Very low accuracy temperaments: Difference between revisions

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Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to temper out very large intervals such as 9/8, 10/9, 32/27, or 15/14, equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.
{{Technical data page}}
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.


== Antitonic ==
== Antitonic ==
This temperament is characterized by [[9/8]] being tempering out, and has been termed a "troll temperament" by its namers. It is named on account of 4/3 and 3/2 both being represented by the 600 cent half octave, which, in terms of diatonic function, serves as an antitonic. Surprisingly, it has extensions all the way up to the 11-limit, as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?limit=11&ets=2_4p&tuning=po x31eq]. The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.  
This temperament is characterized by [[9/8]] being tempered out and has been termed a "troll temperament" by its namers. Its [[ploidacot]] is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 13: Line 14:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3/2 = 600.000, ~5/4 = 386.314 (~6/5 = 213.686)
* [[WE]]: ~3/2 = 615.125{{c}}, ~5/4 = 321.597{{c}} (~6/5 = 293.528{{c}})
: [[error map]]: {{val| 0.000 -101.955 0.000 }}
: [[error map]]: {{val| +30.250 -56.581 -4.217 }}
* [[POTE]]: ~3/2 = 600.000, ~5/4 = 313.690 (~6/5 = 286.310)
* [[CWE]]: ~3/2 = 600.000{{c}}, ~5/4 = 336.527{{c}} (~6/5 = 263.473{{c}})
: error map: {{val| 0.000 -101.955 -72.624 }}
: error map: {{val| 0.000 -101.955 -49.787 }}


{{Optimal ET sequence|legend=1| 2, 4 }}
{{Optimal ET sequence|legend=1| 2, 4 }}


[[Badness]] (Smith): 0.021647
[[Badness]] (Sintel): 0.508


; Music
; Music
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.


=== 7-limit ===
=== Septimal antitonic ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 33: Line 34:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 600.000, ~5/4 = 379.210 (~6/5 = 220.890)
* WE: ~3/2 = 614.759{{c}}, ~7/6 = 309.426{{c}} (~6/5 = 305.334{{c}})
* POTE: ~3/2 = 600.000, ~7/6 = 301.997 (~6/5 = 298.003)
* CWE: ~3/2 = 600.000{{c}}, ~7/6 = 326.047{{c}} (~6/5 = 273.953{{c}})


{{Optimal ET sequence|legend=0| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}


Badness (Smith): 0.019373
Badness (Sintel): 0.490


==== 11-limit ====
==== 11-limit ====
Line 48: Line 49:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 600.000, ~5/4 = 379.210 (~11/10 = 220.890)
* WE: ~3/2 = 606.293{{c}}, ~5/4 = 343.862{{c}} (~8/7 = 262.431{{c}})
* POTE: ~3/2 = 600.000, ~5/4 = 340.293 (~8/7 = 259.707)
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 348.102{{c}} (~8/7 = 251.898{{c}})


{{Optimal ET sequence|legend=0| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}


Badness (Smith): 0.021810
Badness (Sintel): 0.721


==== Antietam ====
==== Antietam ====
Line 63: Line 64:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 600.000, ~5/4 = 415.533 (~12/11 = 184.467)
* WE: ~3/2 = 616.135{{c}}, ~5/4 = 330.741{{c}} (~6/5 = 285.393{{c}})
* POTE: ~3/2 = 600.000, ~7/6 = 322.080 (~6/5 = 277.920)
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 349.843{{c}} (~8/7 = 250.157{{c}})


{{Optimal ET sequence|legend=0| 2, 4e }}
{{Optimal ET sequence|legend=0| 2, 4e }}


Badness (Smith): 0.018993
Badness (Sintel): 0.628


=== Antaeus ===
=== Antaeus ===
Line 78: Line 79:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 600.000, ~5/4 = 323.296 (~7/6 = 276.704)
* WE: ~3/2 = 614.854{{c}}, ~5/4 = 323.784{{c}} (~6/5 = 291.070{{c}})
* POTE: ~3/2 = 600.000, ~5/4 = 315.962 (~7/6 = 284.038)
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 317.349{{c}} (~6/5 = 282.651{{c}})


{{Optimal ET sequence|legend=0| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}


Badness (Smith): 0.037537
Badness (Sintel): 0.950


==== 11-limit ====
==== 11-limit ====
Line 93: Line 94:


Optimal tunings:  
Optimal tunings:  
* CTE: ~3/2 = 600.000, ~5/4 = 323.296 (~7/6 = 276.704)
* WE: ~3/2 = 609.311{{c}}, ~5/4 = 323.110{{c}} (~6/5 = 286.200{{c}})
* POTE: ~3/2 = 600.000, ~5/4 = 318.173 (~7/6 = 281.827)
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 318.904{{c}} (~6/5 = 281.096{{c}})


{{Optimal ET sequence|legend=0| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}


Badness (Smith): 0.033786
Badness (Sintel): 1.12


== Alteraugment ==
== Alteraugment ==
Alteraugment is like [[augmented]], but the generator provides 5/4 instead of 3/2. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.
Alteraugment tempers out the [[32/27|Pythagorean minor third (32/27)]]. It is like [[augmented (temperament)|augmented]], but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 112: Line 113:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~4/3 = 400.000, ~5/4 = 386.314 (~16/15 = 13.686)
* [[WE]]: ~4/3 = 389.212{{c}}, ~5/4 = 447.530{{c}} (~10/9 = 58.318{{c}})
: [[error map]]: {{val| 0.000 +98.045 0.000 }}
: [[error map]]: {{val| -32.364 +44.105 -3.512 }}
* [[POTE]]: ~4/3 = 400.000, ~5/4 = 459.935 (~10/9 = 59.935)
* [[CWE]]: ~4/3 = 400.000{{c}}, ~5/4 = 434.191{{c}} (~15/16 = 34.191{{c}})
: error map: {{val| 0.000 +98.045 +73.621 }}
: error map: {{val| 0.000 +98.045 +47.878 }}


