Amity family: Difference between revisions

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{{Main| Amity }}
{{Main| 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.  
The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[syntonic 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.  
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.  
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{{Mapping|legend=1| 1 3 6 | 0 -5 -13 }}
{{Mapping|legend=1| 1 3 6 | 0 -5 -13 }}
: mapping generators: ~2, ~243/200
: mapping generators: ~2, ~243/200


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~243/200 = 339.537
* [[CTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.537{{c}}
* [[POTE]]: ~2 = 1\1, ~243/200 = 339.519
* [[POTE]]: ~2 = 1200.000{{c}}, ~243/200 = 339.519{{c}}


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


[[Badness]]: 0.021960
[[Badness]] (Smith): 0.021960


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}


Optimal tunings:
[[Optimal tuning]]s:
* CTE: ~2 = 1\1, ~128/105 = 339.446
* [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.446{{c}}
* POTE: ~2 = 1\1, ~128/105 = 339.432
* [[POTE]]: ~2 = 1200.000{{c}}, ~128/105 = 339.432{{c}}


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


[[Badness]]: 0.023649
[[Badness]] (Smith): 0.023649


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.485
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.485{{c}}
* POTE: ~2 = 1\1, ~128/105 = 339.464
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.464{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}
{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 357d, 509dd, 661dd }}


Badness: 0.031506
Badness (Smith): 0.031506


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.508
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.508{{c}}
* POTE: ~2 = 1\1, ~128/105 = 339.481
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.481{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152f, 205 }}
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f, 205 }}


Badness: 0.028008
Badness (Smith): 0.028008


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 3 6 -2 21 17 -1 | 0 -5 -13 17 -62 -47 18 }}
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 tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.496{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46ef, 53, 99ef, 152fg, 205gg }}
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg, 205gg }}


Badness: 0.026201
Badness (Smith): 0.026201


==== 19-limit ====
==== 19-limit ====
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Mapping: {{mapping| 1 3 6 -2 21 17 -1 15 | 0 -5 -13 17 -62 -47 18 -38 }}
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 tuning (CTE): ~2 = 1200.000{{c}}, ~17/14 = 339.501{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46efh, 53, 99ef, 152fg, 205gg }}
{{Optimal ET sequence|legend=0| 46efh, 53, 99ef, 152fg, 205gg }}


Badness: 0.018782
Badness (Smith): 0.018782


=== Hitchcock ===
=== Hitchcock ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.390
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.390
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.390{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25cdde, 32cd, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 25cdde, 32cd, 39, 46, 53, 99 }}


Badness: 0.035187
Badness (Smith): 0.035187


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.411
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.411{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.419
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.419{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddef, 32cd, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 25cddef, 32cd, 39, 46, 53, 99 }}


Badness: 0.022448
Badness (Smith): 0.022448


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.366
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.366
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.366{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgg, 32cdg, 39, 46, 99 }}
{{Optimal ET sequence|legend=0| 7, 25cddefgg, 32cdg, 39, 46, 99 }}


Badness: 0.019395
Badness (Smith): 0.019395


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.415
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.415{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.407
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.407{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}
{{Optimal ET sequence|legend=0| 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h }}


Badness: 0.017513
Badness (Smith): 0.017513


=== Stalagmite ===
=== Stalagmite ===
The stalagmite 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. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is ''down'' from amity.  
The stalagmite 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. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is ''down'' from amity.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.314
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.314{{c}}
* POTE: ~2 = 1\1, ~128/105 = 339.340
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.340{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 99e, 145 }}
{{Optimal ET sequence|legend=0| 46, 99e, 145 }}


Badness: 0.040976
Badness (Smith): 0.040976


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~128/105 = 339.277
* CTE: ~2 = 1200.000{{c}}, ~128/105 = 339.277{{c}}
* POTE: ~2 = 1\1, ~128/105 = 339.313
* POTE: ~2 = 1200.000{{c}}, ~128/105 = 339.313{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cef, 527bccef }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cef, 527bccef }}


Badness: 0.034215
Badness (Smith): 0.034215


==== 17-limit ====
==== 17-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~17/14 = 339.272
* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.272{{c}}
* POTE: ~2 = 1\1, ~17/14 = 339.313
* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.313{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c }}


Badness: 0.021193
Badness (Smith): 0.021193


==== 19-limit ====
==== 19-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~17/14 = 339.282
* CTE: ~2 = 1200.000{{c}}, ~17/14 = 339.282{{c}}
* POTE: ~2 = 1\1, ~17/14 = 339.325
* POTE: ~2 = 1200.000{{c}}, ~17/14 = 339.325{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 99ef, 145, 191c, 336cefg }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145, 191c, 336cefg }}


Badness: 0.018864
Badness (Smith): 0.018864


=== Hemiamity ===
=== Hemiamity ===
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Mapping: {{mapping| 2 1 -1 13 13 | 0 5 13 -17 -14 }}
Mapping: {{mapping| 2 1 -1 13 13 | 0 5 13 -17 -14 }}
: mapping generators: ~99/70, ~64/55
: mapping generators: ~99/70, ~64/55


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 1\2, ~64/55 = 260.566
* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.566{{c}}
* POTE: ~99/70 = 1\2, ~64/55 = 260.561
* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.561{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cde, 32cde, 46, 106, 152, 350 }}
{{Optimal ET sequence|legend=0| 14cde, 32cde, 46, 106, 152, 350 }}


Badness: 0.031307
Badness (Smith): 0.031307


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 1\2, ~64/55 = 260.607
* CTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.607{{c}}
* POTE: ~99/70 = 1\2, ~64/55 = 260.583
* POTE: ~99/70 = 600.000{{c}}, ~64/55 = 260.583{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 106f, 152f, 198 }}
{{Optimal ET sequence|legend=0| 46, 106f, 152f, 198 }}


Badness: 0.025784
Badness (Smith): 0.025784


== Accord ==
== Accord ==
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.154
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.154{{c}}
* POTE: ~2 = 1\1, ~243/200 = 338.993
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 338.993{{c}}


