Mabila family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''mabila family''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|tempers out]] {{monzo| 28 -3 -10 }} = 268435456/263671875 in the 5-limit. This gives a temperament structure superficially similar to [[mavila]], with extremely sharp fourths/flat fifths, three of which make a major third. However, unlike mavila, 10 of these bad fifths reach a more in tune one, which is useful for creating resolutions when using a large enough gamut, such as the [[9L 7s]] [[mos]] which has 3 good major &amp; minor chords.  
The '''mabila family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[mabila comma]] ({{monzo|legend=1| 28 -3 -10 }}, [[ratio]]: 268435456/263671875) in the [[5-limit]].  


== Mabila ==
== Mabila ==
: ''"Mabila" redirects here. For the temperaments that used to go by this name, see [[Semabila]] and [[Mabilic]].''
Mabila has a temperament structure superficially similar to [[mavila]], with extremely sharp fourths/flat fifths, three of which make a major third. However, unlike mavila, 10 of these bad fifths reach a more in tune one, which is useful for creating resolutions when using a large enough gamut, such as the [[9L 7s]] [[mos]] which has 3 good major and minor chords.
Mabila is part of the [[diaschismic–gothmic equivalence continuum]] with {{nowrap| ''n'' {{=}} 5 }}.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 268435456/263671875
[[Comma list]]: 268435456/263671875


{{Mapping|legend=1| 1 6 1 | 0 -10 3 }}
{{Mapping|legend=1| 1 -4 4 | 0 10 -3 }}
: mapping generators: ~2, ~375/256


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~512/375 = 529.685
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3545{{c}}, ~375/256 = 669.9545{{c}}
: [[error map]]: {{val| -0.646 +0.173 +1.240 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/256 = 670.2921{{c}}
: error map: {{val| 0.000 +0.966 +2.810 }}


{{Optimal ET sequence|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
{{Optimal ET sequence|legend=1| 9, 25, 34, 77, 111, 145, 256c }}


[[Badness]]: 0.232481
[[Badness]] (Sintel): 5.45
 
=== Overview to extensions ===
The second comma in the comma list defines which 7-limit family member we are looking at. Amavil (9 & 43) adds [[225/224]], semabila (9 & 25) adds [[49/48]], tuscaloosa (77 & 111) adds [[19683/19600]], muscogee (43 & 77) adds [[126/125]]. These all use the same generators as mabila.  


== Semabila ==
Cohemimabila (25 & 43) tempers out [[3136/3125]], splitting the subfifth in two. Hemimabila (9 & 68) tempers out [[6144/6125]], splitting the subtwelfth in two. Trimabila (9 & 111) tempers out [[1728/1715]], with a 1/3-octave period.
{{See also| Semaphoresmic clan #Semabila }}
 
{{See also| No-threes subgroup temperaments #Mabilic }}
== Amavil ==
Named by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, amavil tempers out [[225/224]] and may be described as the {{nowrap| 9 & 43 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 28672/28125
[[Comma list]]: 225/224, 17496/16807
 
{{Mapping|legend=1| 1 6 1 5 | 0 -10 3 -5 }}


{{Multival|legend=1| 10 -3 5 -28 -20 20 }}
{{Mapping|legend=1| 1 -4 4 -5 | 0 10 -3 14 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/56 = 529.667
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8499{{c}}, ~35/24 = 669.3786{{c}}
: [[error map]]: {{val| -1.150 -3.569 +0.950 +8.224 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 669.9710{{c}}
: error map: {{val| 0.000 -2.245 +3.773 +10.768 }}


{{Optimal ET sequence|legend=1| 9, 25, 34 }}
{{Optimal ET sequence|legend=1| 9, 25d, 34d, 43, 77d }}


[[Badness]]: 0.133638
[[Badness]] (Sintel): 2.77


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 1350/1331
Comma list: 99/98, 176/175, 864/847


Mapping: {{mapping| 1 6 1 5 7 | 0 -10 3 -5 -8 }}
Mapping: {{mapping| 1 -4 4 -5 -1 | 0 10 -3 14 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.729
Optimal tunings:
* WE: ~2 = 1198.5522{{c}}, ~22/15 = 669.2176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 669.9619{{c}}


{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
{{Optimal ET sequence|legend=0| 9, 25de, 34d, 43, 77de }}


Badness: 0.061501
Badness (Sintel): 1.41


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


Comma list: 49/48, 56/55, 91/90, 847/845
Comma list: 78/77, 99/98, 144/143, 176/175


Mapping: {{mapping| 1 6 1 5 7 9 | 0 -10 3 -5 -8 -12 }}
Mapping: {{mapping| 1 -4 4 -5 -1 -3 | 0 10 -3 14 8 12 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.763
Optimal tunings:
* WE: ~2 = 1198.7386{{c}}, ~22/15 = 669.3449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 669.9903{{c}}


{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
{{Optimal ET sequence|legend=0| 9, 25de, 34d, 43, 77de }}


Badness: 0.037270
Badness (Sintel): 1.07


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 56/55, 91/90, 154/153, 375/374
Comma list: 78/77, 99/98, 120/119, 144/143, 176/175


Mapping: {{mapping| 1 6 1 5 7 9 1 | 0 -10 3 -5 -8 -12 7 }}
Mapping: {{mapping| 1 -4 4 -5 -1 -3 8 | 0 10 -3 14 8 12 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.695
Optimal tunings:
* WE: ~2 = 1198.7648{{c}}, ~22/15 = 669.3533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 670.0080{{c}}


{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
{{Optimal ET sequence|legend=0| 9, 25de, 34d, 43, 77de }}


Badness: 0.031888
Badness (Sintel): 1.13


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 49/48, 56/55, 76/75, 91/90, 154/153, 190/187
Comma list: 78/77, 96/95, 99/98, 120/119, 135/133, 144/143
 
Mapping: {{mapping| 1 -4 4 -5 -1 -3 8 -3 | 0 10 -3 14 8 12 -7 13 }}
 
Optimal tunings:
* WE: ~2 = 1198.5939{{c}}, ~22/15 = 669.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 669.9712{{c}}
 
{{Optimal ET sequence|legend=0| 9, 34dh, 43, 77deh }}


Mapping: {{mapping| 1 6 1 5 7 9 1 6 | 0 -10 3 -5 -8 -12 7 -4 }}
Badness (Sintel): 1.09


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.736
== Semabila ==
{{See also| No-threes subgroup temperaments #Mabilic }}


{{Optimal ET sequence|legend=1| 9, 25e, 34 }}
Semabila tempers out [[49/48]] and may be described as the {{nowrap| 9 & 25 }} temperament.


