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Hemimage temperaments: Difference between revisions

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


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


{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
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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| 20cd, 60e, 80, 140, 360 }}
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
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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| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
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Subgroup: 2.3.5.7.11.13.17
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 }}
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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)


{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
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Subgroup: 2.3.5.7.11.13.17.19
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 }}
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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)


{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
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=== 23-limit ===
=== 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
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 }}
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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)


{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
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=== 29-limit ===
=== 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
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 }}
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 tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)


{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
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=== no-31's 37-limit ===
=== 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.
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
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 }}
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 tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)


{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
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=== no-31's 41-limit ===
=== 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:
37:44:54:56:58:60:69:74:82:85
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 Degrees. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
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


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: {{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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