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Hemimage temperaments: Difference between revisions

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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* [[Quasisuper]] (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Stearnsmic clan #Echidna|Stearnsmic clan]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]
* ''[[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]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].  
Considered below are bisupermajor, bicommatic, degrees, 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.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 235298/234375
 
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27
 
[[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]] (Sintel): 1.46
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4356, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
 
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=0| 60e, 99e, 159, 258 }}
 
Badness (Sintel): 1.67
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 625/624, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
 
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=0| 60eff, 99ef, 159, 258, 417d }}
 
Badness (Sintel): 1.90
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
 
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
 
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=0| 99ef, 159, 258, 417dg }}
 
Badness (Sintel): 1.61
 
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
 
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=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.81
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
 
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
 
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=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.54
 
==== Chromic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1875/1859
 
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
 
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=0| 60e, 99ef, 159f }}
 
Badness (Sintel): 2.06
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
 
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
 
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=0| 60e, 99ef, 159f }}
 
Badness (Sintel): 1.58
 
=== Hemichromat ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 10976/10935, 102487/102400
 
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
 
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=0| 39d, 120cd, 159, 198, 357, 912b }}
 
Badness (Sintel): 2.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
 
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=0| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness (Sintel): 1.38


== Bisupermajor ==
== Bisupermajor ==
Line 201: Line 47:
Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 385/384, 3388/3375, 9801/9800


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


Optimal tunings:  
Optimal tunings:  
Line 236: Line 82:
Comma list: 441/440, 3388/3375, 8019/8000
Comma list: 441/440, 3388/3375, 8019/8000


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


Optimal tunings:  
Optimal tunings:  
Line 251: Line 97:
Comma list: 196/195, 352/351, 729/728, 1001/1000
Comma list: 196/195, 352/351, 729/728, 1001/1000


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


Optimal tunings:  
Optimal tunings:  
Line 266: Line 112:
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


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


Optimal tunings:  
Optimal tunings:  
Line 282: Line 128:
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.
An obvious extension to the 23-limit exists by mapping [[23/20]] to 4\20 (1\5), [[69/56]] to 6\20 (3\10), and [[23/18]] to 7\20. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by mapping [[29/22]] to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out [[726/725]], which is the difference between [[33/25]] and [[29/22]], as well as [[784/783]] ({{S|28}}) and [[841/840]] ({{S|29}}). 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 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 for prime [[37/1|37]] agreeing with many [[semiconvergent]]s, tempering out [[481/480]]. By mapping [[60/41]] and [[41/28]] to 11\20 or equivalently [[56/41]] and [[41/30]] to 9\20 and by mapping [[44/41]] to 1\10 (among many other equivalences), there is a very efficient extension for prime [[41/1|41]] tempering out [[451/450]].


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]].
The 80-note generator chain is ideal, so [[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 310: Line 156:
Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 1331/1323, 1375/1372, 2200/2187


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


Optimal tunings:  
Optimal tunings:  
Line 325: Line 171:
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


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


Optimal tunings:  
Optimal tunings:  
Line 340: Line 186:
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
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|legend=0| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


Optimal tunings:  
Optimal tunings:  
Line 355: Line 201:
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
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|legend=0| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 370: Line 216:
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
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|legend=0| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Optimal tunings:  
Optimal tunings:  
Line 385: Line 231:
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
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|legend=0| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Optimal tunings:  
Optimal tunings:  
Line 394: Line 240:


Badness (Sintel): 1.13
Badness (Sintel): 1.13
=== 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: 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 tunings:
* WE: ~29/28 = 60.0001{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}
Badness (Sintel): 1.13
=== 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: 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 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=0| 60el, 80, 140 }}
Badness (Sintel): 1.10


== Squarschmidt ==
== Squarschmidt ==
Line 451: Line 267:
Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 3025/3024, 5632/5625, 10976/10935


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


Optimal tunings:  
Optimal tunings:  
Line 485: Line 301:
Comma list: 540/539, 896/891, 1331/1323
Comma list: 540/539, 896/891, 1331/1323


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


Optimal tunings:  
Optimal tunings:  
Line 500: Line 316:
Comma list: 169/168, 352/351, 364/363, 540/539
Comma list: 169/168, 352/351, 364/363, 540/539


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


Optimal tunings:  
Optimal tunings:  
Line 510: Line 326:
Badness (Sintel): 1.53
Badness (Sintel): 1.53


[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
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