Badness: 0.026981
This temperament used to known as ''septimal mabila''. Semabila, by [[Lériendil]] in 2024, was so named because it was a semaphore temperament.


== Amavil ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 17496/16807
[[Comma list]]: 49/48, 28672/28125


{{Mapping|legend=1| 1 6 1 9 | 0 -10 3 -14 }}
{{Mapping|legend=1| 1 -4 4 0 | 0 10 -3 5 }}


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9854{{c}}, ~112/75 = 670.8838{{c}}
: [[error map]]: {{val| +0.985 +2.941 +4.977 -14.407 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~112/75 = 670.3712{{c}}
: error map: {{val| 0.000 +1.757 +2.573 -16.970 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 529.979
{{Optimal ET sequence|legend=1| 9, 25, 34 }}


{{Optimal ET sequence|legend=1| 9, 25d, 34d, 43, 77d, 120dd }}
[[Badness]] (Sintel): 3.38
 
[[Badness]]: 0.109625


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
Comma list: 49/48, 56/55, 1350/1331


Mapping: {{mapping| 1 6 1 9 7 | 0 -10 3 -14 -8 }}
Mapping: {{mapping| 1 -4 4 0 1 | 0 10 -3 5 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.974
Optimal tunings:
* WE: ~2 = 1200.2248{{c}}, ~22/15 = 670.3965{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 670.2804{{c}}


{{Optimal ET sequence|legend=1| 9, 25de, 34d, 43, 77de, 120dde }}
{{Optimal ET sequence|legend=0| 9, 25e, 34 }}


Badness: 0.042649
Badness (Sintel): 2.03


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


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 49/48, 56/55, 91/90, 847/845


Mapping: {{mapping| 1 6 1 9 7 9 | 0 -10 3 -14 -8 -12 }}
Mapping: {{mapping| 1 -4 4 0 1 -3 | 0 10 -3 5 8 12 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.951
Optimal tunings:
* WE: ~2 = 1200.1265{{c}}, ~22/15 = 670.3078{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 670.2429{{c}}


{{Optimal ET sequence|legend=1| 9, 25de, 34d, 43, 77de, 120dde }}
{{Optimal ET sequence|legend=0| 9, 25e, 34 }}


Badness: 0.025791
Badness (Sintel): 1.54


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 78/77, 99/98, 120/119, 144/143, 176/175
Comma list: 49/48, 56/55, 91/90, 154/153, 375/374


Mapping: {{mapping| 1 6 1 9 7 9 1 | 0 -10 3 -14 -8 -12 7 }}
Mapping: {{mapping| 1 -4 4 0 1 -3 8 | 0 10 -3 5 8 12 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.957
Optimal tunings:
* WE: ~2 = 1199.8798{{c}}, ~22/15 = 670.2382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 670.3021{{c}}


{{Optimal ET sequence|legend=1| 9, 25de, 34d, 43, 77de, 120ddeg }}
{{Optimal ET sequence|legend=0| 9, 25e, 34 }}


Badness: 0.022092
Badness (Sintel): 1.62


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 78/77, 96/95, 99/98, 120/119, 135/133, 144/143
Comma list: 49/48, 56/55, 76/75, 91/90, 154/153, 190/187


Mapping: {{mapping| 1 6 1 9 7 9 1 10 | 0 -10 3 -14 -8 -12 7 -13 }}
Mapping: {{mapping| 1 -4 4 0 1 -3 8 2 | 0 10 -3 5 8 12 -7 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 529.987
Optimal tunings:
* WE: ~2 = 1200.4164{{c}}, ~22/15 = 670.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 670.2749{{c}}


{{Optimal ET sequence|legend=1| 9, 25deh, 34dh, 43, 77deh, 120ddeghh }}
{{Optimal ET sequence|legend=0| 9, 25e, 34 }}


Badness: 0.017955
Badness (Sintel): 1.64


== Tuskaloosa ==
== Tuskaloosa ==
Named by [[Xenllium]] in 2022, tuskaloosa tempers out [[19683/19600]] and may be described as the {{nowrap| 34d & 77 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 110592/109375
[[Comma list]]: 19683/19600, 110592/109375


{{Mapping|legend=1| 1 6 1 24 | 0 -10 3 -48 }}
{{Mapping|legend=1| 1 -4 4 -24 | 0 10 -3 48 }}


{{Multival|legend=1| 10 -3 48 -28 48 120 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4378{{c}}, ~375/256 = 669.9137{{c}}
: [[error map]]: {{val| -0.562 -0.569 +1.696 +0.524 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/256 = 670.2172{{c}}
: error map: {{val| 0.000 +0.217 +3.035 +1.597 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~512/375 = 529.772
{{Optimal ET sequence|legend=1| 34d, 77, 111, 188, 299cd, 487ccd }}


{{Optimal ET sequence|legend=1| 34d, 77, 111, 188, 299cd }}
[[Badness]] (Sintel): 3.67
 
[[Badness]]: 0.145058


=== 11-limit ===
=== 11-limit ===
Line 172: Line 217:
Comma list: 176/175, 1331/1323, 19683/19600
Comma list: 176/175, 1331/1323, 19683/19600


Mapping: {{mapping| 1 6 1 24 22 | 0 -10 3 -48 -42 }}
Mapping: {{mapping| 1 -4 4 -24 -20 | 0 10 -3 48 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~224/165 = 529.749
Optimal tunings:
* WE: ~2 = 1199.4934{{c}}, ~165/112 = 669.9677{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~165/112 = 670.2405{{c}}


{{Optimal ET sequence|legend=1| 34d, 77, 111, 299cd, 410ccd, 521ccdd }}
{{Optimal ET sequence|legend=0| 34d, 77, 111 }}


Badness: 0.061773
Badness (Sintel): 2.04


=== 13-limit ===
=== 13-limit ===
Line 185: Line 232:
Comma list: 176/175, 351/350, 676/675, 1331/1323
Comma list: 176/175, 351/350, 676/675, 1331/1323


Mapping: {{mapping| 1 6 1 24 22 9 | 0 -10 3 -48 -42 -12 }}
Mapping: {{mapping| 1 -4 4 -24 -20 -3 | 0 10 -3 48 42 12 }}


