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== Amity ==
== Amity ==
{{main| Amity }}
{{Main| Amity }}
{{see also| Ragismic microtemperaments #Amity }}
{{See also| Ragismic microtemperaments #Amity }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 1600000/1594323
[[Comma list]]: 1600000/1594323
Line 13: Line 13:
Mapping generators: ~2, ~243/200
Mapping generators: ~2, ~243/200


[[POTE generator]]: ~243/200 = 339.519
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.519


{{Val list|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
{{Val list|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
Line 19: Line 19:
[[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 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.  


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
* ''[[Amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable]]
* ''[[chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[Chromat]]'', {10976/10935, 235298/234375} → [[Hemimage temperaments #Chromat]]
* ''[[witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]
* ''[[Witch]]'', {420175/419904, 1600000/1594323} → [[Wizmic microtemperaments #Witch]]


== Septimal amity ==
== Septimal amity ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103
Line 36: Line 36:
{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
{{Multival|legend=1| 5 13 -17 9 -41 -76 }}


[[POTE generator]]: ~128/105 = 339.432
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432


{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
{{Val list|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
Line 49: Line 49:
Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]
Mapping: [{{val| 1 3 6 -2 21 }}, {{val| 0 -5 -13 17 -62 }}]


POTE generator: ~128/105 = 339.464
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.464


Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
Optimal GPV sequence: {{Val list| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
Line 62: Line 62:
Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
Mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]


POTE generator: ~128/105 = 339.481
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481


Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Optimal GPV sequence: {{Val list| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Line 77: Line 77:
Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]
Mapping: [{{val| 1 3 6 -2 6 }}, {{val| 0 -5 -13 17 -9 }}]


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


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Line 90: Line 90:
Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
Mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]


POTE generator: ~11/9 = 339.419
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.419


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Line 103: Line 103:
Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]
Mapping: [{{val| 1 3 6 -2 6 2 -1 }}, {{val| 0 -5 -13 17 -9 6 18 }}]


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


Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Optimal GPV sequence: {{Val list| 7, 39, 46, 53, 99 }}
Line 116: Line 116:
Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]
Mapping: [{{val| 1 3 6 -2 6 2 -1 0 }}, {{val| 0 -5 -13 17 -9 6 18 15 }}]


POTE generator: ~11/9 = 339.407
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.407


Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}
Optimal GPV sequence: {{Val list| 7, 39h, 46, 53, 99h }}
Line 123: Line 123:


=== Catamite ===
=== Catamite ===
The ''catamite'' temperament (46&amp;99e) 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."
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
Subgroup: 2.3.5.7.11
Line 129: Line 129:
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: [{{val| 1 3 6 -2 -7 }}, {{val| 0 -5 -13 17 37 }}]


POTE generator: ~128/105 = 339.340
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.340


Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}
Optimal GPV sequence: {{Val list| 46, 99e, 145, 244e }}
Line 142: Line 142:
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: [{{val| 1 3 6 -2 -7 -11 }}, {{val| 0 -5 -13 17 37 52 }}]


POTE generator: ~128/105 = 339.313
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.313


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Line 155: Line 155:
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: [{{val| 1 3 6 -2 -7 -11 -1 }}, {{val| 0 -5 -13 17 37 52 18 }}]


POTE generator: ~17/14 = 339.313
Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.313


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Line 168: Line 168:
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: [{{val| 1 3 6 -2 -7 -11 -1 -13 }}, {{val| 0 -5 -13 17 37 52 18 61 }}]


POTE generator: ~17/14 = 339.325
Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.325


Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Optimal GPV sequence: {{Val list| 46, 99ef, 145 }}
Line 185: Line 185:
Mapping generators: ~99/70, ~64/55
Mapping generators: ~99/70, ~64/55


POTE generator: ~64/55 = 260.561
Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561


Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
Optimal GPV sequence: {{Val list| 14cde, 46, 106, 152, 350, 502d }}
Line 198: Line 198:
Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]
Mapping: [{{val| 2 1 -1 13 13 20 }}, {{val| 0 5 13 -17 -14 -29 }}]


