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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]]
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* ''[[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) → [[Quartonic family]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
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{{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
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: 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
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== 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
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: {{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)
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 20cdeijlm, 60el, 80, 140 }}
Badness (Sintel): 1.100


== Squarschmidt ==
== Squarschmidt ==
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{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
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Badness: 0.038186
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: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]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"