Optimal tuning (POTE): ~2 = 1\1, ~65/48 = 529.747
Optimal tunings:
* WE: ~2 = 1199.4539{{c}}, ~96/65 = 669.9476{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/65 = 670.2425{{c}}


{{Optimal ET sequence|legend=1| 34d, 77, 111, 410ccdf, 521ccddff }}
{{Optimal ET sequence|legend=0| 34d, 77, 111 }}


Badness: 0.031480
Badness (Sintel): 1.30


=== 17-limit ===
=== 17-limit ===
Line 198: Line 247:
Comma list: 176/175, 256/255, 351/350, 676/675, 715/714
Comma list: 176/175, 256/255, 351/350, 676/675, 715/714


Mapping: {{mapping| 1 6 1 24 22 9 1 | 0 -10 3 -48 -42 -12 7 }}
Mapping: {{mapping| 1 -4 4 -24 -20 -3 8 | 0 10 -3 48 42 12 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~34/25 = 529.742
Optimal tunings:
* WE: ~2 = 1199.3885{{c}}, ~25/17 = 669.9167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/17 = 670.2500{{c}}


{{Optimal ET sequence|legend=1| 34d, 77, 111 }}
{{Optimal ET sequence|legend=0| 34d, 77, 111 }}


Badness: 0.022765
Badness (Sintel): 1.16


=== 19-limit ===
=== 19-limit ===
Line 211: Line 262:
Comma list: 176/175, 256/255, 286/285, 351/350, 363/361, 476/475
Comma list: 176/175, 256/255, 286/285, 351/350, 363/361, 476/475


Mapping: {{mapping| 1 6 1 24 22 9 1 25 | 0 -10 3 -48 -42 -12 7 -47 }}
Mapping: {{mapping| 1 -4 4 -24 -20 -3 8 -22 | 0 10 -3 48 42 12 -7 47 }}


Optimal tuning (POTE): ~2 = 1\1, ~19/14 = 529.749
Optimal tunings:
* WE: ~2 = 1199.3711{{c}}, ~25/17 = 669.8997{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/17 = 670.2422{{c}}


{{Optimal ET sequence|legend=1| 34dh, 77, 111 }}
{{Optimal ET sequence|legend=0| 34dh, 77, 111 }}


Badness: 0.017924
Badness (Sintel): 1.09


== Muscogee ==
== Muscogee ==
Named by [[Xenllium]] in 2022, muscogee tempers out [[126/125]] and may be described as the {{nowrap| 43 & 77 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 33756345/33554432
[[Comma list]]: 126/125, 33756345/33554432


{{Mapping|legend=1| 1 6 1 -10 | 0 -10 3 29 }}
{{Mapping|legend=1| 1 -4 4 19 | 0 10 -3 -29 }}


{{Multival|legend=1| 10 -3 -29 -28 -74 -59 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9275{{c}}, ~375/256 = 670.0525{{c}}
: [[error map]]: {{val| -0.073 -1.140 +3.239 -1.726 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/256 = 670.0935{{c}}
: error map: {{val| 0.000 -1.020 +3.406 -1.539 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~512/375 = 529.907
{{Optimal ET sequence|legend=1| 34, 43, 77 }}


{{Optimal ET sequence|legend=1| 34, 43, 77, 274c, 351cc, 428ccd }}
[[Badness]] (Sintel): 4.10
 
[[Badness]]: 0.162021


=== 11-limit ===
=== 11-limit ===
Line 239: Line 296:
Comma list: 126/125, 176/175, 264627/262144
Comma list: 126/125, 176/175, 264627/262144


Mapping: {{mapping| 1 6 1 -10 -12 | 0 -10 3 29 35 }}
Mapping: {{mapping| 1 -4 4 19 23 | 0 10 -3 -29 -35 }}


Optimal tuning (POTE): ~2 = 1\1, ~224/165 = 529.955
Optimal tunings:
* WE: ~2 = 1200.0559{{c}}, ~165/112 = 670.0760{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~165/112 = 670.0441{{c}}


{{Optimal ET sequence|legend=1| 34e, 43, 77, 120, 197ce }}
{{Optimal ET sequence|legend=0| 34e, 43, 77, 120 }}


Badness: 0.077552
Badness (Sintel): 2.56


=== 13-limit ===
=== 13-limit ===
Line 252: Line 311:
Comma list: 126/125, 176/175, 676/675, 1287/1280
Comma list: 126/125, 176/175, 676/675, 1287/1280


Mapping: {{mapping| 1 6 1 -10 -12 9 | 0 -10 3 29 35 -12 }}
Mapping: {{mapping| 1 -4 4 19 23 -3 | 0 10 -3 -29 -35 12 }}


Optimal tuning (POTE): ~2 = 1\1, ~65/48 = 529.957
Optimal tunings:
* WE: ~2 = 1200.0428{{c}}, ~96/65 = 670.0673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~96/65 = 670.0431{{c}}


{{Optimal ET sequence|legend=1| 34e, 43, 77, 120, 197ce }}
{{Optimal ET sequence|legend=0| 34e, 43, 77, 120 }}


Badness: 0.043352
Badness (Sintel): 1.79


=== 17-limit ===
=== 17-limit ===
Line 265: Line 326:
Comma list: 126/125, 176/175, 256/255, 273/272, 676/675
Comma list: 126/125, 176/175, 256/255, 273/272, 676/675


Mapping: {{mapping| 1 6 1 -10 -12 9 1 | 0 -10 3 29 35 -12 7 }}
Mapping: {{mapping| 1 -4 4 19 23 -3 8 | 0 10 -3 -29 -35 12 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~34/25 = 529.958
Optimal tunings:
* WE: ~2 = 1199.8666{{c}}, ~25/17 = 669.9675{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/17 = 670.0429{{c}}


{{Optimal ET sequence|legend=1| 34e, 43, 77, 120g, 197ceg }}
{{Optimal ET sequence|legend=0| 34e, 43, 77, 120g }}


Badness: 0.031217
Badness (Sintel): 1.59


=== 19-limit ===
=== 19-limit ===
Line 278: Line 341:
Comma list: 126/125, 171/170, 176/175, 256/255, 273/272, 363/361
Comma list: 126/125, 171/170, 176/175, 256/255, 273/272, 363/361


Mapping: {{mapping| 1 6 1 -10 -12 9 1 -9 | 0 -10 3 29 35 -12 7 30 }}
Mapping: {{mapping| 1 -4 4 19 23 -3 8 21 | 0 10 -3 -29 -35 12 -7 -30 }}
 
Optimal tunings:
* WE: ~2 = 1199.8538{{c}}, ~25/17 = 669.9631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/17 = 670.0460{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~19/14 = 529.955
{{Optimal ET sequence|legend=0| 34e, 43, 77, 120g }}


{{Optimal ET sequence|legend=1| 34e, 43, 77, 120g, 197ceg }}
Badness (Sintel): 1.44


Badness: 0.023670
== Cohemimabila ==
Named by [[Xenllium]] in 2022, cohemimabila tempers out [[3136/3125]] as well as [[65536/64827]] and may be described as the {{nowrap| 43 & 68 }} temperament.  


