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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 10: Line 13:
[[Comma list]]: 1600000/1594323
[[Comma list]]: 1600000/1594323


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


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 amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, 2401/2400 for amicable, and 16875/16807 for familia.  
The second comma to extend 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]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
 
The rest are considered below.


== Septimal amity ==
== Septimal amity ==
Line 36: Line 43:
[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103


[[Mapping]]: [{{val| 1 3 6 -2 }}, {{val| 0 -5 -13 17 }}]
{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
 
{{Multival|legend=1| 5 13 -17 9 -41 -76 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.446
* 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 51: Line 58:
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


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


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
Badness: 0.026201


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}


Badness: 0.028008
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501
 
Optimal ET sequence: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}
 
Badness: 0.018782


=== Hitchcock ===
=== Hitchcock ===
Line 79: Line 114:
Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187


Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
Mapping: {{mapping| 1 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 92: Line 129:
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
Mapping: {{mapping| 1 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 105: Line 144:
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272


Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Mapping: {{mapping| 1 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 118: Line 159:
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189


Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
Mapping: {{mapping| 1 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 133: Line 176:
Comma list: 441/440, 896/891, 4375/4374
Comma list: 441/440, 896/891, 4375/4374


Mapping: [{{val| 1 3 6 -2 -7 }}, {{val| 0 -5 -13 17 37 }}]
Mapping: {{mapping| 1 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 146: Line 191:
Comma list: 196/195, 352/351, 364/363, 4375/4374
Comma list: 196/195, 352/351, 364/363, 4375/4374


Mapping: [{{val| 1 3 6 -2 -7 -11 }}, {{val| 0 -5 -13 17 37 52 }}]
Mapping: {{mapping| 1 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 159: Line 206:
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155


Mapping: [{{val| 1 3 6 -2 -7 -11 -1 }}, {{val| 0 -5 -13 17 37 52 18 }}]
Mapping: {{mapping| 1 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 172: Line 221:
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475


Mapping: [{{val| 1 3 6 -2 -7 -11 -1 -13 }}, {{val| 0 -5 -13 17 37 52 18 61 }}]
Mapping: {{mapping| 1 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 185: Line 236:
Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 3025/3024, 4375/4374, 5120/5103


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


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


Optimal 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 200: Line 253:
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Comma list: 352/351, 847/845, 1716/1715, 3025/3024


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


Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583
Optimal tunings:
* 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 213: Line 268:
[[Comma list]]: 126/125, 100352/98415
[[Comma list]]: 126/125, 100352/98415


[[Mapping]]: [{{val| 1 3 6 11 }}, {{val| 0 -5 -13 -29 }}]
{{Mapping|legend=1| 1 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 228: Line 283:
Comma list: 121/120, 126/125, 896/891
Comma list: 121/120, 126/125, 896/891


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


== Houborizic ==
== Houborizic ==
The ''houborizic'' temperament (53&amp;113) tempers out the [[marvel comma]], 225/224. It is so named because it is closely related to the '''houboriz tuning''' (generator: 339.774971 cents).
The ''houborizic'' temperament (53 &amp; 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
[[Subgroup]]: 2.3.5.7
Line 243: Line 300:
[[Comma list]]: 225/224, 1250000/1240029
[[Comma list]]: 225/224, 1250000/1240029


[[Mapping]]: [{{val| 1 3 6 13 }}, {{val| 0 -5 -13 -36 }}]
{{Mapping|legend=1| 1 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
* 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, 46d, 53, 113, 166 }}


[[Badness]]: 0.066638
[[Badness]]: 0.066638
Line 258: Line 315:
Comma list: 225/224, 385/384, 1250000/1240029
Comma list: 225/224, 385/384, 1250000/1240029


Mapping: [{{val| 1 3 6 13 -9 }}, {{val| 0 -5 -13 -36 44 }}]
Mapping: {{mapping| 1 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 271: Line 330:
Comma list: 225/224, 325/324, 385/384, 2200/2197
Comma list: 225/224, 325/324, 385/384, 2200/2197


Mapping: [{{val| 1 3 6 13 -9 2 }}, {{val| 0 -5 -13 -36 44 6 }}]
Mapping: {{mapping| 1 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 286: 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 299: 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 312: Line 377:
[[Comma list]]: 65625/65536, 1600000/1594323
[[Comma list]]: 65625/65536, 1600000/1594323


[[Mapping]]: [{{val|1 3 6 -17}}, {{val|0 -5 -13 70}}]
{{Mapping|legend=1| 1 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 325: Line 392:
Comma list: 6250/6237, 19712/19683, 41503/41472
Comma list: 6250/6237, 19712/19683, 41503/41472


Mapping: [{{val| 1 3 6 -17 36 }}, {{val| 0 -5 -13 70 -115 }}]
Mapping: {{mapping| 1 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 338: Line 407:
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683


