Amity family: Difference between revisions

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{{Main| Amity }}
{{Main| 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. Its [[ploidacot]] is 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.  
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.  
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.  
Line 13: Line 15:
[[Comma list]]: 1600000/1594323
[[Comma list]]: 1600000/1594323


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


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.537{{c}}
* [[WE]]: ~2 = 1199.9135{{c}}, ~400/243 = 860.4191{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.519{{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, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc }}
{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 152, 205, 463, 668, 873 }}


[[Badness]] (Smith): 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 septimal 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, [[10976/10935]] for chromat, [[703125/702464]] for trinity, [[2401/2400]] for amicable, [[2100875/2097152]] for calamity, [[420175/419904]] for witcher, 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]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
 
The rest are considered below.  


== Septimal amity ==
== Septimal amity ==
Line 42: Line 42:
[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103


{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
{{Mapping|legend=1| 1 -2 -7 15 | 0 5 13 -17 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.446{{c}}
* [[WE]]: ~2 = 1199.6100{{c}}, ~105/64 = 860.2881{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.432{{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 ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99, 152, 251, 350, 601cd, 905bcdd }}


[[Badness]] (Smith): 0.023649
[[Badness]] (Sintel): 0.598


=== 11-limit ===
=== 11-limit ===
Line 57: Line 59:
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.485{{c}}
* WE: ~2 = 1199.5961{{c}}, ~105/64 = 860.2460{{c}}
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.464{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5314{{c}}


{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}
{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}


Badness (Smith): 0.031506
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Line 72: Line 74:
Comma list: 352/351, 540/539, 625/624, 729/728
Comma list: 352/351, 540/539, 625/624, 729/728


Mapping: {{mapping| 1 3 6 -2 21 17 | 0 -5 -13 17 -62 -47 }}
Mapping: {{mapping| 1 -2 -7 15 -41 -30 | 0 5 13 -17 62 47 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.508{{c}}
* WE: ~2 = 1199.5437{{c}}, ~105/64 = 860.1922{{c}}
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.481{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5140{{c}}
 
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f }} *


{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f, 205 }}
<nowiki/>* optimal patent val: 205


Badness (Smith): 0.028008
Badness (Sintel): 1.16


==== 17-limit ====
==== 17-limit ====
Line 87: Line 91:
Comma list: 256/255, 352/351, 375/374, 540/539, 729/728
Comma list: 256/255, 352/351, 375/374, 540/539, 729/728


Mapping: {{mapping| 1 3 6 -2 21 17 -1 | 0 -5 -13 17 -62 -47 18 }}
Mapping: {{mapping| 1 -2 -7 15 -41 -30 17 | 0 5 13 -17 62 47 -18 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.496{{c}}
Optimal tunings:
* WE: ~2 = 1199.3376{{c}}, ~28/17 = 860.0617{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.5317{{c}}


{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg, 205gg }}
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg }}


Badness (Smith): 0.026201
Badness (Sintel): 1.33


==== 19-limit ====
==== 19-limit ====
Line 100: Line 106:
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455


Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}
Mapping: {{mapping| 1 -2 -7 15 -41 -30 17 -23 | 0 5 13 -17 62 47 -18 38 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.501{{c}}
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, 205gg }}
{{Optimal ET sequence|legend=0| 46efh, 53, 99ef, 152fg }}


Badness (Smith): 0.018782
Badness (Sintel): 1.14


=== Hitchcock ===
=== Hitchcock ===
Line 113: Line 121:
Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}
* WE: ~2 = 1199.9979{{c}}, ~18/11 = 860.6089{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6104{{c}}


{{Optimal ET sequence|legend=0| 7, 25cdde, 32cd, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness (Smith): 0.035187
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 128: Line 136:
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.411{{c}}
* WE: ~2 = 1200.2037{{c}}, ~18/11 = 860.5820{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.419{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.7268{{c}}


{{Optimal ET sequence|legend=0| 7, 25cddef, 32cd, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness (Smith): 0.022448
Badness (Sintel): 0.928