== Hemimabila ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 117649/116640
[[Comma list]]: 3136/3125, 65536/64827


{{Mapping|legend=1| 1 6 1 7 | 0 -20 6 -19 }}
{{Mapping|legend=1| 1 -4 4 7 | 0 20 -6 -15 }}
: mapping generators: ~2, ~128/105


{{Multival|legend=1| 20 -6 19 -56 -26 61 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1476{{c}}, ~128/105 = 334.9440{{c}}
: [[error map]]: {{val| -0.852 +0.335 +0.613 +1.047 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 335.1779{{c}}
: error map: {{val| 0.000 +1.603 +2.619 +3.505 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 264.825
{{Optimal ET sequence|legend=1| 25, 43, 68, 111, 179, 290cd, 469bccdd }}


{{Optimal ET sequence|legend=1| 9, 59, 68, 77, 145 }}
[[Badness]] (Sintel): 3.23
 
[[Badness]]: 0.111130


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 67228/66825
Comma list: 176/175, 1375/1372, 16384/16335


Mapping: {{mapping| 1 6 1 7 5 | 0 -20 6 -19 -7 }}
Mapping: {{mapping| 1 -4 4 7 11 | 0 20 -6 -15 -27 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 264.849
Optimal tunings:
* WE: ~2 = 1199.3670{{c}}, ~40/33 = 334.9711{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/33 = 335.1492{{c}}


{{Optimal ET sequence|legend=1| 9, 59, 68, 77, 145e }}
{{Optimal ET sequence|legend=0| 25, 43, 68, 111 }}


Badness: 0.061426
Badness (Sintel): 2.12


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


Comma list: 121/120, 176/175, 196/195, 676/675
Comma list: 176/175, 640/637, 676/675, 1375/1372


Mapping: {{mapping| 1 6 1 7 5 9 | 0 -20 6 -19 -7 -24 }}
Mapping: {{mapping| 1 -4 4 7 11 -3 | 0 20 -6 -15 -27 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 264.861
Optimal tunings:
* WE: ~2 = 1199.3383{{c}}, ~40/33 = 334.9594{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/33 = 335.1431{{c}}


{{Optimal ET sequence|legend=1| 9, 59f, 68, 77, 145e, 222cef }}
{{Optimal ET sequence|legend=0| 25, 43, 68, 111 }}


Badness: 0.034531
Badness (Sintel): 1.47


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 154/153, 176/175, 196/195, 676/675
Comma list: 176/175, 256/255, 442/441, 640/637, 715/714


Mapping: {{mapping| 1 6 1 7 5 9 1 | 0 -20 6 -19 -7 -24 14 }}
Mapping: {{mapping| 1 -4 4 7 11 -3 8 | 0 20 -6 -15 -27 24 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 264.839
Optimal tunings:
* WE: ~2 = 1199.3269{{c}}, ~17/14 = 334.9571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/14 = 335.1451{{c}}


{{Optimal ET sequence|legend=1| 9, 59f, 68, 77, 145e }}
{{Optimal ET sequence|legend=0| 25, 43, 68, 111 }}


Badness: 0.027851
Badness (Sintel): 1.16


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 121/120, 154/153, 176/175, 196/195, 209/208, 361/360
Comma list: 176/175, 256/255, 286/285, 363/361, 442/441, 476/475


Mapping: {{mapping| 1 6 1 7 5 9 1 8 | 0 -20 6 -19 -7 -24 14 -17 }}
Mapping: {{mapping| 1 -4 4 7 11 -3 8 9 | 0 20 -6 -15 -27 24 -14 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 264.839
Optimal tunings:
* WE: ~2 = 1199.2658{{c}}, ~17/14 = 334.9455{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/14 = 335.1516{{c}}


{{Optimal ET sequence|legend=1| 9, 59f, 68, 77, 145e }}
{{Optimal ET sequence|legend=0| 25, 43, 68, 111 }}


Badness: 0.020053
Badness (Sintel): 1.06


== Cohemimabila ==
== Hemimabila ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 65536/64827
[[Comma list]]: 6144/6125, 117649/116640
 
{{Mapping|legend=1| 1 -4 4 7 | 0 20 -6 -15 }}


{{Multival|legend=1| 20 -6 -15 -56 -80 -18 }}
{{Mapping|legend=1| 1 -14 7 -12 | 0 20 -6 19 }}
: mapping generators: ~2, ~12/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 335.182
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5170{{c}}, ~12/7 = 934.7983{{c}}
: [[error map]]: {{val| -0.483 +0.773 +1.516 -1.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 935.1643{{c}}
: error map: {{val| 0.000 +1.330 +2.701 -0.705 }}


{{Optimal ET sequence|legend=1| 25, 43, 68, 111, 179, 290cd, 469bccdd }}
{{Optimal ET sequence|legend=1| 9, …, 59, 68, 77, 145 }}


[[Badness]]: 0.127451
[[Badness]] (Sintel): 2.81


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 1375/1372, 16384/16335
Comma list: 121/120, 176/175, 67228/66825


Mapping: {{mapping| 1 -4 4 7 11 | 0 20 -6 -15 -27 }}
Mapping: {{mapping| 1 -14 7 -12 -2 | 0 20 -6 19 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~40/33 = 335.148
Optimal tunings:
* WE: ~2 = 1199.9930{{c}}, ~12/7 = 935.1459{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 935.1512{{c}}


{{Optimal ET sequence|legend=1| 25, 43, 68, 111 }}
{{Optimal ET sequence|legend=0| 9, 59, 68, 77, 145e }}