{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}
{{Optimal ET sequence|legend=1| 7d, 25cddd, 32cdd, 39d, 46 }}


[[Badness]]: 0.095612
[[Badness]] (Smith): 0.095612


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.136
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.136{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.047
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.047{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 25cddde, 32cdd, 39d, 46 }}
{{Optimal ET sequence|legend=0| 7d, 25cddde, 32cdd, 39d, 46 }}


Badness: 0.042468
Badness (Smith): 0.042468


== Houborizic ==
== 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).
Houborizic tempers out 225/224, the [[marvel comma]], and may be described as the {{nowrap| 53 & 60 }} temperament. 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
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.711
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.711{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.763
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{c}}


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


[[Badness]]: 0.066638
[[Badness]] (Smith): 0.066638


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.751
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.751{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.763
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.763{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166, 551ccee }}
{{Optimal ET sequence|legend=0| 53, 113, 166, 551ccee }}


Badness: 0.067891
Badness (Smith): 0.067891


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~39/32 = 339.754
* CTE: ~2 = 1200.000{{c}}, ~39/32 = 339.754{{c}}
* POTE: ~2 = 1\1, ~39/32 = 339.764
* POTE: ~2 = 1200.000{{c}}, ~39/32 = 339.764{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 113, 166 }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness: 0.032996
Badness (Smith): 0.032996


=== Houbor ===
=== Houbor ===
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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: {{mapping| 1 3 6 13 6 | 0 -5 -13 -36 -9 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.680
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.680{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.814
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.814{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.045232
Badness (Smith): 0.045232


==== 13-limit ====
==== 13-limit ====
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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: {{mapping| 1 3 6 13 6 2 | 0 -5 -13 -36 -9 6 }}


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 339.685
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 339.685{{c}}
* POTE: ~2 = 1\1, ~11/9 = 339.784
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 339.784{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 32cdddd, 39ddd, 46d, 53 }}
{{Optimal ET sequence|legend=0| 7d, 32cdddd, 39ddd, 46d, 53 }}


Badness: 0.031331
Badness (Smith): 0.031331


== Paramity ==
== Paramity ==
The paramity temperament (53 &amp; 311) tempers out the horwell comma (65625/65536) and [[garischisma]] (33554432/33480783).
Paramity tempers out the [[horwell comma]] (65625/65536) and [[garischisma]] (33554432/33480783), and may be described as the {{nowrap| 53 & 311 }} temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 380: Line 378:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.553
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.553{{c}}


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


[[Badness]]: 0.113655
[[Badness]] (Smith): 0.113655


=== 11-limit ===
=== 11-limit ===
Line 395: Line 393:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986 }}
{{Optimal ET sequence|legend=0| 53, 205de, 258, 311, 675, 986 }}


Badness: 0.064853
Badness (Smith): 0.064853


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.554
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.554
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.554{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 205de, 258, 311, 675, 986, 1661cf }}
{{Optimal ET sequence|legend=0| 53, 205de, 258, 311, 675, 986, 1661cf }}


Badness: 0.030347
Badness (Smith): 0.030347


=== 17-limit ===
=== 17-limit ===
Line 425: Line 423:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~243/200 = 339.555
* CTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}
* POTE: ~2 = 1\1, ~243/200 = 339.555
* POTE: ~2 = 1200.000{{c}}, ~243/200 = 339.555{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
{{Optimal ET sequence|legend=0| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}


Badness: 0.024118
Badness (Smith): 0.024118


=== 19-limit ===
=== 19-limit ===
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~208/171 = 339.555
* CTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}
* POTE: ~2 = 1\1, ~208/171 = 339.555
* POTE: ~2 = 1200.000{{c}}, ~208/171 = 339.555{{c}}


Optimal ET sequence: {{Optimal ET sequence| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
{{Optimal ET sequence|legend=0| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}


Badness: 0.017420
Badness (Smith): 0.017420


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


Optimal tunings:
Optimal tunings:
* CTE: ~343/240 = 1\2, ~7/6 = 260.563
* CTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.563{{c}}
* POTE: ~343/240 = 1\2, ~7/6 = 260.402
* POTE: ~343/240 = 600.000{{c}}, ~7/6 = 260.402{{c}}


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


[[Badness]]: 0.083601
[[Badness]] (Smith): 0.083601


=== 11-limit ===
=== 11-limit ===
Line 474: Line 471:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 1\2, ~7/6 = 260.653
* CTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.653{{c}}
* POTE: ~99/70 = 1\2, ~7/6 = 260.393
* POTE: ~99/70 = 600.000{{c}}, ~7/6 = 260.393{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 32c, 46, 152de, 198, 244dee }}
{{Optimal ET sequence|legend=0| 14c, 32c, 46, 152de, 198, 244dee }}


Badness: 0.035504
Badness (Smith): 0.035504


=== 13-limit ===
=== 13-limit ===
Line 489: Line 486:


Optimal tunings:
Optimal tunings:
* CTE: ~55/39 = 1\2, ~7/6 = 260.811
* CTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.811{{c}}
* POTE: ~55/39 = 1\2, ~7/6 = 260.618
* POTE: ~55/39 = 600.000{{c}}, ~7/6 = 260.618{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 32cf, 46 }}
{{Optimal ET sequence|legend=0| 14cf, 32cf, 46 }}


Badness: 0.030885
Badness (Smith): 0.030885


== Hamity ==
== Hamity ==
Line 504: Line 501:


{{Mapping|legend=1| 1 8 19 15 | 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


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.168
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.168{{c}}
* POTE: ~2 = 1\1, ~9/7 = 430.219
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.219{{c}}


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


[[Badness]]: 0.073956
[[Badness]] (Smith): 0.073956


=== 11-limit ===
=== 11-limit ===
Line 523: Line 519:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.220
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.220{{c}}
* POTE: ~2 = 1\1, ~9/7 = 430.192
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.192{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}