POTE generator: ~64/55 = 260.583
Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583


Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}
Optimal GPV sequence: {{Val list| 46, 106f, 152f, 198, 350f, 548cdff }}
Line 205: Line 205:


== Accord ==
== Accord ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 100352/98415
[[Comma list]]: 126/125, 100352/98415
Line 213: Line 213:
{{Multival|legend=1| 5 13 29 9 32 31 }}
{{Multival|legend=1| 5 13 29 9 32 31 }}


[[POTE generator]]: ~243/200 = 338.993
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 338.993


{{Val list|legend=1| 7d, 39d, 46, 131c, 177c }}
{{Val list|legend=1| 7d, 39d, 46, 131c, 177c }}
Line 226: Line 226:
Mapping: [{{val| 1 3 6 11 6 }}, {{val| 0 -5 -13 -29 -9 }}]
Mapping: [{{val| 1 3 6 11 6 }}, {{val| 0 -5 -13 -29 -9 }}]


POTE generator: ~11/9 = 339.047
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.047


Optimal GPV sequence: {{Val list| 7d, 39d, 46, 177c, 223bc, 269bce }}
Optimal GPV sequence: {{Val list| 7d, 39d, 46, 177c, 223bc, 269bce }}
Line 235: Line 235:
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;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).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 1250000/1240029
[[Comma list]]: 225/224, 1250000/1240029
Line 243: Line 243:
{{Multival|legend=1| 5 13 36 9 43 47 }}
{{Multival|legend=1| 5 13 36 9 43 47 }}


[[POTE generator]]: ~243/200 = 339.763
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763


{{Val list|legend=1| 7d, 46d, 53, 113, 166 }}
{{Val list|legend=1| 7d, 46d, 53, 113, 166 }}
Line 256: Line 256:
Mapping: [{{val| 1 3 6 13 -9 }}, {{val| 0 -5 -13 -36 44 }}]
Mapping: [{{val| 1 3 6 13 -9 }}, {{val| 0 -5 -13 -36 44 }}]


POTE generator: ~243/200 = 339.763
Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.763


Optimal GPV sequence: {{Val list| 53, 113, 166 }}
Optimal GPV sequence: {{Val list| 53, 113, 166 }}
Line 269: Line 269:
Mapping: [{{val| 1 3 6 13 -9 2 }}, {{val| 0 -5 -13 -36 44 6 }}]
Mapping: [{{val| 1 3 6 13 -9 2 }}, {{val| 0 -5 -13 -36 44 6 }}]


POTE generator: ~39/32 = 339.764
Optimal tuning (POTE): ~2 = 1\1, ~39/32 = 339.764


Optimal GPV sequence: {{Val list| 53, 113, 166 }}
Optimal GPV sequence: {{Val list| 53, 113, 166 }}
Line 282: Line 282:
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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.814


Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
Line 295: Line 295:
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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.784


Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
Optimal GPV sequence: {{Val list| 7d, 46d, 53, 60e, 113e }}
Line 302: Line 302:


== 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
Line 310: Line 310:
[[Mapping]]: [{{val|1 3 6 -17}}, {{val|0 -5 -13 70}}]
[[Mapping]]: [{{val|1 3 6 -17}}, {{val|0 -5 -13 70}}]


[[POTE generator]]: ~243/200 = 339.553
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.553


{{Val list|legend=1| 53, 205d, 258, 311, 675, 986, 1297c, 2283bc }}
{{Val list|legend=1| 53, 205d, 258, 311, 675, 986, 1297c, 2283bc }}
Line 321: Line 321:
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: [{{val| 1 3 6 -17 36 }}, {{val| 0 -5 -13 70 -115 }}]


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


Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986 }}
Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986 }}
Line 334: Line 334:
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: [{{val| 1 3 6 -17 36 17 }}, {{val| 0 -5 -13 70 -115 -47 }}]


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


Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986, 1661cf }}
Optimal GPV sequence: {{Val list| 53, 205de, 258, 311, 675, 986, 1661cf }}
Line 347: Line 347:
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: [{{val| 1 3 6 -17 36 17 -31 }}, {{val| 0 -5 -13 70 -115 -47 124 }}]