==== 17-limit ====
==== 17-limit ====
Line 143: 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: {{mapping| 1 3 6 -2 6 2 -1 | 0 -5 -13 17 -9 6 18 }}
Mapping: {{mapping| 1 -2 -7 15 -3 8 17 | 0 5 13 -17 9 -6 -18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}
* WE: ~2 = 1200.0179{{c}}, ~18/11 = 860.6472{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6344{{c}}


{{Optimal ET sequence|legend=0| 7, 25cddefgg, 32cdg, 39, 46, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 99 }}


Badness (Smith): 0.019395
Badness (Sintel): 0.988


==== 19-limit ====
==== 19-limit ====
Line 158: 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: {{mapping| 1 3 6 -2 6 2 -1 0 | 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 tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.415{{c}}
* WE: ~2 = 1200.1977{{c}}, ~18/11 = 860.7343{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.407{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.5917{{c}}


{{Optimal ET sequence|legend=0| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
{{Optimal ET sequence|legend=0| 7, 46, 53, 99h }}


Badness (Smith): 0.017513
Badness (Sintel): 1.07


=== Stalagmite ===
=== Stalagmite ===
The stalagmite temperament (46 & 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. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is ''down'' from 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 175: Line 183:
Comma list: 441/440, 896/891, 4375/4374
Comma list: 441/440, 896/891, 4375/4374


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.314{{c}}
* WE: ~2 = 1199.3915{{c}}, ~105/64 = 860.2240{{c}}
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.340{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6649{{c}}


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


Badness (Smith): 0.040976
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 190: Line 198:
Comma list: 196/195, 352/351, 364/363, 4375/4374
Comma list: 196/195, 352/351, 364/363, 4375/4374


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.277{{c}}
* WE: ~2 = 1199.3577{{c}}, ~105/64 = 860.2259{{c}}
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6929{{c}}


{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cef, 527bccef }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness (Smith): 0.034215
Badness (Sintel): 1.41


==== 17-limit ====
==== 17-limit ====
Line 205: 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: {{mapping| 1 3 6 -2 -7 -11 -1 | 0 -5 -13 17 37 52 18 }}
Mapping: {{mapping| 1 -2 -7 15 30 41 17 | 0 5 13 -17 -37 -52 -18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.272{{c}}
* WE: ~2 = 1199.3555{{c}}, ~28/17 = 860.2244{{c}}
* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6932{{c}}


{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness (Smith): 0.021193
Badness (Sintel): 1.08


==== 19-limit ====
==== 19-limit ====
Line 220: 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: {{mapping| 1 3 6 -2 -7 -11 -1 -13 | 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 tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.282{{c}}
* WE: ~2 = 1199.3599{{c}}, ~28/17 = 860.2161{{c}}
* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6820{{c}}


{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cefg }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness (Smith): 0.018864
Badness (Sintel): 1.15


=== Hemiamity ===
=== Hemiamity ===
Line 239: Line 247:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.566{{c}}
* WE: ~99/70 = 599.8218{{c}}, ~64/55 = 260.4833{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.561{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5615{{c}}


{{Optimal ET sequence|legend=0| 14cde, 32cde, 46, 106, 152, 350 }}
{{Optimal ET sequence|legend=0| 46, 106, 152, 350, 502d, 852bcdde }}


Badness (Smith): 0.031307
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Line 254: Line 262:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.607{{c}}
* WE: ~99/70 = 599.7965{{c}}, ~64/55 = 260.4951{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.583{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5872{{c}}


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


Badness (Smith): 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|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}
{{Mapping|legend=1| 1 -2 -7 -18 | 0 5 13 29 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.154{{c}}
* [[WE]]: ~2 = 1198.7458{{c}}, ~224/135 = 860.1071{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 338.993{{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 ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}
{{Optimal ET sequence|legend=1| 7d, …, 39d, 46, 131c }}


[[Badness]] (Smith): 0.095612
[[Badness]] (Sintel): 2.42


=== 11-limit ===
=== 11-limit ===
Line 281: Line 293:
Comma list: 121/120, 126/125, 896/891
Comma list: 121/120, 126/125, 896/891


