Amity family: Difference between revisions

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The '''amity family''' tempers out the 5-limit [[amity comma]], 1600000/1594323. The generator for the [[amity]] temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 comma to 243/200, and from this it derives its name. If you are looking for a different kind of neutral third this could be the temperament for you.
{{Technical data page}}
The '''amity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[amity comma]] ({{monzo|legend=1| 9 -13 15 }}, [[ratio]]: 1600000/1594323).  


== Amity ==
== Amity ==
{{Main| Amity }}
{{Main| Amity }}


In the 5-limit amity is a genuine [[microtemperament]], with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  
The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[syntonic comma]] to [[243/200]], and from this it derives its name. If you are looking for a different kind of neutral third, this could be the temperament for you. However its functional generator is the [[octave complement]] thereof, five of which stack to the [[12/1|12th harmonic]]. Its [[ploidacot]] is thus gamma-pentacot.
 
It is a member of the [[syntonic–chromatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates an apotome with a stack of five syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so unless 53edo is used as a tuning, the schisma is always observed.
 
Amity is a genuine [[microtemperament]] in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 15:
[[Comma list]]: 1600000/1594323
[[Comma list]]: 1600000/1594323


[[Mapping]]: [{{val| 1 3 6 }}, {{val| 0 -5 -13 }}]
{{Mapping|legend=1| 1 -2 -7 | 0 5 13 }}
 
: mapping generators: ~2, ~400/243
Mapping generators: ~2, ~243/200


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.519
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9135{{c}}, ~400/243 = 860.4191{{c}}
: [[error map]]: {{val| -0.087 +0.314 -0.259 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~400/243 = 860.4726{{c}}
: error map: {{val| 0.000 +0.408 -0.169 }}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 152, 205, 463, 668, 873 }}


[[Badness]]: 0.021960
[[Badness]] (Sintel): 0.515


=== Overview to extensions ===
=== Overview to extensions ===
The second comma to extend the 5-limit amity include 4375/4374 for amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, 2401/2400 for amicable, and 16875/16807 for familia.  
The second comma to extend 5-limit amity include [[4375/4374]] for septimal amity, [[65625/65536]] for paramity, [[126/125]] for accord, and [[225/224]] for houborizic. Those all use the same generator as the 5-limit temperament.  


Temperaments discussed elsewhere include:
Bamity adds [[245/243]] with a 1/2-octave period. Hamity adds [[2430/2401]] and splits the generator in two. Gamity adds [[1029/1024]]. Chromat adds [[10976/10935]]. Trinity adds [[703125/702464]]. Those split the generator in three. Amicable adds [[2401/2400]], splitting the generator in four. Witcher adds [[420175/419904]]. Calamity adds [[2100875/2097152]]. Familia adds [[16875/16807]]. Those split the generator in five.
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]


== Septimal amity ==
== Septimal amity ==
{{Main| Amity }}
{{Main| Amity }}


Septimal amity can be described as the 46 &amp; 53 temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo]] is a good tuning, with generator 28\99.  
Septimal amity can be described as the {{nowrap| 46 & 53 }} temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo]] is a good tuning, with generator 28\99.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 36: Line 42:
[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103


[[Mapping]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
{{Mapping|legend=1| 1 -2 -7 15 | 0 5 13 -17 }}


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6100{{c}}, ~105/64 = 860.2881{{c}}
: [[error map]]: {{val| -0.390 +0.266 +0.162 +0.426 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~105/64 = 860.5650{{c}}
: error map: {{val| 0.000 +0.870 +1.032 +1.568 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99, 152, 251, 350, 601cd, 905bcdd }}


{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
[[Badness]] (Sintel): 0.598
 
[[Badness]]: 0.023649


=== 11-limit ===
=== 11-limit ===
Line 51: Line 59:
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
Mapping: {{mapping| 1 -2 -7 15 -41 | 0 5 13 -17 62 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.464
Optimal tunings:
* WE: ~2 = 1199.5961{{c}}, ~105/64 = 860.2460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5314{{c}}


{{Optimal ET sequence|legend=1| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}


Badness: 0.031506
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
Comma list: 352/351, 540/539, 625/624, 729/728
 
Mapping: {{mapping| 1 -2 -7 15 -41 -30 | 0 5 13 -17 62 47 }}
 
Optimal tunings:
* WE: ~2 = 1199.5437{{c}}, ~105/64 = 860.1922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5140{{c}}
 
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f }} *
 
<nowiki/>* optimal patent val: 205
 
Badness (Sintel): 1.16


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481
Comma list: 256/255, 352/351, 375/374, 540/539, 729/728


{{Optimal ET sequence|legend=1| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Mapping: {{mapping| 1 -2 -7 15 -41 -30 17 | 0 5 13 -17 62 47 -18 }}


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Optimal tunings:
* WE: ~2 = 1199.3376{{c}}, ~28/17 = 860.0617{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.5317{{c}}


Badness: 0.028008
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg }}
 
Badness (Sintel): 1.33
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455
 
Mapping: {{mapping| 1 -2 -7 15 -41 -30 17 -23 | 0 5 13 -17 62 47 -18 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.3361{{c}}, ~28/17 = 860.0605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.5310{{c}}
 
{{Optimal ET sequence|legend=0| 46efh, 53, 99ef, 152fg }}
 
Badness (Sintel): 1.14


=== Hitchcock ===
=== Hitchcock ===
Line 79: Line 121:
Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187


Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
Mapping: {{mapping| 1 -2 -7 15 -3 | 0 5 13 -17 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.390
Optimal tunings:
* WE: ~2 = 1199.9979{{c}}, ~18/11 = 860.6089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6104{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness: 0.035187
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 92: Line 136:
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
Mapping: {{mapping| 1 -2 -7 15 -3 8 | 0 5 13 -17 9 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.419
Optimal tunings:
* WE: ~2 = 1200.2037{{c}}, ~18/11 = 860.5820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.7268{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness: 0.022448
Badness (Sintel): 0.928


==== 17-limit ====
==== 17-limit ====
Line 105: Line 151:
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272


Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Mapping: {{mapping| 1 -2 -7 15 -3 8 17 | 0 5 13 -17 9 -6 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.366
Optimal tunings:
* WE: ~2 = 1200.0179{{c}}, ~18/11 = 860.6472{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6344{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 99 }}


Badness: 0.019395
Badness (Sintel): 0.988


==== 19-limit ====
==== 19-limit ====
Line 118: Line 166:
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189


Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
Mapping: {{mapping| 1 -2 -7 15 -3 8 17 15 | 0 5 13 -17 9 -6 -18 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.407
Optimal tunings:
* WE: ~2 = 1200.1977{{c}}, ~18/11 = 860.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.5917{{c}}


{{Optimal ET sequence|legend=1| 7, 39h, 46, 53, 99h }}
{{Optimal ET sequence|legend=0| 7, 46, 53, 99h }}


Badness: 0.017513
Badness (Sintel): 1.07


=== Catamite ===
=== Stalagmite ===
The catamite temperament (46 &amp; 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle) in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit. The word "catamite" itself is a term for male homosexual, but also a play on the words "cata-" (down) and "amity."
The stalagmite temperament (46 & 99ef) tempers out 441/440 ([[werckisma]]) and 896/891 ([[pentacircle comma]]) in the 11-limit; [[196/195]], [[352/351]] and [[364/363]] in the 13-limit. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is ''down'' from amity.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 133: Line 183:
Comma list: 441/440, 896/891, 4375/4374
Comma list: 441/440, 896/891, 4375/4374


Mapping: [{{val| 1 3 6 -2 -7 }}, {{val| 0 -5 -13 17 37 }}]
Mapping: {{mapping| 1 -2 -7 15 30 | 0 5 13 -17 -37 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.340
Optimal tunings:
* WE: ~2 = 1199.3915{{c}}, ~105/64 = 860.2240{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6649{{c}}


{{Optimal ET sequence|legend=1| 46, 99e, 145, 244e }}
{{Optimal ET sequence|legend=0| 46, 99e, 145, 244e }}


Badness: 0.040976
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 146: Line 198:
Comma list: 196/195, 352/351, 364/363, 4375/4374
Comma list: 196/195, 352/351, 364/363, 4375/4374


Mapping: [{{val| 1 3 6 -2 -7 -11 }}, {{val| 0 -5 -13 17 37 52 }}]
Mapping: {{mapping| 1 -2 -7 15 30 41 | 0 5 13 -17 -37 -52 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.313
Optimal tunings:
* WE: ~2 = 1199.3577{{c}}, ~105/64 = 860.2259{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6929{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.034215
Badness (Sintel): 1.41


==== 17-limit ====
==== 17-limit ====
Line 159: Line 213:
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155


Mapping: [{{val| 1 3 6 -2 -7 -11 -1 }}, {{val| 0 -5 -13 17 37 52 18 }}]
Mapping: {{mapping| 1 -2 -7 15 30 41 17 | 0 5 13 -17 -37 -52 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.313
Optimal tunings:
* WE: ~2 = 1199.3555{{c}}, ~28/17 = 860.2244{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6932{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.021193
Badness (Sintel): 1.08


==== 19-limit ====
==== 19-limit ====
Line 172: Line 228:
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475


Mapping: [{{val| 1 3 6 -2 -7 -11 -1 -13 }}, {{val| 0 -5 -13 17 37 52 18 61 }}]
Mapping: {{mapping| 1 -2 -7 15 30 41 17 48 | 0 5 13 -17 -37 -52 -18 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.325
Optimal tunings:
* WE: ~2 = 1199.3599{{c}}, ~28/17 = 860.2161{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6820{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.018864
Badness (Sintel): 1.15


=== Hemiamity ===
=== Hemiamity ===
Line 185: Line 243:
Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 3025/3024, 4375/4374, 5120/5103


Mapping: [{{val| 2 1 -1 13 13 }}, {{val| 0 5 13 -17 -14 }}]
Mapping: {{mapping| 2 1 -1 13 13 | 0 5 13 -17 -14 }}
: mapping generators: ~99/70, ~64/55


Mapping generators: ~99/70, ~64/55
Optimal tunings:
* WE: ~99/70 = 599.8218{{c}}, ~64/55 = 260.4833{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5615{{c}}


Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561
{{Optimal ET sequence|legend=0| 46, 106, 152, 350, 502d, 852bcdde }}


