~~726~~/~~725~~, 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 }}

Optimal ~~tuning~~ (~~CTE~~): ~29/28 = ~~1\20~~, ~3/2 = 703.~~207~~

Optimal tunings:

* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})

* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})

Badness (~~Dirichlet~~): 1.~~100~~

Badness (Sintel): 1.10

== Squarschmidt ==

~~A generator for~~ the ~~squarschimidt temperament is the fourth root of~~ [[~~5/2]],~~ (5~~/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7~~-limit~~. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.~~

: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''

~~[[Subgroup~~]]~~: 2~~.~~3.5~~

~~[[Comma list]]: {{monzo| 61 4 -29 }}~~

~~{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}~~

~~: mapping generators: ~2~~, ~~~98304~~/~~78125~~

Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].

~~[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621~~

~~{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}~~

~~[[Badness]]: 0.218314~~

~~=== 7-limit ===~~

[[Subgroup]]: 2.3.5.7

Line 412:

Line 436:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}

: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}

: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}

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

{{Optimal ET sequence|legend=1| 118, 239, 357, 596~~, 1549bd~~ }}

[[Badness]]: 0.~~132821~~

[[Badness]] (Sintel): 3.36

=== 11-limit ===

Line 427:

Line 453:

Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~44/35 = 396.~~644~~

Optimal tunings:

* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}

Badness (Sintel): 1.26

~~Badness: 0~~.~~038186~~

== Leapmonth ==

Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.

~~== Subfourth ==~~

[[Subgroup]]: 2.3.5.7

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

[[Comma list]]: 10976/10935, 51200/50421

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

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~21/16 = ~~475~~.~~991~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}

: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}

: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}

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

{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}

[[Badness]]: 0.~~140722~~

[[Badness]] (Sintel): 4.79

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

Comma list: 540/539, 896/891, 1331/1323

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

Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~21/16 = ~~475~~.~~995~~

Optimal tunings:

* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}

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

{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}

Badness: 0.~~045323~~

Badness (Sintel): 1.88

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Comma list: 169/168, 352/351, 364/363, 540/539

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

Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~21/16 = ~~475~~.~~996~~

Optimal tunings:

* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}

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

Badness: 0.~~023800~~

Badness (Sintel): 1.53

[[Category:Temperament collections]]

[[Category:Hemimage temperaments| ]]