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.136{{c}}
* WE: ~2 = 1199.3576{{c}}, ~18/11 = 860.4919{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.9224{{c}}


{{Optimal ET sequence|legend=0| 7d, 25cddde, 32cdd, 39d, 46 }}
{{Optimal ET sequence|legend=0| 7d, 39d, 46 }}


Badness (Smith): 0.042468
Badness (Sintel): 1.40


== Houborizic ==
== Houborizic ==
Houborizic tempers out 225/224, the [[marvel comma]], and may be described as the {{nowrap| 53 & 60 }} temperament. 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 298: Line 310:
[[Comma list]]: 225/224, 1250000/1240029
[[Comma list]]: 225/224, 1250000/1240029


{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}
{{Mapping|legend=1| 1 -2 -7 -23 | 0 5 13 36 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.711{{c}}
* [[WE]]: ~2 = 1200.4959{{c}}, ~400/243 = 860.5922{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{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, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}
{{Optimal ET sequence|legend=1| 7d, …, 46d, 53, 113, 166 }}


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


=== 11-limit ===
=== 11-limit ===
Line 313: Line 327:
Comma list: 225/224, 385/384, 1250000/1240029
Comma list: 225/224, 385/384, 1250000/1240029


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.751{{c}}
* WE: ~2 = 1200.4973{{c}}, ~400/243 = 860.5930{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.2386{{c}}


{{Optimal ET sequence|legend=0| 53, 113, 166, 551ccee }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness (Smith): 0.067891
Badness (Sintel): 2.24


==== 13-limit ====
==== 13-limit ====
Line 328: Line 342:
Comma list: 225/224, 325/324, 385/384, 2200/2197
Comma list: 225/224, 325/324, 385/384, 2200/2197


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~39/32 = 339.754{{c}}
* WE: ~2 = 1200.5022{{c}}, ~64/39 = 860.5964{{c}}
* POTE: ~2 = 1200.000{{c}}, ~39/32 = 339.764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/39 = 860.2378{{c}}


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


Badness (Smith): 0.032996
Badness (Sintel): 1.36


=== Houbor ===
=== Houbor ===
Line 343: Line 357:
Comma list: 121/120, 225/224, 2200/2187
Comma list: 121/120, 225/224, 2200/2187


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.680{{c}}
* WE: ~2 = 1201.1684{{c}}, ~18/11 = 861.0240{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.814{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2273{{c}}


{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}
{{Optimal ET sequence|legend=0| 7d, 53, 60e, 113e }}


Badness (Smith): 0.045232
Badness (Sintel): 1.49


==== 13-limit ====
==== 13-limit ====
Line 358: Line 372:
Comma list: 121/120, 225/224, 275/273, 325/324
Comma list: 121/120, 225/224, 275/273, 325/324


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.685{{c}}
* WE: ~2 = 1200.9339{{c}}, ~18/11 = 860.8859{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.784{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2376{{c}}


{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}
{{Optimal ET sequence|legend=0| 7d, 53, 113e }}


Badness (Smith): 0.031331
Badness (Sintel): 1.29


== Paramity ==
== Paramity ==
Paramity tempers out the [[horwell comma]] (65625/65536) and [[garischisma]] (33554432/33480783), and may be described as the {{nowrap| 53 & 311 }} temperament.  
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 375: Line 389:
[[Comma list]]: 65625/65536, 1600000/1594323
[[Comma list]]: 65625/65536, 1600000/1594323


{{Mapping|legend=1| 1 3 6 -17 | 0 -5 -13 70 }}
{{Mapping|legend=1| 1 -2 -7 53 | 0 5 13 -70 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* [[WE]]: ~2 = 1200.0199{{c}}, ~400/243 = 860.4612{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.553{{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 }}
{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986, 1297c }}


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


=== 11-limit ===
=== 11-limit ===
Line 390: Line 406:
Comma list: 6250/6237, 19712/19683, 41503/41472
Comma list: 6250/6237, 19712/19683, 41503/41472


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* WE: ~2 = 1200.0139{{c}}, ~400/243 = 860.4562{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.4462{{c}}


