Xenharmonic Wiki

Amity family: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
Temperament name revised (99&166)
Move chromat and witcher here
 
(16 intermediate revisions by 5 users not shown)
Line 1: Line 1:
The '''amity family''' tempers out the 5-limit [[amity comma]], 1600000/1594323. The generator for the [[amity]] temperament is the acute minor third, which means the 6/5 just minor third raised by an 81/80 comma to 243/200, and from this it derives its name. If you are looking for a different kind of neutral third this could be the temperament for you.
{{Technical data page}}
The '''amity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[amity comma]] ({{monzo|legend=1| 9 -13 15 }}, [[ratio]]: 1600000/1594323).  


== Amity ==
== Amity ==
{{Main| Amity }}
{{Main| Amity }}


In the 5-limit amity is a genuine [[microtemperament]], with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  
The [[generator]] for the amity temperament is the acute minor third, which means the [[6/5]] just minor third raised by a [[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. However its functional generator is the [[octave complement]] thereof, five of which stack to the [[12/1|12th harmonic]]. Its [[ploidacot]] is thus gamma-pentacot.
 
It is a member of the [[syntonic–chromatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so it equates an apotome with a stack of five syntonic commas. It is also in the [[schismic–Mercator equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}, so unless 53edo is used as a tuning, the schisma is always observed.
 
Amity is a genuine [[microtemperament]] in the 5-limit, with 58\205 being a possible tuning. Another good choice is (64/5)<sup>1/13</sup>, which gives a pure classical major third. [[Mos scale]]s of 11, 18, 25, 32, 39, 46 or 53 notes are available.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 15:
[[Comma list]]: 1600000/1594323
[[Comma list]]: 1600000/1594323


{{Mapping|legend=1| 1 3 6 | 0 -5 -13 }}
{{Mapping|legend=1| 1 -2 -7 | 0 5 13 }}
 
: mapping generators: ~2, ~400/243
: mapping generators: ~2, ~243/200


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.519
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9135{{c}}, ~400/243 = 860.4191{{c}}
: [[error map]]: {{val| -0.087 +0.314 -0.259 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~400/243 = 860.4726{{c}}
: error map: {{val| 0.000 +0.408 -0.169 }}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 152, 205, 463, 668, 873 }}
{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 152, 205, 463, 668, 873 }}


[[Badness]]: 0.021960
[[Badness]] (Sintel): 0.515


=== Overview to extensions ===
=== 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.  
The second comma to extend 5-limit amity include [[4375/4374]] for septimal amity, [[65625/65536]] for paramity, [[126/125]] for accord, and [[225/224]] for houborizic. Those all use the same generator as the 5-limit temperament.  
 
Temperaments discussed elsewhere include:
* ''[[Chromat]]'' → [[Hemimage temperaments #Chromat]] (+10976/10935)
* ''[[Witcher]]'' → [[Wizmic microtemperaments #Witcher]] (+420175/419904)


The rest are considered below.  
Bamity adds [[245/243]] with a 1/2-octave period. Hamity adds [[2430/2401]] and splits the generator in two. Gamity adds [[1029/1024]]. Chromat adds [[10976/10935]]. Trinity adds [[703125/702464]]. Those split the generator in three. Amicable adds [[2401/2400]], splitting the generator in four. Witcher adds [[420175/419904]]. Calamity adds [[2100875/2097152]]. Familia adds [[16875/16807]]. Those split the generator in five.  


== Septimal amity ==
== Septimal amity ==
{{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
Line 38: Line 42:
[[Comma list]]: 4375/4374, 5120/5103
[[Comma list]]: 4375/4374, 5120/5103


{{Mapping|legend=1| 1 3 6 -2 | 0 -5 -13 17 }}
{{Mapping|legend=1| 1 -2 -7 15 | 0 5 13 -17 }}


{{Multival|legend=1| 5 13 -17 9 -41 -76 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6100{{c}}, ~105/64 = 860.2881{{c}}
: [[error map]]: {{val| -0.390 +0.266 +0.162 +0.426 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~105/64 = 860.5650{{c}}
: error map: {{val| 0.000 +0.870 +1.032 +1.568 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 339.432
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99, 152, 251, 350, 601cd, 905bcdd }}


{{Optimal ET sequence|legend=1| 7, 32c, 39, 46, 53, 99, 251, 350, 601cd, 951bcdd }}
[[Badness]] (Sintel): 0.598
 
[[Badness]]: 0.023649


=== 11-limit ===
=== 11-limit ===
Line 53: Line 59:
Comma list: 540/539, 4375/4374, 5120/5103
Comma list: 540/539, 4375/4374, 5120/5103


Mapping: {{mapping| 1 3 6 -2 21 | 0 -5 -13 17 -62 }}
Mapping: {{mapping| 1 -2 -7 15 -41 | 0 5 13 -17 62 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.464
Optimal tunings:
* WE: ~2 = 1199.5961{{c}}, ~105/64 = 860.2460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5314{{c}}


{{Optimal ET sequence|legend=1| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}
{{Optimal ET sequence|legend=0| 46e, 53, 99e, 152, 555dee, 707ddee, 859bddee }}


Badness: 0.031506
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 540/539, 625/624, 847/845
Comma list: 352/351, 540/539, 625/624, 729/728
 
Mapping: {{mapping| 1 -2 -7 15 -41 -30 | 0 5 13 -17 62 47 }}
 
Optimal tunings:
* WE: ~2 = 1199.5437{{c}}, ~105/64 = 860.1922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.5140{{c}}
 
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152f }} *
 
<nowiki/>* optimal patent val: 205
 
Badness (Sintel): 1.16
 
==== 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 -2 -7 15 -41 -30 17 | 0 5 13 -17 62 47 -18 }}
 
Optimal tunings:
* WE: ~2 = 1199.3376{{c}}, ~28/17 = 860.0617{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.5317{{c}}
 
{{Optimal ET sequence|legend=0| 46ef, 53, 99ef, 152fg }}
 
Badness (Sintel): 1.33
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


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


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.481
Mapping: {{mapping| 1 -2 -7 15 -41 -30 17 -23 | 0 5 13 -17 62 47 -18 38 }}


{{Optimal ET sequence|legend=1| 46ef, 53, 99ef, 152f }} <nowiki>*</nowiki>
Optimal tunings:
* WE: ~2 = 1199.3361{{c}}, ~28/17 = 860.0605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.5310{{c}}


<nowiki>*</nowiki> optimal patent val: [[205edo|205]]
{{Optimal ET sequence|legend=0| 46efh, 53, 99ef, 152fg }}


Badness: 0.028008
Badness (Sintel): 1.14


=== Hitchcock ===
=== Hitchcock ===
Line 81: Line 121:
Comma list: 121/120, 176/175, 2200/2187
Comma list: 121/120, 176/175, 2200/2187