{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}
{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}


[[Badness]] (Smith): 0.078042
[[Badness]] (Sintel): 1.83


== Yo (2c & 3)  ==
== Antonian ==
This temperament family is characterized by the [[color notation|yo 2nd]] [[10/9]] being tempering out.
{{Main| Antonian }}
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 131: Line 133:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 761.310
* [[WE]]: ~2 = 1187.236{{c}}, ~3/2 = 767.716{{c}}
: [[error map]]: {{val| 0.000 +74.015 -34.374 }}
: [[error map]]: {{val| -12.764 +52.997 -63.645 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 775.970
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: error map: {{val| 0.000 +65.763 -50.878 }}
: error map: {{val| 0.000 +65.763 -50.877 }}
 
{{Optimal ET sequence|legend=1| 2c, 3 }}
 
[[Badness]] (Sintel): 0.392
 
=== Septimal antonian ===
{{See also| Trienstonic clan }}
 
Subgroup: 2.3.5.7
 
Comma list: 10/9, 15/14
 
Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}
 
Optimal tunings:
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}
 
{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}
 
Badness (Sintel): 0.606
 
=== Antonym ===
Subgroup: 2.3.5.7
 
Comma list: 7/6, 10/9
 
Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}
 
Optimal tunings:
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}
 
{{Optimal ET sequence|legend=0| 2cd, 3 }}
 
Badness (Sintel): 0.568
 
=== Antony ===
Subgroup: 2.3.5.7
 
Comma list: 8/7, 10/9
 
Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}
 
Optimal tunings:
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}


{{Optimal ET sequence|legend=1| 3, 11bc, 14bc, 17bb }}
{{Optimal ET sequence|legend=0| 1c, 3d }}


[[Badness]] (Smith): 0.016712
Badness (Sintel): 0.720


=== Brutus ===
=== Brutus ===
Line 148: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 813.116
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 848.221
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}


{{Optimal ET sequence|legend=0| 3, 7bc }}
{{Optimal ET sequence|legend=0| 3, 7bc }}


Badness (Smith): 0.053389
Badness (Sintel): 1.35


=== Phlegyas ===
=== Phlegyas ===
{{See also| Archytas clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 163: Line 214:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 747.225
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 743.135
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}


{{Optimal ET sequence|legend=0| 3, 5c, 8c }}
{{Optimal ET sequence|legend=0| 3, 5c, 8c }}


Badness (Smith): 0.051293
Badness (Sintel): 1.30


=== Charon ===
=== Charon ===
{{See also| Jubilismic clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 176: Line 229:


Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~3/2 = 768.427 (~15/14 = 168.427)
* WE: ~7/5 = 593.832{{c}}, ~3/2 = 774.559{{c}} (~15/14 = 180.726{{c}})
* POTE: ~7/5 = 600.000, ~3/2 = 782.604 (~15/14 = 182.604)
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 774.466{{c}} (~15/14 = 174.466{{c}})


{{Optimal ET sequence|legend=0| 4bcd, 6 }}
{{Optimal ET sequence|legend=0| 4bcd, 6 }}


Badness (Smith): 0.056404
Badness (Sintel): 1.43


=== Nessus ===
=== Nessus ===
Line 193: Line 247:


Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 980.335
* WE: ~2 = 1189.201{{c}}, ~7/4 = 978.002{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 986.883
* CWE: ~2 = 1200.000{{c}}, ~7/4 = 983.918{{c}}


{{Optimal ET sequence|legend=0| 5c, 6 }}
{{Optimal ET sequence|legend=0| 5c, 6 }}


Badness (Smith): 0.059070
Badness (Sintel): 1.49


== Ternary ==
== Ternary ==
Line 214: Line 267:


: mapping generators: ~5/4, ~7
: mapping generators: ~5/4, ~7
{{Multival|legend=1| 0 0 3 0 5 7 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~5/4 = 400.000, ~7/4 = 968.826 (~28/25 = 168.826)
* [[WE]]: ~5/4 = 391.796{{c}}, ~7/4 = 1012.806{{c}} (~15/14 = 162.582{{c}})
: [[error map]]: {{val| 0.000 +98.045 +13.686 0.000 }}
: [[error map]]: {{val| -24.612 +57.026 -43.741 -5.243 }}
* [[POTE]]: ~5/4 = 400.000, ~7/4 = 1034.013 (~15/14 = 165.987)
* [[CWE]]: ~5/4 = 400.000{{c}}, ~7/4 = 1016.378{{c}} (~8/7 = 183.622{{c}})
: error map: {{val| 0.000 +98.045 +13.686 +65.187 }}
: error map: {{val| 0.000 +98.045 +13.686 +47.552 }}


{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}
{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}


[[Badness]] (Smith): 0.028673
[[Badness]] (Sintel): 0.726


=== 11-limit ===
=== 11-limit ===
Line 235: Line 286:


Optimal tunings:  
Optimal tunings:  
* CTE: ~5/4 = 400.000, ~7/4 = 961.874 (~11/10 = 161.874)
* WE: ~5/4 = 391.788{{c}}, ~7/4 = 1011.942{{c}} (~12/11 = 163.422{{c}})
* POTE: ~5/4 = 400.000, ~7/4 = 1033.153 (~12/11 = 166.847)
* CWE: ~5/4 = 400.000{{c}}, ~7/4 = 1013.973{{c}} (~12/11 = 186.027{{c}})


{{Optimal ET sequence|legend=0| 3de, 6 }}
{{Optimal ET sequence|legend=0| 3de, 6 }}


Badness (Smith): 0.025592
Badness (Sintel): 0.846
 
== Quad ==
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 9/8, 25/24
 
{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}}
: [[error map]]: {{val| +32.295 -53.513 -13.650 -5.150 }}
* [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}}
: error map: {{val| 0.000 -101.955 -86.314 -71.236 }}
 
{{Optimal ET sequence|legend=1| 4 }}
 
[[Badness]] (Sintel): 1.16
 
== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16/15, 27/25
 
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
 
: Mapping generators: ~6/5, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}
 
{{Optimal ET sequence|legend=1| 5, 15ccd }}
 
[[Badness]] (Sintel): 1.22
 
== Sept ==
Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.
 
This temperament used to be known as ''mujannab''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 54/49, 64/63
 
{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
: [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
* [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
: error map: {{val| 0.000 -16.241 +6.406 +59.746 }}
 
{{Optimal ET sequence|legend=1| 7, 14d }}
 
[[Badness]] (Sintel): 2.68
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 54/49, 64/63
 
Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}
 
Optimal tunings:
* WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})
 
{{Optimal ET sequence|legend=0| 7, 14de }}
 
Badness (Sintel): 2.02
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 52/49, 64/63
 
Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
 
Optimal tunings:
* WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})
 
{{Optimal ET sequence|legend=0| 7, 14de }}
 
Badness (Sintel): 1.77


== Geryon ==
== Geryon ==
Line 252: Line 394:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 354.664
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
: [[error map]]: {{val| 0.000 +7.373 -31.649 +231.174 }}
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 385.440
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
: error map: {{val| 0.000 +68.926 -0.873 +231.174 }}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}


[[Badness]] (Smith): 0.051009
[[Badness]] (Sintel): 1.29


== Malacoda ==
== Malacoda ==
Line 273: Line 415:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 938.796
* [[WE]]: ~2 = 1223.542{{c}}, ~7/4 = 941.838{{c}}
: [[error map]]: {{val| 0.000 -24.362 -125.110 -30.029 }}
: [[error map]]: {{val| +23.542 -18.278 -57.528 +20.096 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 923.717
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
: error map: {{val| 0.000 -54.521 -110.031 -45.109 }}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}


{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}
{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}


[[Badness]] (Smith): 0.037207
[[Badness]] (Sintel): 0.942


== Ugolino ==
== Ugolino ==
Line 292: Line 434:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/3 = 926.868
* [[WE]]: ~2 = 1206.628{{c}}, ~7/4 = 926.730{{c}}
: [[error map]]: {{val| 0.000 -48.218 -5.708 +65.517 }}
: [[error map]]: {{val| +6.628 -48.494 -6.122 +58.198 }}
* [[POTE]]: ~2 = 1200.000, ~5/3 = 921.640
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
: error map: {{val| 0.000 -58.675 -21.394 +39.374 }}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}


{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}
{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}


[[Badness]] (Smith): 0.043758
[[Badness]] (Sintel): 1.11


== Medusa ==
== Medusa ==
Line 311: Line 453:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 684.722
* [[WE]]: ~2 = 1200.960{{c}}, ~3/2 = 686.181{{c}}
: [[error map]]: {{val| 0.000 -17.233 -40.480 +61.730 }}
: [[error map]]: {{val| +0.960 -14.814 -41.014 +62.655 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 685.632
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.511{{c}}
: error map: {{val| 0.000 -16.323 -43.209 +59.911 }}
: error map: {{val| 0.000 -16.443 -42.849 +60.150 }}


{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
{{Optimal ET sequence|legend=1| 2, 5c, 7 }}


[[Badness]] (Smith): 0.042712
[[Badness]] (Sintel): 1.08


=== 11-limit ===
=== 11-limit ===
Line 328: Line 470:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 683.589
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 685.809
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}


{{Optimal ET sequence|legend=0| 2, 5c, 7 }}
{{Optimal ET sequence|legend=0| 2, 5c, 7 }}


Badness (Smith): 0.026828
Badness (Sintel): 0.887


== Wallaby ==
== Wallaby ==
Line 344: Line 486:
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}


{{Multival|legend=1| 1 -3 3 -7 2 15 }}
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1216.024{{c}}, ~3/2 = 700.583{{c}}
[[Optimal tuning]]s:
: [[error map]]: {{val| +16.024 +14.652 -23.967 -51.053 }}
* [[CTE]]: ~2 = 1200.000, ~3/2 = 691.757
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
: [[error map]]: {{val| 0.000 -8.515 -66.635 -88.505 }}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 691.351
: error map: {{val| 0.000 -10.604 -60.368 -94.772 }}


{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}
{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}


[[Badness]] (Smith): 0.058468
[[Badness]] (Sintel): 1.48


== Plutus ==
== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}
{{See also| Meantone family }}


