Xenharmonic Wiki

Amity family: Difference between revisions

← Older edit
VisualWikitext
+mapping generators
 
(24 intermediate revisions by 7 users not shown)
Line 1: Line 1:
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. It can also be described as the 46&53 temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo|99EDO]] is a good tuning for amity, with generator 28\99, and MOS of 11, 18, 25, 32, 39, 46 or 53 notes are available. 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 }}
{{see also| Ragismic microtemperaments #Amity }}


Subgroup: 2.3.5
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


[[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


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


{{Val list|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


=== 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, 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:  
* ''[[amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
 
The rest are considered below.


== Septimal amity ==
== Septimal amity ==
Subgroup: 2.3.5.7
{{Main| Amity }}
 
Septimal amity can be described as the 46 &amp; 53 temperament, which tempers out [[4375/4374]] and [[5120/5103]] in the 7-limit. [[99edo]] is a good tuning, with generator 28\99.
 
[[Subgroup]]: 2.3.5.7


[[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 }}


[[POTE generator]]: ~128/105 = 339.432
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.446
* POTE: ~2 = 1\1, ~128/105 = 339.432


{{Val list|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 46: 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 }}


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


Vals: {{Val list| 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 57: 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 }}
 
Badness: 0.026201
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455


POTE generator: ~128/105 = 339.481
Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}


Vals: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
Optimal ET sequence: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}


Badness: 0.028008
Badness: 0.018782


=== Hitchcock ===
=== Hitchcock ===
Line 74: 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 }}


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


Vals: {{Val list| 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 87: 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 }}


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


Vals: {{Val list| 7, 39, 46, 53, 99 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}


Badness: 0.022448
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: {{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: {{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: {{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: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
Badness: 0.017513
=== Catamite ===
The catamite temperament (46 &amp; 99ef) tempers out 441/440 (werckisma) and 896/891 (pentacircle) in the 11-limit; 196/195, 352/351 and 364/363 in the 13-limit. The word "catamite" itself is a term for male homosexual, but also a play on the words "cata-" (down) and "amity."
Subgroup: 2.3.5.7.11
Comma list: 441/440, 896/891, 4375/4374
Mapping: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}
Badness: 0.018864


=== Hemiamity ===
=== Hemiamity ===
Line 100: 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


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


Vals: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}


Badness: 0.031307
Badness: 0.031307
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1716/1715, 3025/3024
Mapping: {{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: {{Optimal ET sequence| 46, 106f, 152f, 198 }}
Badness: 0.025784
== Accord ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 100352/98415
{{Mapping|legend=1| 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|legend=1| 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: {{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: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}
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


[[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 }}


[[POTE generator]]: ~243/200 = 339.763
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.711
* POTE: ~2 = 1\1, ~243/200 = 339.763


{{Val list|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 132: 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 }}


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


Vals: {{Val list| 53, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}


Badness: 0.067891
Badness: 0.067891
Line 145: 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 }}


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


Vals: {{Val list| 53, 113, 166 }}
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}


Badness: 0.032996
Badness: 0.032996
Line 158: Line 345:
Comma list: 121/120, 225/224, 2200/2187
Comma list: 121/120, 225/224, 2200/2187


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


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


Vals: {{Val list| 7d, 46d, 53, 60e, 113e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.045232
Badness: 0.045232
Line 171: Line 360:
Comma list: 121/120, 225/224, 275/273, 325/324
Comma list: 121/120, 225/224, 275/273, 325/324


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


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


Vals: {{Val list| 7d, 46d, 53, 60e, 113e }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.031331
Badness: 0.031331


== Paramity ==
== Paramity ==
The ''paramity'' temperament (53&amp;311) tempers out the horwell comma (65625/65536) and [[garischisma]] (33554432/33480783).
The paramity temperament (53 &amp; 311) tempers out the horwell comma (65625/65536) and [[garischisma]] (33554432/33480783).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[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 }}


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


{{Val list|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 199: 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 }}


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


Vals: {{Val list| 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 212: 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 }}