{{Optimal ET sequence|legend=1| 14cde, 46, 106, 152, 350, 502d }}
Badness (Sintel): 1.04
 
Badness: 0.031307


==== 13-limit ====
==== 13-limit ====
Line 200: Line 259:
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Comma list: 352/351, 847/845, 1716/1715, 3025/3024


Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583
Optimal tunings:
* WE: ~99/70 = 599.7965{{c}}, ~64/55 = 260.4951{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5872{{c}}


{{Optimal ET sequence|legend=1| 46, 106f, 152f, 198, 350f, 548cdff }}
{{Optimal ET sequence|legend=0| 46, 106f, 152f, 198, 350f, 548cdff }}


Badness: 0.025784
Badness (Sintel): 1.07


== Accord ==
== Accord ==
Accord tempers out [[126/125]], the starling comma, and may be described as the {{nowrap| 39d & 46 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 100352/98415
[[Comma list]]: 126/125, 100352/98415


[[Mapping]]: [{{val| 1 3 6 11 }}, {{val| 0 -5 -13 -29 }}]
{{Mapping|legend=1| 1 -2 -7 -18 | 0 5 13 29 }}


{{Multival|legend=1| 5 13 29 9 32 31 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7458{{c}}, ~224/135 = 860.1071{{c}}
: [[error map]]: {{val| -1.254 +1.089 +3.858 -3.144 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 860.9433{{c}}
: error map: {{val| 0.000 +2.761 +5.949 -1.471 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 338.993
{{Optimal ET sequence|legend=1| 7d, …, 39d, 46, 131c }}


{{Optimal ET sequence|legend=1| 7d, 39d, 46, 131c, 177c }}
[[Badness]] (Sintel): 2.42
 
[[Badness]]: 0.095612


=== 11-limit ===
=== 11-limit ===
Line 228: Line 293:
Comma list: 121/120, 126/125, 896/891
Comma list: 121/120, 126/125, 896/891


Mapping: [{{val| 1 3 6 11 6 }}, {{val| 0 -5 -13 -29 -9 }}]
Mapping: {{mapping| 1 -2 -7 -18 -3 | 0 5 13 29 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.047
Optimal tunings:
* WE: ~2 = 1199.3576{{c}}, ~18/11 = 860.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.9224{{c}}


{{Optimal ET sequence|legend=1| 7d, 39d, 46, 177c, 223bc, 269bce }}
{{Optimal ET sequence|legend=0| 7d, 39d, 46 }}


Badness: 0.042468
Badness (Sintel): 1.40


== Houborizic ==
== Houborizic ==
The ''houborizic'' temperament (53&amp;113) tempers out the [[marvel comma]], 225/224. It is so named because it is closely related to the '''houboriz tuning''' (generator: 339.774971 cents).
Houborizic tempers out 225/224, the [[marvel comma]], and may be described as the {{nowrap| 53 & 60 }} temperament. It was named by [[Xenllium]] in 2021 for its close relation to the ''houboriz tuning'' (generator: 339.774971 cents).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 243: Line 310:
[[Comma list]]: 225/224, 1250000/1240029
[[Comma list]]: 225/224, 1250000/1240029


[[Mapping]]: [{{val| 1 3 6 13 }}, {{val| 0 -5 -13 -36 }}]
{{Mapping|legend=1| 1 -2 -7 -23 | 0 5 13 36 }}
 
{{Multival|legend=1| 5 13 36 9 43 47 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4959{{c}}, ~400/243 = 860.5922{{c}}
: [[error map]]: {{val| +0.496 +0.014 -2.086 +1.090 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/243 = 860.2558{{c}}
: error map: {{val| 0.000 -0.676 -2.989 +0.382 }}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 113, 166 }}
{{Optimal ET sequence|legend=1| 7d, …, 46d, 53, 113, 166 }}


[[Badness]]: 0.066638
[[Badness]] (Sintel): 1.69


=== 11-limit ===
=== 11-limit ===
Line 258: Line 327:
Comma list: 225/224, 385/384, 1250000/1240029
Comma list: 225/224, 385/384, 1250000/1240029


Mapping: [{{val| 1 3 6 13 -9 }}, {{val| 0 -5 -13 -36 44 }}]
Mapping: {{mapping| 1 -2 -7 -23 35 | 0 5 13 36 -44 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.763
Optimal tunings:
* WE: ~2 = 1200.4973{{c}}, ~400/243 = 860.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.2386{{c}}


{{Optimal ET sequence|legend=1| 53, 113, 166 }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness: 0.067891
Badness (Sintel): 2.24


==== 13-limit ====
==== 13-limit ====
Line 271: Line 342:
Comma list: 225/224, 325/324, 385/384, 2200/2197
Comma list: 225/224, 325/324, 385/384, 2200/2197


Mapping: [{{val| 1 3 6 13 -9 2 }}, {{val| 0 -5 -13 -36 44 6 }}]
Mapping: {{mapping| 1 -2 -7 -23 35 8 | 0 5 13 36 -44 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~39/32 = 339.764
Optimal tunings:
* WE: ~2 = 1200.5022{{c}}, ~64/39 = 860.5964{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/39 = 860.2378{{c}}


{{Optimal ET sequence|legend=1| 53, 113, 166 }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness: 0.032996
Badness (Sintel): 1.36


=== Houbor ===
=== Houbor ===
Line 284: Line 357:
Comma list: 121/120, 225/224, 2200/2187
Comma list: 121/120, 225/224, 2200/2187


Mapping: [{{val| 1 3 6 13 6 }}, {{val| 0 -5 -13 -36 -9 }}]
Mapping: {{mapping| 1 -2 -7 -23 -3 | 0 5 13 36 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.814
Optimal tunings:
* WE: ~2 = 1201.1684{{c}}, ~18/11 = 861.0240{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2273{{c}}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
{{Optimal ET sequence|legend=0| 7d, 53, 60e, 113e }}


Badness: 0.045232
Badness (Sintel): 1.49


==== 13-limit ====
==== 13-limit ====
Line 297: Line 372:
Comma list: 121/120, 225/224, 275/273, 325/324
Comma list: 121/120, 225/224, 275/273, 325/324


Mapping: [{{val| 1 3 6 13 6 2 }}, {{val| 0 -5 -13 -36 -9 6 }}]
Mapping: {{mapping| 1 -2 -7 -23 -3 8 | 0 -5 -13 -36 -9 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.784
Optimal tunings:
* WE: ~2 = 1200.9339{{c}}, ~18/11 = 860.8859{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2376{{c}}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
{{Optimal ET sequence|legend=0| 7d, 53, 113e }}


Badness: 0.031331
Badness (Sintel): 1.29


== Paramity ==
== Paramity ==
The paramity temperament (53 &amp; 311) tempers out the horwell comma (65625/65536) and [[garischisma]] (33554432/33480783).
Paramity tempers out the [[horwell comma]] (65625/65536) and [[garischisma]] (33554432/33480783), and may be described as the {{nowrap| 53 & 311 }} temperament. It was named by [[Xenllium]] in 2021, with the prefix ''para-'' signifying its relation to amity.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 312: Line 389:
[[Comma list]]: 65625/65536, 1600000/1594323
[[Comma list]]: 65625/65536, 1600000/1594323


[[Mapping]]: [{{val|1 3 6 -17}}, {{val|0 -5 -13 70}}]
{{Mapping|legend=1| 1 -2 -7 53 | 0 5 13 -70 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.553
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0199{{c}}, ~400/243 = 860.4612{{c}}
: [[error map]]: {{val| +0.020 +0.311 -0.458 -0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/243 = 860.4468{{c}}
: error map: {{val| 0.000 +0.279 -0.505 -0.103 }}


{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986, 1297c, 2283bc }}
{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986, 1297c }}


[[Badness]]: 0.113655
[[Badness]] (Sintel): 2.88


=== 11-limit ===
=== 11-limit ===
Line 325: Line 406:
Comma list: 6250/6237, 19712/19683, 41503/41472
Comma list: 6250/6237, 19712/19683, 41503/41472


Mapping: [{{val| 1 3 6 -17 36 }}, {{val| 0 -5 -13 70 -115 }}]
Mapping: {{mapping| 1 -2 -7 53 -79 | 0 5 13 -70 115 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* WE: ~2 = 1200.0139{{c}}, ~400/243 = 860.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.4462{{c}}


{{Optimal ET sequence|legend=1| 53, 205de, 258, 311, 675, 986 }}
{{Optimal ET sequence|legend=0| 53, 258, 311, 675, 986 }}


Badness: 0.064853
Badness (Sintel): 2.14


=== 13-limit ===
=== 13-limit ===
Line 338: Line 421:
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683


Mapping: [{{val| 1 3 6 -17 36 17 }}, {{val| 0 -5 -13 70 -115 -47 }}]
Mapping: {{mapping| 1 -2 -7 53 -79 -30 | 0 5 13 -70 115 47 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* WE: ~2 = 1199.9945{{c}}, ~385/234 = 860.4417{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4456{{c}}


{{Optimal ET sequence|legend=1| 53, 205de, 258, 311, 675, 986, 1661cf }}
{{Optimal ET sequence|legend=0| 53, 258, 311, 675, 986 }}


Badness: 0.030347
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 351: Line 436:
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430


Mapping: [{{val| 1 3 6 -17 36 17 -31 }}, {{val| 0 -5 -13 70 -115 -47 124 }}]
Mapping: {{mapping| 1 -2 -7 53 -79 -30 93 | 0 5 13 -70 115 47 -124 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.555
Optimal tunings:
* WE: ~2 = 1200.0026{{c}}, ~385/234 = 860.4468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4450{{c}}


{{Optimal ET sequence|legend=1| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
{{Optimal ET sequence|legend=0| 53, 311, 675 }}


Badness: 0.024118
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 364: Line 451:
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197


Mapping: [{{val| 1 3 6 -17 36 17 -31 15 }}, {{val| 0 -5 -13 70 -115 -47 124 -38 }}]
Mapping: {{mapping| 1 -2 -7 53 -79 -30 93 -23 | 0 5 13 -70 115 47 -124 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~208/171 = 339.555
Optimal tunings:
* WE: ~2 = 1200.0196{{c}}, ~208/171 = 860.4594{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~208/171 = 860.4454{{c}}


{{Optimal ET sequence|legend=1| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
{{Optimal ET sequence|legend=0| 53, 311, 675, 986 }}


