Amity family: Difference between revisions

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


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


In the 5-limit amity is a genuine [[microtemperament]], with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  
The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[81/80|syntonic comma]] 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.
 
Amity is a genuine [[microtemperament]] in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 14: Line 17:
: mapping generators: ~2, ~243/200
: mapping generators: ~2, ~243/200


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~243/200 = 339.537
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1\1, ~243/200 = 339.537
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.519
* [[POTE]]: ~2 = 1\1, ~243/200 = 339.519


{{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, 258, 1085, 1343, 1601, 1859b, 2117bc }}
Line 42: Line 45:
{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~128/105 = 339.446
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~128/105 = 339.446
* POTE: ~2 = 1\1, ~128/105 = 339.432
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432


{{Optimal ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}
{{Optimal ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}
Line 59: Line 60:
Mapping: {{mapping| 1 3 6 -2 21 | 0 -5 -13 17 -62 }}
Mapping: {{mapping| 1 3 6 -2 21 | 0 -5 -13 17 -62 }}


Optimal tuning (CTE): ~2 = 1\1, ~128/105 = 339.485
Optimal tunings:
 
* CTE: ~2 = 1\1, ~128/105 = 339.485
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.464
* POTE: ~2 = 1\1, ~128/105 = 339.464


Optimal ET sequence: {{Optimal ET sequence| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}
Optimal ET sequence: {{Optimal ET sequence| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}
Line 74: Line 75:
Mapping: {{mapping| 1 3 6 -2 21 17 | 0 -5 -13 17 -62 -47 }}
Mapping: {{mapping| 1 3 6 -2 21 17 | 0 -5 -13 17 -62 -47 }}


Optimal tuning (CTE): ~2 = 1\1, ~128/105 = 339.508
Optimal tunings:
 
* CTE: ~2 = 1\1, ~128/105 = 339.508
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481
* POTE: ~2 = 1\1, ~128/105 = 339.481


Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152f, 205 }}
Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152f, 205 }}
Line 115: Line 116:
Mapping: {{mapping| 1 3 6 -2 6 | 0 -5 -13 17 -9 }}
Mapping: {{mapping| 1 3 6 -2 6 | 0 -5 -13 17 -9 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.390
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.390
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.390
* POTE: ~2 = 1\1, ~11/9 = 339.390


Optimal ET sequence: {{Optimal ET sequence| 7, 25cdde, 32cd, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cdde, 32cd, 39, 46, 53, 99 }}
Line 130: Line 131:
Mapping: {{mapping| 1 3 6 -2 6 2 | 0 -5 -13 17 -9 6 }}
Mapping: {{mapping| 1 3 6 -2 6 2 | 0 -5 -13 17 -9 6 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.411
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.411
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.419
* POTE: ~2 = 1\1, ~11/9 = 339.419


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}
Line 145: Line 146:
Mapping: {{mapping| 1 3 6 -2 6 2 -1 | 0 -5 -13 17 -9 6 18 }}
Mapping: {{mapping| 1 3 6 -2 6 2 -1 | 0 -5 -13 17 -9 6 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.366
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.366
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.366
* POTE: ~2 = 1\1, ~11/9 = 339.366


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgg, 32cdg, 39, 46, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgg, 32cdg, 39, 46, 99 }}
Line 160: Line 161:
Mapping: {{mapping| 1 3 6 -2 6 2 -1 0 | 0 -5 -13 17 -9 6 18 15 }}
Mapping: {{mapping| 1 3 6 -2 6 2 -1 0 | 0 -5 -13 17 -9 6 18 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.415
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.415
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.407
* POTE: ~2 = 1\1, ~11/9 = 339.407


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
Line 177: Line 178:
Mapping: {{mapping| 1 3 6 -2 -7 | 0 -5 -13 17 37 }}
Mapping: {{mapping| 1 3 6 -2 -7 | 0 -5 -13 17 37 }}


Optimal tuning (CTE): ~2 = 1\1, ~128/105 = 339.314
Optimal tunings:
 
* CTE: ~2 = 1\1, ~128/105 = 339.314
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.340
* POTE: ~2 = 1\1, ~128/105 = 339.340


Optimal ET sequence: {{Optimal ET sequence| 46, 99e, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99e, 145 }}
Line 192: Line 193:
Mapping: {{mapping| 1 3 6 -2 -7 -11 | 0 -5 -13 17 37 52 }}
Mapping: {{mapping| 1 3 6 -2 -7 -11 | 0 -5 -13 17 37 52 }}


Optimal tuning (CTE): ~2 = 1\1, ~128/105 = 339.277
Optimal tunings:
 
* CTE: ~2 = 1\1, ~128/105 = 339.277
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.313
* POTE: ~2 = 1\1, ~128/105 = 339.313


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cef, 527bccef }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cef, 527bccef }}
Line 207: Line 208:
Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 | 0 -5 -13 17 37 52 18 }}
Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 | 0 -5 -13 17 37 52 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.272
Optimal tunings:
 
* CTE: ~2 = 1\1, ~17/14 = 339.272
Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.313
* POTE: ~2 = 1\1, ~17/14 = 339.313