~~[[Category:Hemimage| ]] ~~

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-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).
+Temperaments discussed elsewhere are:
 * ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
 * ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
@@ Line 9: / Line 12: @@
 * ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
 * [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
-* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
+* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
-* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
+* ''[[Hendecatonic (temperament)|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]]
+Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].
 == Chromat ==
-The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[Amity family|amity extension]] with third-octave period.
+The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[amity family|amity extension]] with third-octave period.
 [[Subgroup]]: 2.3.5.7
@@ Line 23: / Line 29: @@
 {{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
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
+[[Optimal tuning]]s:
+* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
+: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
+* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
+: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}
 {{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
-[[Badness]]: 0.057499
+[[Badness]] (Sintel): 1.46
 === 11-limit ===
@@ Line 41: / Line 48: @@
 Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
+Optimal tunings:
+* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
-{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
+{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}
-Badness: 0.050379
+Badness (Sintel): 1.67
 ==== 13-limit ====
@@ Line 54: / Line 63: @@
 Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428
+Optimal tunings:
+* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}
-{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
+{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}
-Badness: 0.046006
+Badness (Sintel): 1.90
 ===== 17-limit =====
@@ Line 67: / Line 78: @@
 Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
+Optimal tunings:
+* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
-{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
+{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
-Badness: 0.031678
+Badness (Sintel): 1.61
 ==== Catachrome ====
@@ Line 80: / Line 93: @@
 Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
+Optimal tunings:
+* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}
-{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
+{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
-Badness: 0.043844
+Badness (Sintel): 1.81
 ===== 17-limit =====
@@ Line 93: / Line 108: @@
 Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
+Optimal tunings:
+* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}
-{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
+{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
-Badness: 0.030218
+Badness (Sintel): 1.54
 ==== Chromic ====
@@ Line 106: / Line 123: @@
 Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456
+Optimal tunings:
+* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}
-{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
+{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
-Badness: 0.049857
+Badness (Sintel): 2.06
 ===== 17-limit =====
@@ Line 119: / Line 138: @@
 Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
-Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459
+Optimal tunings:
+* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}
-{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
+{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
-Badness: 0.031043
+Badness (Sintel): 1.58
 === Hemichromat ===
@@ Line 132: / Line 153: @@
 Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
-Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
+Optimal tunings:
+* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}
-{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
+{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}
-Badness: 0.067173
+Badness (Sintel): 2.22
 ==== 13-limit ====
@@ Line 145: / Line 168: @@
 Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
-Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
+Optimal tunings:
+* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
+* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}
-{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
+{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
-Badness: 0.033420
+Badness (Sintel): 1.38
 == Bisupermajor ==
-{{See also| Very high accuracy temperaments #Kwazy }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 159: / Line 184: @@
 {{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
 : mapping generators: ~1225/864, ~192/175
-{{Multival|legend=1| 16 -10 34 -53 9 107 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
+: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}
+: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}
 {{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
-[[Badness]]: 0.065492
+[[Badness]] (Sintel): 1.66
 === 11-limit ===
@@ Line 177: / Line 203: @@
 Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
-Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
+Optimal tunings:
+* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}
-{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
+{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}
-Badness: 0.032080
+Badness (Sintel): 1.06
-== 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 191: / Line 219: @@
 {{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
 : mapping generators: ~567/400, ~81/80
-{{Multival|legend=1| 10 38 36 37 29 -23 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}
+: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}
+* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}
+: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
+{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
-{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
+[[Badness]] (Sintel): 2.13
-[[Badness]]: 0.084317
 === 11-limit ===
@@ Line 209: / Line 238: @@
 Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390
+Optimal tunings:
+* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}
-{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
+{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}
-Badness: 0.030461
+Badness (Sintel): 1.01
 === 13-limit ===
@@ Line 222: / Line 253: @@
 Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427
+Optimal tunings:
+* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}
-{{Optimal ET sequence|legend=1| 58, 118, 176f }}
+{{Optimal ET sequence|legend=0| 58, 118, 176f }}
-Badness: 0.026336
+Badness (Sintel): 1.09
 === 17-limit ===
@@ Line 235: / Line 268: @@
 Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
+Optimal tunings:
+* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}
-{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}
+{{Optimal ET sequence|legend=0| 58, 118 }}
-Badness: 0.022396
+Badness (Sintel): 1.14
 == 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.
+{{About|the regular temperament|scale degrees|degree}}
+{{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.
+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.
+By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
 [[Subgroup]]: 2.3.5.7
@@ Line 249: / Line 293: @@
 {{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
 : mapping generators: ~28/27, ~3
-{{Multival|legend=1| 20 40 60 17 39 27 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
-[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015
+: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
+* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
+: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}
-{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
+{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}
-[[Badness]]: 0.106471
+[[Badness]] (Sintel): 2.69
-Badness (Dirichlet): 2.694
 === 11-limit ===
@@ Line 269: / Line 312: @@
 Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
-Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231
+Optimal tunings:
+* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
+* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})
-{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}
-Badness: 0.046770
+Badness (Sintel): 1.55
-Badness (Dirichlet): 1.546
 === 13-limit ===
@@ Line 284: / Line 327: @@
 Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
-Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080
+Optimal tunings:
+* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
+* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})
-{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness: 0.032718
+Badness (Sintel): 1.35
-Badness (Dirichlet): 1.352
 === 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288
+Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
-Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107
+Optimal tunings:
+* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})
+* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})
-{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness (Dirichlet): 1.171
+Badness (Sintel): 1.17
 === 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399
+Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
-Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107
+Optimal tunings:
+* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
+* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})
-{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness (Dirichlet): 1.273
+Badness (Sintel): 1.27
 === 23-limit ===
-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.
 Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459
+Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
-Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169
+Optimal tunings:
+* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})
+* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})
-{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness (Dirichlet): 1.209
+Badness (Sintel): 1.21
 === 29-limit ===
-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 ([[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 doesn't appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
 Subgroup: 2.3.5.7.11.13.17.19.23.29
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725
+Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
-Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171
+Optimal tunings:
+* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
+* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})
-{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness (Dirichlet): 1.134
+Badness (Sintel): 1.13
-=== no-31's 37-limit ===
-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.
+=== 2.3.5.7.11.13.17.19.23.29.37 subgroup ===
 Subgroup: 2.3.5.7.11.13.17.19.23.29.37
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725, 481/480
+Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
 Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
-Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222
+Optimal tunings:
+* WE: ~29/28 = 60.0001{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})
-{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
+* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})
-Badness (Dirichlet): 1.127
-=== no-31's 41-limit ===
-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.
-By looking at the mapping, we observe an 80-note [[MOS]] scale is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament.
-We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding:
+{{Optimal ET sequence|legend=0| 60el, 80, 140 }}
-:44:54:56:58:60:65:69:74:82:85
+Badness (Sintel): 1.13
-Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above.
-The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in Degrees. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
+=== 2.3.5.7.11.13.17.19.23.29.37.41 subgroup ===
 Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725, 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 }}
-Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
+Optimal tunings:
+* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})
+* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})
-{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60el, 80, 140 }}
-Badness (Dirichlet): 1.100
+Badness (Sintel): 1.10
 == Squarschmidt ==
-A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
+: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''
-[[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 61 4 -29 }}
-{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
-: mapping generators: ~2, ~98304/78125
+Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
-{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
-[[Badness]]: 0.218314
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 412: / Line 436: @@
 {{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
-{{Multival|legend=1| 29 4 69 -61 28 149 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}
+: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}
+: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
+{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
-{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
-[[Badness]]: 0.132821
+[[Badness]] (Sintel): 3.36
 === 11-limit ===
@@ Line 427: / Line 453: @@
 Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
-Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
+Optimal tunings:
+* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}
-{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
+{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}
+Badness (Sintel): 1.26
-Badness: 0.038186
+== Leapmonth ==
+Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.
-== Subfourth ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 10976/10935, 65536/64827
+[[Comma list]]: 10976/10935, 51200/50421
-{{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 }}
+{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}
+: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
+: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}
-{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
+{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}
-[[Badness]]: 0.140722
+[[Badness]] (Sintel): 4.79
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 540/539, 896/891, 12005/11979
+Comma list: 540/539, 896/891, 1331/1323
-Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
+Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
+Optimal tunings:
+* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
-{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
+{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}
-Badness: 0.045323
+Badness (Sintel): 1.88
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 352/351, 364/363, 540/539, 676/675
+Comma list: 169/168, 352/351, 364/363, 540/539
-Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
+Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
+Optimal tunings:
+* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}
-{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
+{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}
-Badness: 0.023800
+Badness (Sintel): 1.53
 [[Category:Temperament collections]]
 [[Category:Hemimage temperaments| ]] <!-- main article -->
-[[Category:Hemimage| ]] <!-- key article -->
 [[Category:Rank 2]]