Xenharmonic Wiki

Hemimage temperaments: Difference between revisions

← Older edit
VisualWikitext
Godtone (talk | contribs)
autopatrolled
2,289 edits
m →Degrees: add 41-limit extension that lowers the badness and has an accurate 41 based on convergents of 20 EDO intervals 56/4141/30
 
(14 intermediate revisions by 5 users not shown)
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). 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) → [[Quartonic family]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
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 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
Line 183: Line 181:
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
Line 193: Line 191:


: 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
Line 252: Line 255:
: 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| 20cd, 60, 80, 140, 640b, 780b }}
Line 260: Line 261:
[[Badness]]: 0.106471
[[Badness]]: 0.106471


Badness (Dirichlet): 2.694
Badness (Sintel): 2.694


=== 11-limit ===
=== 11-limit ===
Line 269: Line 270:
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 }}
Line 275: Line 276:
Badness: 0.046770
Badness: 0.046770


Badness (Dirichlet): 1.546
Badness (Sintel): 1.546


=== 13-limit ===
=== 13-limit ===
Line 284: Line 285:
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 }}
Line 290: Line 291:
Badness: 0.032718
Badness: 0.032718


Badness (Dirichlet): 1.352
Badness (Sintel): 1.352


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


Badness (Dirichlet): 1.171
Badness (Sintel): 1.171


=== 19-limit ===
=== 19-limit ===
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 }}


Badness (Dirichlet): 1.273
Badness (Sintel): 1.273


=== 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 }}


Badness (Dirichlet): 1.209
Badness (Sintel): 1.209


=== 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 }}


Badness (Dirichlet): 1.134
Badness (Sintel): 1.134


=== 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 }}


Badness (Dirichlet): 1.127
Badness (Sintel): 1.127


=== 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:
37:44:54:56:58:60:65:69:74:82:85
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, 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 }}
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 }}
Line 386: Line 369:
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}


Badness (Dirichlet): 1.100
Badness (Sintel): 1.100


== Squarschmidt ==
== Squarschmidt ==
Line 411: Line 394:


{{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
Line 432: Line 413:


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"