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c }}
Line 222: Line 223:
Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 -13 | 0 -5 -13 17 37 52 18 61 }}
Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 -13 | 0 -5 -13 17 37 52 18 61 }}


Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.282
Optimal tunings:
 
* CTE: ~2 = 1\1, ~17/14 = 339.282
Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.325
* POTE: ~2 = 1\1, ~17/14 = 339.325


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}
Line 239: Line 240:
: mapping generators: ~99/70, ~64/55
: mapping generators: ~99/70, ~64/55


Optimal tuning (CTE): ~99/70 = 1\2, ~64/55 = 260.566
Optimal tunings:
 
* CTE: ~99/70 = 1\2, ~64/55 = 260.566
Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561
* POTE: ~99/70 = 1\2, ~64/55 = 260.561


Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}
Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}
Line 254: Line 255:
Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}
Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~64/55 = 260.607
Optimal tunings:
 
* CTE: ~99/70 = 1\2, ~64/55 = 260.607
Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583
* POTE: ~99/70 = 1\2, ~64/55 = 260.583


Optimal ET sequence: {{Optimal ET sequence| 46, 106f, 152f, 198 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 106f, 152f, 198 }}
Line 269: Line 270:
{{Mapping|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}
{{Mapping|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}


{{Multival|legend=1| 5 13 29 9 32 31 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.154
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~243/200 = 339.154
* POTE: ~2 = 1\1, ~243/200 = 338.993
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 338.993


{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}
{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}
Line 286: Line 285:
Mapping: {{mapping| 1 3 6 11 6 | 0 -5 -13 -29 -9 }}
Mapping: {{mapping| 1 3 6 11 6 | 0 -5 -13 -29 -9 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.136
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.136
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.047
* POTE: ~2 = 1\1, ~11/9 = 339.047


Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}
Line 303: Line 302:
{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}
{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}


{{Multival|legend=1| 5 13 36 9 43 47 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.711
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~243/200 = 339.711
* POTE: ~2 = 1\1, ~243/200 = 339.763
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763


{{Optimal ET sequence|legend=1| 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}
{{Optimal ET sequence|legend=1| 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}
Line 320: Line 317:
Mapping: {{mapping| 1 3 6 13 -9 | 0 -5 -13 -36 44 }}
Mapping: {{mapping| 1 3 6 13 -9 | 0 -5 -13 -36 44 }}


Optimal tuning (CTE): ~2 = 1\1, ~243/200 = 339.751
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.751
Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.763
* POTE: ~2 = 1\1, ~243/200 = 339.763


Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}
Line 335: Line 332:
Mapping: {{mapping| 1 3 6 13 -9 2 | 0 -5 -13 -36 44 6 }}
Mapping: {{mapping| 1 3 6 13 -9 2 | 0 -5 -13 -36 44 6 }}


Optimal tuning (CTE): ~2 = 1\1, ~39/32 = 339.754
Optimal tunings:
 
* CTE: ~2 = 1\1, ~39/32 = 339.754
Optimal tuning (POTE): ~2 = 1\1, ~39/32 = 339.764
* POTE: ~2 = 1\1, ~39/32 = 339.764


Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}
Line 350: Line 347:
Mapping: [{{val| 1 3 6 13 6 }}, {{val| 0 -5 -13 -36 -9 }}]
Mapping: [{{val| 1 3 6 13 6 }}, {{val| 0 -5 -13 -36 -9 }}]


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.680
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.680
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.814
* POTE: ~2 = 1\1, ~11/9 = 339.814


Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
Line 365: Line 362:
Mapping: [{{val| 1 3 6 13 6 2 }}, {{val| 0 -5 -13 -36 -9 6 }}]
Mapping: [{{val| 1 3 6 13 6 2 }}, {{val| 0 -5 -13 -36 -9 6 }}]


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 339.685
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11/9 = 339.685
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.784
* POTE: ~2 = 1\1, ~11/9 = 339.784


Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
Line 382: Line 379:
{{Mapping|legend=1| 1 3 6 -17 | 0 -5 -13 70 }}
{{Mapping|legend=1| 1 3 6 -17 | 0 -5 -13 70 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.554
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.553
* POTE: ~2 = 1\1, ~243/200 = 339.553


{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986 }}
{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986 }}
Line 397: Line 394:
Mapping: {{mapping| 1 3 6 -17 36 | 0 -5 -13 70 -115 }}
Mapping: {{mapping| 1 3 6 -17 36 | 0 -5 -13 70 -115 }}


Optimal tuning (CTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.554
Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554


Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986 }}
Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986 }}
Line 412: Line 409:
Mapping: {{mapping| 1 3 6 -17 36 17 | 0 -5 -13 70 -115 -47 }}
Mapping: {{mapping| 1 3 6 -17 36 17 | 0 -5 -13 70 -115 -47 }}


Optimal tuning (CTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.554
Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554


Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986, 1661cf }}
Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986, 1661cf }}
Line 427: Line 424:
Mapping: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}
Mapping: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}


Optimal tuning (CTE): ~2 = 1\1, ~243/200 = 339.555
Optimal tunings:
 