Mapping: [{{val| 1 3 6 -17 36 17 }}, {{val| 0 -5 -13 70 -115 -47 }}]
Mapping: {{mapping| 1 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 351: Line 422:
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430


Mapping: [{{val| 1 3 6 -17 36 17 -31 }}, {{val| 0 -5 -13 70 -115 -47 124 }}]
Mapping: {{mapping| 1 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 364: Line 437:
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197


Mapping: [{{val| 1 3 6 -17 36 17 -31 15 }}, {{val| 0 -5 -13 70 -115 -47 124 -38 }}]
Mapping: {{mapping| 1 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


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 379: Line 454:
[[Comma list]]: 245/243, 64827/64000
[[Comma list]]: 245/243, 64827/64000


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


{{Multival|legend=1| 10 26 12 18 -9 -45 }}
: mapping generators: ~343/240, ~7/6


[[Optimal tuning]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402
Optimal tunings:
* CTE: ~343/240 = 1\2, ~7/6 = 260.563
* 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 396: Line 471:
Comma list: 121/120, 245/243, 441/440
Comma list: 121/120, 245/243, 441/440


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


Optimal tuning (POTE): ~99/70 = 1\2, ~7/6 = 260.393
Optimal tunings:
* 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 409: Line 486:
Comma list: 91/90, 121/120, 245/243, 441/440
Comma list: 91/90, 121/120, 245/243, 441/440


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


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


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 424: Line 503:
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


[[Mapping]]: [{{val| 1 -2 -7 -4 }}, {{val| 0 10 26 19 }}]
{{Mapping|legend=1| 1 8 19 15 | 0 -10 -26 -19 }}


{{Multival|legend=1| 10 26 19 18 2 -29 }}
: mapping generators: ~2, ~14/9


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.168
* 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 439: Line 520:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: [{{val| 1 -2 -7 -4 -3 }}, {{val| 0 10 26 19 18 }}]
Mapping: {{mapping| 1 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 452: Line 535:
Comma list: 99/98, 121/120, 275/273, 572/567
Comma list: 99/98, 121/120, 275/273, 572/567


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


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216
Optimal tunings:
* 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 467: Line 552:
[[Comma list]]: 1029/1024, 1071875/1062882
[[Comma list]]: 1029/1024, 1071875/1062882


[[Mapping]]: [{{val| 1 -2 -7 4 }}, {{val| 0 15 39 -5 }}]
{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}


{{Multival|legend=1| 15 39 -5 27 -50 -121 }}
: mapping generators: ~2, ~320/189


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787
Optimal tunings:
* CTE: ~2 = 1\1, ~189/160 = 286.816
* POTE: ~2 = 1\1, ~189/160 = 286.787


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


[[Badness]]: 0.125733
[[Badness]]: 0.125733
Line 482: Line 569:
Comma list: 385/384, 441/440, 1071875/1062882
Comma list: 385/384, 441/440, 1071875/1062882


Mapping: [{{val| 1 -2 -7 4 8 }}, {{val| 0 15 39 -5 -19 }}]
Mapping: {{mapping| 1 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 495: Line 584:
Comma list: 325/324, 364/363, 385/384, 10985/10976
Comma list: 325/324, 364/363, 385/384, 10985/10976


Mapping: [{{val| 1 -2 -7 4 8 8 }}, {{val| 0 15 39 -5 -19 -18 }}]
Mapping: {{mapping| 1 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 508: Line 599:
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757


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


== Trinity ==
== Trinity ==
The ''trinity'' temperament (152 &amp; 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.
The trinity temperament (152 &amp; 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
[[Subgroup]]: 2.3.5.7
Line 523: Line 616:
[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


[[Mapping]]: [{{val| 1 8 19 46 }}, {{val| 0 -15 -39 -101 }}]
{{Mapping|legend=1| 1 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 538: Line 631:
Comma list: 3025/3024, 4000/3993, 19712/19683
Comma list: 3025/3024, 4000/3993, 19712/19683


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


POTE generator: ~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 551: Line 646:
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689


Mapping: [{{val| 1 8 19 46 18 64 }}, {{val| 0 -15 -39 -101 -34 -141 }}]
Mapping: {{mapping| 1 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 564: Line 661:
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619


Mapping: [{{val| 1 8 19 46 18 64 -22 }}, {{val| 0 -15 -39 -101 -34 -141 61 }}]
Mapping: {{mapping| 1 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 577: Line 676:
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573


Mapping: [{{val| 1 8 19 46 18 64 -22 53 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 }}]
Mapping: {{mapping| 1 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 590: Line 691:
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104


Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 }}]
Mapping: {{mapping| 1 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 603: Line 706:
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044


Mapping: [{{val| 1 8 19 46 18 64 -22 53 49 72 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}]
Mapping: {{mapping| 1 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 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


[[Comma list]]: 2401/2400, 1600000/1594323
[[Comma list]]: 2401/2400, 1600000/1594323