Mapping: {{mapping| 1 3 6 -2 6 | 0 -5 -13 17 -9 }}
Mapping: {{mapping| 1 -2 -7 15 -3 | 0 5 13 -17 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.390
Optimal tunings:
* WE: ~2 = 1199.9979{{c}}, ~18/11 = 860.6089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6104{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness: 0.035187
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 94: Line 136:
Comma list: 121/120, 169/168, 176/175, 325/324
Comma list: 121/120, 169/168, 176/175, 325/324


Mapping: {{mapping| 1 3 6 -2 6 2 | 0 -5 -13 17 -9 6 }}
Mapping: {{mapping| 1 -2 -7 15 -3 8 | 0 5 13 -17 9 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.419
Optimal tunings:
* WE: ~2 = 1200.2037{{c}}, ~18/11 = 860.5820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.7268{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 53, 99 }}


Badness: 0.022448
Badness (Sintel): 0.928


==== 17-limit ====
==== 17-limit ====
Line 107: Line 151:
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272
Comma list: 121/120, 154/153, 169/168, 176/175, 273/272


Mapping: {{mapping| 1 3 6 -2 6 2 -1 | 0 -5 -13 17 -9 6 18 }}
Mapping: {{mapping| 1 -2 -7 15 -3 8 17 | 0 5 13 -17 9 -6 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.366
Optimal tunings:
* WE: ~2 = 1200.0179{{c}}, ~18/11 = 860.6472{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.6344{{c}}


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 99 }}
{{Optimal ET sequence|legend=0| 7, 39, 46, 99 }}


Badness: 0.019395
Badness (Sintel): 0.988


==== 19-limit ====
==== 19-limit ====
Line 120: Line 166:
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189
Comma list: 121/120, 154/153, 169/168, 171/170, 176/175, 190/189


Mapping: {{mapping| 1 3 6 -2 6 2 -1 0 | 0 -5 -13 17 -9 6 18 15 }}
Mapping: {{mapping| 1 -2 -7 15 -3 8 17 15 | 0 5 13 -17 9 -6 -18 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.407
Optimal tunings:
* WE: ~2 = 1200.1977{{c}}, ~18/11 = 860.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.5917{{c}}


{{Optimal ET sequence|legend=1| 7, 39h, 46, 53, 99h }}
{{Optimal ET sequence|legend=0| 7, 46, 53, 99h }}


Badness: 0.017513
Badness (Sintel): 1.07


=== Catamite ===
=== Stalagmite ===
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."
The stalagmite temperament (46 & 99ef) tempers out 441/440 ([[werckisma]]) and 896/891 ([[pentacircle comma]]) 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
Line 135: Line 183:
Comma list: 441/440, 896/891, 4375/4374
Comma list: 441/440, 896/891, 4375/4374


Mapping: {{mapping| 1 3 6 -2 -7 | 0 -5 -13 17 37 }}
Mapping: {{mapping| 1 -2 -7 15 30 | 0 5 13 -17 -37 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.340
Optimal tunings:
* WE: ~2 = 1199.3915{{c}}, ~105/64 = 860.2240{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6649{{c}}


{{Optimal ET sequence|legend=1| 46, 99e, 145, 244e }}
{{Optimal ET sequence|legend=0| 46, 99e, 145, 244e }}


Badness: 0.040976
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 148: Line 198:
Comma list: 196/195, 352/351, 364/363, 4375/4374
Comma list: 196/195, 352/351, 364/363, 4375/4374


Mapping: {{mapping| 1 3 6 -2 -7 -11 | 0 -5 -13 17 37 52 }}
Mapping: {{mapping| 1 -2 -7 15 30 41 | 0 5 13 -17 -37 -52 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 339.313
Optimal tunings:
* WE: ~2 = 1199.3577{{c}}, ~105/64 = 860.2259{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~105/64 = 860.6929{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.034215
Badness (Sintel): 1.41


==== 17-limit ====
==== 17-limit ====
Line 161: Line 213:
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155
Comma list: 196/195, 256/255, 352/351, 364/363, 1156/1155


Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 | 0 -5 -13 17 37 52 18 }}
Mapping: {{mapping| 1 -2 -7 15 30 41 17 | 0 5 13 -17 -37 -52 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.313
Optimal tunings:
* WE: ~2 = 1199.3555{{c}}, ~28/17 = 860.2244{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6932{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.021193
Badness (Sintel): 1.08


==== 19-limit ====
==== 19-limit ====
Line 174: Line 228:
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475
Comma list: 196/195, 256/255, 343/342, 352/351, 364/363, 476/475


Mapping: {{mapping| 1 3 6 -2 -7 -11 -1 -13 | 0 -5 -13 17 37 52 18 61 }}
Mapping: {{mapping| 1 -2 -7 15 30 41 17 48 | 0 5 13 -17 -37 -52 -18 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 339.325
Optimal tunings:
* WE: ~2 = 1199.3599{{c}}, ~28/17 = 860.2161{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/17 = 860.6820{{c}}


{{Optimal ET sequence|legend=1| 46, 99ef, 145 }}
{{Optimal ET sequence|legend=0| 46, 99ef, 145 }}


Badness: 0.018864
Badness (Sintel): 1.15


=== Hemiamity ===
=== Hemiamity ===
Line 188: Line 244:


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 tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.561
Optimal tunings:
* WE: ~99/70 = 599.8218{{c}}, ~64/55 = 260.4833{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5615{{c}}


{{Optimal ET sequence|legend=1| 14cde, 46, 106, 152, 350, 502d }}
{{Optimal ET sequence|legend=0| 46, 106, 152, 350, 502d, 852bcdde }}


Badness: 0.031307
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Line 204: Line 261:
Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}
Mapping: {{mapping| 2 1 -1 13 13 20 | 0 5 13 -17 -14 -29 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~64/55 = 260.583
Optimal tunings:
* WE: ~99/70 = 599.7965{{c}}, ~64/55 = 260.4951{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/55 = 260.5872{{c}}


{{Optimal ET sequence|legend=1| 46, 106f, 152f, 198, 350f, 548cdff }}
{{Optimal ET sequence|legend=0| 46, 106f, 152f, 198, 350f, 548cdff }}


Badness: 0.025784
Badness (Sintel): 1.07


== Accord ==
== Accord ==
Accord tempers out [[126/125]], the starling comma, and may be described as the {{nowrap| 39d & 46 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 3 6 11 | 0 -5 -13 -29 }}
{{Mapping|legend=1| 1 -2 -7 -18 | 0 5 13 29 }}
 
{{Multival|legend=1| 5 13 29 9 32 31 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 338.993
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7458{{c}}, ~224/135 = 860.1071{{c}}
: [[error map]]: {{val| -1.254 +1.089 +3.858 -3.144 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 860.9433{{c}}
: error map: {{val| 0.000 +2.761 +5.949 -1.471 }}


