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The generator for [[amity]] temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, [[5120/5103]] and 6144/6125. It can also be described as the 46&53 temperament, or by its wedgie, <<5 13 -17 9 -41 -76||. [[99edo|99EDO]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 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).  


Temperaments not discussed here include [[Breedsmic temperaments|amicable]] and [[Hemimage temperaments|chromat]].
== Amity ==
{{Main| Amity }}


= Amity =
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.
{{see also|Ragismic microtemperaments #Amity}}


[[Comma]]: 1600000/1594323
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.


[[Mapping]]: [&lt;1 3 6|, &lt;0 -5 -13|]
[[Subgroup]]: 2.3.5


[[POTE tuning|POTE generator]]: ~243/200 = 339.519
[[Comma list]]: 1600000/1594323


[[EDO|Vals]]: {{Val list| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
{{Mapping|legend=1| 1 3 6 | 0 -5 -13 }}
 
: mapping generators: ~2, ~243/200
 
[[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, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc }}


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


== 7-limit ==
=== Overview to extensions ===
The second comma to extend the 5-limit amity include [[4375/4374]] for septimal amity, [[225/224]] for houborizic, 65625/65536 for paramity, [[126/125]] for accord, [[245/243]] for bamity, [[2430/2401]] for hamity, [[1029/1024]] for gamity, [[10976/10935]] for chromat, [[703125/702464]] for trinity, [[2401/2400]] for amicable, [[2100875/2097152]] for calamity, [[420175/419904]] for witcher, and [[16875/16807]] for familia.
 
Temperaments discussed elsewhere include:
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)
 
The rest are considered below.
 
== Septimal amity ==
{{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]]: [&lt;1 3 6 -2|, &lt;0 -5 -13 17|]
{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}


[[Wedgie]]: &lt;&lt;5 13 -17 9 -41 -76||
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.446
* POTE: ~2 = 1\1, ~128/105 = 339.432


[[POTE tuning|POTE generator]]: ~128/105 = 339.432
{{Optimal ET sequence|legend=1| 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd }}


[[EDO|Vals]]: {{Val list| 7, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
[[Badness]]: 0.023649


[[Badness]]: 0.023649
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


Mapping: [&lt;1 3 6 -2 21|, &lt;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| 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


=== 13-limit ===
==== 13-limit ====
Comma list: 352/351, 540/539, 625/624, 847/845
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 625/624, 729/728


Mapping: [&lt;1 3 6 -2 21 17|, &lt;0 -5 -13 17 -62 -47|]
Mapping: {{mapping| 1 3 6 -2 21 17 | 0 -5 -13 17 -62 -47 }}


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


Vals: {{Val list| 53, 99ef, 152f }}
Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152f, 205 }}


Badness: 0.028008
Badness: 0.028008


== Hitchcock ==
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 375/374, 540/539, 729/728
 
Mapping: {{mapping| 1 3 6 -2 21 17 -1 | 0 -5 -13 17 -62 -47 18 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.496
 
Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152fg, 205gg }}
 
Badness: 0.026201
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455
 
Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~17/14 = 339.501
 
Optimal ET sequence: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}
 
Badness: 0.018782
 
=== Hitchcock ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187


Mapping: [&lt;1 3 6 -2 6|, &lt;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


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


Mapping: [&lt;1 3 6 -2 6 2|, &lt;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


== Hemiamity ==
==== 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 ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 5120/5103
Comma list: 3025/3024, 4375/4374, 5120/5103


Mapping: [&lt;2 1 -1 13 13|, &lt;0 5 13 -17 -14|]
Mapping: {{mapping| 2 1 -1 13 13 | 0 5 13 -17 -14 }}
 
: mapping generators: ~99/70, ~64/55


POTE generator: ~64/55 = 339.439
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


= Accord =
==== 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
[[Comma list]]: 126/125, 100352/98415


[[Mapping]]: [&lt;1 3 6 11|, &lt;0 -5 -13 -29|]
{{Mapping|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}


[[Wedgie]]: &lt;&lt;5 13 29 9 32 31||
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.154
* POTE: ~2 = 1\1, ~243/200 = 338.993


[[POTE tuning|POTE generator]]: ~243/200 = 338.993
{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}


[[EDO|Vals]]: {{Val list| 7d, 46, 131c, 177c }}
[[Badness]]: 0.095612


[[Badness]]: 0.095612
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 121/120, 126/125, 896/891
Comma list: 121/120, 126/125, 896/891


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


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


Vals: {{Val list| 7d, 46, 177c, 223bc, 269bce }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}


Badness: 0.042468
Badness: 0.042468


= Bamity =
== Houborizic ==
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
 
[[Comma list]]: 225/224, 1250000/1240029
 
{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.711
* POTE: ~2 = 1\1, ~243/200 = 339.763
 
{{Optimal ET sequence|legend=1| 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c }}
 
[[Badness]]: 0.066638
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 1250000/1240029
 
Mapping: {{mapping| 1 3 6 13 -9 | 0 -5 -13 -36 44 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.751
* POTE: ~2 = 1\1, ~243/200 = 339.763
 
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}
 
Badness: 0.067891
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 2200/2197
 