Badness: 0.042947
Badness (Smith): 0.042947


=== 13-limit ===
=== 13-limit ===
Line 538: Line 534:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 430.233
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 430.233{{c}}
* POTE: ~2 = 1\1, ~9/7 = 430.216
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 430.216{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 39df, 53 }}
{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}


Badness: 0.029753
Badness (Smith): 0.029753


== Gamity ==
== Gamity ==
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.
Gamity tempers out 1029/1024, the [[gamelisma]], and may be described as the {{nowrap| 46 & 113 }} temperament. 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
Line 553: Line 549:


{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
: mapping generators: ~2, ~320/189
: mapping generators: ~2, ~320/189


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~189/160 = 286.816
* CTE: ~2 = 1200.000{{c}}, ~189/160 = 286.816{{c}}
* POTE: ~2 = 1\1, ~189/160 = 286.787
* POTE: ~2 = 1200.000{{c}}, ~189/160 = 286.787{{c}}


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


[[Badness]]: 0.125733
[[Badness]] (Smith): 0.125733


=== 11-limit ===
=== 11-limit ===
Line 572: Line 567:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~33/28 = 286.813
* CTE: ~2 = 1200.000{{c}}, ~33/28 = 286.813{{c}}
* POTE: ~2 = 1\1, ~33/28 = 286.797
* POTE: ~2 = 1200.000{{c}}, ~33/28 = 286.797{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 205d, 364d }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 205d, 364d }}


Badness: 0.051111
Badness (Smith): 0.051111


=== 13-limit ===
=== 13-limit ===
Line 587: Line 582:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.803
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.803{{c}}
* POTE: ~2 = 1\1, ~13/11 = 286.789
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.789{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}


Badness: 0.030297
Badness (Smith): 0.030297


=== 17-limit ===
=== 17-limit ===
Line 602: Line 597:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~13/11 = 286.804
* CTE: ~2 = 1200.000{{c}}, ~13/11 = 286.804{{c}}
* POTE: ~2 = 1\1, ~13/11 = 286.795
* POTE: ~2 = 1200.000{{c}}, ~13/11 = 286.795{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 113, 159, 364df, 523ddff }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}


Badness: 0.022036
Badness (Smith): 0.022036


== Trinity ==
== Trinity ==
The trinity temperament (152 &amp; 159) tempers out the [[meter]], 703125/702464. It splits the acute minor tenth (~243/100, an octave plus acute minor third) in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.
Trinity tempers out 703125/702464, the [[meter]], and may be described as the {{nowrap| 152 & 159 }} temperament. 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{{c}} flat), three of which makes acute minor third of ~243/200 with octave reduction.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 619: Line 614:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~168/125 = 513.180
* CTE: ~2 = 1200.000{{c}}, ~168/125 = 513.180{{c}}
* POTE: ~2 = 1\1, ~168/125 = 513.178
* POTE: ~2 = 1200.000{{c}}, ~168/125 = 513.178{{c}}


{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}
{{Optimal ET sequence|legend=1| 7d, …, 145d, 152, 311, 774, 1085 }}


[[Badness]]: 0.119453
[[Badness]] (Smith): 0.119453


=== 11-limit ===
=== 11-limit ===
Line 634: Line 629:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~121/90 = 513.181
* CTE: ~2 = 1200.000{{c}}, ~121/90 = 513.181{{c}}
* POTE: ~2 = 1\1, ~121/90 = 513.177
* POTE: ~2 = 1200.000{{c}}, ~121/90 = 513.177{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}
{{Optimal ET sequence|legend=0| 7d, …, 145d, 152, 311, 774, 1085e, 1396e }}


Badness: 0.031296
Badness (Smith): 0.031296


=== 13-limit ===
=== 13-limit ===
Line 649: Line 644:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* POTE: ~2 = 1\1, ~35/26 = 513.182
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.182{{c}}


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.026418
Badness (Smith): 0.026418


=== 17-limit ===
=== 17-limit ===
Line 664: Line 659:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.185
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* POTE: ~2 = 1\1, ~35/26 = 513.186
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg }}


Badness: 0.025588
Badness (Smith): 0.025588


=== 19-limit ===
=== 19-limit ===
Line 679: Line 674:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* POTE: ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311 }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.018412
Badness (Smith): 0.018412


=== 23-limit ===
=== 23-limit ===
Line 694: Line 689:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.184
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.184{{c}}
* POTE: ~2 = 1\1, ~35/26 = 513.185
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}


Optimal ET sequence: {{Optimal ET sequence| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi }}


Badness: 0.014343
Badness (Smith): 0.014343


=== 29-limit ===
=== 29-limit ===
Line 709: Line 704:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~35/26 = 513.185
* CTE: ~2 = 1200.000{{c}}, ~35/26 = 513.185{{c}}
* POTE: ~2 = 1\1, ~35/26 = 513.186
* POTE: ~2 = 1200.000{{c}}, ~35/26 = 513.186{{c}}


Optimal ET sequence: {{Optimal ET sequence| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}
{{Optimal ET sequence|legend=0| 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij }}


Badness: 0.012038
Badness (Smith): 0.012038


== Amicable ==
== 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.  
Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} 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.  
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 728: Line 723:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8831
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8831{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.880
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 84.880{{c}}


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


[[Badness]]: 0.045473
[[Badness]] (Smith): 0.045473


=== Amical ===
=== Amical ===
Line 743: Line 738:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8843
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.8843
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8843{{c}}


Optimal ET sequence: {{Optimal ET sequence| 99, 212e, 311, 721, 1032, 1343, 2375bc }}
{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343, 2375bc }}


Badness: 0.100668
Badness (Smith): 0.100668


==== 13-limit ====
==== 13-limit ====
Line 758: Line 753:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8837
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8837{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.8838
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8838{{c}}


Optimal ET sequence: {{Optimal ET sequence| 99, 212ef, 311, 721, 1032 }}
{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}