Line 366: Line 507:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 685.837
* [[WE]]: ~2 = 1203.936{{c}}, ~3/2 = 685.135{{c}}
: [[error map]]: {{val| 0.000 -16.118 -42.965 +60.359 }}
: [[error map]]: {{val| +3.936 -12.884 -45.774 +56.849 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 682.895
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 683.935{{c}}
: error map: {{val| 0.000 -19.060 -54.734 +45.649 }}
: error map: {{val| 0.000 -18.020 -50.573 +50.850 }}


{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}


[[Badness]] (Smith): 0.045275
[[Badness]] (Sintel): 1.14


=== 11-limit ===
=== 11-limit ===
Line 383: Line 524:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 687.743
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 685.234
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}


{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}
{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}


Badness (Smith): 0.032521
Badness (Sintel): 1.08


== Involution ==
== Involution ==
Involution was named by [[User:CompactStar|CompactStar]] in 2023.  
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 402: Line 543:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 509.892
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| 0.000 -192.063 -206.097 }}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 520.625
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.330 -227.565 }}
: error map: {{val| 0.000 -181.329 -227.565 }}


{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}


[[Badness]] (Smith): 0.133021
[[Badness]] (Sintel): 3.12


=== 7-limit ===
=== 7-limit ===
Line 419: Line 560:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 509.892
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000, ~3/2 = 515.099
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}


Optimal ET sequence: none
Optimal ET sequence: none


Badness (Smith): 0.118764
Badness (Sintel): 3.01


== Devolution ==
== Devolution ==
Line 438: Line 579:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~3 = 1901.955, ~5/3 = 701.863
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| 0.000 -182.496 -266.779 }}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.995, ~5/3 = 691.468
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}
: error map: {{val| 0.000 -192.891 -256.384 }}


[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]


[[Badness]]:
[[Badness]] (Sintel): 0.751
* Smith: 0.00406
* Dirichlet: 0.751


== Codex ==
== Codex ==
{{See also| Bug family }}
{{See also| Bug family }}
 
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid dicot.  
This temperament was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] in the same way decimal is of [[semaphore]].  


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


[[Comma list]]: [[27/25]], [[54/49]]
[[Comma list]]: [[27/25]], [[50/49]]


{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}
{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~5/3 = 935.289
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: [[error map]]: {{val| 0.000 -31.377 +19.554 +37.041 }}
: [[error map]]: {{val| -2.821 -31.998 +18.621 +34.699 }}
* [[CWE]]: ~7/5 = 600.000, ~5/3 = 936.030
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}


{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}
{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}


[[Badness]] (Smith): 0.077065
[[Badness]] (Sintel): 1.95
 
== Oxygen ==
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 175/162
 
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1213.695{{c}}, ~10/9 = 171.042{{c}}
: error map: {{val| +13.695 +12.309 -0.438 -69.825 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
 
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*
 
<nowiki/>*[[Optimal patent val]]: [[8edo|8]]
 
[[Badness]] (Sintel): 1.52
 
== Bixby ==
{{Main| Bixby }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: [[4/3]]
 
{{Mapping|legend=1| 1 2 0 | 0 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1020.058{{c}}, ~5/4 = 674.394{{c}}
: [[error map]]: {{val| -179.942 +138.161 -71.803 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 629.521{{c}}
: error map: {{val| 0.000 +498.045 +243.208 }}
 
{{Optimal ET sequence|legend=1| 1c, 2b, 3bbcc }}
 
[[Badness]] (Sintel): 0.424
 
== Archon ==
{{Main| Archon }}
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: [[5/4]]
 
{{Mapping|legend=1| 1 0 2 | 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}
 
{{Optimal ET sequence|legend=1| 2c }}
 
[[Badness]] (Sintel): 0.474
 
== Seesaw ==
{{Main| Seesaw }}
Seesaw tempers out the [[6/5|classic minor third (6/5)]], equating the [[5/1|fifth]] and [[6/1|sixth harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 6/5
 
{{Mapping|legend=1| 1 0 1 | 0 1 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1155.569{{c}}, ~3/2 = 643.349{{c}}
: [[error map]]: {{val| -44.431 -103.037 +168.173 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 627.511{{c}}
: error map: {{val| 0.000 -74.444 +241.197 }}
 
{{Optimal ET sequence|legend=1| 2 }}
 
[[Badness]] (Sintel): 0.367
 
=== 2.3.5.11 subgroup ===
This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both [[11/10]] and [[12/11]], undecimal neutral seconds.
 
Subgroup: 2.3.5.11
 
Comma list: 6/5, 11/10
 
Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}
 
Optimal tunings:
* WE: ~2 = 1156.418{{c}}, ~3/2 = 643.202{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 627.023{{c}}
 
{{Optimal ET sequence|legend=0| 2 }}
 
Badness (Sintel): 0.499
 
=== Heavy windmill ===
Heavy windmill tempers out [[9/7]] and [[15/14]] in the 7-limit.
 
Subgroup: 2.3.5.7
 
Comma list: 6/5, 9/7
 
Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}
 
Optimal tunings:
* WE: ~2 = 1161.600{{c}}, ~3/2 = 571.169{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 559.563{{c}}
 
{{Optimal ET sequence|legend=0| 2 }}
 
Badness (Sintel): 0.676
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 6/5, 9/7, 11/10
 
Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}
 
Optimal tunings:
* WE: ~2 = 1166.584{{c}}, ~3/2 = 568.073{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 558.941{{c}}
 
{{Optimal ET sequence|legend=0| 2 }}
 
Badness (Sintel): 0.774
 
=== Light windmill ===
Light windmill tempers out [[8/7]] and [[21/20]] in the 7-limit.
 
