Very low accuracy temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
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.
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 18: Line 18:
* [[CWE]]: ~3/2 = 600.000{{c}}, ~5/4 = 336.527{{c}} (~6/5 = 263.473{{c}})
* [[CWE]]: ~3/2 = 600.000{{c}}, ~5/4 = 336.527{{c}} (~6/5 = 263.473{{c}})
: error map: {{val| 0.000 -101.955 -49.787 }}
: error map: {{val| 0.000 -101.955 -49.787 }}
<!-- * [[CTE]]: ~3/2 = 600.000{{c}}, ~5/4 = 386.314{{c}} (~6/5 = 213.686{{c}})
: [[error map]]: {{val| 0.000 -101.955 0.000 }}
* [[POTE]]: ~3/2 = 600.000{{c}}, ~5/4 = 313.690{{c}} (~6/5 = 286.310{{c}})
: error map: {{val| 0.000 -101.955 -72.624 }} -->


{{Optimal ET sequence|legend=1| 2, 4 }}
{{Optimal ET sequence|legend=1| 2, 4 }}
Line 30: Line 26:
* [[: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 40: Line 36:
* WE: ~3/2 = 614.759{{c}}, ~7/6 = 309.426{{c}} (~6/5 = 305.334{{c}})
* WE: ~3/2 = 614.759{{c}}, ~7/6 = 309.426{{c}} (~6/5 = 305.334{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~7/6 = 326.047{{c}} (~6/5 = 273.953{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~7/6 = 326.047{{c}} (~6/5 = 273.953{{c}})
<!-- * CTE: ~3/2 = 600.000{{c}}, ~5/4 = 379.210{{c}} (~6/5 = 220.890{{c}})
* POTE: ~3/2 = 600.000{{c}}, ~7/6 = 301.997{{c}} (~6/5 = 298.003{{c}}) -->


{{Optimal ET sequence|legend=0| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}
Line 57: Line 51:
* WE: ~3/2 = 606.293{{c}}, ~5/4 = 343.862{{c}} (~8/7 = 262.431{{c}})
* WE: ~3/2 = 606.293{{c}}, ~5/4 = 343.862{{c}} (~8/7 = 262.431{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 348.102{{c}} (~8/7 = 251.898{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 348.102{{c}} (~8/7 = 251.898{{c}})
<!-- * CTE: ~3/2 = 600.000{{c}}, ~5/4 = 379.210{{c}} (~8/7 = 220.890{{c}})
* POTE: ~3/2 = 600.000{{c}}, ~5/4 = 340.293{{c}} (~8/7 = 259.707{{c}}) -->


{{Optimal ET sequence|legend=0| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}
Line 74: Line 66:
* WE: ~3/2 = 616.135{{c}}, ~5/4 = 330.741{{c}} (~6/5 = 285.393{{c}})
* WE: ~3/2 = 616.135{{c}}, ~5/4 = 330.741{{c}} (~6/5 = 285.393{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 349.843{{c}} (~8/7 = 250.157{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 349.843{{c}} (~8/7 = 250.157{{c}})
<!-- * CTE: ~3/2 = 600.000{{c}}, ~5/4 = 415.533{{c}} (~12/11 = 184.467{{c}})
* POTE: ~3/2 = 600.000{{c}}, ~7/6 = 322.080{{c}} (~6/5 = 277.920{{c}}) -->


{{Optimal ET sequence|legend=0| 2, 4e }}
{{Optimal ET sequence|legend=0| 2, 4e }}
Line 91: Line 81:
* WE: ~3/2 = 614.854{{c}}, ~5/4 = 323.784{{c}} (~6/5 = 291.070{{c}})
* WE: ~3/2 = 614.854{{c}}, ~5/4 = 323.784{{c}} (~6/5 = 291.070{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 317.349{{c}} (~6/5 = 282.651{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 317.349{{c}} (~6/5 = 282.651{{c}})
<!-- * CTE: ~3/2 = 600.000{{c}}, ~5/4 = 323.296{{c}} (~7/6 = 276.704{{c}})
* POTE: ~3/2 = 600.000{{c}}, ~5/4 = 315.962{{c}} (~7/6 = 284.038{{c}}) -->