Badness: 0.049893
Badness (Smith): 0.049893


=== Amorous ===
=== Amorous ===
Line 773: Line 768:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8883
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8883{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.8896
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8896{{c}}


Optimal ET sequence: {{Optimal ET sequence| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}
{{Optimal ET sequence|legend=0| 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde }}


Badness: 0.048924
Badness (Smith): 0.048924


==== 13-limit ====
==== 13-limit ====
Line 788: Line 783:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.8895
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8895{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.8910
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8910{{c}}


Optimal ET sequence: {{Optimal ET sequence| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}
{{Optimal ET sequence|legend=0| 99ef, 212, 311, 1145c, 1456cd, 1767cd }}


Badness: 0.034681
Badness (Smith): 0.034681


=== Pseudoamical ===
=== Pseudoamical ===
Line 803: Line 798:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9005
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9005{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.9091
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9091{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85cee, 99, 212 }}
{{Optimal ET sequence|legend=0| 14ce, …, 85cee, 99, 212 }}


Badness: 0.085837
Badness (Smith): 0.085837


==== 13-limit ====
==== 13-limit ====
Line 818: Line 813:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9049
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9049{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.9127
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9127{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14ce, …, 85ceef, 99, 113, 212 }}
{{Optimal ET sequence|legend=0| 14ce, …, 85ceef, 99, 113, 212 }}


Badness: 0.047025
Badness (Smith): 0.047025


=== Pseudoamorous ===
=== Pseudoamorous ===
Line 833: Line 828:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9022
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9022{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.8917
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.8917{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 85ce, 99e, 113, 212e }}
{{Optimal ET sequence|legend=0| 14c, …, 85ce, 99e, 113, 212e }}


Badness: 0.056583
Badness (Smith): 0.056583


==== 13-limit ====
==== 13-limit ====
Line 848: Line 843:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 84.9153
* CTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9153{{c}}
* POTE: ~2 = 1\1, ~21/20 = 84.9164
* POTE: ~2 = 1200.0000{{c}}, ~21/20 = 84.9164{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}
{{Optimal ET sequence|legend=0| 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef }}


Badness: 0.042826
Badness (Smith): 0.042826


=== Floral ===
=== Floral ===
Line 863: Line 858:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 1\2, ~21/20 = 84.8781
* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8781{{c}}
* POTE: ~99/70 = 1\2, ~21/20 = 84.8788
* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8788{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 170bccde, 184c, 198, 212, 410 }}
{{Optimal ET sequence|legend=0| 14c, …, 170bccde, 184c, 198, 212, 410 }}


Badness: 0.065110
Badness (Smith): 0.065110


==== 13-limit ====
==== 13-limit ====
Line 878: Line 873:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 1\2, ~21/20 = 84.8759
* CTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8759{{c}}
* POTE: ~99/70 = 1\2, ~21/20 = 84.8750
* POTE: ~99/70 = 600.0000{{c}}, ~21/20 = 84.8750{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, …, 184cff, 198, 410 }}
{{Optimal ET sequence|legend=0| 14c, …, 184cff, 198, 410 }}


Badness: 0.037013
Badness (Smith): 0.037013


=== Humorous ===
=== Humorous ===
Line 893: Line 888:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~4096/3993 = 42.4414
* CTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4414{{c}}
* POTE: ~2 = 1\1, ~4096/3993 = 42.4391
* POTE: ~2 = 1200.0000{{c}}, ~4096/3993 = 42.4391{{c}}


Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1131, 1442, 1753be }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1131, 1442, 1753be }}


Badness: 0.058249
Badness (Smith): 0.058249


==== 13-limit ====
==== 13-limit ====
Line 908: Line 903:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~40/39 = 42.4425
* CTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4425{{c}}
* POTE: ~2 = 1\1, ~40/39 = 42.4391
* POTE: ~2 = 1200.0000{{c}}, ~40/39 = 42.4391{{c}}


Optimal ET sequence: {{Optimal ET sequence| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff }}


Badness: 0.028267
Badness (Smith): 0.028267


== Calamity ==
== 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.  
Calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of two octaves plus an acute minor third into five.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 924: Line 919:
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~48/35 = 547.909
[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.909{{c}}


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


[[Badness]]: 0.198130
[[Badness]] (Smith): 0.198130


=== 11-limit ===
=== 11-limit ===
Line 937: Line 932:
Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.908
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.908{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 219c, 265, 311, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 219c, 265, 311, 979, 1290 }}


Badness: 0.060408
Badness (Smith): 0.060408


=== 13-limit ===
=== 13-limit ===
Line 950: Line 945:
Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}
Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.907
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~48/35 = 547.907{{c}}


Optimal ET sequence: {{Optimal ET sequence| 46, 265, 311, 668, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 265, 311, 668, 979, 1290 }}


Badness: 0.033617
Badness (Smith): 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.
Familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. It splits the interval of acute minor tenth (~243/100) in five.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 966: Line 961:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~11907/10000 = 307.915
* CTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.915{{c}}
* POTE: ~2 = 1\1, ~11907/10000 = 307.941
* POTE: ~2 = 1200.000{{c}}, ~11907/10000 = 307.941{{c}}


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


[[Badness]]: 0.144551
[[Badness]] (Smith): 0.144551


=== 11-limit ===
=== 11-limit ===
Line 981: Line 976:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3200/2673 = 307.915
* CTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.915{{c}}
* POTE: ~2 = 1\1, ~3200/2673 = 307.906
* POTE: ~2 = 1200.000{{c}}, ~3200/2673 = 307.906{{c}}


Optimal ET sequence: {{Optimal ET sequence| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}
{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 265, 417, 1099cdee }}


Badness: 0.051740
Badness (Smith): 0.051740


=== 13-limit ===
=== 13-limit ===
Line 996: Line 991:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~143/120 = 307.922
* CTE: ~2 = 1200.000{{c}}, ~143/120 = 307.922{{c}}
* POTE: ~2 = 1\1, ~143/120 = 307.913
* POTE: ~2 = 1200.000{{c}}, ~143/120 = 307.913{{c}}


Optimal ET sequence: {{Optimal ET sequence| 39df, 74cdf, 113, 152f, 265 }}
{{Optimal ET sequence|legend=0| 39df, 74cdf, 113, 152f, 265 }}


Badness: 0.038473
Badness (Smith): 0.038473


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity family| ]] <!-- main article -->
[[Category:Amity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 06:53, 1 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The amity family of temperaments tempers out the amity comma (monzo: [9 -13 15⟩, ratio: 1600000/1594323).