{{Optimal ET sequence|legend=1| 7d, 39d, 46, 131c, 177c }}
{{Optimal ET sequence|legend=1| 7d, …, 39d, 46, 131c }}


[[Badness]]: 0.095612
[[Badness]] (Sintel): 2.42


=== 11-limit ===
=== 11-limit ===
Line 230: Line 293:
Comma list: 121/120, 126/125, 896/891
Comma list: 121/120, 126/125, 896/891


Mapping: {{mapping| 1 3 6 11 6 | 0 -5 -13 -29 -9 }}
Mapping: {{mapping| 1 -2 -7 -18 -3 | 0 5 13 29 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.047
Optimal tunings:
* WE: ~2 = 1199.3576{{c}}, ~18/11 = 860.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.9224{{c}}


{{Optimal ET sequence|legend=1| 7d, 39d, 46, 177c, 223bc, 269bce }}
{{Optimal ET sequence|legend=0| 7d, 39d, 46 }}


Badness: 0.042468
Badness (Sintel): 1.40


== 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 was named by [[Xenllium]] in 2021 for its close relation to the ''houboriz tuning'' (generator: 339.774971 cents).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 245: Line 310:
[[Comma list]]: 225/224, 1250000/1240029
[[Comma list]]: 225/224, 1250000/1240029


{{Mapping|legend=1| 1 3 6 13 | 0 -5 -13 -36 }}
{{Mapping|legend=1| 1 -2 -7 -23 | 0 5 13 36 }}
 
{{Multival|legend=1| 5 13 36 9 43 47 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.763
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4959{{c}}, ~400/243 = 860.5922{{c}}
: [[error map]]: {{val| +0.496 +0.014 -2.086 +1.090 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/243 = 860.2558{{c}}
: error map: {{val| 0.000 -0.676 -2.989 +0.382 }}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 113, 166 }}
{{Optimal ET sequence|legend=1| 7d, …, 46d, 53, 113, 166 }}


[[Badness]]: 0.066638
[[Badness]] (Sintel): 1.69


=== 11-limit ===
=== 11-limit ===
Line 260: Line 327:
Comma list: 225/224, 385/384, 1250000/1240029
Comma list: 225/224, 385/384, 1250000/1240029


Mapping: {{mapping| 1 3 6 13 -9 | 0 -5 -13 -36 44 }}
Mapping: {{mapping| 1 -2 -7 -23 35 | 0 5 13 36 -44 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.763
Optimal tunings:
* WE: ~2 = 1200.4973{{c}}, ~400/243 = 860.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.2386{{c}}


{{Optimal ET sequence|legend=1| 53, 113, 166 }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness: 0.067891
Badness (Sintel): 2.24


==== 13-limit ====
==== 13-limit ====
Line 273: Line 342:
Comma list: 225/224, 325/324, 385/384, 2200/2197
Comma list: 225/224, 325/324, 385/384, 2200/2197


Mapping: {{mapping| 1 3 6 13 -9 2 | 0 -5 -13 -36 44 6 }}
Mapping: {{mapping| 1 -2 -7 -23 35 8 | 0 5 13 36 -44 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~39/32 = 339.764
Optimal tunings:
* WE: ~2 = 1200.5022{{c}}, ~64/39 = 860.5964{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/39 = 860.2378{{c}}


{{Optimal ET sequence|legend=1| 53, 113, 166 }}
{{Optimal ET sequence|legend=0| 53, 113, 166 }}


Badness: 0.032996
Badness (Sintel): 1.36


=== Houbor ===
=== Houbor ===
Line 286: Line 357:
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 -2 -7 -23 -3 | 0 5 13 36 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.814
Optimal tunings:
* WE: ~2 = 1201.1684{{c}}, ~18/11 = 861.0240{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2273{{c}}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
{{Optimal ET sequence|legend=0| 7d, 53, 60e, 113e }}


Badness: 0.045232
Badness (Sintel): 1.49


==== 13-limit ====
==== 13-limit ====
Line 299: Line 372:
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 -2 -7 -23 -3 8 | 0 -5 -13 -36 -9 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 339.784
Optimal tunings:
* WE: ~2 = 1200.9339{{c}}, ~18/11 = 860.8859{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 860.2376{{c}}


{{Optimal ET sequence|legend=1| 7d, 46d, 53, 60e, 113e }}
{{Optimal ET sequence|legend=0| 7d, 53, 113e }}


Badness: 0.031331
Badness (Sintel): 1.29


== 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. It was named by [[Xenllium]] in 2021, with the prefix ''para-'' signifying its relation to amity.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 314: Line 389:
[[Comma list]]: 65625/65536, 1600000/1594323
[[Comma list]]: 65625/65536, 1600000/1594323


{{Mapping|legend=1|1 3 6 -17 | 0 -5 -13 70 }}
{{Mapping|legend=1| 1 -2 -7 53 | 0 5 13 -70 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~243/200 = 339.553
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0199{{c}}, ~400/243 = 860.4612{{c}}
: [[error map]]: {{val| +0.020 +0.311 -0.458 -0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/243 = 860.4468{{c}}
: error map: {{val| 0.000 +0.279 -0.505 -0.103 }}


{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986, 1297c, 2283bc }}
{{Optimal ET sequence|legend=1| 53, 205d, 258, 311, 675, 986, 1297c }}


[[Badness]]: 0.113655
[[Badness]] (Sintel): 2.88


=== 11-limit ===
=== 11-limit ===
Line 327: Line 406:
Comma list: 6250/6237, 19712/19683, 41503/41472
Comma list: 6250/6237, 19712/19683, 41503/41472


Mapping: {{mapping| 1 3 6 -17 36 | 0 -5 -13 70 -115 }}
Mapping: {{mapping| 1 -2 -7 53 -79 | 0 5 13 -70 115 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* WE: ~2 = 1200.0139{{c}}, ~400/243 = 860.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/243 = 860.4462{{c}}


{{Optimal ET sequence|legend=1| 53, 205de, 258, 311, 675, 986 }}
{{Optimal ET sequence|legend=0| 53, 258, 311, 675, 986 }}


Badness: 0.064853
Badness (Sintel): 2.14


=== 13-limit ===
=== 13-limit ===
Line 340: Line 421:
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683
Comma list: 625/624, 2080/2079, 2200/2197, 19712/19683


Mapping: {{mapping| 1 3 6 -17 36 17 | 0 -5 -13 70 -115 -47 }}
Mapping: {{mapping| 1 -2 -7 53 -79 -30 | 0 5 13 -70 115 47 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.554
Optimal tunings:
* WE: ~2 = 1199.9945{{c}}, ~385/234 = 860.4417{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4456{{c}}


