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:  
{{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]]
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
Line 12: Line 13:
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


== Chromat ==
== Chromat ==
Line 23: Line 25:


{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
{{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
: mapping generators: ~63/50, ~28/27
Line 136: Line 136:
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.0672
Badness: 0.067173


==== 13-limit ====
==== 13-limit ====
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{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.0334
Badness: 0.033420


== Bisupermajor ==
== Bisupermajor ==
Line 161: Line 161:


: mapping generators: ~1225/864, ~192/175
: 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
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
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Badness: 0.032080
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
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~567/400, ~81/80
: 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
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
Line 242: Line 238:


== Degrees ==
== 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.
{{ 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
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~28/27, ~3
: 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)


[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}


{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b, 920b }}
[[Badness]]: 0.106471


[[Badness]]: 0.106471
Badness (Sintel): 2.694


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)


{{Optimal ET sequence|legend=1| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}


Badness: 0.046770
Badness: 0.046770
Badness (Sintel): 1.546


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
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 tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)


{{Optimal ET sequence|legend=1| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


Badness: 0.032718
Badness: 0.032718


== Squarschmidt ==
Badness (Sintel): 1.352
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.
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.171
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 (~100/99 = 16.893)


[[Subgroup]]: 2.3.5
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


[[Comma list]]: {{monzo| 61 4 -29 }}
Badness (Sintel): 1.273


{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


: mapping generators: ~2, ~98304/78125
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)


[[Badness]]: 0.218314
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}


=== 7-limit ===
Badness (Sintel): 1.209
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 29360128/29296875
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405


{{Multival|legend=1| 29 4 69 -61 28 149 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)


{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}


[[Badness]]: 0.132821
Badness (Sintel): 1.134


=== 11-limit ===
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480


Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
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 (POTE): ~2 = 1\1, ~44/35 = 396.644
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)


{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}


Badness: 0.038186
Badness (Sintel): 1.127


== Subfourth ==
=== no-31's 41-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


[[Comma list]]: 10976/10935, 65536/64827
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|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
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 }}


: mapping generators: ~2, ~21/16
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991
Badness (Sintel): 1.100


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


[[Badness]]: 0.140722
[[Subgroup]]: 2.3.5


=== 11-limit ===
[[Comma list]]: {{monzo| 61 4 -29 }}
Subgroup: 2.3.5.7.11


Comma list: 540/539, 896/891, 12005/11979
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}


Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
: mapping generators: ~2, ~98304/78125


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621


{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


Badness: 0.045323
[[Badness]]: 0.218314


=== 13-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 352/351, 364/363, 540/539, 676/675
[[Comma list]]: 10976/10935, 29360128/29296875


Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643


{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


Badness: 0.023800
[[Badness]]: 0.132821


== Hemihemi ==
=== 11-limit ===
{{See also|Subgroup temperaments#Hemihemi}}
Subgroup: 2.3.5.7.11
[[Subgroup]]: 3/2.5/2.7/2


[[Comma list]]: [[10976/10935]]
Comma list: 3025/3024, 5632/5625, 10976/10935


{{Mapping|legend=2| 1 2 3 | 0 3 1  }}
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644


[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


<nowiki>*</nowiki> wart for 3/2
Badness: 0.038186


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:29, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

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 extension with third-octave period.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

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

Optimal tuning (POTE): ~63/50 = 1\3, ~28/27 = 60.528

Optimal ET sequence: 39d, 60, 99, 258, 357, 456

Badness: 0.057499

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 10976/10935

Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430

Optimal ET sequence: 60e, 99e, 159, 258, 417d

Badness: 0.050379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 10976/10935

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 ET sequence: 99ef, 159, 258, 417d

Badness: 0.046006

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757

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 ET sequence: 99ef, 159, 258, 417dg

Badness: 0.031678

Catachrome

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 1001/1000, 10976/10935

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 ET sequence: 60e, 99e, 159

Badness: 0.043844

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913

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 ET sequence: 60e, 99e, 159

Badness: 0.030218

Chromic

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1875/1859

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 ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.049857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 352/351, 375/374, 595/594

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 ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.031043

Hemichromat

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 10976/10935, 102487/102400

Mapping: [⟨3 4 5 6 10], ⟨0 10 26 32 5]]

Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511

Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b

Badness: 0.067173

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935

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 ET sequence: 39df, 120cdff, 159, 198, 357, 912b

Badness: 0.033420

Bisupermajor

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [⟨2 1 6 1], ⟨0 8 -5 17]]

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

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

Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d

Badness: 0.065492

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [⟨2 1 6 1 8], ⟨0 8 -5 17 -4]]

Optimal tuning (POTE): ~99/70, ~11/10 = 162.773

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de

Badness: 0.032080

Bicommatic

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

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

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

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

Optimal ET sequence: 58, 118, 294, 412d, 530d

Badness: 0.084317

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390

Optimal ET sequence: 58, 118, 294, 412d

Badness: 0.030461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1001/1000

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 ET sequence: 58, 118, 176f

Badness: 0.026336

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

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 ET sequence: 58, 118, 294ffg, 412dffgg

Badness: 0.022396

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 ¢, we get a uniquely elegant extension to the 29-limit which tempers out (33/25)/(29/22) = 726/725, S28 = 784/783 and S29 = 841/840. An edo as large as 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

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

mapping generators: ~28/27, ~3

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

Optimal ET sequence: 20cd, 60, 80, 140, 640b, 780b

Badness: 0.106471

Badness (Sintel): 2.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)

Optimal ET sequence: 20cd, 60e, 80, 140, 360

Badness: 0.046770

Badness (Sintel): 1.546

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness: 0.032718

Badness (Sintel): 1.352

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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 (~100/99 = 16.893)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.171

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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 (~100/99 = 16.893)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.273

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

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 (~100/99 = 16.831)

Optimal ET sequence: 20cdei, 60e, 80, 140

Badness (Sintel): 1.209

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

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 (~100/99 = 16.829)

Optimal ET sequence: 20cdeij, 60e, 80, 140

Badness (Sintel): 1.134

no-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

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

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 (~100/99 = 16.778)

Optimal ET sequence: 20cdeijl, 60el, 80, 140

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, 2871/2870

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 ET sequence: 20cdeijlm, 60el, 80, 140

Badness (Sintel): 1.100

Squarschmidt

A generator for the squarschimidt temperament is the fourth root of 5/2, (5/2)1/4, 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.

Subgroup: 2.3.5

Comma list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

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

Optimal ET sequence: 118, 593, 711, 829, 947

Badness: 0.218314

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]

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

Optimal ET sequence: 118, 239, 357, 596, 1549bd

Badness: 0.132821

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: [⟨1 -8 1 -20 -21], ⟨0 29 4 69 74]]

Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644

Optimal ET sequence: 118, 239, 357, 596

Badness: 0.038186

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