Amity

Main article: Amity

The generator for the amity temperament is the acute minor third, which means the 6/5 just minor third raised by a syntonic comma to 243/200, and from this it derives its name. If you are looking for a different kind of neutral third, this could be the temperament for you. Its ploidacot is gamma-pentacot. It is a member of the syntonic–chromatic equivalence continuum with n = 5, so it equates an apotome with a stack of five syntonic commas. It is also in the schismic–Mercator equivalence continuum with n = 5, so unless 53edo is used as a tuning, the schisma is always observed.

Amity is a genuine microtemperament in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)1/13, which gives a pure classical major third. Mos scales of 11, 18, 25, 32, 39, 46 or 53 notes are available.

Subgroup: 2.3.5

Comma list: 1600000/1594323

Mapping: [⟨1 3 6], ⟨0 -5 -13]]

mapping generators: ~2, ~243/200

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.537 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.519 ¢

Optimal ET sequence: 7, 32c, 39, 46, 53, 152, 205, 258, 1085, 1343, 1601, 1859b, 2117bc

Badness (Smith): 0.021960

Overview to extensions

The second comma to extend the 5-limit amity include 4375/4374 for septimal amity, 225/224 for houborizic, 65625/65536 for paramity, 126/125 for accord, 245/243 for bamity, 2430/2401 for hamity, 1029/1024 for gamity, 10976/10935 for chromat, 703125/702464 for trinity, 2401/2400 for amicable, 2100875/2097152 for calamity, 420175/419904 for witcher, and 16875/16807 for familia.

Temperaments discussed elsewhere include:

The rest are considered below.

Septimal amity

Main article: Amity

Septimal amity can be described as the 46 & 53 temperament, which tempers out 4375/4374 and 5120/5103 in the 7-limit. 99edo is a good tuning, with generator 28\99.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 5120/5103

Mapping: [⟨1 3 6 -2], ⟨0 -5 -13 17]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~128/105 = 339.446 ¢
  • POTE: ~2 = 1200.000 ¢, ~128/105 = 339.432 ¢

Optimal ET sequence: 7, 32cd, 39, 46, 53, 99, 152, 251, 905bcdd

Badness (Smith): 0.023649

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 5120/5103

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~128/105 = 339.485 ¢
  • POTE: ~2 = 1200.000 ¢, ~128/105 = 339.464 ¢

Optimal ET sequence: 46e, 53, 99e, 152, 357d, 509dd, 661dd

Badness (Smith): 0.031506

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~128/105 = 339.508 ¢
  • POTE: ~2 = 1200.000 ¢, ~128/105 = 339.481 ¢

Optimal ET sequence: 46ef, 53, 99ef, 152f, 205

Badness (Smith): 0.028008

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 375/374, 540/539, 729/728

Mapping: [⟨1 3 6 -2 21 17 -1], ⟨0 -5 -13 17 -62 -47 18]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~17/14 = 339.496 ¢

Optimal ET sequence: 46ef, 53, 99ef, 152fg, 205gg

Badness (Smith): 0.026201

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 324/323, 352/351, 375/374, 400/399, 456/455

Mapping: [⟨1 3 6 -2 21 17 -1 15], ⟨0 -5 -13 17 -62 -47 18 -38]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~17/14 = 339.501 ¢

Optimal ET sequence: 46efh, 53, 99ef, 152fg, 205gg

Badness (Smith): 0.018782

Hitchcock

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 2200/2187

Mapping: [⟨1 3 6 -2 6], ⟨0 -5 -13 17 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.390 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.390 ¢

Optimal ET sequence: 7, 25cdde, 32cd, 39, 46, 53, 99

Badness (Smith): 0.035187

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 176/175, 325/324

Mapping: [⟨1 3 6 -2 6 2], ⟨0 -5 -13 17 -9 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.411 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.419 ¢

Optimal ET sequence: 7, 25cddef, 32cd, 39, 46, 53, 99

Badness (Smith): 0.022448

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 169/168, 176/175, 273/272

Mapping: [⟨1 3 6 -2 6 2 -1], ⟨0 -5 -13 17 -9 6 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.366 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.366 ¢

Optimal ET sequence: 7, 25cddefgg, 32cdg, 39, 46, 99

Badness (Smith): 0.019395

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189

Mapping: [⟨1 3 6 -2 6 2 -1 0], ⟨0 -5 -13 17 -9 6 18 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.415 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.407 ¢

Optimal ET sequence: 7, 25cddefgghh, 32cdgh, 39h, 46, 53, 99h

Badness (Smith): 0.017513

Stalagmite

The stalagmite 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. "-mite" in the name references amity, and stalagmites being found in caves underground references how it is down from amity.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 4375/4374

Mapping: [⟨1 3 6 -2 -7], ⟨0 -5 -13 17 37]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~128/105 = 339.314 ¢
  • POTE: ~2 = 1200.000 ¢, ~128/105 = 339.340 ¢

Optimal ET sequence: 46, 99e, 145

Badness (Smith): 0.040976

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 4375/4374

Mapping: [⟨1 3 6 -2 -7 -11], ⟨0 -5 -13 17 37 52]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~128/105 = 339.277 ¢
  • POTE: ~2 = 1200.000 ¢, ~128/105 = 339.313 ¢