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


Badness (Smith): 0.064853
Badness (Sintel): 2.14


=== 13-limit ===
=== 13-limit ===
Line 405: Line 421:
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* WE: ~2 = 1199.9945{{c}}, ~385/234 = 860.4417{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4456{{c}}


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


Badness (Smith): 0.030347
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 420: 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: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}
Mapping: {{mapping| 1 -2 -7 53 -79 -30 93 | 0 5 13 -70 115 47 -124 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}
* WE: ~2 = 1200.0026{{c}}, ~385/234 = 860.4468{{c}}
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4450{{c}}


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


Badness (Smith): 0.024118
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 435: 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: {{mapping| 1 3 6 -17 36 17 -31 15 | 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 tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}
* WE: ~2 = 1200.0196{{c}}, ~208/171 = 860.4594{{c}}
* POTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~208/171 = 860.4454{{c}}


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


Badness (Smith): 0.017420
Badness (Sintel): 1.06


== Bamity ==
== Bamity ==
Line 455: Line 471:
: mapping generators: ~343/240, ~7/6
: mapping generators: ~343/240, ~7/6


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.563{{c}}
* [[WE]]: ~343/240 = 600.3379{{c}}, ~7/6 = 260.5488{{c}}
* POTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.402{{c}}
: [[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, 106d, 152d }}
{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60, 106d }}


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


=== 11-limit ===
=== 11-limit ===
Line 471: Line 489:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.653{{c}}
* WE: ~99/70 = 600.5039{{c}}, ~7/6 = 260.6119{{c}}
* POTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.393{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~7/6 = 260.4948{{c}}


{{Optimal ET sequence|legend=0| 14c, 32c, 46, 152de, 198, 244dee }}
{{Optimal ET sequence|legend=0| 14c, 32c, 46, 60e, 106de }}


Badness (Smith): 0.035504
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 486: Line 504:


Optimal tunings:
Optimal tunings:
* CTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.811{{c}}
* WE: ~55/39 = 600.3896{{c}}, ~7/6 = 260.7874{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.618{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~7/6 = 260.6927{{c}}


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


Badness (Smith): 0.030885
Badness (Sintel): 1.28


== Hamity ==
== Hamity ==
Line 500: Line 518:
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


{{Mapping|legend=1| 1 8 19 15 | 0 -10 -26 -19 }}
{{Mapping|legend=1| 1 -2 -7 -4 | 0 10 26 19 }}
: mapping generators: ~2, ~14/9
: mapping generators: ~2, ~9/7


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.168{{c}}
* [[WE]]: ~2 = 1199.5499{{c}}, ~9/7 = 430.0579{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.219{{c}}
: [[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]] (Smith): 0.073956
[[Badness]] (Sintel): 1.87


=== 11-limit ===
=== 11-limit ===
Line 516: Line 536:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}
Mapping: {{mapping| 1 -2 -7 -4 -3 | 0 10 26 19 18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.220{{c}}
* WE: ~2 = 1200.2215{{c}}, ~9/7 = 430.2713{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2036{{c}}


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


Badness (Smith): 0.042947
Badness (Sintel): 1.42


=== 13-limit ===
=== 13-limit ===
Line 534: Line 554:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.233{{c}}
* WE: ~2 = 1200.1379{{c}}, ~9/7 = 430.2658{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.216{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2227{{c}}


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


Badness (Smith): 0.029753
Badness (Sintel): 1.23


== Gamity ==
== Gamity ==
Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. 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 548: Line 568:
[[Comma list]]: 1029/1024, 1071875/1062882
[[Comma list]]: 1029/1024, 1071875/1062882


{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
{{Mapping|legend=1| 1 -2 -7 4 | 0 15 39 -5 }}
: mapping generators: ~2, ~320/189
: mapping generators: ~2, ~189/160


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~189/160 = 286.816{{c}}
* [[WE]]: ~2 = 1200.4434{{c}}, ~189/160 = 286.8930{{c}}
* POTE: ~2 = 1200.000{{c}}, ~189/160 = 286.787{{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, 205d, 364d }}
{{Optimal ET sequence|legend=1| 46, 113, 159 }}