{{Optimal ET sequence|legend=0| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}
Line 108: Line 96:
* WE: ~3/2 = 609.311{{c}}, ~5/4 = 323.110{{c}} (~6/5 = 286.200{{c}})
* WE: ~3/2 = 609.311{{c}}, ~5/4 = 323.110{{c}} (~6/5 = 286.200{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 318.904{{c}} (~6/5 = 281.096{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 318.904{{c}} (~6/5 = 281.096{{c}})
<!-- * CTE: ~3/2 = 600.000{{c}}, ~5/4 = 323.296{{c}} (~7/6 = 276.704{{c}})
* POTE: ~3/2 = 600.000{{c}}, ~5/4 = 318.173{{c}} (~7/6 = 281.827{{c}}) -->


{{Optimal ET sequence|legend=0| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}
Line 116: Line 102:


== 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 131: Line 117:
* [[CWE]]: ~4/3 = 400.000{{c}}, ~5/4 = 434.191{{c}} (~15/16 = 34.191{{c}})
* [[CWE]]: ~4/3 = 400.000{{c}}, ~5/4 = 434.191{{c}} (~15/16 = 34.191{{c}})
: error map: {{val| 0.000 +98.045 +47.878 }}
: error map: {{val| 0.000 +98.045 +47.878 }}
<!-- * [[CTE]]: ~4/3 = 400.000{{c}}, ~5/4 = 386.314{{c}} (~16/15 = 13.686{{c}})
: [[error map]]: {{val| 0.000 +98.045 0.000 }}
* [[POTE]]: ~4/3 = 400.000{{c}}, ~5/4 = 459.935{{c}} (~10/9 = 59.935{{c}})
: error map: {{val| 0.000 +98.045 +73.621 }} -->


{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}
{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}
Line 140: Line 122:
[[Badness]] (Sintel): 1.83
[[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 154: Line 137:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: error map: {{val| 0.000 +65.763 -50.877 }}
: error map: {{val| 0.000 +65.763 -50.877 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 761.310{{c}}
: [[error map]]: {{val| 0.000 +74.015 -34.374 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 775.970{{c}}
: error map: {{val| 0.000 +65.763 -50.878 }} -->


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


[[Badness]] (Sintel): 0.392
[[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=0| 1c, 3d }}
Badness (Sintel): 0.720


=== Brutus ===
=== Brutus ===
Line 173: Line 199:
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 813.116{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 848.221{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7bc }}
{{Optimal ET sequence|legend=0| 3, 7bc }}
Line 181: Line 205:


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


Line 190: Line 216:
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 747.225{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 743.135{{c}} -->


{{Optimal ET sequence|legend=0| 3, 5c, 8c }}
{{Optimal ET sequence|legend=0| 3, 5c, 8c }}
Line 198: Line 222:


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


Line 203: 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:  
* WE: ~7/5 = 593.832{{c}}, ~3/2 = 774.559{{c}} (~15/14 = 180.726{{c}})
* WE: ~7/5 = 593.832{{c}}, ~3/2 = 774.559{{c}} (~15/14 = 180.726{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 774.466{{c}} (~15/14 = 174.466{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 774.466{{c}} (~15/14 = 174.466{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~3/2 = 768.427{{c}} (~15/14 = 168.427{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~3/2 = 782.604{{c}} (~15/14 = 182.604{{c}}) -->


{{Optimal ET sequence|legend=0| 4bcd, 6 }}
{{Optimal ET sequence|legend=0| 4bcd, 6 }}
Line 222: 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:  
* WE: ~2 = 1189.201{{c}}, ~3/2 = 978.002{{c}}
* WE: ~2 = 1189.201{{c}}, ~7/4 = 978.002{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 983.918{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/4 = 983.918{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/4 = 980.335{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/4 = 986.883{{c}} -->