Optimal ET sequence: 46, 99ef, 145, 191c, 336cef, 527bccef

Badness (Smith): 0.034215

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155

Mapping: [⟨1 3 6 -2 -7 -11 -1], ⟨0 -5 -13 17 37 52 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~17/14 = 339.272 ¢
  • POTE: ~2 = 1200.000 ¢, ~17/14 = 339.313 ¢

Optimal ET sequence: 46, 99ef, 145, 191c

Badness (Smith): 0.021193

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475

Mapping: [⟨1 3 6 -2 -7 -11 -1 -13], ⟨0 -5 -13 17 37 52 18 61]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~17/14 = 339.282 ¢
  • POTE: ~2 = 1200.000 ¢, ~17/14 = 339.325 ¢

Optimal ET sequence: 46, 99ef, 145, 191c, 336cefg

Badness (Smith): 0.018864

Hemiamity

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5120/5103

Mapping: [⟨2 1 -1 13 13], ⟨0 5 13 -17 -14]]

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

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~64/55 = 260.566 ¢
  • POTE: ~99/70 = 600.000 ¢, ~64/55 = 260.561 ¢

Optimal ET sequence: 14cde, 32cde, 46, 106, 152, 350

Badness (Smith): 0.031307

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 3025/3024

Mapping: [⟨2 1 -1 13 13 20], ⟨0 5 13 -17 -14 -29]]

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~64/55 = 260.607 ¢
  • POTE: ~99/70 = 600.000 ¢, ~64/55 = 260.583 ¢

Optimal ET sequence: 46, 106f, 152f, 198

Badness (Smith): 0.025784

Accord

Subgroup: 2.3.5.7

Comma list: 126/125, 100352/98415

Mapping: [⟨1 3 6 11], ⟨0 -5 -13 -29]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.154 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 338.993 ¢

Optimal ET sequence: 7d, 25cddd, 32cdd, 39d, 46

Badness (Smith): 0.095612

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 896/891

Mapping: [⟨1 3 6 11 6], ⟨0 -5 -13 -29 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.136 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.047 ¢

Optimal ET sequence: 7d, 25cddde, 32cdd, 39d, 46

Badness (Smith): 0.042468

Houborizic

Houborizic tempers out 225/224, the marvel comma, and may be described as the 53 & 60 temperament. It is so named because it is closely related to the houboriz tuning (generator: 339.774971 cents).

Subgroup: 2.3.5.7

Comma list: 225/224, 1250000/1240029

Mapping: [⟨1 3 6 13], ⟨0 -5 -13 -36]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.711 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.763 ¢

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53, 166, 219c, 272c

Badness (Smith): 0.066638

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1250000/1240029

Mapping: [⟨1 3 6 13 -9], ⟨0 -5 -13 -36 44]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.751 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.763 ¢

Optimal ET sequence: 53, 113, 166, 551ccee

Badness (Smith): 0.067891

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 2200/2197

Mapping: [⟨1 3 6 13 -9 2], ⟨0 -5 -13 -36 44 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~39/32 = 339.754 ¢
  • POTE: ~2 = 1200.000 ¢, ~39/32 = 339.764 ¢

Optimal ET sequence: 53, 113, 166

Badness (Smith): 0.032996

Houbor

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 2200/2187

Mapping: [⟨1 3 6 13 6], ⟨0 -5 -13 -36 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.680 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.814 ¢

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness (Smith): 0.045232

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 325/324

Mapping: [⟨1 3 6 13 6 2], ⟨0 -5 -13 -36 -9 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11/9 = 339.685 ¢
  • POTE: ~2 = 1200.000 ¢, ~11/9 = 339.784 ¢

Optimal ET sequence: 7d, 32cdddd, 39ddd, 46d, 53

Badness (Smith): 0.031331

Paramity

Paramity tempers out the horwell comma (65625/65536) and garischisma (33554432/33480783), and may be described as the 53 & 311 temperament.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 1600000/1594323

Mapping: [⟨1 3 6 -17], ⟨0 -5 -13 70]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.554 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.553 ¢

Optimal ET sequence: 53, 205d, 258, 311, 675, 986

Badness (Smith): 0.113655

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6250/6237, 19712/19683, 41503/41472

Mapping: [⟨1 3 6 -17 36], ⟨0 -5 -13 70 -115]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.554 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.554 ¢

Optimal ET sequence: 53, 205de, 258, 311, 675, 986

Badness (Smith): 0.064853

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683

Mapping: [⟨1 3 6 -17 36 17], ⟨0 -5 -13 70 -115 -47]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.554 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.554 ¢

Optimal ET sequence: 53, 205de, 258, 311, 675, 986, 1661cf

Badness (Smith): 0.030347

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430

Mapping: [⟨1 3 6 -17 36 17 -31], ⟨0 -5 -13 70 -115 -47 124]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~243/200 = 339.555 ¢
  • POTE: ~2 = 1200.000 ¢, ~243/200 = 339.555 ¢

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf

Badness (Smith): 0.024118

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197

Mapping: [⟨1 3 6 -17 36 17 -31 15], ⟨0 -5 -13 70 -115 -47 124 -38]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~208/171 = 339.555 ¢
  • POTE: ~2 = 1200.000 ¢, ~208/171 = 339.555 ¢

Optimal ET sequence: 53, 205deg, 258g, 311, 675, 986, 1661cfh

Badness (Smith): 0.017420

Bamity

Bamity has a period of half octave and tempers out the sensamagic comma, 245/243. The name bamity is a contraction of bi- and amity.