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


Vals: {{Val list| 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 225: 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 }}


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


Vals: {{Val list| 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 238: 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 }}


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


Vals: {{Val list| 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
== Accord ==
Subgroup: 2.3.5.7
[[Comma list]]: 126/125, 100352/98415
[[Mapping]]: [{{val| 1 3 6 11 }}, {{val| 0 -5 -13 -29 }}
{{Multival|legend=1| 5 13 29 9 32 31 }}
[[POTE generator]]: ~243/200 = 338.993
{{Val list|legend=1| 7d, 39d, 46, 131c, 177c }}
[[Badness]]: 0.095612
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 126/125, 896/891
Mapping: [{{val| 1 3 6 11 6 }}, {{val| 0 -5 -13 -29 -9 }}
POTE generator: ~11/9 = 339.047
Vals: {{Val list| 7d, 39d, 46, 177c, 223bc, 269bce }}
Badness: 0.042468


== 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


[[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 }}


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


[[POTE generator]]: ~7/6 = 260.402
Optimal tunings:
* CTE: ~343/240 = 1\2, ~7/6 = 260.563
* POTE: ~343/240 = 1\2, ~7/6 = 260.402


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


[[Badness]]: 0.083601
[[Badness]]: 0.083601
Line 296: 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 }}


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


Vals: {{Val list| 14c, 32c, 46, 60e, 106de }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}


Badness: 0.035504
Badness: 0.035504
Line 309: 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 }}


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


Vals: {{Val list| 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 "acute minor tenth".
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


[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


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


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
* POTE: ~2 = 1\1, ~9/7 = 430.219


[[POTE generator]]: ~9/7 = 430.219
{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
 
{{Val list|legend=1| 14c, 39d, 53 }}


[[Badness]]: 0.073956
[[Badness]]: 0.073956
Line 341: Line 520:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: [{{val| 1 8 19 15 15 }}, {{val| 0 -10 -26 -19 -18 }}
Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}


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


Vals: {{Val list| 14c, 39d, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 39d, 53 }}


Badness: 0.042947
Badness: 0.042947
Line 354: 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 8 19 15 15 30 }}, {{val| 0 -10 -26 -19 -18 -41 }}
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}


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


Vals: {{Val list| 14cf, 39df, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}


Badness: 0.029753
Badness: 0.029753


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[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 }}


Mapping generators: ~2, ~189/160
: 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


[[POTE generator]]: ~189/160 = 286.787
{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
 
{{Val list|legend=1| 46, 113, 159 }}


[[Badness]]: 0.125733
[[Badness]]: 0.125733
Line 386: 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 }}


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


Vals: {{Val list| 46, 113, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}


Badness: 0.051111
Badness: 0.051111
Line 399: 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 }}


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


Vals: {{Val list| 46, 113, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}


Badness: 0.030297
Badness: 0.030297
Line 412: 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 }}


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


Vals: {{Val list| 46, 113, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}


Badness: 0.022036
Badness: 0.022036
== 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 703125/702464, 1600000/1594323
{{Mapping|legend=1| 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|legend=1| 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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
Badness: 0.012038
== Amicable ==
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 &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
[[Comma list]]: 2401/2400, 1600000/1594323
{{Mapping|legend=1| 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|legend=1| 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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{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: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
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


[[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 }}
 
Mapping generators: ~2, ~11907/10000
 
{{Multival|legend=1| 25 65 67 45 36 -27 }}


[[POTE generator]]: ~11907/10000 = 307.941
Optimal tunings:
* CTE: ~2 = 1\1, ~11907/10000 = 307.915
* POTE: ~2 = 1\1, ~11907/10000 = 307.941


{{Val list|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 444: 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 }}


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


Vals: {{Val list| 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


[[Category:Regular temperament theory]]
=== 13-limit ===
[[Category:Temperament family]]
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
 
Mapping: {{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: {{Optimal ET sequence| 39df, 74cdf, 113, 152f, 265 }}
 
Badness: 0.038473
 
[[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"