* CTE: ~2 = 1\1, ~243/200 = 339.555
Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.555
* POTE: ~2 = 1\1, ~243/200 = 339.555


Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
Line 442: Line 439:
Mapping: {{mapping| 1 3 6 -17 36 17 -31 15 | 0 -5 -13 70 -115 -47 124 -38 }}
Mapping: {{mapping| 1 3 6 -17 36 17 -31 15 | 0 -5 -13 70 -115 -47 124 -38 }}


Optimal tuning (CTE): ~2 = 1\1, ~208/171 = 339.555
Optimal tunings:
 
* CTE: ~2 = 1\1, ~208/171 = 339.555
Optimal tuning (POTE): ~2 = 1\1, ~208/171 = 339.555
* POTE: ~2 = 1\1, ~208/171 = 339.555


Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
Line 461: Line 458:
: mapping generators: ~343/240, ~7/6
: mapping generators: ~343/240, ~7/6


{{Multival|legend=1| 10 26 12 18 -9 -45 }}
Optimal tunings:
 
* CTE: ~343/240 = 1\2, ~7/6 = 260.563
[[Optimal tuning]] ([[CTE]]): ~343/240 = 1\2, ~7/6 = 260.563
* POTE: ~343/240 = 1\2, ~7/6 = 260.402
 
[[Optimal tuning]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 106d, 152d }}
{{Optimal ET sequence|legend=1| 14c, 32c, 46, 106d, 152d }}
Line 478: Line 473:
Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}
Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~7/6 = 260.653
Optimal tunings:
 
* CTE: ~99/70 = 1\2, ~7/6 = 260.653
Optimal tuning (POTE): ~99/70 = 1\2, ~7/6 = 260.393
* POTE: ~99/70 = 1\2, ~7/6 = 260.393


Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}
Line 493: Line 488:
Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}
Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~7/6 = 260.811
Optimal tunings:
 
* CTE: ~55/39 = 1\2, ~7/6 = 260.811
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 260.618
* POTE: ~55/39 = 1\2, ~7/6 = 260.618


Optimal ET sequence: {{Optimal ET sequence| 14cf, 32cf, 46 }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, 32cf, 46 }}
Line 512: Line 507:
: mapping generators: ~2, ~14/9
: mapping generators: ~2, ~14/9


{{Multival|legend=1| 10 26 19 18 2 -29 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~9/7 = 430.168
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~9/7 = 430.168
* POTE: ~2 = 1\1, ~9/7 = 430.219
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219


{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
Line 529: Line 522:
Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}
Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 430.220
Optimal tunings:
 
* CTE: ~2 = 1\1, ~9/7 = 430.220
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.192
* POTE: ~2 = 1\1, ~9/7 = 430.192


Optimal ET sequence: {{Optimal ET sequence| 14c, 39d, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 39d, 53 }}
Line 544: Line 537:
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 430.233
Optimal tunings:
 
* CTE: ~2 = 1\1, ~9/7 = 430.233
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216
* POTE: ~2 = 1\1, ~9/7 = 430.216


Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}
Line 563: Line 556:
: mapping generators: ~2, ~320/189
: mapping generators: ~2, ~320/189


{{Multival|legend=1| 15 39 -5 27 -50 -121 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~189/160 = 286.816
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~189/160 = 286.816
* POTE: ~2 = 1\1, ~189/160 = 286.787
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787


{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
Line 580: Line 571:
Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/28 = 286.813
Optimal tunings:
 
* CTE: ~2 = 1\1, ~33/28 = 286.813
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 286.797
* POTE: ~2 = 1\1, ~33/28 = 286.797


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}
Line 595: Line 586:
Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~13/11 = 286.803
Optimal tunings:
 
* CTE: ~2 = 1\1, ~13/11 = 286.803
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.789
* POTE: ~2 = 1\1, ~13/11 = 286.789


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
Line 610: Line 601:
Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}


Optimal tuning (CTE): ~2 = 1\1, ~13/11 = 286.804
Optimal tunings:
 
* CTE: ~2 = 1\1, ~13/11 = 286.804
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.795
* POTE: ~2 = 1\1, ~13/11 = 286.795


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
Line 627: Line 618:
{{Mapping|legend=1| 1 8 19 46 | 0 -15 -39 -101 }}
{{Mapping|legend=1| 1 8 19 46 | 0 -15 -39 -101 }}


{{Multival|legend=1| 15 39 101 27 118 125 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~168/125 = 513.180
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~168/125 = 513.180
* POTE: ~2 = 1\1, ~168/125 = 513.178
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~168/125 = 513.178


{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
Line 644: Line 633:
Mapping: {{mapping| 1 8 19 46 18 | 0 -15 -39 -101 -34 }}
Mapping: {{mapping| 1 8 19 46 18 | 0 -15 -39 -101 -34 }}


Optimal tuning (CTE): ~2 = 1\1, ~121/90 = 513.181
Optimal tunings:
 
* CTE: ~2 = 1\1, ~121/90 = 513.181
Optimal tuning (POTE): ~2 = 1\1, ~121/90 = 513.177
* POTE: ~2 = 1\1, ~121/90 = 513.177