Mapping: {{mapping| 1 3 6 13 -9 2 | 0 -5 -13 -36 44 6 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~39/32 = 339.754
* POTE: ~2 = 1\1, ~39/32 = 339.764
 
Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}
 
Badness: 0.032996
 
=== Houbor ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 225/224, 2200/2187
 
Mapping: [{{val| 1 3 6 13 6 }}, {{val| 0 -5 -13 -36 -9 }}]
 
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.680
* POTE: ~2 = 1\1, ~11/9 = 339.814
 
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
 
Badness: 0.045232
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 225/224, 275/273, 325/324
 
Mapping: [{{val| 1 3 6 13 6 2 }}, {{val| 0 -5 -13 -36 -9 6 }}]
 
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.685
* POTE: ~2 = 1\1, ~11/9 = 339.784
 
Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
 
Badness: 0.031331
 
== Paramity ==
The paramity temperament (53 &amp; 311) tempers out the horwell comma (65625/65536) and [[garischisma]] (33554432/33480783).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 1600000/1594323
 
{{Mapping|legend=1| 1 3 6 -17 | 0 -5 -13 70 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.553
 
{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986 }}
 
[[Badness]]: 0.113655
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 6250/6237, 19712/19683, 41503/41472
 
Mapping: {{mapping| 1 3 6 -17 36 | 0 -5 -13 70 -115 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554
 
Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986 }}
 
Badness: 0.064853
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
 
Mapping: {{mapping| 1 3 6 -17 36 17 | 0 -5 -13 70 -115 -47 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1\1, ~243/200 = 339.554
 
Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986, 1661cf }}
 
Badness: 0.030347
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
 
Mapping: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.555
* POTE: ~2 = 1\1, ~243/200 = 339.555
 
Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
 
Badness: 0.024118
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
 
Mapping: {{mapping| 1 3 6 -17 36 17 -31 15 | 0 -5 -13 70 -115 -47 124 -38 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~208/171 = 339.555
* POTE: ~2 = 1\1, ~208/171 = 339.555
 
Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
 
Badness: 0.017420
 
== Bamity ==
Bamity has a period of half octave and tempers out the sensamagic comma, [[245/243]]. The name ''bamity'' is a contraction of ''bi-'' and ''amity''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 64827/64000
[[Comma list]]: 245/243, 64827/64000


[[Mapping]]: [&lt;2 1 -1 3|, &lt;0 5 13 6|]
{{Mapping|legend=1| 2 1 -1 3 | 0 5 13 6 }}


[[Wedgie]]: &lt;&lt;10 26 12 18 -9 -45||
: mapping generators: ~343/240, ~7/6


[[POTE tuning|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


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


[[Badness]]: 0.083601
[[Badness]]: 0.083601


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 245/243, 441/440
Comma list: 121/120, 245/243, 441/440


Mapping: [&lt;2 1 -1 3 3|, &lt;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, 46, 60e, 106de }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}


Badness: 0.035504
Badness: 0.035504


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 245/243, 441/440
Comma list: 91/90, 121/120, 245/243, 441/440


Mapping: [&lt;2 1 -1 3 3 0|, &lt;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| 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 contraction of ''half'' and ''amity''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


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


[[Wedgie]]: &lt;&lt;10 26 19 18 2 -29||
: mapping generators: ~2, ~14/9


[[POTE tuning|POTE generator]]: ~9/7 = 430.219
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.168
* POTE: ~2 = 1\1, ~9/7 = 430.219


[[EDO|Vals]]: {{Val list| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}


[[Badness]]: 0.073956
[[Badness]]: 0.073956


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: [&lt;1 8 19 15 15|, &lt;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


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 121/120, 275/273, 572/567
Comma list: 99/98, 121/120, 275/273, 572/567


Mapping: [&lt;1 8 19 15 15 30|, &lt;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, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}


Badness: 0.029753
Badness: 0.029753


[[Category:Theory]]
== Gamity ==
[[Category:Temperament family]]
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.
[[Category:Amity]]
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 1071875/1062882
 
{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
 
: mapping generators: ~2, ~320/189
 
Optimal tunings:
* CTE: ~2 = 1\1, ~189/160 = 286.816
* POTE: ~2 = 1\1, ~189/160 = 286.787
 
{{Optimal ET sequence|legend=1| 46, 113, 159, 205d, 364d }}
 
[[Badness]]: 0.125733
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 1071875/1062882
 
Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~33/28 = 286.813
* POTE: ~2 = 1\1, ~33/28 = 286.797
 
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}
 
Badness: 0.051111
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 385/384, 10985/10976
 
Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.803
* POTE: ~2 = 1\1, ~13/11 = 286.789
 
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
 
Badness: 0.030297
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
 
Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.804
* POTE: ~2 = 1\1, ~13/11 = 286.795
 
Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
 
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 ==
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
 
[[Comma list]]: 16875/16807, 1600000/1594323
 
{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~11907/10000 = 307.915
* POTE: ~2 = 1\1, ~11907/10000 = 307.941
 
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd }}
 
[[Badness]]: 0.144551
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 1600000/1594323
 
Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}
 
Optimal tunings:
* CTE: ~2 = 1\1, ~3200/2673 = 307.915
* POTE: ~2 = 1\1, ~3200/2673 = 307.906
 
Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
 
Badness: 0.051740
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
 
Mapping: {{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| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Amity_family"