{{Optimal ET sequence|legend=1| 53, 205de, 258, 311, 675, 986, 1661cf }}
{{Optimal ET sequence|legend=0| 53, 258, 311, 675, 986 }}


Badness: 0.030347
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 353: Line 436:
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430
Comma list: 625/624, 1225/1224, 2080/2079, 2200/2197, 2431/2430


Mapping: {{mapping| 1 3 6 -17 36 17 -31 | 0 -5 -13 70 -115 -47 124 }}
Mapping: {{mapping| 1 -2 -7 53 -79 -30 93 | 0 5 13 -70 115 47 -124 }}


Optimal tuning (POTE): ~2 = 1\1, ~243/200 = 339.555
Optimal tunings:
* WE: ~2 = 1200.0026{{c}}, ~385/234 = 860.4468{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/234 = 860.4450{{c}}


{{Optimal ET sequence|legend=1| 53, 205deg, 258g, 311, 675, 1661cf, 2336bccf, 3011bccf }}
{{Optimal ET sequence|legend=0| 53, 311, 675 }}


Badness: 0.024118
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 366: Line 451:
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197
Comma list: 625/624, 1225/1224, 1540/1539, 1729/1728, 2080/2079, 2200/2197


Mapping: {{mapping| 1 3 6 -17 36 17 -31 15 | 0 -5 -13 70 -115 -47 124 -38 }}
Mapping: {{mapping| 1 -2 -7 53 -79 -30 93 -23 | 0 5 13 -70 115 47 -124 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~208/171 = 339.555
Optimal tunings:
* WE: ~2 = 1200.0196{{c}}, ~208/171 = 860.4594{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~208/171 = 860.4454{{c}}


{{Optimal ET sequence|legend=1| 53, 205deg, 258g, 311, 675, 986, 1661cfh }}
{{Optimal ET sequence|legend=0| 53, 311, 675, 986 }}


Badness: 0.017420
Badness (Sintel): 1.06


== Bamity ==
== Bamity ==
Line 382: Line 469:


{{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


{{Multival|legend=1| 10 26 12 18 -9 -45 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/240 = 600.3379{{c}}, ~7/6 = 260.5488{{c}}
[[Optimal tuning]] ([[POTE]]): ~343/240 = 1\2, ~7/6 = 260.402
: [[error map]]: {{val| +0.676 +1.127 +0.483 -4.519 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~7/6 = 260.4686{{c}}
: error map: {{val| 0.000 +0.388 -0.222 -6.014 }}


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


[[Badness]]: 0.083601
[[Badness]] (Sintel): 2.12


=== 11-limit ===
=== 11-limit ===
Line 400: Line 488:
Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}
Mapping: {{mapping| 2 1 -1 3 3 | 0 5 13 6 9 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~7/6 = 260.393
Optimal tunings:
* WE: ~99/70 = 600.5039{{c}}, ~7/6 = 260.6119{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~7/6 = 260.4948{{c}}


{{Optimal ET sequence|legend=1| 14c, 32c, 46, 60e, 106de }}
{{Optimal ET sequence|legend=0| 14c, 32c, 46, 60e, 106de }}


Badness: 0.035504
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 413: Line 503:
Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}
Mapping: {{mapping| 2 1 -1 3 3 0 | 0 5 13 6 9 17 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 260.618
Optimal tunings:
* WE: ~55/39 = 600.3896{{c}}, ~7/6 = 260.7874{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~7/6 = 260.6927{{c}}


{{Optimal ET sequence|legend=1| 14cf, 32cf, 46, 106def, 152def }}
{{Optimal ET sequence|legend=0| 14cf, 32cf, 46, 106deff }}


Badness: 0.030885
Badness (Sintel): 1.28


== Hamity ==
== Hamity ==
Line 426: Line 518:
[[Comma list]]: 2430/2401, 4000/3969  
[[Comma list]]: 2430/2401, 4000/3969  


{{Mapping|legend=1| 1 8 19 15 | 0 -10 -26 -19 }}
{{Mapping|legend=1| 1 -2 -7 -4 | 0 10 26 19 }}
 
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~14/9


{{Multival|legend=1| 10 26 19 18 2 -29 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.5499{{c}}, ~9/7 = 430.0579{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 430.219
: [[error map]]: {{val| -0.450 -0.476 -1.657 +4.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 430.1958{{c}}
: error map: {{val| 0.000 +0.003 -1.222 +4.895 }}


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


[[Badness]]: 0.073956
[[Badness]] (Sintel): 1.87


=== 11-limit ===
=== 11-limit ===
Line 443: Line 536:
Comma list: 99/98, 121/120, 2200/2187
Comma list: 99/98, 121/120, 2200/2187


Mapping: {{mapping| 1 8 19 15 15 | 0 -10 -26 -19 -18 }}
Mapping: {{mapping| 1 -2 -7 -4 -3 | 0 10 26 19 18 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.192
Optimal tunings:
* WE: ~2 = 1200.2215{{c}}, ~9/7 = 430.2713{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2036{{c}}


{{Optimal ET sequence|legend=1| 14c, 39d, 53 }}
{{Optimal ET sequence|legend=0| 14c, 39d, 53 }}


Badness: 0.042947
Badness (Sintel): 1.42


=== 13-limit ===
=== 13-limit ===
Line 458: Line 553:
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}
Mapping: {{mapping| 1 8 19 15 15 30 | 0 -10 -26 -19 -18 -41 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 430.216
Optimal tunings:
* WE: ~2 = 1200.1379{{c}}, ~9/7 = 430.2658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 430.2227{{c}}


{{Optimal ET sequence|legend=1| 14cf, 39df, 53 }}
{{Optimal ET sequence|legend=0| 14cf, 39df, 53 }}


Badness: 0.029753
Badness (Sintel): 1.23


== 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 the grave major sixth in three. It was named by [[Xenllium]] in 2021 as a contraction of ''gamelismic'' and ''amity''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 471: Line 568:
[[Comma list]]: 1029/1024, 1071875/1062882
[[Comma list]]: 1029/1024, 1071875/1062882


{{Mapping|legend=1| 1 13 32 -1 | 0 -15 -39 5 }}
{{Mapping|legend=1| 1 -2 -7 4 | 0 15 39 -5 }}
 
: mapping generators: ~2, ~189/160
: mapping generators: ~2, ~320/189


{{Multival|legend=1| 15 39 -5 27 -50 -121 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.4434{{c}}, ~189/160 = 286.8930{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 286.787
: [[error map]]: {{val| +0.443 +0.553 -0.591 -1.571 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/160 = 286.7955{{c}}
: error map: {{val| 0.000 -0.022 -1.288 -2.804 }}