Optimal ET sequence: {{Optimal ET sequence| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
Line 659: Line 648:
Mapping: {{mapping| 1 8 19 46 18 64 | 0 -15 -39 -101 -34 -141 }}
Mapping: {{mapping| 1 8 19 46 18 64 | 0 -15 -39 -101 -34 -141 }}


Optimal tuning (CTE): ~2 = 1\1, ~35/26 = 513.184
Optimal tunings:
 
* CTE: ~2 = 1\1, ~35/26 = 513.184
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.182
* POTE: ~2 = 1\1, ~35/26 = 513.182


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
Line 674: Line 663:
Mapping: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}
Mapping: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}


Optimal tuning (CTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
 
* CTE: ~2 = 1\1, ~35/26 = 513.185
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
* POTE: ~2 = 1\1, ~35/26 = 513.186


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
Line 689: Line 678:
Mapping: {{mapping| 1 8 19 46 18 64 -22 53 | 0 -15 -39 -101 -34 -141 61 -114 }}
Mapping: {{mapping| 1 8 19 46 18 64 -22 53 | 0 -15 -39 -101 -34 -141 61 -114 }}


Optimal tuning (CTE): ~2 = 1\1, ~35/26 = 513.184
Optimal tunings:
 
* CTE: ~2 = 1\1, ~35/26 = 513.184
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1\1, ~35/26 = 513.185


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
Line 704: Line 693:
Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 | 0 -15 -39 -101 -34 -141 61 -114 -104 }}
Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 | 0 -15 -39 -101 -34 -141 61 -114 -104 }}


Optimal tuning (CTE): ~2 = 1\1, ~35/26 = 513.184
Optimal tunings:
 
* CTE: ~2 = 1\1, ~35/26 = 513.184
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1\1, ~35/26 = 513.185


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
Line 719: Line 708:
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 8 19 46 18 64 -22 53 49 72 | 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}


Optimal tuning (CTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
 
* CTE: ~2 = 1\1, ~35/26 = 513.185
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
* POTE: ~2 = 1\1, ~35/26 = 513.186


Optimal ET sequence: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
Optimal ET sequence: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
Line 738: Line 727:
{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}
{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}


{{Multival|legend=1| 20 52 31 36 -7 -74 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.8831
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~21/20 = 84.8831
* POTE: ~2 = 1\1, ~21/20 = 84.880
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 84.880


{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}
{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}
Line 755: Line 742:
Mapping: {{mapping| 1 3 6 5 -8 | 0 -20 -52 -31 162 }}
Mapping: {{mapping| 1 3 6 5 -8 | 0 -20 -52 -31 162 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.8843
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.8843
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8843
* POTE: ~2 = 1\1, ~21/20 = 84.8843


Optimal ET sequence: {{Optimal ET sequence| 99, 212e, 311, 721, 1032, 1343, 2375bc }}
Optimal ET sequence: {{Optimal ET sequence| 99, 212e, 311, 721, 1032, 1343, 2375bc }}
Line 770: Line 757:
Mapping: {{mapping| 1 3 6 5 -8 -5 | 0 -20 -52 -31 162 123 }}
Mapping: {{mapping| 1 3 6 5 -8 -5 | 0 -20 -52 -31 162 123 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.8837
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.8837
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8838
* POTE: ~2 = 1\1, ~21/20 = 84.8838


Optimal ET sequence: {{Optimal ET sequence| 99, 212ef, 311, 721, 1032 }}
Optimal ET sequence: {{Optimal ET sequence| 99, 212ef, 311, 721, 1032 }}
Line 785: Line 772:
Mapping: {{mapping| 1 3 6 5 14 | 0 -20 -52 -31 -149 }}
Mapping: {{mapping| 1 3 6 5 14 | 0 -20 -52 -31 -149 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.8883
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.8883
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8896
* POTE: ~2 = 1\1, ~21/20 = 84.8896


Optimal ET sequence: {{Optimal ET sequence| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}
Optimal ET sequence: {{Optimal ET sequence| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}
Line 800: Line 787:
Mapping: {{mapping| 1 3 6 5 14 17 | 0 -20 -52 -31 -149 -188 }}
Mapping: {{mapping| 1 3 6 5 14 17 | 0 -20 -52 -31 -149 -188 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.8895
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.8895
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8910
* POTE: ~2 = 1\1, ~21/20 = 84.8910


Optimal ET sequence: {{Optimal ET sequence| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}
Optimal ET sequence: {{Optimal ET sequence| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}
Line 815: Line 802:
Mapping: {{mapping| 1 3 6 5 -1 | 0 -20 -52 -31 63 }}
Mapping: {{mapping| 1 3 6 5 -1 | 0 -20 -52 -31 63 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.9005
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.9005
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9091
* POTE: ~2 = 1\1, ~21/20 = 84.9091


Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85cee, 99, 212 }}
Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85cee, 99, 212 }}
Line 830: Line 817:
Mapping: {{mapping| 1 3 6 5 -1 2 | 0 -20 -52 -31 63 24 }}
Mapping: {{mapping| 1 3 6 5 -1 2 | 0 -20 -52 -31 63 24 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.9049
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.9049
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9127
* POTE: ~2 = 1\1, ~21/20 = 84.9127


Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85ceef, 99, 113, 212 }}
Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85ceef, 99, 113, 212 }}
Line 845: Line 832:
Mapping: {{mapping| 1 3 6 5 7 | 0 -20 -52 -31 -50 }}
Mapping: {{mapping| 1 3 6 5 7 | 0 -20 -52 -31 -50 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.9022
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.9022
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8917
* POTE: ~2 = 1\1, ~21/20 = 84.8917


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 85ce, 99e, 113, 212e }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 85ce, 99e, 113, 212e }}
Line 860: Line 847:
Mapping: {{mapping| 1 3 6 5 7 10 | 0 -20 -52 -31 -50 -89 }}
Mapping: {{mapping| 1 3 6 5 7 10 | 0 -20 -52 -31 -50 -89 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 84.9153
Optimal tunings:
 
* CTE: ~2 = 1\1, ~21/20 = 84.9153
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9164
* POTE: ~2 = 1\1, ~21/20 = 84.9164


Optimal ET sequence: {{Optimal ET sequence| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}
Line 875: Line 862:
Mapping: {{mapping| 2 6 12 10 13 | 0 -20 -52 -31 -43 }}
Mapping: {{mapping| 2 6 12 10 13 | 0 -20 -52 -31 -43 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~21/20 = 84.8781
Optimal tunings:
 
* CTE: ~99/70 = 1\2, ~21/20 = 84.8781
Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8788
* POTE: ~99/70 = 1\2, ~21/20 = 84.8788


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 170bccde, 184c, 198, 212, 410 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 170bccde, 184c, 198, 212, 410 }}
Line 890: Line 877:
Mapping: {{mapping| 2 6 12 10 13 19 | 0 -20 -52 -31 -43 -82 }}
Mapping: {{mapping| 2 6 12 10 13 19 | 0 -20 -52 -31 -43 -82 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~21/20 = 84.8759
Optimal tunings:
 
* CTE: ~99/70 = 1\2, ~21/20 = 84.8759
Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8750
* POTE: ~99/70 = 1\2, ~21/20 = 84.8750


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 184cff, 198, 410 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 184cff, 198, 410 }}
Line 905: Line 892:
Mapping: {{mapping| 1 3 6 5 3 | 0 -40 -104 -62 13 }}
Mapping: {{mapping| 1 3 6 5 3 | 0 -40 -104 -62 13 }}


Optimal tuning (CTE): ~2 = 1\1, ~4096/3993 = 42.4414
Optimal tunings:
 
* CTE: ~2 = 1\1, ~4096/3993 = 42.4414
Optimal tuning (POTE): ~2 = 1\1, ~4096/3993 = 42.4391
* POTE: ~2 = 1\1, ~4096/3993 = 42.4391


Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1131, 1442, 1753be }}
Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1131, 1442, 1753be }}
Line 920: Line 907:
Mapping: {{mapping| 1 3 6 5 3 6 | 0 -40 -104 -62 13 -65 }}
Mapping: {{mapping| 1 3 6 5 3 6 | 0 -40 -104 -62 13 -65 }}


Optimal tuning (CTE): ~2 = 1\1, ~40/39 = 42.4425
Optimal tunings:
 
* CTE: ~2 = 1\1, ~40/39 = 42.4425
Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 42.4391
* POTE: ~2 = 1\1, ~40/39 = 42.4391


Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
Line 937: Line 924:
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~48/35 = 547.909
[[Optimal tuning]] (CTE): ~2 = 1\1, ~48/35 = 547.909


{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
Line 978: Line 965:
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}


{{Multival|legend=1| 25 65 67 45 36 -27 }}
Optimal tunings:
 
* CTE: ~2 = 1\1, ~11907/10000 = 307.915
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~11907/10000 = 307.915
* POTE: ~2 = 1\1, ~11907/10000 = 307.941
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
Line 995: Line 980:
Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}
Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}


Optimal tuning (CTE): ~2 = 1\1, ~3200/2673 = 307.915
Optimal tunings:
 
* CTE: ~2 = 1\1, ~3200/2673 = 307.915
Optimal tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906
* POTE: ~2 = 1\1, ~3200/2673 = 307.906


Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
Line 1,010: Line 995:
Mapping: {{mapping| 1 8 19 20 5 25 | 0 -25 -65 -67 -6 -83 }}
Mapping: {{mapping| 1 8 19 20 5 25 | 0 -25 -65 -67 -6 -83 }}


Optimal tuning (CTE): ~2 = 1\1, ~143/120 = 307.922
Optimal tunings:
 
* CTE: ~2 = 1\1, ~143/120 = 307.922
Optimal tuning (POTE): ~2 = 1\1, ~143/120 = 307.913
* POTE: ~2 = 1\1, ~143/120 = 307.913


Optimal ET sequence: {{Optimal ET sequence| 39df, 74cdf, 113, 152f, 265 }}
Optimal ET sequence: {{Optimal ET sequence| 39df, 74cdf, 113, 152f, 265 }}
Line 1,019: Line 1,004:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity| ]] <!-- key article -->
[[Category:Amity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:26, 24 June 2025

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 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 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 3 6], ⟨0 -5 -13]]

mapping generators: ~2, ~243/200

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.537
  • POTE: ~2 = 1\1, ~243/200 = 339.519

Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc

Badness: 0.021960

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.

