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


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc }}


[[Badness]]: 0.021960
[[Badness]]: 0.021960


=== 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 witch, and [[16875/16807]] for familia.  
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:  
Temperaments discussed elsewhere include:  
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witch]]'' → [[Wizmic microtemperaments #Witch]] (+420175/419904)
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)


The rest are considered below.  
The rest are considered below.  
Line 40: 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]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432
* POTE: ~2 = 1\1, ~128/105 = 339.432


{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
{{Optimal ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}


[[Badness]]: 0.023649
[[Badness]]: 0.023649
Line 55: 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 (POTE): ~2 = 1\1, ~128/105 = 339.464
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.485
* POTE: ~2 = 1\1, ~128/105 = 339.464


{{Optimal ET sequence|legend=1| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
Optimal ET sequence: {{Optimal ET sequence| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}


Badness: 0.031506
Badness: 0.031506
Line 64: Line 71:
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
Comma list: 352/351, 540/539, 625/624, 729/728


Mapping: {{mapping| 1 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 (POTE): ~2 = 1\1, ~128/105 = 339.481
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.508
* POTE: ~2 = 1\1, ~128/105 = 339.481
 
Optimal ET sequence: {{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: {{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: {{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


{{Optimal ET sequence|legend=1| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501


Badness: 0.028008
Optimal ET sequence: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}
 
Badness: 0.018782


=== Hitchcock ===
=== Hitchcock ===
Line 83: 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 (POTE): ~2 = 1\1, ~11/9 = 339.390
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.390
* POTE: ~2 = 1\1, ~11/9 = 339.390


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cdde, 32cd, 39, 46, 53, 99 }}


Badness: 0.035187
Badness: 0.035187
Line 96: 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 (POTE): ~2 = 1\1, ~11/9 = 339.419
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.411
* POTE: ~2 = 1\1, ~11/9 = 339.419


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}


Badness: 0.022448
Badness: 0.022448
Line 109: 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 (POTE): ~2 = 1\1, ~11/9 = 339.366
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.366
* POTE: ~2 = 1\1, ~11/9 = 339.366


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgg, 32cdg, 39, 46, 99 }}


Badness: 0.019395
Badness: 0.019395
Line 122: 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 (POTE): ~2 = 1\1, ~11/9 = 339.407
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.415
* POTE: ~2 = 1\1, ~11/9 = 339.407


{{Optimal ET sequence|legend=1| 7, 39h, 46, 53, 99h }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}


Badness: 0.017513
Badness: 0.017513
Line 137: 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 (POTE): ~2 = 1\1, ~128/105 = 339.340
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.314
* POTE: ~2 = 1\1, ~128/105 = 339.340


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


Badness: 0.040976
Badness: 0.040976
Line 150: 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 (POTE): ~2 = 1\1, ~128/105 = 339.313
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.277
* POTE: ~2 = 1\1, ~128/105 = 339.313


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cef, 527bccef }}


Badness: 0.034215
Badness: 0.034215
Line 163: 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 (POTE): ~2 = 1\1, ~17/14 = 339.313
Optimal tunings:
* CTE: ~2 = 1\1, ~17/14 = 339.272
* POTE: ~2 = 1\1, ~17/14 = 339.313


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c }}


Badness: 0.021193
Badness: 0.021193
Line 176: 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 (POTE): ~2 = 1\1, ~17/14 = 339.325
Optimal tunings:
* CTE: ~2 = 1\1, ~17/14 = 339.282
* POTE: ~2 = 1\1, ~17/14 = 339.325


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}


Badness: 0.018864
Badness: 0.018864
Line 191: Line 240:
: mapping generators: ~99/70, ~64/55
: mapping generators: ~99/70, ~64/55


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


{{Optimal ET sequence|legend=1| 14cde, 46, 106, 152, 350, 502d }}
Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}


Badness: 0.031307
Badness: 0.031307
Line 204: 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 (POTE): ~99/70 = 1\2, ~64/55 = 260.583
Optimal tunings:
* CTE: ~99/70 = 1\2, ~64/55 = 260.607
* POTE: ~99/70 = 1\2, ~64/55 = 260.583


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


Badness: 0.025784
Badness: 0.025784
Line 217: 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
* 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, 39d, 46, 131c, 177c }}