Subgroup: 2.3.5.7
 
Comma list: 6/5, 8/7
 
Mapping: {{mapping| 1 0 1 3 | 0 1 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1134.018{{c}}, ~3/2 = 670.285{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 667.893{{c}}
 
{{Optimal ET sequence|legend=0| 2 }}
 
Badness (Sintel): 0.629
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 6/5, 8/7, 11/10
 
Mapping: {{mapping| 1 0 1 3 2 | 0 1 1 0 1 }}
 
Optimal tunings:
* WE: ~2 = 1136.109{{c}}, ~3/2 = 672.403{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 668.374{{c}}
 
{{Optimal ET sequence|legend=0| 2 }}
 
Badness (Sintel): 0.681


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Latest revision as of 10:55, 29 May 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.

Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large intervals such as 9/8, 10/9, 32/27, or 15/14, equating wildly different interval sizes with semitone-level or even greater errors, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are exotemperaments.

Antitonic

This temperament is characterized by 9/8 being tempered out and has been termed a "troll temperament" by its namers. Its ploidacot is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of 2edo with the addition of harmonic 5 represented by an independent generator.

Subgroup: 2.3.5

Comma list: 9/8

Mapping: [⟨2 3 0], ⟨0 0 1]]

mapping generators: ~3/2, ~5

Optimal tunings:

  • WE: ~3/2 = 615.125 ¢, ~5/4 = 321.597 ¢ (~6/5 = 293.528 ¢)
error map: ⟨+30.250 -56.581 -4.217]
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 336.527 ¢ (~6/5 = 263.473 ¢)
error map: ⟨0.000 -101.955 -49.787]

Optimal ET sequence: 2, 4

Badness (Sintel): 0.508

Music
  • Antitonic(8) (2024) – short composition by Wensik in POTE-tuned 5-limit antitonic using an 8-note ternary scale.

Septimal antitonic

Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

Mapping: [⟨2 3 0 1], ⟨0 0 1 1]]

Optimal tunings:

  • WE: ~3/2 = 614.759 ¢, ~7/6 = 309.426 ¢ (~6/5 = 305.334 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~7/6 = 326.047 ¢ (~6/5 = 273.953 ¢)

Optimal ET sequence: 2, 4

Badness (Sintel): 0.490

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 15/14

Mapping: [⟨2 3 0 1 7], ⟨0 0 1 1 0]]

Optimal tunings:

  • WE: ~3/2 = 606.293 ¢, ~5/4 = 343.862 ¢ (~8/7 = 262.431 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 348.102 ¢ (~8/7 = 251.898 ¢)

Optimal ET sequence: 2, 4

Badness (Sintel): 0.721

Antietam

Subgroup: 2.3.5.7.11

Comma list: 9/8, 11/10, 15/14

Mapping: [⟨2 3 0 1 2], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~3/2 = 616.135 ¢, ~5/4 = 330.741 ¢ (~6/5 = 285.393 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 349.843 ¢ (~8/7 = 250.157 ¢)

Optimal ET sequence: 2, 4e

Badness (Sintel): 0.628

Antaeus

Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: [⟨2 3 0 10], ⟨0 0 1 -1]]

Optimal tunings:

  • WE: ~3/2 = 614.854 ¢, ~5/4 = 323.784 ¢ (~6/5 = 291.070 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 317.349 ¢ (~6/5 = 282.651 ¢)

Optimal ET sequence: 2d, 4

Badness (Sintel): 0.950

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 35/32

Mapping: [⟨2 3 0 10 7], ⟨0 0 1 -1 0]]

Optimal tunings:

  • WE: ~3/2 = 609.311 ¢, ~5/4 = 323.110 ¢ (~6/5 = 286.200 ¢)
  • CWE: ~3/2 = 600.000 ¢, ~5/4 = 318.904 ¢ (~6/5 = 281.096 ¢)

Optimal ET sequence: 2d, 4

Badness (Sintel): 1.12

Alteraugment

Alteraugment tempers out the Pythagorean minor third (32/27). It is like augmented, but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. Vector Graphics suggests the name kinsborough for this temperament.

Subgroup: 2.3.5

Comma list: 32/27

Mapping: [⟨3 5 0], ⟨0 0 1]]

mapping generators: ~4/3, ~5

Optimal tunings:

  • WE: ~4/3 = 389.212 ¢, ~5/4 = 447.530 ¢ (~10/9 = 58.318 ¢)
error map: ⟨-32.364 +44.105 -3.512]
  • CWE: ~4/3 = 400.000 ¢, ~5/4 = 434.191 ¢ (~15/16 = 34.191 ¢)
error map: ⟨0.000 +98.045 +47.878]

Optimal ET sequence: 3, 12bcc, 15bbcc

Badness (Sintel): 1.83

Antonian

Main article: Antonian

This temperament family is characterized by the yo 2nd (10/9) being tempered out. It identifies 3/2 with 5/3, 4/3 with 6/5, and 5/4 with 9/8.