Temperaments discussed elsewhere include:

The rest are considered below.

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 3 6 -2], ⟨0 -5 -13 17]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.446
  • POTE: ~2 = 1\1, ~128/105 = 339.432

Optimal ET sequence: 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd

Badness: 0.023649

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 -2 21], ⟨0 -5 -13 17 -62]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.485
  • POTE: ~2 = 1\1, ~128/105 = 339.464

Optimal ET sequence: 46e, 53, 99e, 152, 357d, 509dd, 661dd

Badness: 0.031506

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 -2 21 17], ⟨0 -5 -13 17 -62 -47]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.508
  • POTE: ~2 = 1\1, ~128/105 = 339.481

Optimal ET sequence: 46ef, 53, 99ef, 152f, 205

Badness: 0.028008

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 6 -2 21 17 -1], ⟨0 -5 -13 17 -62 -47 18]]

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.496

Optimal ET sequence: 46ef, 53, 99ef, 152fg, 205gg

Badness: 0.026201

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 3 6 -2 21 17 -1 15], ⟨0 -5 -13 17 -62 -47 18 -38]]

Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501

Optimal ET sequence: 46efh, 53, 99ef, 152fg, 205gg

Badness: 0.018782

Hitchcock

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.390
  • POTE: ~2 = 1\1, ~11/9 = 339.390

Optimal ET sequence: 7, 25cdde, 32cd, 39, 46, 53, 99

Badness: 0.035187

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.411
  • POTE: ~2 = 1\1, ~11/9 = 339.419

Optimal ET sequence: 7, 25cddef, 32cd, 39, 46, 53, 99

Badness: 0.022448

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.366
  • POTE: ~2 = 1\1, ~11/9 = 339.366

Optimal ET sequence: 7, 25cddefgg, 32cdg, 39, 46, 99

Badness: 0.019395

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 3 6 -2 6 2 -1 0], ⟨0 -5 -13 17 -9 6 18 15]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.415
  • POTE: ~2 = 1\1, ~11/9 = 339.407

Optimal ET sequence: 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h

Badness: 0.017513

Catamite

The catamite 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. The word "catamite" itself is a term for male homosexual, but also a play on the words "cata-" (down) and "amity."

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 -2 -7], ⟨0 -5 -13 17 37]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.314
  • POTE: ~2 = 1\1, ~128/105 = 339.340

Optimal ET sequence: 46, 99e, 145

Badness: 0.040976

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 -2 -7 -11], ⟨0 -5 -13 17 37 52]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~128/105 = 339.277
  • POTE: ~2 = 1\1, ~128/105 = 339.313

Optimal ET sequence: 46, 99ef, 145, 191c, 336cef, 527bccef

Badness: 0.034215

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 6 -2 -7 -11 -1], ⟨0 -5 -13 17 37 52 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~17/14 = 339.272
  • POTE: ~2 = 1\1, ~17/14 = 339.313

Optimal ET sequence: 46, 99ef, 145, 191c

Badness: 0.021193

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 3 6 -2 -7 -11 -1 -13], ⟨0 -5 -13 17 37 52 18 61]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~17/14 = 339.282
  • POTE: ~2 = 1\1, ~17/14 = 339.325

Optimal ET sequence: 46, 99ef, 145, 191c, 336cefg

Badness: 0.018864

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:

  • CTE: ~99/70 = 1\2, ~64/55 = 260.566
  • POTE: ~99/70 = 1\2, ~64/55 = 260.561

Optimal ET sequence: 14cde, 32cde, 46, 106, 152, 350

Badness: 0.031307

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:

  • CTE: ~99/70 = 1\2, ~64/55 = 260.607
  • POTE: ~99/70 = 1\2, ~64/55 = 260.583

Optimal ET sequence: 46, 106f, 152f, 198

Badness: 0.025784

Accord

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 3 6 11], ⟨0 -5 -13 -29]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.154
  • POTE: ~2 = 1\1, ~243/200 = 338.993

Optimal ET sequence: 7d, 25cddd, 32cdd, 39d, 46

Badness: 0.095612

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 11 6], ⟨0 -5 -13 -29 -9]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.136
  • POTE: ~2 = 1\1, ~11/9 = 339.047

Optimal ET sequence: 7d, 25cddde, 32cdd, 39d, 46

Badness: 0.042468

Houborizic

The houborizic temperament (53 & 60) tempers out the marvel comma, 225/224. It is so named because it is closely related to the houboriz tuning (generator: 339.774971 cents).

