Xenharmonic Wiki

Hemimage temperaments: Difference between revisions

← Older edit
VisualWikitext
+bisupermajor from temperament finder
 
(64 intermediate revisions by 10 users not shown)
Line 1: Line 1:
Hemimage temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935.
{{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]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn 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) → [[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) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[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]]


= Commatic =
== Chromat ==
Commas: 10976/10935, 50421/50000
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.


POTE generator: ~81/80 = 20.377
[[Subgroup]]: 2.3.5.7


Map: [<2 3 4 5|, <0 5 19 18|]
[[Comma list]]: 10976/10935, 235298/234375


Wedgie: <<10 38 36 37 29 -23||
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}


EDOs: 58, 118, 294, 412d, 530d
: mapping generators: ~63/50, ~28/27


Badness: 0.0843
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528


== 11-limit ==
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
Commas: 441/440, 3388/3375, 8019/8000


POTE generator: ~81/80 = 20.390
[[Badness]]: 0.057499


Map: [<2 3 4 5 6|, <0 5 19 18 27|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 58, 118, 294, 412d
Comma list: 441/440, 4375/4356, 10976/10935


Badness: 0.0305
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


= Chromat =
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
{{see also|Amity family}}


Commas: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}


POTE generator: ~28/27 = 60.528
Badness: 0.050379


Map: [<3 4 5 6|, <0 5 13 16|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Wedgie: <<15 39 48 27 34 2||
Comma list: 364/363, 441/440, 625/624, 10976/10935


EDOs: 60, 99, 258, 357, 456
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Badness: 0.0575
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428


= Degrees =
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
Commas: 10976/10935, 390625/388962


POTE generator: ~3/2 = 703.015
Badness: 0.046006


Map: [<20 0 -17 -39|, <0 1 2 3|]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Wedgie: <<20 40 60 17 39 27||
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


EDOs: 60, 80, 140, 640b, 780b, 920b
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


Badness: 0.1065
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438


== 11-limit ==
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
Commas: 1331/1323, 1375/1372, 2200/2187


POTE generator: ~3/2 = 703.231
Badness: 0.031678


Map: [<20 0 -17 -39 -26|, <0 1 2 3 3|]
==== Catachrome ====
Subgroup: 2.3.5.7.11.13


EDOs: 80, 140, 360, 500be, 860bde
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


Badness: 0.0468
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


== 13-limit ==
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
Commas: 325/324, 352/351, 1001/1000, 1331/1323


POTE generator: ~3/2 = 703.080
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Map: [<20 0 -17 -39 -26 74|, <0 1 2 3 3 0|]
Badness: 0.043844


EDOs: 60e, 80, 140, 500be, 640be, 780be
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0327
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


= Subfourth =
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
Commas: 10976/10935, 65536/64827


POTE generator: ~21/16 = 475.991
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377


Map: [<1 0 17 4|, <0 4 -37 -3|]
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


EDOs: 58, 121, 179, 300bd, 479bcd
Badness: 0.030218


Badness: 0.1407
==== Chromic ====
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 196/195, 352/351, 729/728, 1875/1859
Commas: 540/539, 896/891, 12005/11979


POTE generator: ~21/16 = 475.995
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


Map: [<1 0 17 4 11|, <0 4 -37 -3 -19|]
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456


EDOs: 58, 121, 179e, 300bde
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.0453
Badness: 0.049857


== 13-limit ==
===== 17-limit =====
Commas: 352/351, 364/363, 540/539, 676/675
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~21/16 = 475.996
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


Map: [<1 0 17 4 11 16|, <0 4 -37 -3 -19 -31|]
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


EDOs: 58, 121, 179ef, 300bdef
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459


Badness: 0.0238
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


= Bisupermajor =
Badness: 0.031043
{{see also| Very high accuracy temperaments #Kwazy }}


Commas: 10976/10935, 65625/65536
=== Hemichromat ===
Subgroup: 2.3.5.7.11


POTE generator: ~192/175 = 162.8061
Comma list: 3025/3024, 10976/10935, 102487/102400


Map: [<2 1 6 1|, <0 8 -5 17|]
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


EDOs: 22, 74d, 96d, 118, 140, 258, 398
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511


Badness: 0.0655
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}


== 11-limit ==
Badness: 0.067173
Commas: 385/384, 3388/3375, 9801/9800


POTE generators: ~11/10 = 162.7733
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [<2 1 6 1 8|, <0 8 -5 17 -4|]
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935


EDOs: 22, 74d, 96d, 118, 258e, 376de
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


Badness: 0.0321
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527


[[Category:Theory]]
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
[[Category:Temperament]]
 
[[Category:Hemimage]]
Badness: 0.033420
 
== Bisupermajor ==
{{See also| Very high accuracy temperaments #Kwazy }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 65625/65536
 
{{Mapping|legend=1| 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|legend=1| 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: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
 
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
 
{{Optimal ET sequence|legend=1| 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|legend=1| 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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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{{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
 
[[Comma list]]: 10976/10935, 390625/388962
 
{{Mapping|legend=1| 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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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 ==
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.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}
 
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
 
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
 
[[Badness]]: 0.218314
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 29360128/29296875
 
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
 
Badness: 0.038186
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"