Badness: 0.017420
Badness (Sintel): 1.06


== Bamity ==
== Bamity ==
Bamity has a period of half octave and tempers out the sensamagic comma, [[245/243]]. The name ''bamity'' is a play on the words ''bi-'' and ''amity''.
Bamity has a period of half octave and tempers out the sensamagic comma, [[245/243]]. The name ''bamity'' is a contraction of ''bi-'' and ''amity''.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 379: Line 468:
[[Comma list]]: 245/243, 64827/64000
[[Comma list]]: 245/243, 64827/64000


[[Mapping]]: [{{val| 2 1 -1 3 }}, {{val| 0 5 13 6 }}]
{{Mapping|legend=1| 2 1 -1 3 | 0 5 13 6 }}
 
: mapping generators: ~343/240, ~7/6
Mapping generators: ~343/240, ~7/6


{{Multival|legend=1| 10 26 12 18 -9 -45 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/240 = 600.3379{{c}}, ~7/6 = 260.5488{{c}}
[[Optimal tuning]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402
: [[error map]]: {{val| +0.676 +1.127 +0.483 -4.519 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~7/6 = 260.4686{{c}}
: error map: {{val| 0.000 +0.388 -0.222 -6.014 }}


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60, 106d }}
{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60, 106d }}


[[Badness]]: 0.083601
[[Badness]] (Sintel): 2.12


=== 11-limit ===
=== 11-limit ===
Line 396: Line 486:
Comma list: 121/120, 245/243, 441/440
Comma list: 121/120, 245/243, 441/440


Mapping: [{{val| 2 1 -1 3 3 }}, {{val| 0 5 13 6 9 }}]
Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~7/6 = 260.393
Optimal tunings:
* WE: ~99/70 = 600.5039{{c}}, ~7/6 = 260.6119{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~7/6 = 260.4948{{c}}


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60e, 106de }}
{{Optimal ET sequence|legend=0| 14c, 32c, 46, 60e, 106de }}


Badness: 0.035504
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 409: Line 501:
Comma list: 91/90, 121/120, 245/243, 441/440
Comma list: 91/90, 121/120, 245/243, 441/440


Mapping: [{{val| 2 1 -1 3 3 0 }}, {{val| 0 5 13 6 9 17 }}]
Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 260.618
Optimal tunings:
* WE: ~55/39 = 600.3896{{c}}, ~7/6 = 260.7874{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~7/6 = 260.6927{{c}}


{{Optimal ET sequence|legend=1| 14cf, 32cf, 46, 106def, 152def }}
{{Optimal ET sequence|legend=0| 14cf, 32cf, 46, 106deff }}


Badness: 0.030885
Badness (Sintel): 1.28


== Hamity ==
== Hamity ==
Hamity has a generator of about 430 cents which represents [[9/7]]. It is also generated by half of acute minor "tenth" (acute minor third of 243/200 plus an octave), and its name is a play on the words ''half'' and ''amity''.
Hamity has a generator of about 430 cents which represents [[9/7]]. It is also generated by half of acute minor "tenth" (acute minor third of 243/200 plus an octave), and its name is a contraction of ''half'' and ''amity''.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 424: Line 518:
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


[[Mapping]]: [{{val| 1 -2 -7 -4 }}, {{val| 0 10 26 19 }}]
{{Mapping|legend=1| 1 -2 -7 -4 | 0 10 26 19 }}
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 10 26 19 18 2 -29 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.5499{{c}}, ~9/7 = 430.0579{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219
: [[error map]]: {{val| -0.450 -0.476 -1.657 +4.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 430.1958{{c}}
: error map: {{val| 0.000 +0.003 -1.222 +4.895 }}


{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}


[[Badness]]: 0.073956
[[Badness]] (Sintel): 1.87


=== 11-limit ===
=== 11-limit ===
Line 439: Line 536:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: [{{val| 1 -2 -7 -4 -3 }}, {{val| 0 10 26 19 18 }}]
Mapping: {{mapping| 1 -2 -7 -4 -3 | 0 10 26 19 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.192
Optimal tunings:
* WE: ~2 = 1200.2215{{c}}, ~9/7 = 430.2713{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2036{{c}}


{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}


Badness: 0.042947
Badness (Sintel): 1.42


=== 13-limit ===
=== 13-limit ===
Line 452: Line 551:
Comma list: 99/98, 121/120, 275/273, 572/567
Comma list: 99/98, 121/120, 275/273, 572/567


Mapping: [{{val| 1 -2 -7 -4 -3 -11 }}, {{val| 0 10 26 19 18 41 }}]
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216
Optimal tunings:
* WE: ~2 = 1200.1379{{c}}, ~9/7 = 430.2658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2227{{c}}


{{Optimal ET sequence|legend=1| 14cf, 39df, 53 }}
{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}


Badness: 0.029753
Badness (Sintel): 1.23


== Gamity ==
== Gamity ==
The gamity temperament (46 &amp; 113) tempers out the [[gamelisma]], 1029/1024. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.
Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. It splits the interval of the grave major sixth in three. It was named by [[Xenllium]] in 2021 as a contraction of ''gamelismic'' and ''amity''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 467: Line 568:
[[Comma list]]: 1029/1024, 1071875/1062882
[[Comma list]]: 1029/1024, 1071875/1062882


[[Mapping]]: [{{val| 1 -2 -7 4 }}, {{val| 0 15 39 -5 }}]
{{Mapping|legend=1| 1 -2 -7 4 | 0 15 39 -5 }}
 
: mapping generators: ~2, ~189/160
{{Multival|legend=1| 15 39 -5 27 -50 -121 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4434{{c}}, ~189/160 = 286.8930{{c}}
: [[error map]]: {{val| +0.443 +0.553 -0.591 -1.571 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/160 = 286.7955{{c}}
: error map: {{val| 0.000 -0.022 -1.288 -2.804 }}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=1| 46, 113, 159 }}


[[Badness]]: 0.125733
[[Badness]] (Sintel): 3.18


=== 11-limit ===
=== 11-limit ===
Line 482: Line 586:
Comma list: 385/384, 441/440, 1071875/1062882
Comma list: 385/384, 441/440, 1071875/1062882


Mapping: [{{val| 1 -2 -7 4 8 }}, {{val| 0 15 39 -5 -19 }}]
Mapping: {{mapping| 1 -2 -7 4 8 | 0 15 39 -5 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 286.797
Optimal tunings:
* WE: ~2 = 1200.3315{{c}}, ~33/28 = 286.8764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/28 = 286.8002{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness: 0.051111
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 495: Line 601:
Comma list: 325/324, 364/363, 385/384, 10985/10976
Comma list: 325/324, 364/363, 385/384, 10985/10976


Mapping: [{{val| 1 -2 -7 4 8 8 }}, {{val| 0 15 39 -5 -19 -18 }}]
Mapping: {{mapping| 1 -2 -7 4 8 8 | 0 15 39 -5 -19 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.789
Optimal tunings:
* WE: ~2 = 1200.4248{{c}}, ~13/11 = 286.8906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7912{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}


Badness: 0.030297
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 508: Line 616:
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757


Mapping: [{{val| 1 -2 -7 4 8 8 6 }}, {{val| 0 15 39 -5 -19 -18 -8 }}]
Mapping: {{mapping| 1 -2 -7 4 8 8 6 | 0 15 39 -5 -19 -18 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.795
Optimal tunings:
* WE: ~2 = 1200.3287{{c}}, ~13/11 = 286.8731{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7957{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness: 0.022036
Badness (Sintel): 1.12


== Trinity ==
== Trinity ==
The ''trinity'' temperament (152 &amp; 159) tempers out the [[meter]], 703125/702464. It splits the interval of acute minor tenth (~243/100, an octave plus acute minor third) in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.
Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. [[311edo|178\311]] is a recommendable generator. It was named by [[Xenllium]] in 2022 for the fact that it splits amity's generator in three.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 523: Line 633:
[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


[[Mapping]]: [{{val| 1 8 19 46 }}, {{val| 0 -15 -39 -101 }}]
{{Mapping|legend=1| 1 -7 -20 -55 | 0 15 39 101 }}
: mapping generators: ~2, ~125/84


{{Multival|legend=1| 15 39 101 27 118 125 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9474{{c}}, ~125/84 = 686.7921{{c}}
: [[error map]]: {{val| -0.053 +0.295 -0.369 +0.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~125/84 = 686.8215{{c}}
: error map: {{val| 0.000 +0.367 -0.276 +0.144 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~168/125 = 513.178
{{Optimal ET sequence|legend=1| 7d, …, 152, 311, 463, 774 }}


{{Optimal ET sequence|legend=1| 152, 311, 463, 774 }}
[[Badness]] (Sintel): 3.02
 
[[Badness]]: 0.119453


=== 11-limit ===
=== 11-limit ===
Line 538: Line 651:
Comma list: 3025/3024, 4000/3993, 19712/19683
Comma list: 3025/3024, 4000/3993, 19712/19683


Mapping: [{{val| 1 8 19 46 18 }}, {{val| 0 -15 -39 -101 -34 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 | 0 15 39 101 34 }}


POTE generator: ~121/90 = 513.177
Optimal tunings:
* WE: ~2 = 1199.9090{{c}}, ~125/84 = 686.7710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~125/84 = 686.8218{{c}}


{{Optimal ET sequence|legend=1| 152, 311, 463, 774, 1237e }}
{{Optimal ET sequence|legend=0| 7d, …, 152, 311, 463, 774, 1237e }}


Badness: 0.031296
Badness (Sintel): 1.03


=== 13-limit ===
=== 13-limit ===
Line 551: Line 666:
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689


Mapping: [{{val| 1 8 19 46 18 64 }}, {{val| 0 -15 -39 -101 -34 -141 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 | 0 15 39 101 34 141 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.182
Optimal tunings:
* WE: ~2 = 1199.9531{{c}}, ~52/35 = 686.7909{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8172{{c}}


{{Optimal ET sequence|legend=1| 152f, 311 }}
{{Optimal ET sequence|legend=0| 7df, …, 152f, 311 }}


Badness: 0.026418
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 564: Line 681:
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619


Mapping: [{{val| 1 8 19 46 18 64 -22 }}, {{val| 0 -15 -39 -101 -34 -141 61 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 | 0 15 39 101 34 141 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
Optimal tunings:
* WE: ~2 = 1200.0347{{c}}, ~52/35 = 686.8342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8146{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1092cdg }}


Badness: 0.025588
Badness (Sintel): 1.30


=== 19-limit ===
=== 19-limit ===
Line 577: Line 696:
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573


Mapping: [{{val| 1 8 19 46 18 64 -22 53 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 | 0 15 39 101 34 141 -61 114 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* WE: ~2 = 1200.0406{{c}}, ~52/35 = 686.8382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8152{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1403cdgh, 1714cdegh, 2025cdefgghh, 2336bccdefgghh }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.018412
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 590: Line 711:
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104


Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 | 0 15 39 101 34 141 -61 114 104 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8148{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.014343
Badness (Sintel): 1.03


=== 29-limit ===
=== 29-limit ===
Line 603: Line 726:
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044


Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 72 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}]
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 -85 | 0 15 39 101 34 141 -61 114 104 157 }}
 
Optimal tunings:
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8142{{c}}
 
{{Optimal ET sequence|legend=0| 152fj, 159, 311, 781dh }}
 
Badness (Sintel): 1.01
 
== Chromat ==
Chromat has a period of 1/3 octave and tempers out the [[hemimage comma]] (10976/10935) and the [[triwellisma]].
 