Subgroup: 2.3.5

Comma list: 10/9

Mapping: [⟨1 0 -1], ⟨0 1 2]]

Optimal tunings:

  • WE: ~2 = 1187.236 ¢, ~3/2 = 767.716 ¢
error map: ⟨-12.764 +52.997 -63.645]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 767.718 ¢
error map: ⟨0.000 +65.763 -50.877]

Optimal ET sequence: 2c, 3

Badness (Sintel): 0.392

Septimal antonian

See also: Trienstonic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 15/14

Mapping: [⟨1 0 -1 -2], ⟨0 1 2 3]]

Optimal tunings:

  • WE: ~2 = 1193.691 ¢, ~3/2 = 742.509 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 743.086 ¢

Optimal ET sequence: 2cd, 3d, 5c

Badness (Sintel): 0.606

Antonym

Subgroup: 2.3.5.7

Comma list: 7/6, 10/9

Mapping: [⟨1 0 -1 1], ⟨0 1 2 1]]

Optimal tunings:

  • WE: ~2 = 1209.795 ¢, ~3/2 = 765.995 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 765.949 ¢

Optimal ET sequence: 2cd, 3

Badness (Sintel): 0.568

Antony

Subgroup: 2.3.5.7

Comma list: 8/7, 10/9

Mapping: [⟨1 0 -1 3], ⟨0 1 2 0]]

Optimal tunings:

  • WE: ~2 = 1151.235 ¢, ~3/2 = 789.399 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 800.996 ¢

Optimal ET sequence: 1c, 3d

Badness (Sintel): 0.720

Brutus

Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: [⟨1 0 -1 -4], ⟨0 1 2 4]]

Optimal tunings:

  • WE: ~2 = 1158.982 ¢, ~3/2 = 819.228 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 831.346 ¢

Optimal ET sequence: 3, 7bc

Badness (Sintel): 1.35

Phlegyas

See also: Archytas clan

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: [⟨1 0 -1 6], ⟨0 1 2 -2]]

Optimal tunings:

  • WE: ~2 = 1206.510 ¢, ~3/2 = 747.166 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 743.797 ¢

Optimal ET sequence: 3, 5c, 8c

Badness (Sintel): 1.30

Charon

See also: Jubilismic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 593.832 ¢, ~3/2 = 774.559 ¢ (~15/14 = 180.726 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 774.466 ¢ (~15/14 = 174.466 ¢)

Optimal ET sequence: 4bcd, 6

Badness (Sintel): 1.43

Nessus

See also: Semaphoresmic clan

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

Mapping: [⟨1 0 -1 2], ⟨0 2 4 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1189.201 ¢, ~7/4 = 978.002 ¢
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 983.918 ¢

Optimal ET sequence: 5c, 6

Badness (Sintel): 1.49

Ternary

Ternary is identical to 3et in the 5-limit, but has an independent generator for prime 7.

Subgroup: 2.3.5.7

Comma list: 10/9, 16/15

Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Optimal tunings:

  • WE: ~5/4 = 391.796 ¢, ~7/4 = 1012.806 ¢ (~15/14 = 162.582 ¢)
error map: ⟨-24.612 +57.026 -43.741 -5.243]
  • CWE: ~5/4 = 400.000 ¢, ~7/4 = 1016.378 ¢ (~8/7 = 183.622 ¢)
error map: ⟨0.000 +98.045 +13.686 +47.552]

Optimal ET sequence: 3d, 6, 9bd

Badness (Sintel): 0.726

11-limit

Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~5/4 = 391.788 ¢, ~7/4 = 1011.942 ¢ (~12/11 = 163.422 ¢)
  • CWE: ~5/4 = 400.000 ¢, ~7/4 = 1013.973 ¢ (~12/11 = 186.027 ¢)

Optimal ET sequence: 3de, 6

Badness (Sintel): 0.846

Quad

Quad is identical to 4et in the 5-limit, but has an independent generator for prime 7.

Subgroup: 2.3.5.7

Comma list: 9/8, 25/24

Mapping: [⟨4 6 9 0], ⟨0 0 0 1]]

Optimal tunings:

  • WE: ~6/5 = 308.074 ¢, ~7/4 = 963.670 ¢
error map: ⟨+32.295 -53.513 -13.650 -5.150]
  • CWE: ~6/5 = 300.000 ¢, ~7/4 = 897.589 ¢
error map: ⟨0.000 -101.955 -86.314 -71.236]

Optimal ET sequence: 4

Badness (Sintel): 1.16

Quint

Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what blackwood does.

Subgroup: 2.3.5.7

Comma list: 16/15, 27/25

Mapping: [⟨5 8 12 0], ⟨0 0 0 1]]

Mapping generators: ~6/5, ~7

Optimal tunings:

  • WE: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038)
error map: ⟨-17.921 -10.628 +50.676 -1.966]
  • CWE: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
error map: ⟨0.000 +18.045 +93.686 +36.309]

Optimal ET sequence: 5, 15ccd

Badness (Sintel): 1.22

Sept

Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what jamesbond does.

This temperament used to be known as mujannab.