[[Badness]]: 0.095612
[[Badness]]: 0.095612
Line 232: 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 (POTE): ~2 = 1\1, ~11/9 = 339.047
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.136
* POTE: ~2 = 1\1, ~11/9 = 339.047


{{Optimal ET sequence|legend=1| 7d, 39d, 46, 177c, 223bc, 269bce }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}


Badness: 0.042468
Badness: 0.042468
Line 247: 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]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763
* POTE: ~2 = 1\1, ~243/200 = 339.763


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 113, 166 }}
{{Optimal ET sequence|legend=1| 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}


[[Badness]]: 0.066638
[[Badness]]: 0.066638
Line 262: 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 (POTE): ~2 = 1\1, ~243/200 = 339.763
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.751
* POTE: ~2 = 1\1, ~243/200 = 339.763


{{Optimal ET sequence|legend=1| 53, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}


Badness: 0.067891
Badness: 0.067891
Line 275: 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 (POTE): ~2 = 1\1, ~39/32 = 339.764
Optimal tunings:
* CTE: ~2 = 1\1, ~39/32 = 339.754
* POTE: ~2 = 1\1, ~39/32 = 339.764


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


Badness: 0.032996
Badness: 0.032996
Line 288: 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 (POTE): ~2 = 1\1, ~11/9 = 339.814
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.680
* POTE: ~2 = 1\1, ~11/9 = 339.814


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.045232
Badness: 0.045232
Line 301: 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 (POTE): ~2 = 1\1, ~11/9 = 339.784
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.685
* POTE: ~2 = 1\1, ~11/9 = 339.784


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.031331
Badness: 0.031331
Line 314: Line 377:
[[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 3 6 -17 | 0 -5 -13 70 }}


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


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


[[Badness]]: 0.113655
[[Badness]]: 0.113655
Line 329: 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 (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554


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


Badness: 0.064853
Badness: 0.064853
Line 342: 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 (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554


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


Badness: 0.030347
Badness: 0.030347
Line 355: 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 (POTE): ~2 = 1\1, ~243/200 = 339.555
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.555
* POTE: ~2 = 1\1, ~243/200 = 339.555


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


Badness: 0.024118
Badness: 0.024118
Line 368: 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 (POTE): ~2 = 1\1, ~208/171 = 339.555
Optimal tunings:
* CTE: ~2 = 1\1, ~208/171 = 339.555
* POTE: ~2 = 1\1, ~208/171 = 339.555


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


Badness: 0.017420
Badness: 0.017420
Line 385: 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]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402
* POTE: ~343/240 = 1\2, ~7/6 = 260.402


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60, 106d }}
{{Optimal ET sequence|legend=1| 14c, 32c, 46, 106d, 152d }}


[[Badness]]: 0.083601
[[Badness]]: 0.083601
Line 400: 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 (POTE): ~99/70 = 1\2, ~7/6 = 260.393
Optimal tunings:
* CTE: ~99/70 = 1\2, ~7/6 = 260.653
* POTE: ~99/70 = 1\2, ~7/6 = 260.393


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60e, 106de }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}


Badness: 0.035504
Badness: 0.035504
Line 413: 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 (POTE): ~55/39 = 1\2, ~7/6 = 260.618
Optimal tunings:
* CTE: ~55/39 = 1\2, ~7/6 = 260.811
* POTE: ~55/39 = 1\2, ~7/6 = 260.618


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


Badness: 0.030885
Badness: 0.030885
Line 430: 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]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219
* 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 445: 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 (POTE): ~2 = 1\1, ~9/7 = 430.192
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.220
* POTE: ~2 = 1\1, ~9/7 = 430.192


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


Badness: 0.042947
Badness: 0.042947
Line 458: 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 (POTE): ~2 = 1\1, ~9/7 = 430.216
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.233
* POTE: ~2 = 1\1, ~9/7 = 430.216


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


Badness: 0.029753
Badness: 0.029753
Line 475: 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
* 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 }}


[[Badness]]: 0.125733
[[Badness]]: 0.125733
Line 490: 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 (POTE): ~2 = 1\1, ~33/28 = 286.797
Optimal tunings:
* CTE: ~2 = 1\1, ~33/28 = 286.813
* POTE: ~2 = 1\1, ~33/28 = 286.797


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}


Badness: 0.051111
Badness: 0.051111
Line 503: 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 (POTE): ~2 = 1\1, ~13/11 = 286.789
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.803
* POTE: ~2 = 1\1, ~13/11 = 286.789