[[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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


{{Optimal ET sequence|legend=1| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
[[Badness]] (Sintel): 1.46


Badness: 0.012038
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Amicable ==
Comma list: 441/440, 4375/4356, 10976/10935
The amicable temperament tempers out the [[amity comma]] and the [[canousma]] in addition to the breedsma, and is closely associated with the canou temperament.  
 
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}}


While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Subgroup: 2.3.5.7
Badness (Sintel): 2.06


[[Comma list]]: 2401/2400, 1600000/1594323
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 1 3 6 5 }}, {{val| 0 -20 -52 -31 }}]
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


{{Multival|legend=1| 20 52 31 36 -7 -74 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 84.880
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=1| 99, 212, 311, 410, 1131, 1541b }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


[[Badness]]: 0.045473
Badness (Sintel): 1.58


=== Amical ===
=== Hemichromat ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 131072/130977, 1600000/1594323
Comma list: 3025/3024, 10976/10935, 102487/102400


Mapping: [{{val| 1 3 6 5 -8 }}, {{val| 0 -20 -52 -31 162 }}]
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8843
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=1| 99, 212e, 311, 410, 721, 1032, 1343 }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.100668
Badness (Sintel): 2.22


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
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
 
== Amicable ==
Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} temperament.
 
There are multiple reasonable places for primes 11 and 13 in the interval chain. Amorous (99ef & 311) has these primes placed on the same side of the interval chain so it blends smarter with the lower harmonics. Amical (99 & 311) shows a slightly more accurate but much more complex alternative. Pseudoamorous (99ef & 113) and pseudoamical (99 & 113) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semi-octave period and the ~21/20 generator with [[harry]], but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1600000/1594323


Mapping: [{{val| 1 3 6 5 -8 -5 }}, {{val| 0 -20 -52 -31 162 123 }}]
{{Mapping|legend=1| 1 -17 -46 -26 | 0 20 52 31 }}
: mapping generators: ~2, ~40/21


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8838
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9463{{c}}, ~40/21 = 1115.0701{{c}}
: [[error map]]: {{val| -0.054 +0.361 -0.195 -0.254 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1187{{c}}
: error map: {{val| 0.000 +0.419 -0.142 -0.146 }}


{{Optimal ET sequence|legend=1| 99, 212ef, 311, 410, 721, 1032 }}
{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 410, 1131, 1541b }}


Badness: 0.049893
[[Badness]] (Sintel): 1.15


=== Amorous ===
=== Amorous ===
Line 661: Line 918:
Comma list: 2401/2400, 6250/6237, 19712/19683
Comma list: 2401/2400, 6250/6237, 19712/19683


Mapping: [{{val| 1 3 6 5 14 }}, {{val| 0 -20 -52 -31 -149 }}]
Mapping: {{mapping| 1 -17 -46 -26 -135 | 0 20 52 31 149 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8896
Optimal tunings:
* WE: ~2 = 1200.0380{{c}}, ~40/21 = 1115.1458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1109{{c}}


{{Optimal ET sequence|legend=1| 99e, 212, 311, 1145c, 1456cd }}
{{Optimal ET sequence|legend=0| 99e, 212, 311, 1145c, 1456cd }}


Badness: 0.048924
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 674: Line 933:
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647


Mapping: [{{val| 1 3 6 5 14 17 }}, {{val| 0 -20 -52 -31 -149 -188 }}]
apping: {{mapping| 1 -17 -46 -26 -135 -171 | 0 20 52 31 149 188 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8910
Optimal tunings:
* WE: ~2 = 1200.0543{{c}}, ~40/21 = 1115.1594{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1095{{c}}


{{Optimal ET sequence|legend=1| 99ef, 212, 311, 834, 1145c }}
{{Optimal ET sequence|legend=0| 99ef, 212, 311, 834, 1145c }}


Badness: 0.034681
Badness (Sintel): 1.43


=== Pseudoamical ===
=== Amical ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 1600000/1594323
Comma list: 2401/2400, 131072/130977, 1600000/1594323


Mapping: [{{val| 1 3 6 5 -1 }}, {{val| 0 -20 -52 -31 63 }}]
Mapping: {{mapping| 1 -17 -46 -26 154 | 0 20 52 31 -162 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9091
Optimal tunings:
* WE: ~2 = 1200.0008{{c}}, ~40/21 = 1115.1165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1157{{c}}


{{Optimal ET sequence|legend=1| 99, 113, 212, 961ccdeee }}
{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343 }}


Badness: 0.085837
Badness (Sintel): 3.33


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206


Mapping: [{{val| 1 3 6 5 -1 2 }}, {{val| 0 -20 -52 -31 63 24 }}]
Mapping: {{mapping| 1 -17 -46 -26 154 118 | 0 20 52 31 -162 -123 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9127
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~40/21 = 1115.1094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1162{{c}}


{{Optimal ET sequence|legend=1| 99, 113, 212, 537cdeff, 749ccdeefff }}
{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}


Badness: 0.047025
Badness (Sintel): 2.06


=== Pseudoamorous ===
=== Pseudoamorous ===
Line 713: Line 978:
Comma list: 243/242, 441/440, 980000/970299
Comma list: 243/242, 441/440, 980000/970299


Mapping: [{{val| 1 3 6 5 7 }}, {{val| 0 -20 -52 -31 -50 }}]
Mapping: {{mapping| 1 -17 -46 -26 -43 | 0 20 52 31 50 }}
 
Optimal tunings:
* WE: ~2 = 1199.8269{{c}}, ~21/11 = 1114.9475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1115.1038{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8917
{{Optimal ET sequence|legend=0| 99e, 212e }} *


{{Optimal ET sequence|legend=1| 99e, 212e }}
<nowiki/>* optimal patent val: [[113edo|113]]


Badness: 0.056583
Badness (Sintel): 1.87


==== 13-limit ====
==== 13-limit ====
Line 726: Line 995:
Comma list: 243/242, 364/363, 441/440, 1875/1859
Comma list: 243/242, 364/363, 441/440, 1875/1859


Mapping: [{{val| 1 3 6 5 7 10 }}, {{val| 0 -20 -52 -31 -50 -89 }}]
Mapping: {{mapping| 1 -17 -46 -26 -43 -79 | 0 20 52 31 50 89 }}
 
Optimal tunings:
* WE: ~2 = 1200.0182{{c}}, ~40/21 = 1115.1005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0840{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 113, 212ef }}
 
Badness (Sintel): 1.77
 
=== Pseudoamical ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 1600000/1594323
 
Mapping: {{mapping| 1 -17 -46 -26 62 | 0 20 52 31 -63 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9164
Optimal tunings:
* WE: ~2 = 1200.3436{{c}}, ~40/21 = 1115.4102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0925{{c}}


{{Optimal ET sequence|legend=1| 99ef, 113, 212ef }}
{{Optimal ET sequence|legend=0| 99, 113, 212, 961ccdeee }}


Badness: 0.042826
Badness (Sintel): 2.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
 
Mapping: {{mapping| 1 -17 -46 -26 62 26 | 0 20 52 31 -63 -24 }}
 
Optimal tunings:
* WE: ~2 = 1200.4212{{c}}, ~40/21 = 1115.4787{{c}}
* CWE: ~2 = 1200.0000{{c}}, 40/21 = 1115.0885{{c}}
 
{{Optimal ET sequence|legend=0| 99, 113, 212, 537cdeff, 749ccdeefff }}
 
Badness (Sintel): 1.94


=== Floral ===
=== Floral ===
Line 739: Line 1,040:
Comma list: 2401/2400, 9801/9800, 14641/14580
Comma list: 2401/2400, 9801/9800, 14641/14580


Mapping: [{{val| 2 6 12 10 13 }}, {{val| 0 -20 -52 -31 -43 }}]
Mapping: {{mapping| 2 -14 -40 -21 -30 | 0 20 52 31 43 }}
: mapping generators: ~99/70, ~66/49


Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8788
Optimal tunings:
* WE: ~99/70 = 600.0052{{c}}, ~66/49 = 515.1257{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1215{{c}}


{{Optimal ET sequence|legend=1| 198, 212, 410 }}
{{Optimal ET sequence|legend=0| 14c, … 198, 212, 410 }}


Badness: 0.065110
Badness (Sintel): 2.15


==== 13-limit ====
==== 13-limit ====
Line 752: Line 1,056:
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580


Mapping: [{{val| 2 6 12 10 13 19 }}, {{val| 0 -20 -52 -31 -43 -82 }}]
Mapping: {{mapping| 2 -14 -40 -21 -30 -63 | 0 20 52 31 43 82 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8750
Optimal tunings:
* WE: ~99/70 = 599.9923{{c}}, ~66/49 = 515.1184{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1246{{c}}


{{Optimal ET sequence|legend=1| 198, 410 }}
{{Optimal ET sequence|legend=0| 14cf, …, 198, 410 }}


Badness: 0.037013
Badness (Sintel): 1.53


=== Humorous ===
=== Humorous ===
Line 765: Line 1,071:
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Comma list: 2401/2400, 3025/3024, 1600000/1594323


Mapping: [{{val| 1 3 6 5 3 }}, {{val| 0 -40 -104 -62 13 }}]
Mapping: {{mapping| 1 -37 -98 -57 16 | 0 40 104 62 -13 }}
: mapping generators: ~2, ~3993/2048