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


[[Badness]]: 0.125733
[[Badness]] (Sintel): 3.18


=== 11-limit ===
=== 11-limit ===
Line 488: Line 586:
Comma list: 385/384, 441/440, 1071875/1062882
Comma list: 385/384, 441/440, 1071875/1062882


Mapping: {{mapping| 1 13 32 -1 -11 | 0 -15 -39 5 19 }}
Mapping: {{mapping| 1 -2 -7 4 8 | 0 15 39 -5 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 286.797
Optimal tunings:
* WE: ~2 = 1200.3315{{c}}, ~33/28 = 286.8764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/28 = 286.8002{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness: 0.051111
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 501: Line 601:
Comma list: 325/324, 364/363, 385/384, 10985/10976
Comma list: 325/324, 364/363, 385/384, 10985/10976


Mapping: {{mapping| 1 13 32 -1 -11 -10 | 0 -15 -39 5 19 18 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 | 0 15 39 -5 -19 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.789
Optimal tunings:
* WE: ~2 = 1200.4248{{c}}, ~13/11 = 286.8906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7912{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159, 364df, 523ddff }}


Badness: 0.030297
Badness (Sintel): 1.25


=== 17-limit ===
=== 17-limit ===
Line 514: Line 616:
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757
Comma list: 273/272, 325/324, 364/363, 385/384, 3773/3757


Mapping: {{mapping| 1 13 32 -1 -11 -10 -2 | 0 -15 -39 5 19 18 8 }}
Mapping: {{mapping| 1 -2 -7 4 8 8 6 | 0 15 39 -5 -19 -18 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 286.795
Optimal tunings:
* WE: ~2 = 1200.3287{{c}}, ~13/11 = 286.8731{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 286.7957{{c}}


{{Optimal ET sequence|legend=1| 46, 113, 159 }}
{{Optimal ET sequence|legend=0| 46, 113, 159 }}


Badness: 0.022036
Badness (Sintel): 1.12


== 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. [[311edo|178\311]] is a recommendable generator. It was named by [[Xenllium]] in 2022 for the fact that it splits amity's generator in three.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 529: Line 633:
[[Comma list]]: 703125/702464, 1600000/1594323
[[Comma list]]: 703125/702464, 1600000/1594323


{{Mapping|legend=1| 1 8 19 46 | 0 -15 -39 -101 }}
{{Mapping|legend=1| 1 -7 -20 -55 | 0 15 39 101 }}
 
: mapping generators: ~2, ~125/84
{{Multival|legend=1| 15 39 101 27 118 125 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~168/125 = 513.178
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9474{{c}}, ~125/84 = 686.7921{{c}}
: [[error map]]: {{val| -0.053 +0.295 -0.369 +0.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~125/84 = 686.8215{{c}}
: error map: {{val| 0.000 +0.367 -0.276 +0.144 }}


{{Optimal ET sequence|legend=1| 152, 311, 463, 774 }}
{{Optimal ET sequence|legend=1| 7d, …, 152, 311, 463, 774 }}


[[Badness]]: 0.119453
[[Badness]] (Sintel): 3.02


=== 11-limit ===
=== 11-limit ===
Line 544: Line 651:
Comma list: 3025/3024, 4000/3993, 19712/19683
Comma list: 3025/3024, 4000/3993, 19712/19683


Mapping: {{mapping| 1 8 19 46 18 | 0 -15 -39 -101 -34 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 | 0 15 39 101 34 }}


Optimal tuning (POTE): ~2 = 1\1, ~121/90 = 513.177
Optimal tunings:
* WE: ~2 = 1199.9090{{c}}, ~125/84 = 686.7710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~125/84 = 686.8218{{c}}


{{Optimal ET sequence|legend=1| 152, 311, 463, 774, 1237e }}
{{Optimal ET sequence|legend=0| 7d, …, 152, 311, 463, 774, 1237e }}


Badness: 0.031296
Badness (Sintel): 1.03


=== 13-limit ===
=== 13-limit ===
Line 557: Line 666:
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689
Comma list: 625/624, 1575/1573, 2080/2079, 13720/13689


Mapping: {{mapping| 1 8 19 46 18 64 | 0 -15 -39 -101 -34 -141 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 | 0 15 39 101 34 141 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.182
Optimal tunings:
* WE: ~2 = 1199.9531{{c}}, ~52/35 = 686.7909{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8172{{c}}


{{Optimal ET sequence|legend=1| 152f, 311 }}
{{Optimal ET sequence|legend=0| 7df, …, 152f, 311 }}


Badness: 0.026418
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 570: Line 681:
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619
Comma list: 595/594, 625/624, 833/832, 1575/1573, 8624/8619


Mapping: {{mapping| 1 8 19 46 18 64 -22 | 0 -15 -39 -101 -34 -141 61 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 | 0 15 39 101 34 141 -61 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
Optimal tunings:
* WE: ~2 = 1200.0347{{c}}, ~52/35 = 686.8342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8146{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdg, 1403cdg, 1714cdeg }}
{{Optimal ET sequence|legend=0| 152f, 159, 311, 1092cdg }}


Badness: 0.025588
Badness (Sintel): 1.30


=== 19-limit ===
=== 19-limit ===
Line 583: Line 696:
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573
Comma list: 595/594, 625/624, 833/832, 969/968, 1216/1215, 1575/1573


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 | 0 -15 -39 -101 -34 -141 61 -114 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 | 0 15 39 101 34 141 -61 114 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* WE: ~2 = 1200.0406{{c}}, ~52/35 = 686.8382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8152{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1403cdgh, 1714cdegh, 2025cdefgghh, 2336bccdefgghh }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.018412
Badness (Sintel): 1.12


=== 23-limit ===
=== 23-limit ===
Line 596: Line 711:
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104
Comma list: 595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 | 0 -15 -39 -101 -34 -141 61 -114 -104 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 | 0 15 39 101 34 141 -61 114 104 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.185
Optimal tunings:
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8148{{c}}


{{Optimal ET sequence|legend=1| 152f, 159, 311, 1092cdgh, 1403cdgh, 1714cdeghi }}
{{Optimal ET sequence|legend=0| 152f, 159, 311 }}


Badness: 0.014343
Badness (Sintel): 1.03


=== 29-limit ===
=== 29-limit ===
Line 609: Line 726:
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044
Comma list: 595/594, 625/624, 760/759, 784/783, 833/832, 875/874, 969/968, 1045/1044


Mapping: {{mapping| 1 8 19 46 18 64 -22 53 49 72 | 0 -15 -39 -101 -34 -141 61 -114 -104 -157 }}
Mapping: {{mapping| 1 -7 -20 -55 -16 -77 39 -61 -55 -85 | 0 15 39 101 34 141 -61 114 104 157 }}
 
Optimal tunings:
* WE: ~2 = 1200.0311{{c}}, ~52/35 = 686.8318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~52/35 = 686.8142{{c}}
 
{{Optimal ET sequence|legend=0| 152fj, 159, 311, 781dh }}
 
Badness (Sintel): 1.01
 
== Chromat ==
Chromat has a period of 1/3 octave and tempers out the [[hemimage comma]] (10976/10935) and the [[triwellisma]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 235298/234375
 
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/26 = 513.186
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


{{Optimal ET sequence|legend=1| 152fj, 159, 311, 781dh, 1092cdgh, 1403cdgh }}
[[Badness]] (Sintel): 1.46


Badness: 0.012038
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Amicable ==
Comma list: 441/440, 4375/4356, 10976/10935
The amicable temperament (99 & 311) tempers out the [[amity comma]] and the [[canousma]] in addition to the breedsma, and is closely associated with the canou temperament.  
 