Badness: 0.064164
Badness (Sintel): 2.03


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


Comma list: 176/175, 640/637, 676/675, 1375/1372
Comma list: 121/120, 176/175, 196/195, 676/675


Mapping: {{mapping| 1 -4 4 7 11 -3 | 0 20 -6 -15 -27 24 }}
Mapping: {{mapping| 1 -14 7 -12 -2 -15 | 0 20 -6 19 7 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~40/33 = 335.144
Optimal tunings:
* WE: ~2 = 1199.9061{{c}}, ~12/7 = 935.0656{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 935.1367{{c}}


{{Optimal ET sequence|legend=1| 25, 43, 68, 111 }}
{{Optimal ET sequence|legend=0| 9, 59f, 68, 77 }}


Badness: 0.035463
Badness (Sintel): 1.43


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 176/175, 256/255, 442/441, 640/637, 715/714
Comma list: 121/120, 154/153, 176/175, 196/195, 676/675


Mapping: {{mapping| 1 -4 4 7 11 -3 8 | 0 20 -6 -15 -27 24 -14 }}
Mapping: {{mapping| 1 -14 7 -12 -2 -15 15 | 0 20 -6 19 7 24 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 335.145
Optimal tunings:
* WE: ~2 = 1199.7422{{c}}, ~12/7 = 934.9596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 935.1572{{c}}


{{Optimal ET sequence|legend=1| 25, 43, 68, 111 }}
{{Optimal ET sequence|legend=0| 9, 59f, 68, 77 }}


Badness: 0.022728
Badness (Sintel): 1.42


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 256/255, 286/285, 363/361, 442/441, 476/475
Comma list: 121/120, 154/153, 176/175, 196/195, 209/208, 361/360


Mapping: {{mapping| 1 -4 4 7 11 -3 8 9 | 0 20 -6 -15 -27 24 -14 -17 }}
Mapping: {{mapping| 1 -14 7 -12 -2 -15 15 -9 | 0 20 -6 19 7 24 -14 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/14 = 335.151
Optimal tunings:
* WE: ~2 = 1199.7650{{c}}, ~12/7 = 934.9782{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 935.1581{{c}}


{{Optimal ET sequence|legend=1| 25, 43, 68, 111 }}
{{Optimal ET sequence|legend=0| 9, 59f, 68, 77 }}


Badness: 0.017450
Badness (Sintel): 1.22


== Trimabila ==
== Trimabila ==
Named by [[Xenllium]] in 2022, trimabila tempers out [[1728/1715]] and may be described as the {{nowrap| 9 & 111 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 268435456/263671875
[[Comma list]]: 1728/1715, 268435456/263671875


{{Mapping|legend=1| 3 8 6 12 | 0 -10 3 -11 }}
{{Mapping|legend=1| 3 -2 9 1 | 0 10 -3 11 }}
: mapping generators: ~1125/896, ~7/6


{{Multival|legend=1| 30 -9 33 -84 -32 102 }}
[[Optimal tuning]]s:
* [[WE]]: ~1125/896 = 399.7349{{c}}, ~7/6 = 270.0900{{c}}
: [[error map]]: {{val| -0.795 -0.525 +1.030 +1.899 }}
* [[CWE]]: ~1125/896 = 400.0000{{c}}, ~7/6 = 270.2343{{c}}
: error map: {{val| 0.000 +0.388 +2.983 +3.752 }}


[[Optimal tuning]] ([[POTE]]): ~1125/896 = 1\3, ~7/6 = 270.269
{{Optimal ET sequence|legend=1| 9, …, 102d, 111 }}


{{Optimal ET sequence|legend=1| 9, 102d, 111 }}
[[Badness]] (Sintel): 6.76
 
[[Badness]]: 0.267168


=== 11-limit ===
=== 11-limit ===
Line 440: Line 534:
Comma list: 176/175, 540/539, 805255/802816
Comma list: 176/175, 540/539, 805255/802816


Mapping: {{mapping| 3 8 6 12 12 | 0 -10 3 -11 -5 }}
Mapping: {{mapping| 3 -2 9 1 7 | 0 10 -3 11 5 }}


Optimal tuning (POTE): ~495/392 = 1\3, ~7/6 = 270.256
Optimal tunings:
* WE: ~495/392 = 399.7963{{c}}, ~7/6 = 270.1183{{c}}
* CWE: ~495/392 = 400.0000{{c}}, ~7/6 = 270.2301{{c}}


{{Optimal ET sequence|legend=1| 9, 102d, 111 }}
{{Optimal ET sequence|legend=0| 9, …, 102d, 111 }}


Badness: 0.081946
Badness (Sintel): 2.71


=== 13-limit ===
=== 13-limit ===
Line 453: Line 549:
Comma list: 176/175, 540/539, 676/675, 1573/1568
Comma list: 176/175, 540/539, 676/675, 1573/1568


Mapping: {{mapping| 3 8 6 12 12 15 | 0 -10 3 -11 -5 -12 }}
Mapping: {{mapping| 3 -2 9 1 7 3 | 0 10 -3 11 5 12 }}


Optimal tuning (POTE): ~495/392 = 1\3, ~7/6 = 270.254 (or ~14/13 = 129.746)
Optimal tunings:
* WE: ~495/392 = 399.7935{{c}}, ~7/6 = 270.1144{{c}}
* CWE: ~495/392 = 400.0000{{c}}, ~7/6 = 270.2258{{c}}


{{Optimal ET sequence|legend=1| 9, 102df, 111 }}
{{Optimal ET sequence|legend=0| 9, …, 102df, 111 }}


Badness: 0.040102
Badness (Sintel): 1.66


=== 17-limit ===
=== 17-limit ===
Line 466: Line 564:
Comma list: 176/175, 256/255, 540/539, 676/675, 715/714
Comma list: 176/175, 256/255, 540/539, 676/675, 715/714


Mapping: {{mapping| 3 8 6 12 12 15 10 | 0 -10 3 -11 -5 -12 7 }}
Mapping: {{mapping| 3 -2 9 1 7 3 17 | 0 10 -3 11 5 12 -7 }}


Optimal tuning (POTE): ~495/392 = 1\3, ~7/6 = 270.266 (or ~14/13 = 129.734)
Optimal tunings:
* WE: ~495/392 = 399.7781{{c}}, ~7/6 = 270.1159{{c}}
* CWE: ~495/392 = 400.0000{{c}}, ~7/6 = 270.2476{{c}}