Optimal tuning (POTE): ~2 = 1\1, ~4096/3993 = 42.4391
Optimal tunings:
* WE: ~2 = 1199.9189{{c}}, ~3993/2048 = 1157.4826{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3993/2048 = 1157.5602{{c}}


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820 }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820 }}


Badness: 0.058249
Badness (Sintel): 1.93


==== 13-limit ====
==== 13-limit ====
Line 778: Line 1,087:
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024


Mapping: [{{val| 1 3 6 5 3 6 }}, {{val| 0 -40 -104 -62 13 -65 }}]
Mapping: {{mapping| 1 -37 -98 -57 16 -59 | 0 40 104 62 -13 65 }}


Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 42.4391
Optimal tunings:
* WE: ~2 = 1199.8903{{c}}, ~39/20 = 1157.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~39/20 = 1157.5600{{c}}


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820f }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820f }}


Badness: 0.028267
Badness (Sintel): 1.17
 
== Witcher ==
Witcher tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 99 & 166 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 1600000/1594323
 
{{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }}
: mapping generators: ~2, ~80/63
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8698{{c}}, ~80/63 = 412.0531{{c}}
: [[error map]]: {{val| -0.130 +0.283 -0.260 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/63 = 412.0956{{c}}
: error map: {{val| 0.000 +0.435 -0.100 +0.512 }}
 
{{Optimal ET sequence|legend=1| 32c, 67c, 99, 265, 364, 463, 562, 1025 }}
 
[[Badness]] (Sintel): 2.04
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4356, 1240029/1239040
 
Mapping: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }}
 
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~80/63 = 412.0758{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/63 = 412.0788{{c}}
 
{{Optimal ET sequence|legend=0| 99e, 166, 265 }}
 
Badness (Sintel): 2.73
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4375/4356, 10648/10647
 
Mapping: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }}
 
Optimal tunings:
* WE: ~2 = 1200.0660{{c}}, ~33/26 = 412.0928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/26 = 412.0709{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 166, 265 }}
 
Badness (Sintel): 2.40
 
=== Semiwitcher ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 41503/41472, 585640/583443
 
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
: mapping generators: ~99/70, ~80/63
 
Optimal tunings:
* WE: ~99/70 = 599.9717{{c}}, ~80/63 = 412.0766{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~80/63 = 412.0951{{c}}
 
{{Optimal ET sequence|legend=0| 32c, 166, 198, 364, 562, 926 }}
 
Badness (Sintel): 2.92
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280
 
Mapping: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9666{{c}}, ~33/26 = 412.0726{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/26 = 412.0944{{c}}
 
{{Optimal ET sequence|legend=0| 32cf, 166, 198, 364, 562, 926 }}
 
Badness (Sintel): 1.57
 
=== Hitch ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 117649/117612, 1600000/1594323
 
Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560
 
Optimal tunings:
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}
 
{{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }}
 
Badness (Sintel): 3.09
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
 
Mapping: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }}
 
Optimal tunings:
* WE: ~2 = 1199.8415{{c}}, ~891/560 = 805.9451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0508{{c}}
 
{{Optimal ET sequence|legend=0| 198, 463, 661 }}
 
Badness (Sintel): 2.51
 
==== Hitcher ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
 
Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}
 
Optimal tunings:
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}
 
{{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }}
 
Badness (Sintel): 2.46
 
== Calamity ==
Named by [[Flora Canou]] in 2023, calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of grave major sixth plus an octave into five.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1600000/1594323, 2100875/2097152
 
{{Mapping|legend=1| 1 -12 -33 24 | 0 25 65 -39 }}
: mapping generators: ~2, ~35/24
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0332{{c}}, ~35/24 = 652.1081{{c}}
: [[error map]]: {{val| +0.033 +0.348 -0.387 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 652.0902{{c}}
: error map: {{val| 0.000 +0.300 -0.452 -0.343 }}
 
{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
 
[[Badness]] (Sintel): 5.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 12005/11979, 131072/130977
 
Mapping: {{mapping| 1 -12 -33 24 29 | 0 25 65 -39 -47 }}
 
Optimal tunings:
* WE: ~2 = 1199.9938{{c}}, ~35/24 = 652.0887{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/24 = 652.0920{{c}}
 
{{Optimal ET sequence|legend=0| 46, 219c, 265, 311, 979, 1290 }}
 
Badness (Sintel): 2.00
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979
 
Mapping: {{mapping| 1 -12 -33 24 29 45 | 0 25 65 -39 -47 -76 }}
 
Optimal tunings:
* WE: ~2 = 1199.9742{{c}}, ~35/24 = 652.0794{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/24 = 652.0934{{c}}
 
{{Optimal ET sequence|legend=0| 46, 265, 311, 668, 979, 1290 }}
 
Badness (Sintel): 1.39


== Familia ==
== Familia ==
The familia temperament (113 &amp; 152) tempers out the mirkwai comma, 16875/16807. It splits the interval of acute minor tenth (~243/100) in five.
Named by [[Xenllium]] in 2021, familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. It is generated by a major sixth, five of which octave reduced make the original generator of amity.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 793: Line 1,281:
[[Comma list]]: 16875/16807, 1600000/1594323
[[Comma list]]: 16875/16807, 1600000/1594323


[[Mapping]]: [{{val| 1 8 19 20 }}, {{val| 0 -25 -65 -67 }}]
{{Mapping|legend=1| 1 -17 -46 -47 | 0 25 65 67 }}
 
: mapping generators: ~2, ~6561/3920
{{Multival|legend=1| 25 65 67 45 36 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9165{{c}}, ~6561/3920 = 892.0267{{c}}
: [[error map]]: {{val| -0.084 +0.132 -0.736 +0.888 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6561/3920 = 892.0871{{c}}
: error map: {{val| 0.000 +0.224 -0.649 +1.013 }}


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417 }}


[[Badness]]: 0.144551
[[Badness]] (Sintel): 3.66


=== 11-limit ===
=== 11-limit ===
Line 808: Line 1,299:
Comma list: 540/539, 1375/1372, 1600000/1594323
Comma list: 540/539, 1375/1372, 1600000/1594323


Mapping: [{{val| 1 8 19 20 5 }}, {{val| 0 -25 -65 -67 -6 }}]
Mapping: {{mapping| 1 -17 -46 -47 -1 | 0 25 65 67 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906
Optimal tunings:
* WE: ~2 = 1199.7927{{c}}, ~2673/1600 = 891.9397{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2673/1600 = 892.0909{{c}}


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 417, 569de, 721de }}
{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 417, 569de, 721de }}


Badness: 0.051740
Badness (Sintel): 1.71


=== 13-limit ===
=== 13-limit ===
Line 821: Line 1,314:
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
Comma list: 540/539, 729/728, 1375/1372, 2205/2197


Mapping: [{{val| 1 8 19 20 5 25 }}, {{val| 0 -25 -65 -67 -6 -83 }}]
Mapping: {{mapping| 1 -17 -46 -47 -1 -58 | 0 25 65 67 6 83 }}


Optimal tuning (POTE): ~2 = 1\1, ~143/120 = 307.913
Optimal tunings:
* WE: ~2 = 1199.7844{{c}}, ~240/143 = 892.9268{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~240/143 = 892.0839{{c}}


{{Optimal ET sequence|legend=1| 39df, 74cdf, 113, 152f, 265, 417f }}
{{Optimal ET sequence|legend=0| 39df, 74cdf, 113, 152f, 265 }}


Badness: 0.038473
Badness (Sintel): 1.59


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity|#]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 14:49, 3 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The amity family of temperaments tempers out the amity comma (monzo: [9 -13 15⟩, ratio: 1600000/1594323).

Amity

Main article: Amity

The generator for the amity temperament is the acute minor third, which means the 6/5 just minor third raised by a syntonic comma to 243/200, and from this it derives its name. If you are looking for a different kind of neutral third, this could be the temperament for you. However its functional generator is the octave complement thereof, five of which stack to the 12th harmonic. Its ploidacot is thus gamma-pentacot.

It is a member of the syntonic–chromatic equivalence continuum with n = 5, so it equates an apotome with a stack of five syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 5, so unless 53edo is used as a tuning, the schisma is always observed.

Amity is a genuine microtemperament in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)1/13, which gives a pure classical major third. Mos scales of 11, 18, 25, 32, 39, 46 or 53 notes are available.

Subgroup: 2.3.5

Comma list: 1600000/1594323

Mapping: [⟨1 -2 -7], ⟨0 5 13]]

mapping generators: ~2, ~400/243

Optimal tunings:

  • WE: ~2 = 1199.9135 ¢, ~400/243 = 860.4191 ¢
error map: ⟨-0.087 +0.314 -0.259]
  • CWE: ~2 = 1200.000 ¢, ~400/243 = 860.4726 ¢
error map: ⟨0.000 +0.408 -0.169]

Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 463, 668, 873

Badness (Sintel): 0.515

Overview to extensions

The second comma to extend 5-limit amity include 4375/4374 for septimal amity, 65625/65536 for paramity, 126/125 for accord, and 225/224 for houborizic. Those all use the same generator as the 5-limit temperament.

Bamity adds 245/243 with a 1/2-octave period. Hamity adds 2430/2401 and splits the generator in two. Gamity adds 1029/1024. Chromat adds 10976/10935. Trinity adds 703125/702464. Those split the generator in three. Amicable adds 2401/2400, splitting the generator in four. Witcher adds 420175/419904. Calamity adds 2100875/2097152. Familia adds 16875/16807. Those split the generator in five.