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}
 
Badness (Sintel): 1.67
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 625/624, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}
 
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}
 
Badness (Sintel): 1.90
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
 
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
 
Badness (Sintel): 1.61
 
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.81
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
 
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
 
Badness (Sintel): 1.54
 
==== Chromic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1875/1859
 
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}


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.
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


[[Subgroup]]: 2.3.5.7
Badness (Sintel): 2.06


[[Comma list]]: 2401/2400, 1600000/1594323
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Mapping|legend=1| 1 3 6 5 | 0 -20 -52 -31 }}
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


{{Multival|legend=1| 20 52 31 36 -7 -74 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 84.880
Optimal tunings:
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}


{{Optimal ET sequence|legend=1| 99, 212, 311, 410, 1131, 1541b }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


[[Badness]]: 0.045473
Badness (Sintel): 1.58


=== Amical ===
=== Hemichromat ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 131072/130977, 1600000/1594323
Comma list: 3025/3024, 10976/10935, 102487/102400


Mapping: {{mapping| 1 3 6 5 -8 | 0 -20 -52 -31 162 }}
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8843
Optimal tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}


{{Optimal ET sequence|legend=1| 99, 212e, 311, 410, 721, 1032, 1343 }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.100668
Badness (Sintel): 2.22


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}
 
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness (Sintel): 1.38
 
== Amicable ==
Amicable tempers out the [[breedsma]] as well as the [[canousma]], and may be described as the {{nowrap| 99 & 311 }} temperament.
 
There are multiple reasonable places for primes 11 and 13 in the interval chain. Amorous (99ef & 311) has these primes placed on the same side of the interval chain so it blends smarter with the lower harmonics. Amical (99 & 311) shows a slightly more accurate but much more complex alternative. Pseudoamorous (99ef & 113) and pseudoamical (99 & 113) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semi-octave 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: {{mapping| 1 3 6 5 -8 -5 | 0 -20 -52 -31 162 123 }}
{{Mapping|legend=1| 1 -17 -46 -26 | 0 20 52 31 }}
: mapping generators: ~2, ~40/21


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8838
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9463{{c}}, ~40/21 = 1115.0701{{c}}
: [[error map]]: {{val| -0.054 +0.361 -0.195 -0.254 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1187{{c}}
: error map: {{val| 0.000 +0.419 -0.142 -0.146 }}


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


Badness: 0.049893
[[Badness]] (Sintel): 1.15


=== Amorous ===
=== Amorous ===
Line 667: Line 918:
Comma list: 2401/2400, 6250/6237, 19712/19683
Comma list: 2401/2400, 6250/6237, 19712/19683


Mapping: {{mapping| 1 3 6 5 14 | 0 -20 -52 -31 -149 }}
Mapping: {{mapping| 1 -17 -46 -26 -135 | 0 20 52 31 149 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8896
Optimal tunings:
* WE: ~2 = 1200.0380{{c}}, ~40/21 = 1115.1458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1109{{c}}


{{Optimal ET sequence|legend=1| 99e, 212, 311, 1145c, 1456cd }}
{{Optimal ET sequence|legend=0| 99e, 212, 311, 1145c, 1456cd }}


Badness: 0.048924
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 680: Line 933:
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647
Comma list: 625/624, 2080/2079, 2401/2400, 10648/10647


Mapping: {{mapping| 1 3 6 5 14 17 | 0 -20 -52 -31 -149 -188 }}
apping: {{mapping| 1 -17 -46 -26 -135 -171 | 0 20 52 31 149 188 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8910
Optimal tunings:
* WE: ~2 = 1200.0543{{c}}, ~40/21 = 1115.1594{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1095{{c}}


{{Optimal ET sequence|legend=1| 99ef, 212, 311, 834, 1145c }}
{{Optimal ET sequence|legend=0| 99ef, 212, 311, 834, 1145c }}


Badness: 0.034681
Badness (Sintel): 1.43


=== Pseudoamical ===
=== Amical ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1375/1372, 1600000/1594323
Comma list: 2401/2400, 131072/130977, 1600000/1594323


Mapping: {{mapping| 1 3 6 5 -1 | 0 -20 -52 -31 63 }}
Mapping: {{mapping| 1 -17 -46 -26 154 | 0 20 52 31 -162 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9091
Optimal tunings:
* WE: ~2 = 1200.0008{{c}}, ~40/21 = 1115.1165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1157{{c}}


{{Optimal ET sequence|legend=1| 99, 113, 212, 961ccdeee }}
{{Optimal ET sequence|legend=0| 99, 212e, 311, 721, 1032, 1343 }}


Badness: 0.085837
Badness (Sintel): 3.33


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 385/384, 1375/1372, 19773/19712
Comma list: 2080/2079, 2401/2400, 4096/4095, 741125/739206


Mapping: {{mapping| 1 3 6 5 -1 2 | 0 -20 -52 -31 63 24 }}
Mapping: {{mapping| 1 -17 -46 -26 154 118 | 0 20 52 31 -162 -123 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9127
Optimal tunings:
* WE: ~2 = 1199.9926{{c}}, ~40/21 = 1115.1094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.1162{{c}}


{{Optimal ET sequence|legend=1| 99, 113, 212, 537cdeff, 749ccdeefff }}
{{Optimal ET sequence|legend=0| 99, 212ef, 311, 721, 1032 }}


Badness: 0.047025
Badness (Sintel): 2.06


=== Pseudoamorous ===
=== Pseudoamorous ===
Line 719: Line 978:
Comma list: 243/242, 441/440, 980000/970299
Comma list: 243/242, 441/440, 980000/970299


Mapping: {{mapping| 1 3 6 5 7 | 0 -20 -52 -31 -50 }}
Mapping: {{mapping| 1 -17 -46 -26 -43 | 0 20 52 31 50 }}
 