Subgroup: 2.3.5.7

Comma list: 245/243, 64827/64000

Mapping: [⟨2 1 -1 3], ⟨0 5 13 6]]

mapping generators: ~343/240, ~7/6

Optimal tunings:

  • CTE: ~343/240 = 600.000 ¢, ~7/6 = 260.563 ¢
  • POTE: ~343/240 = 600.000 ¢, ~7/6 = 260.402 ¢

Optimal ET sequence: 14c, 32c, 46, 106d, 152d

Badness (Smith): 0.083601

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3], ⟨0 5 13 6 9]]

Optimal tunings:

  • CTE: ~99/70 = 600.000 ¢, ~7/6 = 260.653 ¢
  • POTE: ~99/70 = 600.000 ¢, ~7/6 = 260.393 ¢

Optimal ET sequence: 14c, 32c, 46, 152de, 198, 244dee

Badness (Smith): 0.035504

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 245/243, 441/440

Mapping: [⟨2 1 -1 3 3 0], ⟨0 5 13 6 9 17]]

Optimal tunings:

  • CTE: ~55/39 = 600.000 ¢, ~7/6 = 260.811 ¢
  • POTE: ~55/39 = 600.000 ¢, ~7/6 = 260.618 ¢

Optimal ET sequence: 14cf, 32cf, 46

Badness (Smith): 0.030885

Hamity

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

Subgroup: 2.3.5.7

Comma list: 2430/2401, 4000/3969

Mapping: [⟨1 8 19 15], ⟨0 -10 -26 -19]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~9/7 = 430.168 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.219 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Smith): 0.073956

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 8 19 15 15], ⟨0 -10 -26 -19 -18]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~9/7 = 430.220 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.192 ¢

Optimal ET sequence: 14c, 39d, 53

Badness (Smith): 0.042947

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 8 19 15 15 30], ⟨0 -10 -26 -19 -18 -41]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~9/7 = 430.233 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 430.216 ¢

Optimal ET sequence: 14cf, 39df, 53

Badness (Smith): 0.029753

Gamity

Gamity tempers out 1029/1024, the gamelisma, and may be described as the 46 & 113 temperament. It splits the interval of grave major sixth (~400/243, an octave minus acute minor third) in three.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 1071875/1062882

Mapping: [⟨1 13 32 -1], ⟨0 -15 -39 5]]

mapping generators: ~2, ~320/189

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~189/160 = 286.816 ¢
  • POTE: ~2 = 1200.000 ¢, ~189/160 = 286.787 ¢

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness (Smith): 0.125733

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 13 32 -1 -11], ⟨0 -15 -39 5 19]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~33/28 = 286.813 ¢
  • POTE: ~2 = 1200.000 ¢, ~33/28 = 286.797 ¢

Optimal ET sequence: 46, 113, 159, 205d, 364d

Badness (Smith): 0.051111

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 13 32 -1 -11 -10], ⟨0 -15 -39 5 19 18]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/11 = 286.803 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/11 = 286.789 ¢

Optimal ET sequence: 46, 113, 159, 364df, 523ddff

Badness (Smith): 0.030297

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757

Mapping: [⟨1 13 32 -1 -11 -10 -2], ⟨0 -15 -39 5 19 18 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/11 = 286.804 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/11 = 286.795 ¢

Optimal ET sequence: 46, 113, 159, 364df, 523ddff

Badness (Smith): 0.022036

Trinity

Trinity tempers out 703125/702464, the meter, and may be described as the 152 & 159 temperament. It splits the acute minor tenth (~243/100, an octave plus acute minor third) in three. It was so named for the following reason – 133\311 (133 steps of 311edo) is a possible generator, which is placed around 3\7 (1.1 ¢ flat), three of which makes acute minor third of ~243/200 with octave reduction.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 1600000/1594323

Mapping: [⟨1 8 19 46], ⟨0 -15 -39 -101]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~168/125 = 513.180 ¢
  • POTE: ~2 = 1200.000 ¢, ~168/125 = 513.178 ¢

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085

Badness (Smith): 0.119453

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 19712/19683

Mapping: [⟨1 8 19 46 18], ⟨0 -15 -39 -101 -34]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~121/90 = 513.181 ¢
  • POTE: ~2 = 1200.000 ¢, ~121/90 = 513.177 ¢

Optimal ET sequence: 7d, …, 145d, 152, 311, 774, 1085e, 1396e

Badness (Smith): 0.031296

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689

Mapping: [⟨1 8 19 46 18 64], ⟨0 -15 -39 -101 -34 -141]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.182 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Smith): 0.026418

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619

Mapping: [⟨1 8 19 46 18 64 -22], ⟨0 -15 -39 -101 -34 -141 61]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.186 ¢

Optimal ET sequence: 152f, 159, 311, 1714cdeg, 2025cdefgg, 2336bccdefgg

Badness (Smith): 0.025588

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573

Mapping: [⟨1 8 19 46 18 64 -22 53], ⟨0 -15 -39 -101 -34 -141 61 -114]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢

Optimal ET sequence: 152f, 159, 311

Badness (Smith): 0.018412

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104

Mapping: [⟨1 8 19 46 18 64 -22 53 49], ⟨0 -15 -39 -101 -34 -141 61 -114 -104]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~35/26 = 513.184 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢

Optimal ET sequence: 152f, 159, 311, 1714cdeghi, 2025cdefgghhi, 2336bccdefgghhi

Badness (Smith): 0.014343

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~35/26 = 513.185 ¢
  • POTE: ~2 = 1200.000 ¢, ~35/26 = 513.186 ¢

Optimal ET sequence: 152fj, 159, 311, 1403cdgh, 1714cdeghi, 2025cdefgghhij, 2336bccdefgghhij

Badness (Smith): 0.012038

Amicable

Amicable tempers out the breedsma as well as the canousma, and may be described as the 99 & 311 temperament.