{{Optimal ET sequence|legend=0| 5c, 6 }}
{{Optimal ET sequence|legend=0| 5c, 6 }}
Line 251: Line 273:
* [[CWE]]: ~5/4 = 400.000{{c}}, ~7/4 = 1016.378{{c}} (~8/7 = 183.622{{c}})
* [[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 +47.552 }}
: error map: {{val| 0.000 +98.045 +13.686 +47.552 }}
<!-- * [[CTE]]: ~5/4 = 400.000{{c}}, ~7/4 = 968.826{{c}} (~28/25 = 168.826{{c}})
: [[error map]]: {{val| 0.000 +98.045 +13.686 0.000 }}
* [[POTE]]: ~5/4 = 400.000{{c}}, ~7/4 = 1034.013{{c}} (~15/14 = 165.987{{c}})
: error map: {{val| 0.000 +98.045 +13.686 +65.187 }} -->


{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}
{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}
Line 270: Line 288:
* WE: ~5/4 = 391.788{{c}}, ~7/4 = 1011.942{{c}} (~12/11 = 163.422{{c}})
* WE: ~5/4 = 391.788{{c}}, ~7/4 = 1011.942{{c}} (~12/11 = 163.422{{c}})
* CWE: ~5/4 = 400.000{{c}}, ~7/4 = 1013.973{{c}} (~12/11 = 186.027{{c}})
* CWE: ~5/4 = 400.000{{c}}, ~7/4 = 1013.973{{c}} (~12/11 = 186.027{{c}})
<!-- * CTE: ~5/4 = 400.000{{c}}, ~7/4 = 961.874{{c}} (~11/10 = 161.874{{c}})
* POTE: ~5/4 = 400.000{{c}}, ~7/4 = 1033.153{{c}} (~12/11 = 166.847{{c}}) -->


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


Badness (Sintel): 0.846
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


== Geryon ==
== Geryon ==
Line 287: Line 343:


[[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 308: Line 364:


[[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 327: Line 383:


[[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 346: Line 402:


[[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 363: Line 419:


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 379: Line 435:
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}


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


{{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 399: Line 456:


[[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 416: Line 473:


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 435: Line 492:


[[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 452: Line 509:


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 471: Line 528:


[[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]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]].  


[[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 ==
== Oxygen ==
{{See also| Porcupine family }}
{{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
[[Subgroup]]: 2.3.5.7
Line 513: Line 568:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.341
* [[WE]]: ~2 = 1213.695{{c}}, ~10/9 = 171.042{{c}}
: [[error map]]: {{val| 0.000 +14.023 +6.982 -91.507 }}
: error map: {{val| +13.695 +12.309 -0.438 -69.825 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 169.112
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
: error map: {{val| 0.000 -9.291 -31.873 -107.050 }}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
 
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}
 
[[Badness]] (Smith): 0.059866
 
== Bixby ==
''Main article: [[Bixby]]''
 
''See also: [[Circular temperament]]''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: [[4/3]]
 
[[Mapping]]: [[1 2 2][0 0 1]]
 
[[Optimal tuning|Optimal tunings]]:
 
* CWE: ~2 = 1200.000, ~5/4 = 629.521
 
== Archon ==
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 5/4
 
[[Mapping]]: [[1 1 2][0 1 0]]


[[Optimal tuning|Optimal tunings]]:
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*


* WE: ~2 = 1268.274, ~3/2 = 612.921
<nowiki/>*[[Optimal patent val]]: [[8edo|8]]
** Error map: (68.274, -20.760, -249.765)
* CWE: ~2 = 1200.000, ~3/2 = 614.055
** Error map: (0.000, -87.900, -386.314)


[[Category:Temperament collections]]
[[Badness]] (Sintel): 1.52
[[Category:Pages with mostly numerical content]]

Latest revision as of 03:58, 16 June 2026

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

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

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

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