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


Badness: 0.030297
Badness: 0.030297
Line 516: 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 (POTE): ~2 = 1\1, ~13/11 = 286.795
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.804
* POTE: ~2 = 1\1, ~13/11 = 286.795


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


Badness: 0.022036
Badness: 0.022036
Line 531: 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
* 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| 152, 311, 463, 774 }}


[[Badness]]: 0.119453
[[Badness]]: 0.119453
Line 546: 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 (POTE): ~2 = 1\1, ~121/90 = 513.177
Optimal tunings:
* CTE: ~2 = 1\1, ~121/90 = 513.181
* POTE: ~2 = 1\1, ~121/90 = 513.177


{{Optimal ET sequence|legend=1| 152, 311, 463, 774, 1237e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}


Badness: 0.031296
Badness: 0.031296
Line 559: 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 (POTE): ~2 = 1\1, ~35/26 = 513.182
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* POTE: ~2 = 1\1, ~35/26 = 513.182


{{Optimal ET sequence|legend=1| 152f, 311 }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}


Badness: 0.026418
Badness: 0.026418
Line 572: 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 (POTE): ~2 = 1\1, ~35/26 = 513.186
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1\1, ~35/26 = 513.186


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}


Badness: 0.025588
Badness: 0.025588
Line 585: 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 (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* POTE: ~2 = 1\1, ~35/26 = 513.185


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


Badness: 0.018412
Badness: 0.018412
Line 598: 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 (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* POTE: ~2 = 1\1, ~35/26 = 513.185


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}


Badness: 0.014343
Badness: 0.014343
Line 611: 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 (POTE): ~2 = 1\1, ~35/26 = 513.186
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1\1, ~35/26 = 513.186


{{Optimal ET sequence|legend=1| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
Optimal ET sequence: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}


Badness: 0.012038
Badness: 0.012038


== Amicable ==
== 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.  
The amicable temperament (99 &amp; 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.  
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 &amp; 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 &amp; 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 &amp; 113) and pseudoamorous (14cf &amp; 99ef) are the corresponding low-complexity interpretations. Floral (198 &amp; 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 &amp; 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 628: 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
* 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| 99, 212, 311, 410, 1131, 1541b }}


[[Badness]]: 0.045473
[[Badness]]: 0.045473
Line 643: 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 (POTE): ~2 = 1\1, ~21/20 = 84.8843
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8843
* POTE: ~2 = 1\1, ~21/20 = 84.8843


{{Optimal ET sequence|legend=1| 99, 212e, 311, 410, 721, 1032, 1343 }}
Optimal ET sequence: {{Optimal ET sequence| 99, 212e, 311, 721, 1032, 1343, 2375bc }}


Badness: 0.100668
Badness: 0.100668
Line 656: 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 (POTE): ~2 = 1\1, ~21/20 = 84.8838
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8837
* POTE: ~2 = 1\1, ~21/20 = 84.8838


{{Optimal ET sequence|legend=1| 99, 212ef, 311, 410, 721, 1032 }}
Optimal ET sequence: {{Optimal ET sequence| 99, 212ef, 311, 721, 1032 }}


Badness: 0.049893
Badness: 0.049893
Line 669: 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 (POTE): ~2 = 1\1, ~21/20 = 84.8896
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8883
* POTE: ~2 = 1\1, ~21/20 = 84.8896


{{Optimal ET sequence|legend=1| 99e, 212, 311, 1145c, 1456cd }}
Optimal ET sequence: {{Optimal ET sequence| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}


Badness: 0.048924
Badness: 0.048924
Line 682: 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 (POTE): ~2 = 1\1, ~21/20 = 84.8910
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8895
* POTE: ~2 = 1\1, ~21/20 = 84.8910


{{Optimal ET sequence|legend=1| 99ef, 212, 311, 834, 1145c }}
Optimal ET sequence: {{Optimal ET sequence| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}


Badness: 0.034681
Badness: 0.034681
Line 695: 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 (POTE): ~2 = 1\1, ~21/20 = 84.9091
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9005
* POTE: ~2 = 1\1, ~21/20 = 84.9091


{{Optimal ET sequence|legend=1| 99, 113, 212, 961ccdeee }}
Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85cee, 99, 212 }}