Septimal amity

Main article: Amity

Septimal amity can be described as the 46 & 53 temperament, which tempers out 4375/4374 and 5120/5103 in the 7-limit. 99edo is a good tuning, with generator 28\99.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 5120/5103

Mapping: [⟨1 -2 -7 15], ⟨0 5 13 -17]]

Optimal tunings:

  • WE: ~2 = 1199.6100 ¢, ~105/64 = 860.2881 ¢
error map: ⟨-0.390 +0.266 +0.162 +0.426]
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5650 ¢
error map: ⟨0.000 +0.870 +1.032 +1.568]

Optimal ET sequence: 7, 39, 46, 53, 99, 152, 251, 350, 601cd, 905bcdd

Badness (Sintel): 0.598

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 5120/5103

Mapping: [⟨1 -2 -7 15 -41], ⟨0 5 13 -17 62]]

Optimal tunings:

  • WE: ~2 = 1199.5961 ¢, ~105/64 = 860.2460 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5314 ¢

Optimal ET sequence: 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 625/624, 729/728

Mapping: [⟨1 -2 -7 15 -41 -30], ⟨0 5 13 -17 62 47]]

Optimal tunings:

  • WE: ~2 = 1199.5437 ¢, ~105/64 = 860.1922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.5140 ¢

Optimal ET sequence: 46ef, 53, 99ef, 152f *

* optimal patent val: 205

Badness (Sintel): 1.16

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 375/374, 540/539, 729/728

Mapping: [⟨1 -2 -7 15 -41 -30 17], ⟨0 5 13 -17 62 47 -18]]

Optimal tunings:

  • WE: ~2 = 1199.3376 ¢, ~28/17 = 860.0617 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.5317 ¢

Optimal ET sequence: 46ef, 53, 99ef, 152fg

Badness (Sintel): 1.33

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455

Mapping: [⟨1 -2 -7 15 -41 -30 17 -23], ⟨0 5 13 -17 62 47 -18 38]]

Optimal tunings:

  • WE: ~2 = 1199.3361 ¢, ~28/17 = 860.0605 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.5310 ¢

Optimal ET sequence: 46efh, 53, 99ef, 152fg

Badness (Sintel): 1.14

Hitchcock

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 2200/2187

Mapping: [⟨1 -2 -7 15 -3], ⟨0 5 13 -17 9]]

Optimal tunings:

  • WE: ~2 = 1199.9979 ¢, ~18/11 = 860.6089 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.6104 ¢

Optimal ET sequence: 7, 39, 46, 53, 99

Badness (Sintel): 1.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 325/324

Mapping: [⟨1 -2 -7 15 -3 8], ⟨0 5 13 -17 9 -6]]

Optimal tunings:

  • WE: ~2 = 1200.2037 ¢, ~18/11 = 860.5820 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.7268 ¢

Optimal ET sequence: 7, 39, 46, 53, 99

Badness (Sintel): 0.928

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 169/168, 176/175, 273/272

Mapping: [⟨1 -2 -7 15 -3 8 17], ⟨0 5 13 -17 9 -6 -18]]

Optimal tunings:

  • WE: ~2 = 1200.0179 ¢, ~18/11 = 860.6472 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.6344 ¢

Optimal ET sequence: 7, 39, 46, 99

Badness (Sintel): 0.988

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189

Mapping: [⟨1 -2 -7 15 -3 8 17 15], ⟨0 5 13 -17 9 -6 -18 -15]]

Optimal tunings:

  • WE: ~2 = 1200.1977 ¢, ~18/11 = 860.7343 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.5917 ¢

Optimal ET sequence: 7, 46, 53, 99h

Badness (Sintel): 1.07

Stalagmite

The stalagmite temperament (46 & 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle comma) in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is down from amity.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 4375/4374

Mapping: [⟨1 -2 -7 15 30], ⟨0 5 13 -17 -37]]

Optimal tunings:

  • WE: ~2 = 1199.3915 ¢, ~105/64 = 860.2240 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.6649 ¢

Optimal ET sequence: 46, 99e, 145, 244e

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 4375/4374

Mapping: [⟨1 -2 -7 15 30 41], ⟨0 5 13 -17 -37 -52]]

Optimal tunings:

  • WE: ~2 = 1199.3577 ¢, ~105/64 = 860.2259 ¢
  • CWE: ~2 = 1200.0000 ¢, ~105/64 = 860.6929 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.41

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155

Mapping: [⟨1 -2 -7 15 30 41 17], ⟨0 5 13 -17 -37 -52 -18]]

Optimal tunings:

  • WE: ~2 = 1199.3555 ¢, ~28/17 = 860.2244 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.6932 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475

Mapping: [⟨1 -2 -7 15 30 41 17 48], ⟨0 5 13 -17 -37 -52 -18 -61]]

Optimal tunings:

  • WE: ~2 = 1199.3599 ¢, ~28/17 = 860.2161 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/17 = 860.6820 ¢

Optimal ET sequence: 46, 99ef, 145

Badness (Sintel): 1.15

Hemiamity

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5120/5103

Mapping: [⟨2 1 -1 13 13], ⟨0 5 13 -17 -14]]

mapping generators: ~99/70, ~64/55

Optimal tunings:

  • WE: ~99/70 = 599.8218 ¢, ~64/55 = 260.4833 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/55 = 260.5615 ¢

Optimal ET sequence: 46, 106, 152, 350, 502d, 852bcdde

Badness (Sintel): 1.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 3025/3024

Mapping: [⟨2 1 -1 13 13 20], ⟨0 5 13 -17 -14 -29]]

Optimal tunings:

  • WE: ~99/70 = 599.7965 ¢, ~64/55 = 260.4951 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~64/55 = 260.5872 ¢

Optimal ET sequence: 46, 106f, 152f, 198, 350f, 548cdff

Badness (Sintel): 1.07

Accord

Accord tempers out 126/125, the starling comma, and may be described as the 39d & 46 temperament.

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 -2 -7 -18], ⟨0 5 13 29]]

Optimal tunings:

  • WE: ~2 = 1198.7458 ¢, ~224/135 = 860.1071 ¢
error map: ⟨-1.254 +1.089 +3.858 -3.144]
  • CWE: ~2 = 1200.0000 ¢, ~224/135 = 860.9433 ¢
error map: ⟨0.000 +2.761 +5.949 -1.471]

Optimal ET sequence: 7d, …, 39d, 46, 131c

Badness (Sintel): 2.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 896/891

Mapping: [⟨1 -2 -7 -18 -3], ⟨0 5 13 29 9]]

Optimal tunings:

  • WE: ~2 = 1199.3576 ¢, ~18/11 = 860.4919 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.9224 ¢

Optimal ET sequence: 7d, 39d, 46

Badness (Sintel): 1.40

Houborizic

Houborizic tempers out 225/224, the marvel comma, and may be described as the 53 & 60 temperament. It was named by Xenllium in 2021 for its close relation to the houboriz tuning (generator: 339.774971 cents).

Subgroup: 2.3.5.7

Comma list: 225/224, 1250000/1240029

Mapping: [⟨1 -2 -7 -23], ⟨0 5 13 36]]

Optimal tunings:

  • WE: ~2 = 1200.4959 ¢, ~400/243 = 860.5922 ¢
error map: ⟨+0.496 +0.014 -2.086 +1.090]
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.2558 ¢
error map: ⟨0.000 -0.676 -2.989 +0.382]

Optimal ET sequence: 7d, …, 46d, 53, 113, 166

Badness (Sintel): 1.69

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1250000/1240029

Mapping: [⟨1 -2 -7 -23 35], ⟨0 5 13 36 -44]]

Optimal tunings:

  • WE: ~2 = 1200.4973 ¢, ~400/243 = 860.5930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.2386 ¢

Optimal ET sequence: 53, 113, 166

Badness (Sintel): 2.24

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 2200/2197

Mapping: [⟨1 -2 -7 -23 35 8], ⟨0 5 13 36 -44 -6]]

Optimal tunings:

  • WE: ~2 = 1200.5022 ¢, ~64/39 = 860.5964 ¢
  • CWE: ~2 = 1200.0000 ¢, ~64/39 = 860.2378 ¢

Optimal ET sequence: 53, 113, 166

Badness (Sintel): 1.36

Houbor

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 2200/2187

Mapping: [⟨1 -2 -7 -23 -3], ⟨0 5 13 36 9]]

Optimal tunings:

  • WE: ~2 = 1201.1684 ¢, ~18/11 = 861.0240 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.2273 ¢

Optimal ET sequence: 7d, 53, 60e, 113e

Badness (Sintel): 1.49

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 325/324

Mapping: [⟨1 -2 -7 -23 -3 8], ⟨0 -5 -13 -36 -9 6]]

Optimal tunings:

  • WE: ~2 = 1200.9339 ¢, ~18/11 = 860.8859 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 860.2376 ¢

Optimal ET sequence: 7d, 53, 113e

Badness (Sintel): 1.29

Paramity

Paramity tempers out the horwell comma (65625/65536) and garischisma (33554432/33480783), and may be described as the 53 & 311 temperament. It was named by Xenllium in 2021, with the prefix para- signifying its relation to amity.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1600000/1594323

Mapping: [⟨1 -2 -7 53], ⟨0 5 13 -70]]

Optimal tunings:

  • WE: ~2 = 1200.0199 ¢, ~400/243 = 860.4612 ¢
error map: ⟨+0.020 +0.311 -0.458 -0.055]
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.4468 ¢
error map: ⟨0.000 +0.279 -0.505 -0.103]

Optimal ET sequence: 53, 205d, 258, 311, 675, 986, 1297c

Badness (Sintel): 2.88

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6250/6237, 19712/19683, 41503/41472

Mapping: [⟨1 -2 -7 53 -79], ⟨0 5 13 -70 115]]

Optimal tunings:

  • WE: ~2 = 1200.0139 ¢, ~400/243 = 860.4562 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/243 = 860.4462 ¢

Optimal ET sequence: 53, 258, 311, 675, 986

Badness (Sintel): 2.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683

Mapping: [⟨1 -2 -7 53 -79 -30], ⟨0 5 13 -70 115 47]]

Optimal tunings:

  • WE: ~2 = 1199.9945 ¢, ~385/234 = 860.4417 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/234 = 860.4456 ¢

Optimal ET sequence: 53, 258, 311, 675, 986

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430

Mapping: [⟨1 -2 -7 53 -79 -30 93], ⟨0 5 13 -70 115 47 -124]]

Optimal tunings:

  • WE: ~2 = 1200.0026 ¢, ~385/234 = 860.4468 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/234 = 860.4450 ¢

Optimal ET sequence: 53, 311, 675

Badness (Sintel): 1.23

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197

Mapping: [⟨1 -2 -7 53 -79 -30 93 -23], ⟨0 5 13 -70 115 47 -124 38]]

Optimal tunings:

  • WE: ~2 = 1200.0196 ¢, ~208/171 = 860.4594 ¢
  • CWE: ~2 = 1200.0000 ¢, ~208/171 = 860.4454 ¢

Optimal ET sequence: 53, 311, 675, 986

Badness (Sintel): 1.06

Bamity

Bamity has a period of half octave and tempers out the sensamagic comma, 245/243. The name bamity is a contraction of bi- and amity.