Optimal tunings:
* WE: ~2 = 1199.8269{{c}}, ~21/11 = 1114.9475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1115.1038{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.8917
{{Optimal ET sequence|legend=0| 99e, 212e }} *


{{Optimal ET sequence|legend=1| 99e, 212e }}
<nowiki/>* optimal patent val: [[113edo|113]]


Badness: 0.056583
Badness (Sintel): 1.87


==== 13-limit ====
==== 13-limit ====
Line 732: Line 995:
Comma list: 243/242, 364/363, 441/440, 1875/1859
Comma list: 243/242, 364/363, 441/440, 1875/1859


Mapping: {{mapping| 1 3 6 5 7 10 | 0 -20 -52 -31 -50 -89 }}
Mapping: {{mapping| 1 -17 -46 -26 -43 -79 | 0 20 52 31 50 89 }}
 
Optimal tunings:
* WE: ~2 = 1200.0182{{c}}, ~40/21 = 1115.1005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0840{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 113, 212ef }}
 
Badness (Sintel): 1.77
 
=== Pseudoamical ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 1600000/1594323
 
Mapping: {{mapping| 1 -17 -46 -26 62 | 0 20 52 31 -63 }}
 
Optimal tunings:
* WE: ~2 = 1200.3436{{c}}, ~40/21 = 1115.4102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1115.0925{{c}}
 
{{Optimal ET sequence|legend=0| 99, 113, 212, 961ccdeee }}
 
Badness (Sintel): 2.84
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 1375/1372, 19773/19712
 
Mapping: {{mapping| 1 -17 -46 -26 62 26 | 0 20 52 31 -63 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 84.9164
Optimal tunings:
* WE: ~2 = 1200.4212{{c}}, ~40/21 = 1115.4787{{c}}
* CWE: ~2 = 1200.0000{{c}}, 40/21 = 1115.0885{{c}}


{{Optimal ET sequence|legend=1| 99ef, 113, 212ef }}
{{Optimal ET sequence|legend=0| 99, 113, 212, 537cdeff, 749ccdeefff }}


Badness: 0.042826
Badness (Sintel): 1.94


=== Floral ===
=== Floral ===
Line 745: Line 1,040:
Comma list: 2401/2400, 9801/9800, 14641/14580
Comma list: 2401/2400, 9801/9800, 14641/14580


Mapping: {{mapping| 2 6 12 10 13 | 0 -20 -52 -31 -43 }}
Mapping: {{mapping| 2 -14 -40 -21 -30 | 0 20 52 31 43 }}
: mapping generators: ~99/70, ~66/49


Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8788
Optimal tunings:
* WE: ~99/70 = 600.0052{{c}}, ~66/49 = 515.1257{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1215{{c}}


{{Optimal ET sequence|legend=1| 198, 212, 410 }}
{{Optimal ET sequence|legend=0| 14c, … 198, 212, 410 }}


Badness: 0.065110
Badness (Sintel): 2.15


==== 13-limit ====
==== 13-limit ====
Line 758: Line 1,056:
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580
Comma list: 676/675, 1001/1000, 1716/1715, 14641/14580


Mapping: {{mapping| 2 6 12 10 13 19 | 0 -20 -52 -31 -43 -82 }}
Mapping: {{mapping| 2 -14 -40 -21 -30 -63 | 0 20 52 31 43 82 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~21/20 = 84.8750
Optimal tunings:
* WE: ~99/70 = 599.9923{{c}}, ~66/49 = 515.1184{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/49 = 515.1246{{c}}


{{Optimal ET sequence|legend=1| 198, 410 }}
{{Optimal ET sequence|legend=0| 14cf, …, 198, 410 }}


Badness: 0.037013
Badness (Sintel): 1.53


=== Humorous ===
=== Humorous ===
Line 771: Line 1,071:
Comma list: 2401/2400, 3025/3024, 1600000/1594323
Comma list: 2401/2400, 3025/3024, 1600000/1594323


Mapping: {{mapping| 1 3 6 5 3 | 0 -40 -104 -62 13 }}
Mapping: {{mapping| 1 -37 -98 -57 16 | 0 40 104 62 -13 }}
: mapping generators: ~2, ~3993/2048


Optimal tuning (POTE): ~2 = 1\1, ~4096/3993 = 42.4391
Optimal tunings:
* WE: ~2 = 1199.9189{{c}}, ~3993/2048 = 1157.4826{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3993/2048 = 1157.5602{{c}}


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820 }}
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820 }}


Badness: 0.058249
Badness (Sintel): 1.93


==== 13-limit ====
==== 13-limit ====
Line 784: Line 1,087:
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024
Comma list: 2080/2079, 2200/2197, 2401/2400, 3025/3024


Mapping: {{mapping| 1 3 6 5 3 6 | 0 -40 -104 -62 13 -65 }}
Mapping: {{mapping| 1 -37 -98 -57 16 -59 | 0 40 104 62 -13 65 }}
 
Optimal tunings:
* WE: ~2 = 1199.8903{{c}}, ~39/20 = 1157.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~39/20 = 1157.5600{{c}}
 
{{Optimal ET sequence|legend=0| 85c, 113, 198, 311, 509, 820f }}
 
Badness (Sintel): 1.17
 
== Witcher ==
Witcher tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 99 & 166 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 1600000/1594323
 
{{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }}
: mapping generators: ~2, ~80/63
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8698{{c}}, ~80/63 = 412.0531{{c}}
: [[error map]]: {{val| -0.130 +0.283 -0.260 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/63 = 412.0956{{c}}
: error map: {{val| 0.000 +0.435 -0.100 +0.512 }}
 
{{Optimal ET sequence|legend=1| 32c, 67c, 99, 265, 364, 463, 562, 1025 }}
 
[[Badness]] (Sintel): 2.04
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 42.4391
Comma list: 540/539, 4375/4356, 1240029/1239040


{{Optimal ET sequence|legend=1| 85c, 113, 198, 311, 509, 820f }}
Mapping: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }}


Badness: 0.028267
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~80/63 = 412.0758{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/63 = 412.0788{{c}}
 
{{Optimal ET sequence|legend=0| 99e, 166, 265 }}
 
Badness (Sintel): 2.73
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 4375/4356, 10648/10647
 
Mapping: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }}
 
Optimal tunings:
* WE: ~2 = 1200.0660{{c}}, ~33/26 = 412.0928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/26 = 412.0709{{c}}
 
{{Optimal ET sequence|legend=0| 99ef, 166, 265 }}
 
Badness (Sintel): 2.40
 
=== Semiwitcher ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 41503/41472, 585640/583443
 
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
: mapping generators: ~99/70, ~80/63
 
Optimal tunings:
* WE: ~99/70 = 599.9717{{c}}, ~80/63 = 412.0766{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~80/63 = 412.0951{{c}}
 