{{Optimal ET sequence|legend=1| 9, 102df, 111 }}
{{Optimal ET sequence|legend=0| 9, 102df, 111 }}


Badness: 0.030657
Badness (Sintel): 1.56


=== 19-limit ===
=== 19-limit ===
Line 479: Line 579:
Comma list: 176/175, 256/255, 286/285, 363/361, 476/475, 540/539
Comma list: 176/175, 256/255, 286/285, 363/361, 476/475, 540/539


Mapping: {{mapping| 3 8 6 12 12 15 10 16 | 0 -10 3 -11 -5 -12 7 -10 }}
Mapping: {{mapping| 3 -2 9 1 7 3 17 6 | 0 10 -3 11 5 12 -7 10 }}


Optimal tuning (POTE): ~208/165 = 1\3, ~7/6 = 270.260 (or ~14/13 = 129.740)
Optimal tunings:
* WE: ~208/165 = 399.7588{{c}}, ~7/6 = 270.0969{{c}}
* CWE: ~208/165 = 400.0000{{c}}, ~7/6 = 270.2391{{c}}


{{Optimal ET sequence|legend=1| 9, 102dfh, 111 }}
{{Optimal ET sequence|legend=0| 9, 102dfh, 111 }}


Badness: 0.022851
Badness (Sintel): 1.39


== References ==
[[Category:Mabila family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Mabila family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Mabila| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 12:05, 14 August 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 mabila family of temperaments tempers out the mabila comma (monzo: [28 -3 -10⟩, ratio: 268435456/263671875) in the 5-limit.

Mabila

"Mabila" redirects here. For the temperaments that used to go by this name, see Semabila and Mabilic.

Mabila has a temperament structure superficially similar to mavila, with extremely sharp fourths/flat fifths, three of which make a major third. However, unlike mavila, 10 of these bad fifths reach a more in tune one, which is useful for creating resolutions when using a large enough gamut, such as the 9L 7s mos which has 3 good major and minor chords.

Mabila is part of the diaschismic–gothmic equivalence continuum with n = 5.

Subgroup: 2.3.5

Comma list: 268435456/263671875

Mapping: [⟨1 -4 4], ⟨0 10 -3]]

mapping generators: ~2, ~375/256

Optimal tunings:

  • WE: ~2 = 1199.3545 ¢, ~375/256 = 669.9545 ¢
error map: ⟨-0.646 +0.173 +1.240]
  • CWE: ~2 = 1200.0000 ¢, ~375/256 = 670.2921 ¢
error map: ⟨0.000 +0.966 +2.810]

Optimal ET sequence: 9, 25, 34, 77, 111, 145, 256c

Badness (Sintel): 5.45

Overview to extensions

The second comma in the comma list defines which 7-limit family member we are looking at. Amavil (9 & 43) adds 225/224, semabila (9 & 25) adds 49/48, tuscaloosa (77 & 111) adds 19683/19600, muscogee (43 & 77) adds 126/125. These all use the same generators as mabila.

Cohemimabila (25 & 43) tempers out 3136/3125, splitting the subfifth in two. Hemimabila (9 & 68) tempers out 6144/6125, splitting the subtwelfth in two. Trimabila (9 & 111) tempers out 1728/1715, with a 1/3-octave period.

Amavil

Named by Petr Pařízek in 2011[1], amavil tempers out 225/224 and may be described as the 9 & 43 temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 17496/16807

Mapping: [⟨1 -4 4 -5], ⟨0 10 -3 14]]

Optimal tunings:

  • WE: ~2 = 1198.8499 ¢, ~35/24 = 669.3786 ¢
error map: ⟨-1.150 -3.569 +0.950 +8.224]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 669.9710 ¢
error map: ⟨0.000 -2.245 +3.773 +10.768]

Optimal ET sequence: 9, 25d, 34d, 43, 77d

Badness (Sintel): 2.77

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 864/847

Mapping: [⟨1 -4 4 -5 -1], ⟨0 10 -3 14 8]]

Optimal tunings:

  • WE: ~2 = 1198.5522 ¢, ~22/15 = 669.2176 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 669.9619 ¢

Optimal ET sequence: 9, 25de, 34d, 43, 77de

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 144/143, 176/175

Mapping: [⟨1 -4 4 -5 -1 -3], ⟨0 10 -3 14 8 12]]

Optimal tunings:

  • WE: ~2 = 1198.7386 ¢, ~22/15 = 669.3449 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 669.9903 ¢

Optimal ET sequence: 9, 25de, 34d, 43, 77de

Badness (Sintel): 1.07

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 144/143, 176/175

Mapping: [⟨1 -4 4 -5 -1 -3 8], ⟨0 10 -3 14 8 12 -7]]

Optimal tunings:

  • WE: ~2 = 1198.7648 ¢, ~22/15 = 669.3533 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 670.0080 ¢

Optimal ET sequence: 9, 25de, 34d, 43, 77de

Badness (Sintel): 1.13

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 96/95, 99/98, 120/119, 135/133, 144/143

Mapping: [⟨1 -4 4 -5 -1 -3 8 -3], ⟨0 10 -3 14 8 12 -7 13]]

Optimal tunings:

  • WE: ~2 = 1198.5939 ¢, ~22/15 = 669.2282 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 669.9712 ¢

Optimal ET sequence: 9, 34dh, 43, 77deh

Badness (Sintel): 1.09

Semabila

See also: No-threes subgroup temperaments § Mabilic

Semabila tempers out 49/48 and may be described as the 9 & 25 temperament.

This temperament used to known as septimal mabila. Semabila, by Lériendil in 2024, was so named because it was a semaphore temperament.