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


Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
Line 360: Line 360:
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: [{{val| 1 3 6 -17 36 17 -31 15 }}, {{val| 0 -5 -13 70 -115 -47 124 -38 }}]


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


Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
Optimal GPV sequence: {{Val list| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
Line 369: Line 369:


== 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 play on the words ''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
Line 381: Line 381:
{{Multival|legend=1| 10 26 12 18 -9 -45 }}
{{Multival|legend=1| 10 26 12 18 -9 -45 }}


[[POTE generator]]: ~7/6 = 260.402
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 260.402


{{Val list|legend=1| 14c, 32c, 46, 60, 106d }}
{{Val list|legend=1| 14c, 32c, 46, 60, 106d }}
Line 418: Line 418:


== 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 play on the words ''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  
Line 428: Line 428:
{{Multival|legend=1| 10 26 19 18 2 -29 }}
{{Multival|legend=1| 10 26 19 18 2 -29 }}


[[POTE generator]]: ~9/7 = 430.219
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219


{{Val list|legend=1| 14c, 39d, 53 }}
{{Val list|legend=1| 14c, 39d, 53 }}
Line 441: Line 441:
Mapping: [{{val| 1 -2 -7 -4 -3 }}, {{val| 0 10 26 19 18 }}]
Mapping: [{{val| 1 -2 -7 -4 -3 }}, {{val| 0 10 26 19 18 }}]


POTE generator: ~9/7 = 430.192
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.192


Optimal GPV sequence: {{Val list| 14c, 39d, 53 }}
Optimal GPV sequence: {{Val list| 14c, 39d, 53 }}
Line 454: Line 454:
Mapping: [{{val| 1 -2 -7 -4 -3 -11 }}, {{val| 0 10 26 19 18 41 }}]
Mapping: [{{val| 1 -2 -7 -4 -3 -11 }}, {{val| 0 10 26 19 18 41 }}]


POTE generator: ~9/7 = 430.216
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216


Optimal GPV sequence: {{Val list| 14cf, 39df, 53 }}
Optimal GPV sequence: {{Val list| 14cf, 39df, 53 }}
Line 461: Line 461:


== 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
Line 471: Line 471:
{{Multival|legend=1| 15 39 -5 27 -50 -121 }}
{{Multival|legend=1| 15 39 -5 27 -50 -121 }}


[[POTE generator]]: ~189/160 = 286.787
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787


{{Val list|legend=1| 46, 113, 159 }}
{{Val list|legend=1| 46, 113, 159 }}
Line 484: Line 484:
Mapping: [{{val| 1 -2 -7 4 8 }}, {{val| 0 15 39 -5 -19 }}]
Mapping: [{{val| 1 -2 -7 4 8 }}, {{val| 0 15 39 -5 -19 }}]


POTE generator: ~33/28 = 286.797
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 286.797


Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Line 497: Line 497:
Mapping: [{{val| 1 -2 -7 4 8 8 }}, {{val| 0 15 39 -5 -19 -18 }}]
Mapping: [{{val| 1 -2 -7 4 8 8 }}, {{val| 0 15 39 -5 -19 -18 }}]


POTE generator: ~13/11 = 286.789
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.789


Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Line 510: Line 510:
Mapping: [{{val| 1 -2 -7 4 8 8 6 }}, {{val| 0 15 39 -5 -19 -18 -8 }}]
Mapping: [{{val| 1 -2 -7 4 8 8 6 }}, {{val| 0 15 39 -5 -19 -18 -8 }}]


POTE generator: ~13/11 = 286.795
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.795


Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Optimal GPV sequence: {{Val list| 46, 113, 159 }}
Line 517: Line 517:


== Trinity ==
== Trinity ==
The ''trinity'' temperament (152&amp;159) tempers out the [[meter comma]], 703125/702464. It splits the interval of 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 interval of 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


[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


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


{{Multival|legend=1| 15 39 101 27 118 125 }}
{{Multival|legend=1| 15 39 101 27 118 125 }}