Badness: 0.085837
Badness: 0.085837
Line 708: 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 (POTE): ~2 = 1\1, ~21/20 = 84.9127
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9049
* POTE: ~2 = 1\1, ~21/20 = 84.9127


{{Optimal ET sequence|legend=1| 99, 113, 212, 537cdeff, 749ccdeefff }}
Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85ceef, 99, 113, 212 }}


Badness: 0.047025
Badness: 0.047025
Line 721: 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 (POTE): ~2 = 1\1, ~21/20 = 84.8917
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9022
* POTE: ~2 = 1\1, ~21/20 = 84.8917


{{Optimal ET sequence|legend=1| 99e, 212e }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 85ce, 99e, 113, 212e }}


Badness: 0.056583
Badness: 0.056583
Line 734: 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 (POTE): ~2 = 1\1, ~21/20 = 84.9164
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9153
* POTE: ~2 = 1\1, ~21/20 = 84.9164


{{Optimal ET sequence|legend=1| 99ef, 113, 212ef }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}


Badness: 0.042826
Badness: 0.042826
Line 747: 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 (POTE): ~99/70 = 1\2, ~21/20 = 84.8788
Optimal tunings:
* CTE: ~99/70 = 1\2, ~21/20 = 84.8781
* POTE: ~99/70 = 1\2, ~21/20 = 84.8788


{{Optimal ET sequence|legend=1| 198, 212, 410 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 170bccde, 184c, 198, 212, 410 }}


Badness: 0.065110
Badness: 0.065110
Line 760: 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 (POTE): ~99/70 = 1\2, ~21/20 = 84.8750
Optimal tunings:
* CTE: ~99/70 = 1\2, ~21/20 = 84.8759
* POTE: ~99/70 = 1\2, ~21/20 = 84.8750


{{Optimal ET sequence|legend=1| 198, 410 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, …, 184cff, 198, 410 }}


Badness: 0.037013
Badness: 0.037013
Line 773: 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 (POTE): ~2 = 1\1, ~4096/3993 = 42.4391
Optimal tunings:
* CTE: ~2 = 1\1, ~4096/3993 = 42.4414
* POTE: ~2 = 1\1, ~4096/3993 = 42.4391


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820 }}
Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1131, 1442, 1753be }}


Badness: 0.058249
Badness: 0.058249
Line 786: 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 (POTE): ~2 = 1\1, ~40/39 = 42.4391
Optimal tunings:
* CTE: ~2 = 1\1, ~40/39 = 42.4425
* POTE: ~2 = 1\1, ~40/39 = 42.4391


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820f }}
Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}


Badness: 0.028267
Badness: 0.028267


== Calamity ==
== 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.  
The calamity temperament (46 &amp; 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
[[Subgroup]]: 2.3.5.7
Line 801: 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 = 547.9092
[[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 }}


[[Badness]]: 0.198
[[Badness]]: 0.198130


=== 11-limit ===
=== 11-limit ===
Line 814: Line 937:
Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.9080
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908


{{Optimal ET sequence|legend=1| 46, 219c, 265, 311, 979, 1290 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 219c, 265, 311, 979, 1290 }}


Badness: 0.0604
Badness: 0.060408


=== 13-limit ===
=== 13-limit ===
Line 827: Line 950:
Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}
Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.9066
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.907


{{Optimal ET sequence|legend=1| 46, 265, 311, 668, 979, 1290 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 265, 311, 668, 979, 1290 }}


Badness: 0.0336
Badness: 0.033617


== Familia ==
== Familia ==
Line 842: 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]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941
* POTE: ~2 = 1\1, ~11907/10000 = 307.941


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}


[[Badness]]: 0.144551
[[Badness]]: 0.144551
Line 857: 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 (POTE): ~2 = 1\1, ~3200/2673 = 307.906
Optimal tunings:
* CTE: ~2 = 1\1, ~3200/2673 = 307.915
* POTE: ~2 = 1\1, ~3200/2673 = 307.906


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 417, 569de, 721de }}
Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}


Badness: 0.051740
Badness: 0.051740
Line 870: 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 (POTE): ~2 = 1\1, ~143/120 = 307.913
Optimal tunings:
* CTE: ~2 = 1\1, ~143/120 = 307.922
* POTE: ~2 = 1\1, ~143/120 = 307.913


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


Badness: 0.038473
Badness: 0.038473


[[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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