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


=== 11-limit ===
=== 11-limit ===
Line 564: Line 586:
Comma list: 385/384, 441/440, 1071875/1062882
Comma list: 385/384, 441/440, 1071875/1062882


Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
Mapping: {{mapping| 1 -2 -7 4 8 | 0 15 39 -5 -19 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~33/28 = 286.813{{c}}
* WE: ~2 = 1200.3315{{c}}, ~33/28 = 286.8764{{c}}
* POTE: ~2 = 1200.000{{c}}, ~33/28 = 286.797{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/28 = 286.8002{{c}}


{{Optimal ET sequence|legend=0| 46, 113, 159, 205d, 364d }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness (Smith): 0.051111
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 579: Line 601:
Comma list: 325/324, 364/363, 385/384, 10985/10976
Comma list: 325/324, 364/363, 385/384, 10985/10976


Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 | 0 15 39 -5 -19 -18 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.803{{c}}
* WE: ~2 = 1200.4248{{c}}, ~13/11 = 286.8906{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.789{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7912{{c}}


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


Badness (Smith): 0.030297
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 594: 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: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 6 | 0 15 39 -5 -19 -18 -8 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.804{{c}}
* WE: ~2 = 1200.3287{{c}}, ~13/11 = 286.8731{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.795{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7957{{c}}


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


Badness (Smith): 0.022036
Badness (Sintel): 1.12


== Trinity ==
== Trinity ==
Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. It splits the 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{{c}} 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 611: Line 633:
[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


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


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~168/125 = 513.180{{c}}
* [[WE]]: ~2 = 1199.9474{{c}}, ~125/84 = 686.7921{{c}}
* POTE: ~2 = 1200.000{{c}}, ~168/125 = 513.178{{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 ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
{{Optimal ET sequence|legend=1| 7d, …, 152, 311, 463, 774 }}


[[Badness]] (Smith): 0.119453
[[Badness]] (Sintel): 3.02


=== 11-limit ===
=== 11-limit ===
Line 626: Line 651:
Comma list: 3025/3024, 4000/3993, 19712/19683
Comma list: 3025/3024, 4000/3993, 19712/19683


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~121/90 = 513.181{{c}}
* WE: ~2 = 1199.9090{{c}}, ~125/84 = 686.7710{{c}}
* POTE: ~2 = 1200.000{{c}}, ~121/90 = 513.177{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~125/84 = 686.8218{{c}}


{{Optimal ET sequence|legend=0| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
{{Optimal ET sequence|legend=0| 7d, …, 152, 311, 463, 774, 1237e }}


Badness (Smith): 0.031296
Badness (Sintel): 1.03


=== 13-limit ===
=== 13-limit ===
Line 641: Line 666:
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1199.9531{{c}}, ~52/35 = 686.7909{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8172{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 7df, …, 152f, 311 }}


Badness (Smith): 0.026418
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 656: 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: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 | 0 15 39 101 34 141 -61 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* WE: ~2 = 1200.0347{{c}}, ~52/35 = 686.8342{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8146{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1092cdg }}


Badness (Smith): 0.025588
Badness (Sintel): 1.30


=== 19-limit ===
=== 19-limit ===
Line 671: 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: {{mapping| 1 8 19 46 18 64 -22 53 | 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 tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1200.0406{{c}}, ~52/35 = 686.8382{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8152{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness (Smith): 0.018412
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 686: 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: {{mapping| 1 8 19 46 18 64 -22 53 49 | 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 tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8324{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8148{{c}}


{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness (Smith): 0.014343
Badness (Sintel): 1.03


=== 29-limit ===
=== 29-limit ===
Line 701: 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: {{mapping| 1 8 19 46 18 64 -22 53 49 72 | 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:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8318{{c}}
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{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 ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
 
[[Badness]] (Sintel): 1.46
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4356, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}
 
Badness (Sintel): 1.67
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 625/624, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}
 
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}
 
Badness (Sintel): 1.90
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
 
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
 
Badness (Sintel): 1.61
 
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.81
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
 
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.54
 
==== Chromic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1875/1859
 
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
 
Badness (Sintel): 2.06
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
 
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
 
Optimal tunings:
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}
 
Badness (Sintel): 1.58
 
=== Hemichromat ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 10976/10935, 102487/102400
 
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}
 
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}
 
Badness (Sintel): 2.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


{{Optimal ET sequence|legend=0| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
Optimal tunings:
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}


Badness (Smith): 0.012038
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness (Sintel): 1.38


== Amicable ==
== Amicable ==
Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} temperament.  
Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} temperament.  