[[POTE generator]]: ~168/125 = 513.178
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~168/125 = 513.178


{{Val list|legend=1| 152, 311, 463, 774 }}
{{Val list|legend=1| 152, 311, 463, 774 }}
Line 538: Line 538:
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: [{{val| 1 8 19 46 18 }}, {{val| 0 -15 -39 -101 -34 }}]


POTE generator: ~121/90 = 513.177
POTE generator: ~121/90 = 513.177
Line 551: Line 551:
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: [{{val| 1 8 19 46 18 64 }}, {{val| 0 -15 -39 -101 -34 -141 }}]


POTE generator: ~35/26 = 513.182
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.182


Optimal GPV sequence: {{Val list| 152f, 311 }}
Optimal GPV sequence: {{Val list| 152f, 311 }}
Line 564: Line 564:
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: [{{val| 1 8 19 46 18 64 -22 }}, {{val| 0 -15 -39 -101 -34 -141 61 }}]


POTE generator: ~35/26 = 513.186
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186


Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
Line 577: Line 577:
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: [{{val| 1 8 19 46 18 64 -22 53 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 }}]


POTE generator: ~35/26 = 513.185
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185


Optimal GPV sequence: {{Val list| 152f, 159, 311, 1403cdgh, 1714cdegh, 2025cdefgghh, 2336bccdefgghh }}
Optimal GPV sequence: {{Val list| 152f, 159, 311, 1403cdgh, 1714cdegh, 2025cdefgghh, 2336bccdefgghh }}
Line 590: Line 590:
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: [{{val| 1 8 19 46 18 64 -22 53 49 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 }}]


POTE generator: ~35/26 = 513.185
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185


Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
Optimal GPV sequence: {{Val list| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
Line 603: Line 603:
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: [{{val| 1 8 19 46 18 64 -22 53 49 72 }}, {{val| 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}]


POTE generator: ~35/26 = 513.186
Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186


Optimal GPV sequence: {{Val list| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
Optimal GPV sequence: {{Val list| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
Line 612: Line 612:


== 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
Line 622: Line 622:
{{Multival|legend=1| 25 65 67 45 36 -27 }}
{{Multival|legend=1| 25 65 67 45 36 -27 }}


[[POTE generator]]: ~11907/10000 = 307.941
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941


{{Val list|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
{{Val list|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
Line 635: Line 635:
Mapping: [{{val| 1 8 19 20 5 }}, {{val| 0 -25 -65 -67 -6 }}]
Mapping: [{{val| 1 8 19 20 5 }}, {{val| 0 -25 -65 -67 -6 }}]


POTE generator: ~3200/2673 = 307.906
Optimal tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906


Optimal GPV sequence: {{Val list| 39d, 74cd, 113, 152, 417, 569de, 721de }}
Optimal GPV sequence: {{Val list| 39d, 74cd, 113, 152, 417, 569de, 721de }}
Line 648: Line 648:
Mapping: [{{val| 1 8 19 20 5 25 }}, {{val| 0 -25 -65 -67 -6 -83 }}]
Mapping: [{{val| 1 8 19 20 5 25 }}, {{val| 0 -25 -65 -67 -6 -83 }}]


POTE generator: ~143/120 = 307.913
Optimal tuning (POTE): ~2 = 1\1, ~143/120 = 307.913


Optimal GPV sequence: {{Val list| 39df, 74cdf, 113, 152f, 265, 417f }}
Optimal GPV sequence: {{Val list| 39df, 74cdf, 113, 152f, 265, 417f }}

Revision as of 12:30, 24 January 2023

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 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.

Amity

Main article: Amity
See also: Ragismic microtemperaments § Amity

Subgroup: 2.3.5

Comma list: 1600000/1594323

Mapping: [⟨1 3 6], ⟨0 -5 -13]]

Mapping generators: ~2, ~243/200

Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.519

Template:Val list

Badness: 0.021960

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.