Subgroup: 2.3.5.7

Comma list: 49/48, 28672/28125

Mapping: [⟨1 -4 4 0], ⟨0 10 -3 5]]

Optimal tunings:

  • WE: ~2 = 1200.9854 ¢, ~112/75 = 670.8838 ¢
error map: ⟨+0.985 +2.941 +4.977 -14.407]
  • CWE: ~2 = 1200.0000 ¢, ~112/75 = 670.3712 ¢
error map: ⟨0.000 +1.757 +2.573 -16.970]

Optimal ET sequence: 9, 25, 34

Badness (Sintel): 3.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 1350/1331

Mapping: [⟨1 -4 4 0 1], ⟨0 10 -3 5 8]]

Optimal tunings:

  • WE: ~2 = 1200.2248 ¢, ~22/15 = 670.3965 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 670.2804 ¢

Optimal ET sequence: 9, 25e, 34

Badness (Sintel): 2.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 847/845

Mapping: [⟨1 -4 4 0 1 -3], ⟨0 10 -3 5 8 12]]

Optimal tunings:

  • WE: ~2 = 1200.1265 ¢, ~22/15 = 670.3078 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 670.2429 ¢

Optimal ET sequence: 9, 25e, 34

Badness (Sintel): 1.54

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 56/55, 91/90, 154/153, 375/374

Mapping: [⟨1 -4 4 0 1 -3 8], ⟨0 10 -3 5 8 12 -7]]

Optimal tunings:

  • WE: ~2 = 1199.8798 ¢, ~22/15 = 670.2382 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 670.3021 ¢

Optimal ET sequence: 9, 25e, 34

Badness (Sintel): 1.62

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 49/48, 56/55, 76/75, 91/90, 154/153, 190/187

Mapping: [⟨1 -4 4 0 1 -3 8 2], ⟨0 10 -3 5 8 12 -7 4]]

Optimal tunings:

  • WE: ~2 = 1200.4164 ¢, ~22/15 = 670.4966 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 670.2749 ¢

Optimal ET sequence: 9, 25e, 34

Badness (Sintel): 1.64

Tuskaloosa

Named by Xenllium in 2022, tuskaloosa tempers out 19683/19600 and may be described as the 34d & 77 temperament.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 110592/109375

Mapping: [⟨1 -4 4 -24], ⟨0 10 -3 48]]

Optimal tunings:

  • WE: ~2 = 1199.4378 ¢, ~375/256 = 669.9137 ¢
error map: ⟨-0.562 -0.569 +1.696 +0.524]
  • CWE: ~2 = 1200.0000 ¢, ~375/256 = 670.2172 ¢
error map: ⟨0.000 +0.217 +3.035 +1.597]

Optimal ET sequence: 34d, 77, 111, 188, 299cd, 487ccd

Badness (Sintel): 3.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323, 19683/19600

Mapping: [⟨1 -4 4 -24 -20], ⟨0 10 -3 48 42]]

Optimal tunings:

  • WE: ~2 = 1199.4934 ¢, ~165/112 = 669.9677 ¢
  • CWE: ~2 = 1200.0000 ¢, ~165/112 = 670.2405 ¢

Optimal ET sequence: 34d, 77, 111

Badness (Sintel): 2.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 676/675, 1331/1323

Mapping: [⟨1 -4 4 -24 -20 -3], ⟨0 10 -3 48 42 12]]

Optimal tunings:

  • WE: ~2 = 1199.4539 ¢, ~96/65 = 669.9476 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/65 = 670.2425 ¢

Optimal ET sequence: 34d, 77, 111

Badness (Sintel): 1.30

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 676/675, 715/714

Mapping: [⟨1 -4 4 -24 -20 -3 8], ⟨0 10 -3 48 42 12 -7]]

Optimal tunings:

  • WE: ~2 = 1199.3885 ¢, ~25/17 = 669.9167 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/17 = 670.2500 ¢

Optimal ET sequence: 34d, 77, 111

Badness (Sintel): 1.16

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 351/350, 363/361, 476/475

Mapping: [⟨1 -4 4 -24 -20 -3 8 -22], ⟨0 10 -3 48 42 12 -7 47]]

Optimal tunings:

  • WE: ~2 = 1199.3711 ¢, ~25/17 = 669.8997 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/17 = 670.2422 ¢

Optimal ET sequence: 34dh, 77, 111

Badness (Sintel): 1.09

Muscogee

Named by Xenllium in 2022, muscogee tempers out 126/125 and may be described as the 43 & 77 temperament.

Subgroup: 2.3.5.7

Comma list: 126/125, 33756345/33554432

Mapping: [⟨1 -4 4 19], ⟨0 10 -3 -29]]

Optimal tunings:

  • WE: ~2 = 1199.9275 ¢, ~375/256 = 670.0525 ¢
error map: ⟨-0.073 -1.140 +3.239 -1.726]
  • CWE: ~2 = 1200.0000 ¢, ~375/256 = 670.0935 ¢
error map: ⟨0.000 -1.020 +3.406 -1.539]

Optimal ET sequence: 34, 43, 77

Badness (Sintel): 4.10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 264627/262144

Mapping: [⟨1 -4 4 19 23], ⟨0 10 -3 -29 -35]]

Optimal tunings:

  • WE: ~2 = 1200.0559 ¢, ~165/112 = 670.0760 ¢
  • CWE: ~2 = 1200.0000 ¢, ~165/112 = 670.0441 ¢

Optimal ET sequence: 34e, 43, 77, 120

Badness (Sintel): 2.56

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 676/675, 1287/1280

Mapping: [⟨1 -4 4 19 23 -3], ⟨0 10 -3 -29 -35 12]]

Optimal tunings:

  • WE: ~2 = 1200.0428 ¢, ~96/65 = 670.0673 ¢
  • CWE: ~2 = 1200.0000 ¢, ~96/65 = 670.0431 ¢

Optimal ET sequence: 34e, 43, 77, 120

Badness (Sintel): 1.79

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 256/255, 273/272, 676/675

Mapping: [⟨1 -4 4 19 23 -3 8], ⟨0 10 -3 -29 -35 12 -7]]

Optimal tunings:

  • WE: ~2 = 1199.8666 ¢, ~25/17 = 669.9675 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/17 = 670.0429 ¢

Optimal ET sequence: 34e, 43, 77, 120g

Badness (Sintel): 1.59

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 256/255, 273/272, 363/361

Mapping: [⟨1 -4 4 19 23 -3 8 21], ⟨0 10 -3 -29 -35 12 -7 -30]]

Optimal tunings:

  • WE: ~2 = 1199.8538 ¢, ~25/17 = 669.9631 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/17 = 670.0460 ¢

Optimal ET sequence: 34e, 43, 77, 120g

Badness (Sintel): 1.44

Cohemimabila

Named by Xenllium in 2022, cohemimabila tempers out 3136/3125 as well as 65536/64827 and may be described as the 43 & 68 temperament.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 65536/64827

Mapping: [⟨1 -4 4 7], ⟨0 20 -6 -15]]

mapping generators: ~2, ~128/105

Optimal tunings:

  • WE: ~2 = 1199.1476 ¢, ~128/105 = 334.9440 ¢
error map: ⟨-0.852 +0.335 +0.613 +1.047]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 335.1779 ¢
error map: ⟨0.000 +1.603 +2.619 +3.505]

Optimal ET sequence: 25, 43, 68, 111, 179, 290cd, 469bccdd

Badness (Sintel): 3.23

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 16384/16335

Mapping: [⟨1 -4 4 7 11], ⟨0 20 -6 -15 -27]]

Optimal tunings:

  • WE: ~2 = 1199.3670 ¢, ~40/33 = 334.9711 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/33 = 335.1492 ¢

Optimal ET sequence: 25, 43, 68, 111

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 640/637, 676/675, 1375/1372

Mapping: [⟨1 -4 4 7 11 -3], ⟨0 20 -6 -15 -27 24]]

Optimal tunings:

  • WE: ~2 = 1199.3383 ¢, ~40/33 = 334.9594 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/33 = 335.1431 ¢

Optimal ET sequence: 25, 43, 68, 111

Badness (Sintel): 1.47

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 442/441, 640/637, 715/714

Mapping: [⟨1 -4 4 7 11 -3 8], ⟨0 20 -6 -15 -27 24 -14]]

Optimal tunings:

  • WE: ~2 = 1199.3269 ¢, ~17/14 = 334.9571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.1451 ¢

Optimal ET sequence: 25, 43, 68, 111

Badness (Sintel): 1.16

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 363/361, 442/441, 476/475

Mapping: [⟨1 -4 4 7 11 -3 8 9], ⟨0 20 -6 -15 -27 24 -14 -17]]

Optimal tunings:

  • WE: ~2 = 1199.2658 ¢, ~17/14 = 334.9455 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.1516 ¢

Optimal ET sequence: 25, 43, 68, 111

Badness (Sintel): 1.06

Hemimabila

Subgroup: 2.3.5.7

Comma list: 6144/6125, 117649/116640

Mapping: [⟨1 -14 7 -12], ⟨0 20 -6 19]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • WE: ~2 = 1199.5170 ¢, ~12/7 = 934.7983 ¢
error map: ⟨-0.483 +0.773 +1.516 -1.862]
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 935.1643 ¢
error map: ⟨0.000 +1.330 +2.701 -0.705]

Optimal ET sequence: 9, …, 59, 68, 77, 145

Badness (Sintel): 2.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 67228/66825

Mapping: [⟨1 -14 7 -12 -2], ⟨0 20 -6 19 7]]

Optimal tunings:

  • WE: ~2 = 1199.9930 ¢, ~12/7 = 935.1459 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 935.1512 ¢

Optimal ET sequence: 9, 59, 68, 77, 145e

Badness (Sintel): 2.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 676/675

Mapping: [⟨1 -14 7 -12 -2 -15], ⟨0 20 -6 19 7 24]]

Optimal tunings:

  • WE: ~2 = 1199.9061 ¢, ~12/7 = 935.0656 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 935.1367 ¢

Optimal ET sequence: 9, 59f, 68, 77

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 176/175, 196/195, 676/675

Mapping: [⟨1 -14 7 -12 -2 -15 15], ⟨0 20 -6 19 7 24 -14]]

Optimal tunings:

  • WE: ~2 = 1199.7422 ¢, ~12/7 = 934.9596 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 935.1572 ¢

Optimal ET sequence: 9, 59f, 68, 77

Badness (Sintel): 1.42

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 154/153, 176/175, 196/195, 209/208, 361/360

Mapping: [⟨1 -14 7 -12 -2 -15 15 -9], ⟨0 20 -6 19 7 24 -14 17]]

Optimal tunings:

  • WE: ~2 = 1199.7650 ¢, ~12/7 = 934.9782 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/7 = 935.1581 ¢

Optimal ET sequence: 9, 59f, 68, 77

Badness (Sintel): 1.22

Trimabila

Named by Xenllium in 2022, trimabila tempers out 1728/1715 and may be described as the 9 & 111 temperament.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 268435456/263671875

Mapping: [⟨3 -2 9 1], ⟨0 10 -3 11]]

mapping generators: ~1125/896, ~7/6

Optimal tunings:

  • WE: ~1125/896 = 399.7349 ¢, ~7/6 = 270.0900 ¢
error map: ⟨-0.795 -0.525 +1.030 +1.899]
  • CWE: ~1125/896 = 400.0000 ¢, ~7/6 = 270.2343 ¢
error map: ⟨0.000 +0.388 +2.983 +3.752]

Optimal ET sequence: 9, …, 102d, 111

Badness (Sintel): 6.76

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 805255/802816

Mapping: [⟨3 -2 9 1 7], ⟨0 10 -3 11 5]]

Optimal tunings:

  • WE: ~495/392 = 399.7963 ¢, ~7/6 = 270.1183 ¢
  • CWE: ~495/392 = 400.0000 ¢, ~7/6 = 270.2301 ¢

Optimal ET sequence: 9, …, 102d, 111

Badness (Sintel): 2.71

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 540/539, 676/675, 1573/1568

Mapping: [⟨3 -2 9 1 7 3], ⟨0 10 -3 11 5 12]]

Optimal tunings:

  • WE: ~495/392 = 399.7935 ¢, ~7/6 = 270.1144 ¢
  • CWE: ~495/392 = 400.0000 ¢, ~7/6 = 270.2258 ¢

Optimal ET sequence: 9, …, 102df, 111

Badness (Sintel): 1.66

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 540/539, 676/675, 715/714

Mapping: [⟨3 -2 9 1 7 3 17], ⟨0 10 -3 11 5 12 -7]]

Optimal tunings:

  • WE: ~495/392 = 399.7781 ¢, ~7/6 = 270.1159 ¢
  • CWE: ~495/392 = 400.0000 ¢, ~7/6 = 270.2476 ¢

Optimal ET sequence: 9, 102df, 111

Badness (Sintel): 1.56

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 363/361, 476/475, 540/539

Mapping: [⟨3 -2 9 1 7 3 17 6], ⟨0 10 -3 11 5 12 -7 10]]

Optimal tunings:

  • WE: ~208/165 = 399.7588 ¢, ~7/6 = 270.0969 ¢
  • CWE: ~208/165 = 400.0000 ¢, ~7/6 = 270.2391 ¢

Optimal ET sequence: 9, 102dfh, 111

Badness (Sintel): 1.39

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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