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.  
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
[[Subgroup]]: 2.3.5.7
Line 720: Line 900:
[[Comma list]]: 2401/2400, 1600000/1594323
[[Comma list]]: 2401/2400, 1600000/1594323


{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}
{{Mapping|legend=1| 1 -17 -46 -26 | 0 20 52 31 }}
: mapping generators: ~2, ~40/21
 
[[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| 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: {{mapping| 1 -17 -46 -26 -135 | 0 20 52 31 149 }}
 
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=0| 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: {{mapping| 1 -17 -46 -26 -135 -171 | 0 20 52 31 149 188 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8831{{c}}
* WE: ~2 = 1200.0543{{c}}, ~40/21 = 1115.1594{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 84.880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1095{{c}}


{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}
{{Optimal ET sequence|legend=0| 99ef, 212, 311, 834, 1145c }}


[[Badness]] (Smith): 0.045473
Badness (Sintel): 1.43


=== Amical ===
=== Amical ===
Line 735: Line 948:
Comma list: 2401/2400, 131072/130977, 1600000/1594323
Comma list: 2401/2400, 131072/130977, 1600000/1594323


Mapping: {{mapping| 1 3 6 5 -8 | 0 -20 -52 -31 162 }}
Mapping: {{mapping| 1 -17 -46 -26 154 | 0 20 52 31 -162 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}
* WE: ~2 = 1200.0008{{c}}, ~40/21 = 1115.1165{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1157{{c}}


{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343, 2375bc }}
{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343 }}


Badness (Smith): 0.100668
Badness (Sintel): 3.33


==== 13-limit ====
==== 13-limit ====
Line 750: Line 963:
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206


Mapping: {{mapping| 1 3 6 5 -8 -5 | 0 -20 -52 -31 162 123 }}
Mapping: {{mapping| 1 -17 -46 -26 154 118 | 0 20 52 31 -162 -123 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8837{{c}}
* WE: ~2 = 1199.9926{{c}}, ~40/21 = 1115.1094{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8838{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1162{{c}}


{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}
{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}


Badness (Smith): 0.049893
Badness (Sintel): 2.06


=== Amorous ===
=== Pseudoamorous ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 6250/6237, 19712/19683
Comma list: 243/242, 441/440, 980000/970299


Mapping: {{mapping| 1 3 6 5 14 | 0 -20 -52 -31 -149 }}
Mapping: {{mapping| 1 -17 -46 -26 -43 | 0 20 52 31 50 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8883{{c}}
* WE: ~2 = 1199.8269{{c}}, ~21/11 = 1114.9475{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8896{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1115.1038{{c}}
 
{{Optimal ET sequence|legend=0| 99e, 212e }} *


{{Optimal ET sequence|legend=0| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}
<nowiki/>* optimal patent val: [[113edo|113]]


Badness (Smith): 0.048924
Badness (Sintel): 1.87


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


Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Comma list: 243/242, 364/363, 441/440, 1875/1859


Mapping: {{mapping| 1 3 6 5 14 17 | 0 -20 -52 -31 -149 -188 }}
Mapping: {{mapping| 1 -17 -46 -26 -43 -79 | 0 20 52 31 50 89 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8895{{c}}
* WE: ~2 = 1200.0182{{c}}, ~40/21 = 1115.1005{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8910{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0840{{c}}


{{Optimal ET sequence|legend=0| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}
{{Optimal ET sequence|legend=0| 99ef, 113, 212ef }}


Badness (Smith): 0.034681
Badness (Sintel): 1.77


=== Pseudoamical ===
=== Pseudoamical ===
Line 795: Line 1,010:
Comma list: 385/384, 1375/1372, 1600000/1594323
Comma list: 385/384, 1375/1372, 1600000/1594323