Subgroup: 2.3.5.7

Comma list: 245/243, 64827/64000

Mapping: [⟨2 1 -1 3], ⟨0 5 13 6]]

mapping generators: ~343/240, ~7/6

Optimal tunings:

  • WE: ~343/240 = 600.3379 ¢, ~7/6 = 260.5488 ¢
error map: ⟨+0.676 +1.127 +0.483 -4.519]
  • CWE: ~343/240 = 600.0000 ¢, ~7/6 = 260.4686 ¢
error map: ⟨0.000 +0.388 -0.222 -6.014]

Optimal ET sequence: 14c, 32c, 46, 60, 106d

Badness (Sintel): 2.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3], ⟨0 5 13 6 9]]

Optimal tunings:

  • WE: ~99/70 = 600.5039 ¢, ~7/6 = 260.6119 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~7/6 = 260.4948 ¢

Optimal ET sequence: 14c, 32c, 46, 60e, 106de

Badness (Sintel): 1.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3 0], ⟨0 5 13 6 9 17]]

Optimal tunings:

  • WE: ~55/39 = 600.3896 ¢, ~7/6 = 260.7874 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~7/6 = 260.6927 ¢

Optimal ET sequence: 14cf, 32cf, 46, 106deff

Badness (Sintel): 1.28

Hamity

Hamity has a generator of about 430 cents which represents 9/7. It is also generated by half of acute minor "tenth" (acute minor third of 243/200 plus an octave), and its name is a contraction of half and amity.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 4000/3969

Mapping: [⟨1 -2 -7 -4], ⟨0 10 26 19]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1199.5499 ¢, ~9/7 = 430.0579 ¢
error map: ⟨-0.450 -0.476 -1.657 +4.075]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.1958 ¢
error map: ⟨0.000 +0.003 -1.222 +4.895]

Optimal ET sequence: 14c, 39d, 53

Badness (Sintel): 1.87

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 2200/2187

Mapping: [⟨1 -2 -7 -4 -3], ⟨0 10 26 19 18]]

Optimal tunings:

  • WE: ~2 = 1200.2215 ¢, ~9/7 = 430.2713 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.2036 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Sintel): 1.42

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 275/273, 572/567

Mapping: [⟨1 8 19 15 15 30], ⟨0 -10 -26 -19 -18 -41]]

Optimal tunings:

  • WE: ~2 = 1200.1379 ¢, ~9/7 = 430.2658 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 430.2227 ¢

Optimal ET sequence: 14cf, 39df, 53

Badness (Sintel): 1.23

Gamity

Gamity tempers out 1029/1024, the gamelisma, and may be described as the 46 & 113 temperament. It splits the interval of the grave major sixth in three. It was named by Xenllium in 2021 as a contraction of gamelismic and amity.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1071875/1062882

Mapping: [⟨1 -2 -7 4], ⟨0 15 39 -5]]

mapping generators: ~2, ~189/160

Optimal tunings:

  • WE: ~2 = 1200.4434 ¢, ~189/160 = 286.8930 ¢
error map: ⟨+0.443 +0.553 -0.591 -1.571]
  • CWE: ~2 = 1200.0000 ¢, ~189/160 = 286.7955 ¢
error map: ⟨0.000 -0.022 -1.288 -2.804]

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 3.18

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 1071875/1062882

Mapping: [⟨1 -2 -7 4 8], ⟨0 15 39 -5 -19]]

Optimal tunings:

  • WE: ~2 = 1200.3315 ¢, ~33/28 = 286.8764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/28 = 286.8002 ¢

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 10985/10976

Mapping: [⟨1 -2 -7 4 8 8], ⟨0 15 39 -5 -19 -18]]

Optimal tunings:

  • WE: ~2 = 1200.4248 ¢, ~13/11 = 286.8906 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 286.7912 ¢

Optimal ET sequence: 46, 113, 159, 364df, 523ddff

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757

Mapping: [⟨1 -2 -7 4 8 8 6], ⟨0 15 39 -5 -19 -18 -8]]

Optimal tunings:

  • WE: ~2 = 1200.3287 ¢, ~13/11 = 286.8731 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 286.7957 ¢

Optimal ET sequence: 46, 113, 159

Badness (Sintel): 1.12

Trinity

Trinity tempers out 703125/702464, the meter, and may be described as the 152 & 159 temperament. 178\311 is a recommendable generator. It was named by Xenllium in 2022 for the fact that it splits amity's generator in three.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 1600000/1594323

Mapping: [⟨1 -7 -20 -55], ⟨0 15 39 101]]

mapping generators: ~2, ~125/84

Optimal tunings:

  • WE: ~2 = 1199.9474 ¢, ~125/84 = 686.7921 ¢
error map: ⟨-0.053 +0.295 -0.369 +0.071]
  • CWE: ~2 = 1200.0000 ¢, ~125/84 = 686.8215 ¢
error map: ⟨0.000 +0.367 -0.276 +0.144]

Optimal ET sequence: 7d, …, 152, 311, 463, 774

Badness (Sintel): 3.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 19712/19683

Mapping: [⟨1 -7 -20 -55 -16], ⟨0 15 39 101 34]]

Optimal tunings:

  • WE: ~2 = 1199.9090 ¢, ~125/84 = 686.7710 ¢
  • CWE: ~2 = 1200.0000 ¢, ~125/84 = 686.8218 ¢

Optimal ET sequence: 7d, …, 152, 311, 463, 774, 1237e

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689

Mapping: [⟨1 -7 -20 -55 -16 -77], ⟨0 15 39 101 34 141]]

Optimal tunings:

  • WE: ~2 = 1199.9531 ¢, ~52/35 = 686.7909 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8172 ¢

Optimal ET sequence: 7df, …, 152f, 311

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619

Mapping: [⟨1 -7 -20 -55 -16 -77 39], ⟨0 15 39 101 34 141 -61]]

Optimal tunings:

  • WE: ~2 = 1200.0347 ¢, ~52/35 = 686.8342 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8146 ¢

Optimal ET sequence: 152f, 159, 311, 1092cdg

Badness (Sintel): 1.30

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573

Mapping: [⟨1 -7 -20 -55 -16 -77 39 -61], ⟨0 15 39 101 34 141 -61 114]]

Optimal tunings:

  • WE: ~2 = 1200.0406 ¢, ~52/35 = 686.8382 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8152 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Sintel): 1.12

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104

Mapping: [⟨1 -7 -20 -55 -16 -77 39 -61 -55], ⟨0 15 39 101 34 141 -61 114 104]]

Optimal tunings:

  • WE: ~2 = 1200.0311 ¢, ~52/35 = 686.8324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8148 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Sintel): 1.03

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044

Mapping: [⟨1 -7 -20 -55 -16 -77 39 -61 -55 -85], ⟨0 15 39 101 34 141 -61 114 104 157]]

Optimal tunings:

  • WE: ~2 = 1200.0311 ¢, ~52/35 = 686.8318 ¢
  • CWE: ~2 = 1200.0000 ¢, ~52/35 = 686.8142 ¢

Optimal ET sequence: 152fj, 159, 311, 781dh

Badness (Sintel): 1.01

Chromat

Chromat has a period of 1/3 octave and tempers out the hemimage comma (10976/10935) and the triwellisma.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

mapping generators: ~63/50, ~28/27

Optimal tunings:

  • WE: ~63/50 = 399.9549 ¢, ~28/27 = 60.5216 ¢
error map: ⟨-0.135 +0.473 +0.241 -0.751]
  • CWE: ~63/50 = 400.0000 ¢, ~28/27 = 60.5162 ¢
error map: ⟨0.000 +0.626 +0.397 -0.567]

Optimal ET sequence: 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: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

Optimal tunings:

  • WE: ~44/35 = 400.0359 ¢, ~28/27 = 60.4357 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]

Optimal tunings:

  • WE: ~44/35 = 400.0382 ¢, ~28/27 = 60.4342 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4331 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]

Optimal tunings:

  • WE: ~44/35 = 399.9982 ¢, ~28/27 = 60.4374 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]

Optimal tunings:

  • WE: ~44/35 = 400.1386 ¢, ~28/27 = 60.3986 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3929 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]

Optimal tunings:

  • WE: ~44/35 = 400.1115 ¢, ~28/27 = 60.3935 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3893 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]

Optimal tunings:

  • WE: ~44/35 = 399.9082 ¢, ~28/27 = 60.4425 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4380 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]

Optimal tunings:

  • WE: ~44/35 = 399.8948 ¢, ~28/27 = 60.4435 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4385 ¢

Optimal ET sequence: 60e, 99ef, 159f

Badness (Sintel): 1.58

Hemichromat

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 10976/10935, 102487/102400

Mapping: [⟨3 4 5 6 10], ⟨0 10 26 32 5]]

Optimal tunings:

  • WE: ~63/50 = 399.9750 ¢, ~55/54 = 30.2568 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2561 ¢

Optimal ET sequence: 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: [⟨3 4 5 6 10 8], ⟨0 10 26 32 5 41]]

Optimal tunings:

  • WE: ~63/50 = 399.9741 ¢, ~55/54 = 30.2584 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2577 ¢

Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b

Badness (Sintel): 1.38

Amicable

Amicable tempers out the breedsma as well as the canousma, and may be described as the 99 & 311 temperament.