Subgroup: 2.3.5.7

Comma list: 225/224, 1250000/1240029

Mapping: [⟨1 3 6 13], ⟨0 -5 -13 -36]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.711
  • POTE: ~2 = 1\1, ~243/200 = 339.763

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c

Badness: 0.066638

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 13 -9], ⟨0 -5 -13 -36 44]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.751
  • POTE: ~2 = 1\1, ~243/200 = 339.763

Optimal ET sequence: 53, 113, 166, 551ccee

Badness: 0.067891

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 13 -9 2], ⟨0 -5 -13 -36 44 6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~39/32 = 339.754
  • POTE: ~2 = 1\1, ~39/32 = 339.764

Optimal ET sequence: 53, 113, 166

Badness: 0.032996

Houbor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 13 6], ⟨0 -5 -13 -36 -9]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.680
  • POTE: ~2 = 1\1, ~11/9 = 339.814

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness: 0.045232

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 339.685
  • POTE: ~2 = 1\1, ~11/9 = 339.784

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness: 0.031331

Paramity

The paramity temperament (53 & 311) tempers out the horwell comma (65625/65536) and garischisma (33554432/33480783).

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1600000/1594323

Mapping: [⟨1 3 6 -17], ⟨0 -5 -13 70]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.553

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

Badness: 0.113655

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 -17 36], ⟨0 -5 -13 70 -115]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.554

Optimal ET sequence: 53, 205de, 258, 311, 675, 986

Badness: 0.064853

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 -17 36 17], ⟨0 -5 -13 70 -115 -47]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.554
  • POTE: ~2 = 1\1, ~243/200 = 339.554

Optimal ET sequence: 53, 205de, 258, 311, 675, 986, 1661cf

Badness: 0.030347

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 6 -17 36 17 -31], ⟨0 -5 -13 70 -115 -47 124]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~243/200 = 339.555
  • POTE: ~2 = 1\1, ~243/200 = 339.555

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf

Badness: 0.024118

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 3 6 -17 36 17 -31 15], ⟨0 -5 -13 70 -115 -47 124 -38]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~208/171 = 339.555
  • POTE: ~2 = 1\1, ~208/171 = 339.555

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 986, 1661cfh

Badness: 0.017420

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:

  • CTE: ~343/240 = 1\2, ~7/6 = 260.563
  • POTE: ~343/240 = 1\2, ~7/6 = 260.402

Optimal ET sequence: 14c, 32c, 46, 106d, 152d

Badness: 0.083601

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:

  • CTE: ~99/70 = 1\2, ~7/6 = 260.653
  • POTE: ~99/70 = 1\2, ~7/6 = 260.393

Optimal ET sequence: 14c, 32c, 46, 152de, 198, 244dee

Badness: 0.035504

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:

  • CTE: ~55/39 = 1\2, ~7/6 = 260.811
  • POTE: ~55/39 = 1\2, ~7/6 = 260.618

Optimal ET sequence: 14cf, 32cf, 46

Badness: 0.030885

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 8 19 15], ⟨0 -10 -26 -19]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/7 = 430.168
  • POTE: ~2 = 1\1, ~9/7 = 430.219

Optimal ET sequence: 14c, 39d, 53

Badness: 0.073956

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • CTE: ~2 = 1\1, ~9/7 = 430.220
  • POTE: ~2 = 1\1, ~9/7 = 430.192

Optimal ET sequence: 14c, 39d, 53

Badness: 0.042947

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:

  • CTE: ~2 = 1\1, ~9/7 = 430.233
  • POTE: ~2 = 1\1, ~9/7 = 430.216

Optimal ET sequence: 14cf, 39df, 53

Badness: 0.029753

Gamity

The gamity temperament (46 & 113) tempers out the gamelisma, 1029/1024. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1071875/1062882

Mapping: [⟨1 13 32 -1], ⟨0 -15 -39 5]]

mapping generators: ~2, ~320/189

Optimal tunings:

  • CTE: ~2 = 1\1, ~189/160 = 286.816
  • POTE: ~2 = 1\1, ~189/160 = 286.787

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness: 0.125733

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 32 -1 -11], ⟨0 -15 -39 5 19]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~33/28 = 286.813
  • POTE: ~2 = 1\1, ~33/28 = 286.797

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness: 0.051111

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 13 32 -1 -11 -10], ⟨0 -15 -39 5 19 18]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~13/11 = 286.803
  • POTE: ~2 = 1\1, ~13/11 = 286.789

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

Badness: 0.030297

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 13 32 -1 -11 -10 -2], ⟨0 -15 -39 5 19 18 8]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~13/11 = 286.804
  • POTE: ~2 = 1\1, ~13/11 = 286.795

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

Badness: 0.022036

Trinity

The trinity temperament (152 & 159) tempers out the meter, 703125/702464. 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¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 1600000/1594323

Mapping: [⟨1 8 19 46], ⟨0 -15 -39 -101]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~168/125 = 513.180
  • POTE: ~2 = 1\1, ~168/125 = 513.178

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085

Badness: 0.119453

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 19 46 18], ⟨0 -15 -39 -101 -34]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~121/90 = 513.181
  • POTE: ~2 = 1\1, ~121/90 = 513.177

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085e, 1396e

Badness: 0.031296

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 19 46 18 64], ⟨0 -15 -39 -101 -34 -141]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.182