Subgroup: 2.3.5.7

Comma list: 54/49, 64/63

Mapping: [⟨7 11 0 20], ⟨0 0 1 0]]

Optimal tunings:

  • WE: ~9/8 = 170.823 ¢, ~5/4 = 393.792 ¢ (~15/14 = 52.145 ¢)
error map: ⟨-4.236 -22.898 -0.994 +47.642]
  • CWE: ~9/8 = 171.429 ¢, ~5/4 = 392.719 ¢ (~15/14 = 49.862 ¢)
error map: ⟨0.000 -16.241 +6.406 +59.746]

Optimal ET sequence: 7, 14d

Badness (Sintel): 2.68

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 54/49, 64/63

Mapping: [⟨7 11 0 20 8], ⟨0 0 1 0 1]]

Optimal tunings:

  • WE: ~11/10 = 170.817 ¢, ~5/4 = 393.252 ¢ (~33/32 = 51.619 ¢)
  • CWE: ~11/10 = 171.429 ¢, ~5/4 = 391.840 ¢ (~33/32 = 48.983 ¢)

Optimal ET sequence: 7, 14de

Badness (Sintel): 2.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 52/49, 64/63

Mapping: [⟨7 11 0 20 8 26], ⟨0 0 1 0 1 0]]

Optimal tunings:

  • WE: ~11/10 = 170.795 ¢, ~5/4 = 393.611 ¢ (~33/32 = 52.021 ¢)
  • CWE: ~11/10 = 171.429 ¢, ~5/4 = 392.725 ¢ (~33/32 = 49.868 ¢)

Optimal ET sequence: 7, 14de

Badness (Sintel): 1.77

Geryon

See also: Dicot family

Subgroup: 2.3.5.7

Comma list: 8/7, 25/21

Mapping: [⟨1 1 2 3], ⟨0 2 1 0]]

Optimal tunings:

  • WE: ~2 = 1164.885 ¢, ~5/4 = 374.162 ¢
error map: ⟨-35.115 +11.253 -82.382 +125.830]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 375.277 ¢
error map: ⟨0.000 +48.600 -11.036 +231.174]

Optimal ET sequence: 1b, 2b, 3d

Badness (Sintel): 1.29

Malacoda

See also: Semaphoresmic clan

Subgroup: 2.3.5.7

Comma list: 15/14, 35/32

Mapping: [⟨1 0 3 2], ⟨0 2 -1 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1223.542 ¢, ~7/4 = 941.838 ¢
error map: ⟨+23.542 -18.278 -57.528 +20.096]
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 927.096 ¢
error map: ⟨0.000 -47.763 -113.410 -41.730]

Optimal ET sequence: 1, 3b, 4, 9c, 13bcc

Badness (Sintel): 0.942

Ugolino

See also: Bug family

Subgroup: 2.3.5.7

Comma list: 15/14, 27/25

Mapping: [⟨1 0 0 -1], ⟨0 2 3 5]]

Optimal tunings:

  • WE: ~2 = 1206.628 ¢, ~7/4 = 926.730 ¢
error map: ⟨+6.628 -48.494 -6.122 +58.198]
  • CWE: ~2 = 1200.000 ¢, ~7/4 = 923.776 ¢
error map: ⟨0.000 -54.403 -14.986 +50.054]

Optimal ET sequence: 1cdd, 3bcdd, 4, 9d

Badness (Sintel): 1.11

Medusa

See also: Archytas clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 15/14, 64/63

Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]

Optimal tunings:

  • WE: ~2 = 1200.960 ¢, ~3/2 = 686.181 ¢
error map: ⟨+0.960 -14.814 -41.014 +62.655]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 685.511 ¢
error map: ⟨0.000 -16.443 -42.849 +60.150]

Optimal ET sequence: 2, 5c, 7

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 33/32

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

Optimal tunings:

  • WE: ~2 = 1202.757 ¢, ~3/2 = 687.384 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 685.462 ¢

Optimal ET sequence: 2, 5c, 7

Badness (Sintel): 0.887

Wallaby

See also: Trienstonic clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Optimal tunings:

  • WE: ~2 = 1216.024 ¢, ~3/2 = 700.583 ¢
error map: ⟨+16.024 +14.652 -23.967 -51.053]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 691.757 ¢
error map: ⟨0.000 -10.198 -61.585 -93.555]

Optimal ET sequence: 2d, 5c, 7d, 19ccdd

Badness (Sintel): 1.48

Plutus

Not to be confused with Pluto.
See also: Meantone family

Subgroup: 2.3.5.7

Comma list: 15/14, 81/80

Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]

Optimal tunings:

  • WE: ~2 = 1203.936 ¢, ~3/2 = 685.135 ¢
error map: ⟨+3.936 -12.884 -45.774 +56.849]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 683.935 ¢
error map: ⟨0.000 -18.020 -50.573 +50.850]

Optimal ET sequence: 2cd, 5d, 7

Badness (Sintel): 1.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

Mapping: [⟨1 0 -4 -5 -6], ⟨0 1 4 5 6]]

Optimal tunings:

  • WE: ~2 = 1203.293 ¢, ~3/2 = 687.114 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 686.078 ¢

Optimal ET sequence: 2cde, 5de, 7

Badness (Sintel): 1.08

Involution

Involution tempers out the ptolemaic augmented fourth (45/32). Its ploidacot is monocot. It was named by CompactStar in 2023.

Subgroup: 2.3.5

Comma list: 45/32

Mapping: [⟨1 0 5], ⟨0 1 -2]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1265.406 ¢, ~3/2 = 552.346 ¢
error map: ⟨+65.405 -84.203 -94.789]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 520.626 ¢
error map: ⟨0.000 -181.329 -227.565]

Optimal ET sequence: 2c, 5bc, 7bbcc

Badness (Sintel): 3.12

7-limit

Subgroup: 2.3.5.7

Comma list: 8/7, 45/28

Mapping: [⟨1 0 5 3], ⟨0 1 -2 0]]

Optimal tunings:

  • WE: ~2 = 1205.230 ¢, ~3/2 = 517.557 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 515.099 ¢

Optimal ET sequence: none

Badness (Sintel): 3.01

Devolution

Devolution was named by Akselai in 2024.