[[Mapping]]: [{{val| 1 3 6 5 }}, {{val| 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 tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 84.880
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8831
* POTE: ~2 = 1\1, ~21/20 = 84.880


{{Optimal ET sequence|legend=1| 99, 212, 311, 410, 1131, 1541b }}
{{Optimal ET sequence|legend=1| 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b }}


[[Badness]]: 0.045473
[[Badness]]: 0.045473
Line 635: Line 740:
Comma list: 2401/2400, 131072/130977, 1600000/1594323
Comma list: 2401/2400, 131072/130977, 1600000/1594323


Mapping: [{{val| 1 3 6 5 -8 }}, {{val| 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 648: Line 755:
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206


Mapping: [{{val| 1 3 6 5 -8 -5 }}, {{val| 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 661: Line 770:
Comma list: 2401/2400, 6250/6237, 19712/19683
Comma list: 2401/2400, 6250/6237, 19712/19683


Mapping: [{{val| 1 3 6 5 14 }}, {{val| 0 -20 -52 -31 -149 }}]
Mapping: {{mapping| 1 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 674: Line 785:
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647


Mapping: [{{val| 1 3 6 5 14 17 }}, {{val| 0 -20 -52 -31 -149 -188 }}]
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 687: Line 800:
Comma list: 385/384, 1375/1372, 1600000/1594323
Comma list: 385/384, 1375/1372, 1600000/1594323


Mapping: [{{val| 1 3 6 5 -1 }}, {{val| 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 700: Line 815:
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Comma list: 325/324, 385/384, 1375/1372, 19773/19712


Mapping: [{{val| 1 3 6 5 -1 2 }}, {{val| 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 713: Line 830:
Comma list: 243/242, 441/440, 980000/970299
Comma list: 243/242, 441/440, 980000/970299


Mapping: [{{val| 1 3 6 5 7 }}, {{val| 0 -20 -52 -31 -50 }}]
Mapping: {{mapping| 1 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 726: Line 845:
Comma list: 243/242, 364/363, 441/440, 1875/1859
Comma list: 243/242, 364/363, 441/440, 1875/1859


Mapping: [{{val| 1 3 6 5 7 10 }}, {{val| 0 -20 -52 -31 -50 -89 }}]
Mapping: {{mapping| 1 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 739: Line 860:
Comma list: 2401/2400, 9801/9800, 14641/14580
Comma list: 2401/2400, 9801/9800, 14641/14580


Mapping: [{{val| 2 6 12 10 13 }}, {{val| 0 -20 -52 -31 -43 }}]
Mapping: {{mapping| 2 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 752: Line 875:
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580


Mapping: [{{val| 2 6 12 10 13 19 }}, {{val| 0 -20 -52 -31 -43 -82 }}]
Mapping: {{mapping| 2 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 765: Line 890:
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Comma list: 2401/2400, 3025/3024, 1600000/1594323


Mapping: [{{val| 1 3 6 5 3 }}, {{val| 0 -40 -104 -62 13 }}]
Mapping: {{mapping| 1 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 778: Line 905:
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024


Mapping: [{{val| 1 3 6 5 3 6 }}, {{val| 0 -40 -104 -62 13 -65 }}]
Mapping: {{mapping| 1 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 ==
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
[[Comma list]]: 1600000/1594323, 2100875/2097152
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
[[Optimal tuning]] (CTE): ~2 = 1\1, ~48/35 = 547.909
{{Optimal ET sequence|legend=1| 46, 219c, 265, 311 }}
[[Badness]]: 0.198130
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 12005/11979, 131072/130977
Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908
Optimal ET sequence: {{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: {{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: {{Optimal ET sequence| 46, 265, 311, 668, 979, 1290 }}
Badness: 0.033617


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 793: Line 963:
[[Comma list]]: 16875/16807, 1600000/1594323
[[Comma list]]: 16875/16807, 1600000/1594323


[[Mapping]]: [{{val| 1 8 19 20 }}, {{val| 0 -25 -65 -67 }}]
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
 
{{Multival|legend=1| 25 65 67 45 36 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941
Optimal tunings:
* CTE: ~2 = 1\1, ~11907/10000 = 307.915
* 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 808: Line 978:
Comma list: 540/539, 1375/1372, 1600000/1594323
Comma list: 540/539, 1375/1372, 1600000/1594323


Mapping: [{{val| 1 8 19 20 5 }}, {{val| 0 -25 -65 -67 -6 }}]
Mapping: {{mapping| 1 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 821: Line 993:
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
Comma list: 540/539, 729/728, 1375/1372, 2205/2197


Mapping: [{{val| 1 8 19 20 5 25 }}, {{val| 0 -25 -65 -67 -6 -83 }}]
Mapping: {{mapping| 1 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]]
Retrieved from "https://en.xen.wiki/w/Amity_family"