Temperaments discussed elsewhere include:

Septimal amity

Subgroup: 2.3.5.7

Comma list: 4375/4374, 5120/5103

Mapping: [⟨1 3 6 -2], ⟨0 -5 -13 17]]

Wedgie: ⟨⟨ 5 13 -17 9 -41 -76 ]]

Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.432

Template:Val list

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 tuning (POTE): ~2 = 1\1, ~128/105 = 339.464

Optimal GPV sequence: Template:Val list

Badness: 0.031506

13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal GPV sequence: Template:Val list *

* optimal patent val: 205

Badness: 0.028008

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.390

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.419

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.366

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.407

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~128/105 = 339.340

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~128/105 = 339.313

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~17/14 = 339.313

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~17/14 = 339.325

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583

Optimal GPV sequence: Template:Val list

Badness: 0.025784

Accord

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 3 6 11], ⟨0 -5 -13 -29]]

Wedgie: ⟨⟨ 5 13 29 9 32 31 ]]

Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 338.993

Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.047

Optimal GPV sequence: Template:Val list

Badness: 0.042468

Houborizic

The houborizic temperament (53&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).

Subgroup: 2.3.5.7

Comma list: 225/224, 1250000/1240029

Mapping: [⟨1 3 6 13], ⟨0 -5 -13 -36]]

Wedgie: ⟨⟨ 5 13 36 9 43 47 ]]

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

Template:Val list

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 tuning (POTE): ~2 = 1\1, ~243/200 = 339.763

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~39/32 = 339.764

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.814

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~11/9 = 339.784

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~243/200 = 339.553

Template:Val list

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 tuning (POTE): ~2 = 1\1, ~243/200 = 339.554

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~243/200 = 339.554

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~243/200 = 339.555

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~208/171 = 339.555

Optimal GPV sequence: Template:Val list

Badness: 0.017420

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.

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

Wedgie: ⟨⟨ 10 26 12 18 -9 -45 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 260.402

Template:Val list

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]]

Mapping generators: ~99/70, ~7/6

POTE generator: ~7/6 = 260.393

Optimal GPV sequence: Template:Val list

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]]

Mapping generators: ~55/39, ~7/6

POTE generator: ~7/6 = 260.618

Optimal GPV sequence: Template:Val list

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 play on the words half and amity.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 4000/3969

Mapping: [⟨1 -2 -7 -4], ⟨0 10 26 19]]

Wedgie: ⟨⟨ 10 26 19 18 2 -29 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.219

Template:Val list

Badness: 0.073956

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 -4 -3], ⟨0 10 26 19 18]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.042947

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 275/273, 572/567

Mapping: [⟨1 -2 -7 -4 -3 -11], ⟨0 10 26 19 18 41]]

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

Optimal GPV sequence: Template:Val list

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 -2 -7 4], ⟨0 15 39 -5]]

Wedgie: ⟨⟨ 15 39 -5 27 -50 -121 ]]

Optimal tuning (POTE): ~2 = 1\1, ~189/160 = 286.787

Template:Val list

Badness: 0.125733

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -2 -7 4 8], ⟨0 15 39 -5 -19]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.051111

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -2 -7 4 8 8], ⟨0 15 39 -5 -19 -18]]

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

Optimal GPV sequence: Template:Val list

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 -2 -7 4 8 8 6], ⟨0 15 39 -5 -19 -18 -8]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.022036

Trinity

The trinity temperament (152 & 159) tempers out the meter, 703125/702464. It splits the interval of 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]]

Wedgie: ⟨⟨ 15 39 101 27 118 125 ]]

Optimal tuning (POTE): ~2 = 1\1, ~168/125 = 513.178

Template:Val list

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]]

POTE generator: ~121/90 = 513.177

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.182

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.186

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.185

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.185

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~35/26 = 513.186

Optimal GPV sequence: Template:Val list

Badness: 0.012038

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]]

Wedgie: ⟨⟨ 25 65 67 45 36 -27 ]]

Optimal tuning (POTE): ~2 = 1\1, ~11907/10000 = 307.941

Template:Val list

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 tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906

Optimal GPV sequence: Template:Val list

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 tuning (POTE): ~2 = 1\1, ~143/120 = 307.913

Optimal GPV sequence: Template:Val list

Badness: 0.038473

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