While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1600000/1594323

Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8831 ¢
  • POTE: ~2 = 1200.000 ¢, ~21/20 = 84.880 ¢

Optimal ET sequence: 14c, …, 85c, 99, 212, 311, 721, 1032, 1753b

Badness (Smith): 0.045473

Amical

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 131072/130977, 1600000/1594323

Mapping: [⟨1 3 6 5 -8], ⟨0 -20 -52 -31 162]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8843 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8843 ¢

Optimal ET sequence: 99, 212e, 311, 721, 1032, 1343, 2375bc

Badness (Smith): 0.100668

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206

Mapping: [⟨1 3 6 5 -8 -5], ⟨0 -20 -52 -31 162 123]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8837 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8838 ¢

Optimal ET sequence: 99, 212ef, 311, 721, 1032

Badness (Smith): 0.049893

Amorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 19712/19683

Mapping: [⟨1 3 6 5 14], ⟨0 -20 -52 -31 -149]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8883 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8896 ¢

Optimal ET sequence: 99e, 212, 311, 2389bccd, 2700bccde, 3011bccde, 3322bccdde

Badness (Smith): 0.048924

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647

Mapping: [⟨1 3 6 5 14 17], ⟨0 -20 -52 -31 -149 -188]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8895 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8910 ¢

Optimal ET sequence: 99ef, 212, 311, 1145c, 1456cd, 1767cd

Badness (Smith): 0.034681

Pseudoamical

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 1600000/1594323

Mapping: [⟨1 3 6 5 -1], ⟨0 -20 -52 -31 63]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9005 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9091 ¢

Optimal ET sequence: 14ce, …, 85cee, 99, 212

Badness (Smith): 0.085837

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1375/1372, 19773/19712

Mapping: [⟨1 3 6 5 -1 2], ⟨0 -20 -52 -31 63 24]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9049 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9127 ¢

Optimal ET sequence: 14ce, …, 85ceef, 99, 113, 212

Badness (Smith): 0.047025

Pseudoamorous

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 980000/970299

Mapping: [⟨1 3 6 5 7], ⟨0 -20 -52 -31 -50]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9022 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.8917 ¢

Optimal ET sequence: 14c, …, 85ce, 99e, 113, 212e

Badness (Smith): 0.056583

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 1875/1859

Mapping: [⟨1 3 6 5 7 10], ⟨0 -20 -52 -31 -50 -89]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9153 ¢
  • POTE: ~2 = 1200.0000 ¢, ~21/20 = 84.9164 ¢

Optimal ET sequence: 14cf, …, 85ceff, 99ef, 113, 212ef, 325ce, 537cdeef

Badness (Smith): 0.042826

Floral

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 14641/14580

Mapping: [⟨2 6 12 10 13], ⟨0 -20 -52 -31 -43]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8781 ¢
  • POTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8788 ¢

Optimal ET sequence: 14c, …, 170bccde, 184c, 198, 212, 410

Badness (Smith): 0.065110

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580

Mapping: [⟨2 6 12 10 13 19], ⟨0 -20 -52 -31 -43 -82]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8759 ¢
  • POTE: ~99/70 = 600.0000 ¢, ~21/20 = 84.8750 ¢

Optimal ET sequence: 14c, …, 184cff, 198, 410

Badness (Smith): 0.037013

Humorous

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 1600000/1594323

Mapping: [⟨1 3 6 5 3], ⟨0 -40 -104 -62 13]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~4096/3993 = 42.4414 ¢
  • POTE: ~2 = 1200.0000 ¢, ~4096/3993 = 42.4391 ¢

Optimal ET sequence: 85c, 113, 198, 311, 1131, 1442, 1753be

Badness (Smith): 0.058249

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024

Mapping: [⟨1 3 6 5 3 6], ⟨0 -40 -104 -62 13 -65]]

Optimal tunings:

  • CTE: ~2 = 1200.0000 ¢, ~40/39 = 42.4425 ¢
  • POTE: ~2 = 1200.0000 ¢, ~40/39 = 42.4391 ¢

Optimal ET sequence: 85c, 113, 198, 311, 1753beff, 2064beff, 2375bceff

Badness (Smith): 0.028267

Calamity

Calamity tempers out 2100875/2097152, the rainy comma, and may be described as the 46 & 311 temperament. It splits the interval of two octaves plus an acute minor third into five.

Subgroup: 2.3.5.7

Comma list: 1600000/1594323, 2100875/2097152

Mapping: [⟨1 13 32 -15], ⟨0 -25 -65 39]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~48/35 = 547.909 ¢

Optimal ET sequence: 46, 219c, 265, 311

Badness (Smith): 0.198130

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 12005/11979, 131072/130977

Mapping: [⟨1 13 32 -15 -18], ⟨0 -25 -65 39 47]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~48/35 = 547.908 ¢

Optimal ET sequence: 46, 219c, 265, 311, 979, 1290

Badness (Smith): 0.060408

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979

Mapping: [⟨1 13 32 -15 -18 -31], ⟨0 -25 -65 39 47 76]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~48/35 = 547.907 ¢

Optimal ET sequence: 46, 265, 311, 668, 979, 1290

Badness (Smith): 0.033617

Familia

Familia tempers out 16875/16807, the mirkwai comma, and may be described as the 113 & 152 temperament. It splits the interval of acute minor tenth (~243/100) in five.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 1600000/1594323

Mapping: [⟨1 8 19 20], ⟨0 -25 -65 -67]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~11907/10000 = 307.915 ¢
  • POTE: ~2 = 1200.000 ¢, ~11907/10000 = 307.941 ¢

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1516ccdd, 1933ccdd

Badness (Smith): 0.144551

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 1600000/1594323

Mapping: [⟨1 8 19 20 5], ⟨0 -25 -65 -67 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3200/2673 = 307.915 ¢
  • POTE: ~2 = 1200.000 ¢, ~3200/2673 = 307.906 ¢

Optimal ET sequence: 39d, 74cd, 113, 152, 265, 417, 1099cdee

Badness (Smith): 0.051740

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 2205/2197

Mapping: [⟨1 8 19 20 5 25], ⟨0 -25 -65 -67 -6 -83]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~143/120 = 307.922 ¢
  • POTE: ~2 = 1200.000 ¢, ~143/120 = 307.913 ¢

Optimal ET sequence: 39df, 74cdf, 113, 152f, 265

Badness (Smith): 0.038473

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