Subgroup: 3.5.7

Comma list: 35/27

Mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~3, ~5

Optimal tunings:

  • WE: ~3 = 1988.549 ¢, ~5/3 = 719.679 ¢
error map: ⟨+86.594 -78.085 -111.407]
  • CWE: ~3 = 1901.955 ¢, ~5/3 = 691.468 ¢
error map: ⟨0.000 -192.891 -256.384]

Optimal ET sequence: b1, b2d, b3, b8cdd

Badness (Sintel): 0.751

Codex

See also: Bug family

Codex was named by Jerdle as an abbreviation of "co-decimal", as it resembles decimal in many ways, but exchanges the roles of 5's and 7's. While decimal equates 8/7 and 7/6, as well as 6/5 and 5/4, this equates 10/9 and 6/5, as well as 7/6 and 9/7. It is an extension of bug and mujannabic in the same way decimal is of semaphore and dicot. Its ploidacot is diploid dicot.

Subgroup: 2.3.5.7

Comma list: 27/25, 50/49

Mapping: [⟨2 0 0 1], ⟨0 2 3 3]]

Optimal tunings:

  • WE: ~7/5 = 598.589 ¢, ~5/3 = 934.978 ¢
error map: ⟨-2.821 -31.998 +18.621 +34.699]
  • CWE: ~7/5 = 600.000 ¢, ~5/3 = 936.030 ¢
error map: ⟨0.000 -29.895 +21.776 +39.264]

Optimal ET sequence: 4, 10cd, 14d

Badness (Sintel): 1.95

Oxygen

See also: Porcupine family

Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct Fokker blocks.

Subgroup: 2.3.5.7

Comma list: 21/20, 175/162

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -2]]

Optimal tunings:

  • WE: ~2 = 1213.695 ¢, ~10/9 = 171.042 ¢
error map: ⟨+13.695 +12.309 -0.438 -69.825]
  • CWE: ~2 = 1200.000 ¢, ~10/9 = 166.042 ¢
error map: ⟨0.000 -0.083 -16.526 -100.911]

Optimal ET sequence: 1c, …, 6bcd, 7d*

*Optimal patent val: 8

Badness (Sintel): 1.52

Bixby

Main article: Bixby

Subgroup: 2.3.5

Comma list: 4/3

Mapping: [⟨1 2 0], ⟨0 0 1]]

Optimal tunings:

  • WE: ~2 = 1020.058 ¢, ~5/4 = 674.394 ¢
error map: ⟨-179.942 +138.161 -71.803]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 629.521 ¢
error map: ⟨0.000 +498.045 +243.208]

Optimal ET sequence: 1c, 2b, 3bbcc

Badness (Sintel): 0.424

Archon

Main article: Archon

Subgroup: 2.3.5

Comma list: 5/4

Mapping: [⟨1 0 2], ⟨0 1 0]]

Optimal tunings:

  • WE: ~2 = 1268.274 ¢, ~3/2 = 612.921 ¢
error map: ⟨+68.274 -20.760 -249.765]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 614.055 ¢
error map: ⟨0.000 -87.900 -386.314]

Optimal ET sequence: 2c

Badness (Sintel): 0.474

Seesaw

Main article: Seesaw

Seesaw tempers out the classic minor third (6/5), equating the fifth and sixth harmonics. It was named by Xenllium in 2026.

Subgroup: 2.3.5

Comma list: 6/5

Mapping: [⟨1 0 1], ⟨0 1 1]]

Optimal tunings:

  • WE: ~2 = 1155.569 ¢, ~3/2 = 643.349 ¢
error map: ⟨-44.431 -103.037 +168.173]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 627.511 ¢
error map: ⟨0.000 -74.444 +241.197]

Optimal ET sequence: 2

Badness (Sintel): 0.367

2.3.5.11 subgroup

This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both 11/10 and 12/11, undecimal neutral seconds.

Subgroup: 2.3.5.11

Comma list: 6/5, 11/10

Mapping: [⟨1 0 1 2], ⟨0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1156.418 ¢, ~3/2 = 643.202 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 627.023 ¢

Optimal ET sequence: 2

Badness (Sintel): 0.499

Heavy windmill

Heavy windmill tempers out 9/7 and 15/14 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 9/7

Mapping: [⟨1 0 1 0], ⟨0 1 1 2]]

Optimal tunings:

  • WE: ~2 = 1161.600 ¢, ~3/2 = 571.169 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 559.563 ¢

Optimal ET sequence: 2

Badness (Sintel): 0.676

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6/5, 9/7, 11/10

Mapping: [⟨1 0 1 0 2], ⟨0 1 1 2 1]]

Optimal tunings:

  • WE: ~2 = 1166.584 ¢, ~3/2 = 568.073 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 558.941 ¢

Optimal ET sequence: 2

Badness (Sintel): 0.774

Light windmill

Light windmill tempers out 8/7 and 21/20 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 8/7

Mapping: [⟨1 0 1 3], ⟨0 1 1 0]]

Optimal tunings:

  • WE: ~2 = 1134.018 ¢, ~3/2 = 670.285 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 667.893 ¢

Optimal ET sequence: 2

Badness (Sintel): 0.629

11-limit

Subgroup: 2.3.5.7.11

Comma list: 6/5, 8/7, 11/10

Mapping: [⟨1 0 1 3 2], ⟨0 1 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1136.109 ¢, ~3/2 = 672.403 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 668.374 ¢

Optimal ET sequence: 2

Badness (Sintel): 0.681

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