Mapping: {{mapping| 1 3 6 5 -1 | 0 -20 -52 -31 63 }}
Mapping: {{mapping| 1 -17 -46 -26 62 | 0 20 52 31 -63 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9005{{c}}
* WE: ~2 = 1200.3436{{c}}, ~40/21 = 1115.4102{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9091{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0925{{c}}


{{Optimal ET sequence|legend=0| 14ce, …, 85cee, 99, 212 }}
{{Optimal ET sequence|legend=0| 99, 113, 212, 961ccdeee }}


Badness (Smith): 0.085837
Badness (Sintel): 2.84


==== 13-limit ====
==== 13-limit ====
Line 810: Line 1,025:
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Comma list: 325/324, 385/384, 1375/1372, 19773/19712


Mapping: {{mapping| 1 3 6 5 -1 2 | 0 -20 -52 -31 63 24 }}
Mapping: {{mapping| 1 -17 -46 -26 62 26 | 0 20 52 31 -63 -24 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9049{{c}}
* WE: ~2 = 1200.4212{{c}}, ~40/21 = 1115.4787{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9127{{c}}
* CWE: ~2 = 1200.0000{{c}}, 40/21 = 1115.0885{{c}}


{{Optimal ET sequence|legend=0| 14ce, …, 85ceef, 99, 113, 212 }}
{{Optimal ET sequence|legend=0| 99, 113, 212, 537cdeff, 749ccdeefff }}


Badness (Smith): 0.047025
Badness (Sintel): 1.94


=== Pseudoamorous ===
=== Floral ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 980000/970299
Comma list: 2401/2400, 9801/9800, 14641/14580


Mapping: {{mapping| 1 3 6 5 7 | 0 -20 -52 -31 -50 }}
Mapping: {{mapping| 2 -14 -40 -21 -30 | 0 20 52 31 43 }}
: mapping generators: ~99/70, ~66/49


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9022{{c}}
* WE: ~99/70 = 600.0052{{c}}, ~66/49 = 515.1257{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8917{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1215{{c}}


{{Optimal ET sequence|legend=0| 14c, …, 85ce, 99e, 113, 212e }}
{{Optimal ET sequence|legend=0| 14c, … 198, 212, 410 }}


Badness (Smith): 0.056583
Badness (Sintel): 2.15


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


Comma list: 243/242, 364/363, 441/440, 1875/1859
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580


Mapping: {{mapping| 1 3 6 5 7 10 | 0 -20 -52 -31 -50 -89 }}
Mapping: {{mapping| 2 -14 -40 -21 -30 -63 | 0 20 52 31 43 82 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9153{{c}}
* WE: ~99/70 = 599.9923{{c}}, ~66/49 = 515.1184{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9164{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1246{{c}}


{{Optimal ET sequence|legend=0| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}
{{Optimal ET sequence|legend=0| 14cf, …, 198, 410 }}


Badness (Smith): 0.042826
Badness (Sintel): 1.53


=== Floral ===
=== Humorous ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 9801/9800, 14641/14580
Comma list: 2401/2400, 3025/3024, 1600000/1594323


Mapping: {{mapping| 2 6 12 10 13 | 0 -20 -52 -31 -43 }}
Mapping: {{mapping| 1 -37 -98 -57 16 | 0 40 104 62 -13 }}
: mapping generators: ~2, ~3993/2048


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8781{{c}}
* WE: ~2 = 1199.9189{{c}}, ~3993/2048 = 1157.4826{{c}}
* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8788{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3993/2048 = 1157.5602{{c}}


{{Optimal ET sequence|legend=0| 14c, …, 170bccde, 184c, 198, 212, 410 }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820 }}


Badness (Smith): 0.065110
Badness (Sintel): 1.93


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


Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024


Mapping: {{mapping| 2 6 12 10 13 19 | 0 -20 -52 -31 -43 -82 }}
Mapping: {{mapping| 1 -37 -98 -57 16 -59 | 0 40 104 62 -13 65 }}