There are multiple reasonable places for primes 11 and 13 in the interval chain. Amorous (99ef & 311) has these primes placed on the same side of the interval chain so it blends smarter with the lower harmonics. Amical (99 & 311) shows a slightly more accurate but much more complex alternative. Pseudoamorous (99ef & 113) and pseudoamical (99 & 113) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semi-octave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 -17 -46 -26], ⟨0 20 52 31]]

mapping generators: ~2, ~40/21

Optimal tunings:

  • WE: ~2 = 1199.9463 ¢, ~40/21 = 1115.0701 ¢
error map: ⟨-0.054 +0.361 -0.195 -0.254]
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1187 ¢
error map: ⟨0.000 +0.419 -0.142 -0.146]

Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 410, 1131, 1541b

Badness (Sintel): 1.15

Amorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: [⟨1 -17 -46 -26 -135], ⟨0 20 52 31 149]]

Optimal tunings:

  • WE: ~2 = 1200.0380 ¢, ~40/21 = 1115.1458 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1109 ¢

Optimal ET sequence: 99e, 212, 311, 1145c, 1456cd

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647

apping: [⟨1 -17 -46 -26 -135 -171], ⟨0 20 52 31 149 188]]

Optimal tunings:

  • WE: ~2 = 1200.0543 ¢, ~40/21 = 1115.1594 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1095 ¢

Optimal ET sequence: 99ef, 212, 311, 834, 1145c

Badness (Sintel): 1.43

Amical

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: [⟨1 -17 -46 -26 154], ⟨0 20 52 31 -162]]

Optimal tunings:

  • WE: ~2 = 1200.0008 ¢, ~40/21 = 1115.1165 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1157 ¢

Optimal ET sequence: 99, 212e, 311, 721, 1032, 1343

Badness (Sintel): 3.33

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: [⟨1 -17 -46 -26 154 118], ⟨0 20 52 31 -162 -123]]

Optimal tunings:

  • WE: ~2 = 1199.9926 ¢, ~40/21 = 1115.1094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.1162 ¢

Optimal ET sequence: 99, 212ef, 311, 721, 1032

Badness (Sintel): 2.06

Pseudoamorous

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 980000/970299

Mapping: [⟨1 -17 -46 -26 -43], ⟨0 20 52 31 50]]

Optimal tunings:

  • WE: ~2 = 1199.8269 ¢, ~21/11 = 1114.9475 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/11 = 1115.1038 ¢

Optimal ET sequence: 99e, 212e *

* optimal patent val: 113

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 1875/1859

Mapping: [⟨1 -17 -46 -26 -43 -79], ⟨0 20 52 31 50 89]]

Optimal tunings:

  • WE: ~2 = 1200.0182 ¢, ~40/21 = 1115.1005 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.0840 ¢

Optimal ET sequence: 99ef, 113, 212ef

Badness (Sintel): 1.77

Pseudoamical

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: [⟨1 -17 -46 -26 62], ⟨0 20 52 31 -63]]

Optimal tunings:

  • WE: ~2 = 1200.3436 ¢, ~40/21 = 1115.4102 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1115.0925 ¢

Optimal ET sequence: 99, 113, 212, 961ccdeee

Badness (Sintel): 2.84

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: [⟨1 -17 -46 -26 62 26], ⟨0 20 52 31 -63 -24]]

Optimal tunings:

  • WE: ~2 = 1200.4212 ¢, ~40/21 = 1115.4787 ¢
  • CWE: ~2 = 1200.0000 ¢, 40/21 = 1115.0885 ¢

Optimal ET sequence: 99, 113, 212, 537cdeff, 749ccdeefff

Badness (Sintel): 1.94

Floral

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [⟨2 -14 -40 -21 -30], ⟨0 20 52 31 43]]

mapping generators: ~99/70, ~66/49

Optimal tunings:

  • WE: ~99/70 = 600.0052 ¢, ~66/49 = 515.1257 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/49 = 515.1215 ¢

Optimal ET sequence: 14c, … 198, 212, 410

Badness (Sintel): 2.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [⟨2 -14 -40 -21 -30 -63], ⟨0 20 52 31 43 82]]

Optimal tunings:

  • WE: ~99/70 = 599.9923 ¢, ~66/49 = 515.1184 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/49 = 515.1246 ¢

Optimal ET sequence: 14cf, …, 198, 410

Badness (Sintel): 1.53

Humorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 1600000/1594323

Mapping: [⟨1 -37 -98 -57 16], ⟨0 40 104 62 -13]]

mapping generators: ~2, ~3993/2048

Optimal tunings:

  • WE: ~2 = 1199.9189 ¢, ~3993/2048 = 1157.4826 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3993/2048 = 1157.5602 ¢

Optimal ET sequence: 85c, 113, 198, 311, 509, 820

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [⟨1 -37 -98 -57 16 -59], ⟨0 40 104 62 -13 65]]

Optimal tunings:

  • WE: ~2 = 1199.8903 ¢, ~39/20 = 1157.4551 ¢
  • CWE: ~2 = 1200.0000 ¢, ~39/20 = 1157.5600 ¢

Optimal ET sequence: 85c, 113, 198, 311, 509, 820f

Badness (Sintel): 1.17

Witcher

Witcher tempers out 420175/419904, the wizma, and may be described as the 99 & 166 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 1600000/1594323

Mapping: [⟨1 -7 -20 -2], ⟨0 25 65 14]]

mapping generators: ~2, ~80/63

Optimal tunings:

  • WE: ~2 = 1199.8698 ¢, ~80/63 = 412.0531 ¢
error map: ⟨-0.130 +0.283 -0.260 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0956 ¢
error map: ⟨0.000 +0.435 -0.100 +0.512]

Optimal ET sequence: 32c, 67c, 99, 265, 364, 463, 562, 1025

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 1240029/1239040

Mapping: [⟨1 -7 -20 -2 -35], ⟨0 25 65 14 112]]

Optimal tunings:

  • WE: ~2 = 1199.9909 ¢, ~80/63 = 412.0758 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0788 ¢

Optimal ET sequence: 99e, 166, 265

Badness (Sintel): 2.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4375/4356, 10648/10647

Mapping: [⟨1 -7 -20 -2 -35 -43], ⟨0 25 65 14 112 136]]

Optimal tunings:

  • WE: ~2 = 1200.0660 ¢, ~33/26 = 412.0928 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/26 = 412.0709 ¢

Optimal ET sequence: 99ef, 166, 265

Badness (Sintel): 2.40

Semiwitcher

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472, 585640/583443

Mapping: [⟨2 -14 -40 -4 -13], ⟨0 25 65 14 29]]

mapping generators: ~99/70, ~80/63

Optimal tunings:

  • WE: ~99/70 = 599.9717 ¢, ~80/63 = 412.0766 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~80/63 = 412.0951 ¢

Optimal ET sequence: 32c, 166, 198, 364, 562, 926

Badness (Sintel): 2.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280

Mapping: [⟨2 11 25 10 16 24], ⟨0 25 65 14 29 53]]

Optimal tunings:

  • WE: ~99/70 = 599.9666 ¢, ~33/26 = 412.0726 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/26 = 412.0944 ¢

Optimal ET sequence: 32cf, 166, 198, 364, 562, 926

Badness (Sintel): 1.57

Hitch

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 117649/117612, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31], ⟨0 50 130 28 -41]]

mapping generators: ~2, ~891/560

Optimal tunings:

  • WE: ~2 = 1199.8455 ¢, ~891/560 = 805.9457 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0488 ¢

Optimal ET sequence: 67c, 198, 463, 661, 1124bde

Badness (Sintel): 3.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31 -134], ⟨0 50 130 28 -41 205]]

Optimal tunings:

  • WE: ~2 = 1199.8415 ¢, ~891/560 = 805.9451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0508 ¢

Optimal ET sequence: 198, 463, 661

Badness (Sintel): 2.51

Hitcher

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561

Mapping: [⟨1 -32 -85 -16 31 -1], ⟨0 50 130 28 -41 7]]

Optimal tunings:

  • WE: ~2 = 1199.7690 ¢, ~78/49 = 805.8958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~78/49 = 806.0500 ¢

Optimal ET sequence: 67c, 198, 463f, 661ff

Badness (Sintel): 2.46

Calamity

Named by Flora Canou in 2023, calamity tempers out 2100875/2097152, the rainy comma, and may be described as the 46 & 311 temperament. It splits the interval of grave major sixth plus an octave into five.

Subgroup: 2.3.5.7

Comma list: 1600000/1594323, 2100875/2097152

Mapping: [⟨1 -12 -33 24], ⟨0 25 65 -39]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1200.0332 ¢, ~35/24 = 652.1081 ¢
error map: ⟨+0.033 +0.348 -0.387 -0.243]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0902 ¢
error map: ⟨0.000 +0.300 -0.452 -0.343]

Optimal ET sequence: 46, 219c, 265, 311

Badness (Sintel): 5.01

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 12005/11979, 131072/130977

Mapping: [⟨1 -12 -33 24 29], ⟨0 25 65 -39 -47]]

Optimal tunings:

  • WE: ~2 = 1199.9938 ¢, ~35/24 = 652.0887 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0920 ¢

Optimal ET sequence: 46, 219c, 265, 311, 979, 1290

Badness (Sintel): 2.00

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979

Mapping: [⟨1 -12 -33 24 29 45], ⟨0 25 65 -39 -47 -76]]

Optimal tunings:

  • WE: ~2 = 1199.9742 ¢, ~35/24 = 652.0794 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 652.0934 ¢

Optimal ET sequence: 46, 265, 311, 668, 979, 1290

Badness (Sintel): 1.39

Familia

Named by Xenllium in 2021, familia tempers out 16875/16807, the mirkwai comma, and may be described as the 113 & 152 temperament. It is generated by a major sixth, five of which octave reduced make the original generator of amity.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1600000/1594323

Mapping: [⟨1 -17 -46 -47], ⟨0 25 65 67]]

mapping generators: ~2, ~6561/3920

Optimal tunings:

  • WE: ~2 = 1199.9165 ¢, ~6561/3920 = 892.0267 ¢
error map: ⟨-0.084 +0.132 -0.736 +0.888]
  • CWE: ~2 = 1200.0000 ¢, ~6561/3920 = 892.0871 ¢
error map: ⟨0.000 +0.224 -0.649 +1.013]

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417

Badness (Sintel): 3.66

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 1600000/1594323

Mapping: [⟨1 -17 -46 -47 -1], ⟨0 25 65 67 6]]

Optimal tunings:

  • WE: ~2 = 1199.7927 ¢, ~2673/1600 = 891.9397 ¢
  • CWE: ~2 = 1200.0000 ¢, ~2673/1600 = 892.0909 ¢

Optimal ET sequence: 39d, 74cd, 113, 152, 417, 569de, 721de

Badness (Sintel): 1.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2205/2197

Mapping: [⟨1 -17 -46 -47 -1 -58], ⟨0 25 65 67 6 83]]

Optimal tunings:

  • WE: ~2 = 1199.7844 ¢, ~240/143 = 892.9268 ¢
  • CWE: ~2 = 1200.0000 ¢, ~240/143 = 892.0839 ¢

Optimal ET sequence: 39df, 74cdf, 113, 152f, 265

Badness (Sintel): 1.59

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