{{Optimal ET sequence|legend=0| 32c, 166, 198, 364, 562, 926 }}
 
Badness (Sintel): 2.92
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280
 
Mapping: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9666{{c}}, ~33/26 = 412.0726{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/26 = 412.0944{{c}}
 
{{Optimal ET sequence|legend=0| 32cf, 166, 198, 364, 562, 926 }}
 
Badness (Sintel): 1.57
 
=== Hitch ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 117649/117612, 1600000/1594323
 
Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560
 
Optimal tunings:
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}
 
{{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }}
 
Badness (Sintel): 3.09
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
 
Mapping: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }}
 
Optimal tunings:
* WE: ~2 = 1199.8415{{c}}, ~891/560 = 805.9451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0508{{c}}
 
{{Optimal ET sequence|legend=0| 198, 463, 661 }}
 
Badness (Sintel): 2.51
 
==== Hitcher ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
 
Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}
 
Optimal tunings:
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}
 
{{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }}
 
Badness (Sintel): 2.46


== Calamity ==
== Calamity ==
The calamity temperament (46 & 311) tempers out the [[rainy comma]], 2100875/2097152, splitting the interval of two octaves plus an acute minor third into five.  
Named by [[Flora Canou]] in 2023, calamity tempers out 2100875/2097152, the [[rainy comma]], and may be described as the {{nowrap| 46 & 311 }} temperament. It splits the interval of grave major sixth plus an octave into five.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 799: Line 1,231:
[[Comma list]]: 1600000/1594323, 2100875/2097152
[[Comma list]]: 1600000/1594323, 2100875/2097152


{{Mapping|legend=1| 1 13 32 -15 | 0 -25 -65 39 }}
{{Mapping|legend=1| 1 -12 -33 24 | 0 25 65 -39 }}
: mapping generators: ~2, ~35/24


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~48/35 = 547.9092
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0332{{c}}, ~35/24 = 652.1081{{c}}
: [[error map]]: {{val| +0.033 +0.348 -0.387 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 652.0902{{c}}
: error map: {{val| 0.000 +0.300 -0.452 -0.343 }}


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


[[Badness]]: 0.198
[[Badness]] (Sintel): 5.01


=== 11-limit ===
=== 11-limit ===
Line 812: Line 1,249:
Comma list: 3025/3024, 12005/11979, 131072/130977
Comma list: 3025/3024, 12005/11979, 131072/130977


Mapping: {{mapping| 1 13 32 -15 -18 | 0 -25 -65 39 47 }}
Mapping: {{mapping| 1 -12 -33 24 29 | 0 25 65 -39 -47 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.9080
Optimal tunings:
* WE: ~2 = 1199.9938{{c}}, ~35/24 = 652.0887{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/24 = 652.0920{{c}}


{{Optimal ET sequence|legend=1| 46, 219c, 265, 311, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 219c, 265, 311, 979, 1290 }}


Badness: 0.0604
Badness (Sintel): 2.00


=== 13-limit ===
=== 13-limit ===
Line 825: Line 1,264:
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979
Comma list: 2080/2079, 3025/3024, 4096/4095, 12005/11979


Mapping: {{mapping| 1 13 32 -15 -18 -31 | 0 -25 -65 39 47 76 }}
Mapping: {{mapping| 1 -12 -33 24 29 45 | 0 25 65 -39 -47 -76 }}


Optimal tuning (CTE): ~2 = 1\1, ~48/35 = 547.9066
Optimal tunings:
* WE: ~2 = 1199.9742{{c}}, ~35/24 = 652.0794{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/24 = 652.0934{{c}}


{{Optimal ET sequence|legend=1| 46, 265, 311, 668, 979, 1290 }}
{{Optimal ET sequence|legend=0| 46, 265, 311, 668, 979, 1290 }}


Badness: 0.0336
Badness (Sintel): 1.39


== 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.
Named by [[Xenllium]] in 2021, familia tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 113 & 152 }} temperament. It is generated by a major sixth, five of which octave reduced make the original generator of amity.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 840: Line 1,281:
[[Comma list]]: 16875/16807, 1600000/1594323
[[Comma list]]: 16875/16807, 1600000/1594323


{{Mapping|legend=1| 1 8 19 20 | 0 -25 -65 -67 }}
{{Mapping|legend=1| 1 -17 -46 -47 | 0 25 65 67 }}
 
: mapping generators: ~2, ~6561/3920
{{Multival|legend=1| 25 65 67 45 36 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11907/10000 = 307.941
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9165{{c}}, ~6561/3920 = 892.0267{{c}}
: [[error map]]: {{val| -0.084 +0.132 -0.736 +0.888 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6561/3920 = 892.0871{{c}}
: error map: {{val| 0.000 +0.224 -0.649 +1.013 }}


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417, 986d }}
{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 265, 417 }}


[[Badness]]: 0.144551
[[Badness]] (Sintel): 3.66


=== 11-limit ===
=== 11-limit ===
Line 855: Line 1,299:
Comma list: 540/539, 1375/1372, 1600000/1594323
Comma list: 540/539, 1375/1372, 1600000/1594323


Mapping: {{mapping| 1 8 19 20 5 | 0 -25 -65 -67 -6 }}
Mapping: {{mapping| 1 -17 -46 -47 -1 | 0 25 65 67 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3200/2673 = 307.906
Optimal tunings:
* WE: ~2 = 1199.7927{{c}}, ~2673/1600 = 891.9397{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2673/1600 = 892.0909{{c}}


{{Optimal ET sequence|legend=1| 39d, 74cd, 113, 152, 417, 569de, 721de }}
{{Optimal ET sequence|legend=0| 39d, 74cd, 113, 152, 417, 569de, 721de }}


Badness: 0.051740
Badness (Sintel): 1.71


=== 13-limit ===
=== 13-limit ===
Line 868: Line 1,314:
Comma list: 540/539, 729/728, 1375/1372, 2205/2197
Comma list: 540/539, 729/728, 1375/1372, 2205/2197


Mapping: {{mapping| 1 8 19 20 5 25 | 0 -25 -65 -67 -6 -83 }}
Mapping: {{mapping| 1 -17 -46 -47 -1 -58 | 0 25 65 67 6 83 }}


Optimal tuning (POTE): ~2 = 1\1, ~143/120 = 307.913
Optimal tunings:
* WE: ~2 = 1199.7844{{c}}, ~240/143 = 892.9268{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~240/143 = 892.0839{{c}}


{{Optimal ET sequence|legend=1| 39df, 74cdf, 113, 152f, 265, 417f }}
{{Optimal ET sequence|legend=0| 39df, 74cdf, 113, 152f, 265 }}


Badness: 0.038473
Badness (Sintel): 1.59


[[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]]
Retrieved from "https://en.xen.wiki/w/Amity_family"