Optimal ET sequence: 152f, 159, 311

Badness: 0.026418

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 8 19 46 18 64 -22], ⟨0 -15 -39 -101 -34 -141 61]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.185
  • POTE: ~2 = 1\1, ~35/26 = 513.186

Optimal ET sequence: 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg

Badness: 0.025588

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 8 19 46 18 64 -22 53], ⟨0 -15 -39 -101 -34 -141 61 -114]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.185

Optimal ET sequence: 152f, 159, 311

Badness: 0.018412

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 8 19 46 18 64 -22 53 49], ⟨0 -15 -39 -101 -34 -141 61 -114 -104]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.184
  • POTE: ~2 = 1\1, ~35/26 = 513.185

Optimal ET sequence: 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi

Badness: 0.014343

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 8 19 46 18 64 -22 53 49 72], ⟨0 -15 -39 -101 -34 -141 61 -114 -104 -157]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~35/26 = 513.185
  • POTE: ~2 = 1\1, ~35/26 = 513.186

Optimal ET sequence: 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij

Badness: 0.012038

Amicable

The amicable temperament (99 & 311) tempers out the amity comma and the canousma in addition to the breedsma, and is closely associated with the canou 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.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8831
  • POTE: ~2 = 1\1, ~21/20 = 84.880

Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b

Badness: 0.045473

Amical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8843
  • POTE: ~2 = 1\1, ~21/20 = 84.8843

Optimal ET sequence: 99, 212e, 311, 721, 1032, 1343, 2375bc

Badness: 0.100668

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8837
  • POTE: ~2 = 1\1, ~21/20 = 84.8838

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

Badness: 0.049893

Amorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8883
  • POTE: ~2 = 1\1, ~21/20 = 84.8896

Optimal ET sequence: 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde

Badness: 0.048924

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.8895
  • POTE: ~2 = 1\1, ~21/20 = 84.8910

Optimal ET sequence: 99ef, 212, 311, 1145c, 1456cd, 1767cd

Badness: 0.034681

Pseudoamical

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9005
  • POTE: ~2 = 1\1, ~21/20 = 84.9091

Optimal ET sequence: 14ce, …, 85cee, 99, 212

Badness: 0.085837

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9049
  • POTE: ~2 = 1\1, ~21/20 = 84.9127

Optimal ET sequence: 14ce, …, 85ceef, 99, 113, 212

Badness: 0.047025

Pseudoamorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9022
  • POTE: ~2 = 1\1, ~21/20 = 84.8917

Optimal ET sequence: 14c, …, 85ce, 99e, 113, 212e

Badness: 0.056583

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~21/20 = 84.9153
  • POTE: ~2 = 1\1, ~21/20 = 84.9164

Optimal ET sequence: 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef

Badness: 0.042826

Floral

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~21/20 = 84.8781
  • POTE: ~99/70 = 1\2, ~21/20 = 84.8788

Optimal ET sequence: 14c, …, 170bccde, 184c, 198, 212, 410

Badness: 0.065110

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]

Optimal tunings:

  • CTE: ~99/70 = 1\2, ~21/20 = 84.8759
  • POTE: ~99/70 = 1\2, ~21/20 = 84.8750

Optimal ET sequence: 14c, …, 184cff, 198, 410

Badness: 0.037013

Humorous

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~4096/3993 = 42.4414
  • POTE: ~2 = 1\1, ~4096/3993 = 42.4391

Optimal ET sequence: 85c, 113, 198, 311, 1131, 1442, 1753be

Badness: 0.058249

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~40/39 = 42.4425
  • POTE: ~2 = 1\1, ~40/39 = 42.4391

Optimal ET sequence: 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff

Badness: 0.028267

Calamity

The calamity temperament (46 & 311) tempers out the rainy comma, 2100875/2097152, splitting the interval of two octaves plus an acute minor third into five.

Subgroup: 2.3.5.7

Comma list: 1600000/1594323, 2100875/2097152

Mapping: [⟨1 13 32 -15], ⟨0 -25 -65 39]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.909

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

Badness: 0.198130

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 32 -15 -18], ⟨0 -25 -65 39 47]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908

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

Badness: 0.060408

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 13 32 -15 -18 -31], ⟨0 -25 -65 39 47 76]]

Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.907

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

Badness: 0.033617

Familia

The familia temperament (113 & 152) tempers out the mirkwai comma, 16875/16807. It splits the interval of acute minor tenth (~243/100) in five.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1600000/1594323

Mapping: [⟨1 8 19 20], ⟨0 -25 -65 -67]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11907/10000 = 307.915
  • POTE: ~2 = 1\1, ~11907/10000 = 307.941

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd

Badness: 0.144551

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 19 20 5], ⟨0 -25 -65 -67 -6]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3200/2673 = 307.915
  • POTE: ~2 = 1\1, ~3200/2673 = 307.906

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1099cdee

Badness: 0.051740

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 19 20 5 25], ⟨0 -25 -65 -67 -6 -83]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~143/120 = 307.922
  • POTE: ~2 = 1\1, ~143/120 = 307.913

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

Badness: 0.038473

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