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8759{{c}}
* WE: ~2 = 1199.8903{{c}}, ~39/20 = 1157.4551{{c}}
* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8750{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~39/20 = 1157.5600{{c}}
 
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820f }}
 
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


{{Optimal ET sequence|legend=0| 14c, …, 184cff, 198, 410 }}
Comma list: 9801/9800, 41503/41472, 585640/583443


Badness (Smith): 0.037013
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
: mapping generators: ~99/70, ~80/63


=== Humorous ===
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
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3025/3024, 1600000/1594323
Comma list: 3025/3024, 117649/117612, 1600000/1594323


Mapping: {{mapping| 1 3 6 5 3 | 0 -40 -104 -62 13 }}
Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4414{{c}}
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}


{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1131, 1442, 1753be }}
{{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }}


Badness (Smith): 0.058249
Badness (Sintel): 3.09


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


Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
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 3 6 5 3 6 | 0 -40 -104 -62 13 -65 }}
Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4425{{c}}
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* POTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4391{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}


{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
{{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }}


Badness (Smith): 0.028267
Badness (Sintel): 2.46


== Calamity ==
== Calamity ==
Calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of two octaves plus an acute minor third into five.  
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
[[Subgroup]]: 2.3.5.7
Line 917: Line 1,231:
[[Comma list]]: 1600000/1594323, 2100875/2097152
[[Comma list]]: 1600000/1594323, 2100875/2097152


{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
{{Mapping|legend=1| 1 -12 -33 24 | 0 25 65 -39 }}
: mapping generators: ~2, ~35/24


[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.909{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}


[[Badness]] (Smith): 0.198130
[[Badness]] (Sintel): 5.01


=== 11-limit ===
=== 11-limit ===
Line 930: Line 1,249:
Comma list: 3025/3024, 12005/11979, 131072/130977
Comma list: 3025/3024, 12005/11979, 131072/130977


Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
Mapping: {{mapping| 1 -12 -33 24 29 | 0 25 65 -39 -47 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.908{{c}}
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 }}
{{Optimal ET sequence|legend=0| 46, 219c, 265, 311, 979, 1290 }}


Badness (Smith): 0.060408
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Line 943: Line 1,264:
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979


Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}
Mapping: {{mapping| 1 -12 -33 24 29 45 | 0 25 65 -39 -47 -76 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.907{{c}}
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 }}
{{Optimal ET sequence|legend=0| 46, 265, 311, 668, 979, 1290 }}


Badness (Smith): 0.033617
Badness (Sintel): 1.39


== Familia ==
== Familia ==
Familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. 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 958: Line 1,281:
[[Comma list]]: 16875/16807, 1600000/1594323
[[Comma list]]: 16875/16807, 1600000/1594323


{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
{{Mapping|legend=1| 1 -17 -46 -47 | 0 25 65 67 }}
: mapping generators: ~2, ~6561/3920


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.915{{c}}
* [[WE]]: ~2 = 1199.9165{{c}}, ~6561/3920 = 892.0267{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.941{{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, 1516ccdd, 1933ccdd }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417 }}


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


=== 11-limit ===
=== 11-limit ===
Line 973: Line 1,299:
Comma list: 540/539, 1375/1372, 1600000/1594323
Comma list: 540/539, 1375/1372, 1600000/1594323


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.915{{c}}
* WE: ~2 = 1199.7927{{c}}, ~2673/1600 = 891.9397{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2673/1600 = 892.0909{{c}}


{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 417, 569de, 721de }}


Badness (Smith): 0.051740
Badness (Sintel): 1.71


=== 13-limit ===
=== 13-limit ===
Line 988: Line 1,314:
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
Comma list: 540/539, 729/728, 1375/1372, 2205/2197


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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~143/120 = 307.922{{c}}
* WE: ~2 = 1199.7844{{c}}, ~240/143 = 892.9268{{c}}
* POTE: ~2 = 1200.000{{c}}, ~143/120 = 307.913{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~240/143 = 892.0839{{c}}


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


Badness (Smith): 0.038473
Badness (Sintel): 1.59


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity family| ]] <!-- main 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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