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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 “commas” such as 9/8, 32/27, or 15/14, equating wildly different interval sizes. All temperaments listed temper out an interval of at least size 15/14.
{{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". 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
 
[[Comma list]]: [[9/8]]


[[Comma]]: [[9/8]]
{{Mapping|legend=1| 2 3 0 | 0 0 1 }}


[[Mapping]]: [{{val| 2 3 0 }}, {{val| 0 0 1 }}]
: mapping generators: ~3/2, ~5


Mapping generators: ~3/2, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3/2 = 615.125{{c}}, ~5/4 = 321.597{{c}} (~6/5 = 293.528{{c}})
: [[error map]]: {{val| +30.250 -56.581 -4.217 }}
* [[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 }}


[[POTE generator]]: ~5/4 = 313.690 or ~6/5 = 286.310
{{Optimal ET sequence|legend=1| 2, 4 }}


{{Val list|legend=1| 2, 4 }}
[[Badness]] (Sintel): 0.508


[[Badness]]: 0.021647
; Music
* [[: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


[[Comma list]]: 9/8, 15/14
Comma list: 9/8, 15/14


[[Mapping]]: [{{val| 2 3 0 1 }}, {{val| 0 0 1 1 }}]
Mapping: {{mapping| 2 3 0 1 | 0 0 1 1 }}


[[POTE generator]]: ~5/4 = 301.997 or ~6/5 = 298.003
Optimal tunings:  
* 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}})


{{Val list|legend=1| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}


[[Badness]]: 0.019373
Badness (Sintel): 0.490


==== 11-limit ====
==== 11-limit ====
Line 36: Line 46:
Comma list: 9/8, 12/11, 15/14
Comma list: 9/8, 12/11, 15/14


Mapping: [{{val| 2 3 0 1 7 }}, {{val| 0 0 1 1 0 }}]
Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}


POTE generator: ~5/4 = 340.293 or ~6/5 = 259.707
Optimal tunings:  
* 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}})


Optimal GPV sequence: {{Val list| 2, 4 }}
{{Optimal ET sequence|legend=0| 2, 4 }}


Badness: 0.021810
Badness (Sintel): 0.721


==== Antietam ====
==== Antietam ====
Line 49: Line 61:
Comma list: 9/8, 11/10, 15/14
Comma list: 9/8, 11/10, 15/14


Mapping: [{{val| 2 3 0 1 2 }}, {{val | 0 0 1 1 1}}]
Mapping: {{mapping| 2 3 0 1 2 | 0 0 1 1 1 }}


POTE generator: ~5/4 = 322.080 or ~6/5 = 277.920
Optimal tunings:  
* 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}})


Optimal GPV sequence: {{Val list| 2, 4e }}
{{Optimal ET sequence|legend=0| 2, 4e }}


[[TE error]]: 27.061497 [[cpo]]
Badness (Sintel): 0.628


=== Antaeus ===
=== Antaeus ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 9/8, 35/32
Comma list: 9/8, 35/32


[[Mapping]]: [{{val| 2 3 0 10 }}, {{val| 0 0 1 -1 }}]
Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}


[[POTE generator]]: ~5/4 = 315.962 or ~6/5 = 284.038
Optimal tunings:  
* 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}})


{{Val list|legend=1| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}


[[Badness]]: 0.037537
Badness (Sintel): 0.950


==== 11-limit ====
==== 11-limit ====
Line 75: Line 91:
Comma list: 9/8, 12/11, 35/32
Comma list: 9/8, 12/11, 35/32


Mapping: [{{val| 2 3 0 10 7 }}, {{val| 0 0 1 -1 0 }}]
Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}


POTE generator: ~5/4 = 318.173 or ~6/5 = 281.827
Optimal tunings:  
* 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}})


Optimal GPV sequence: {{Val list| 2d, 4 }}
{{Optimal ET sequence|legend=0| 2d, 4 }}


Badness: 0.033786
Badness (Sintel): 1.12


== Alteraugment ==
== Alteraugment ==
Like [[augmented]], but the generator provides 5/4 instead of 3/2.
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


[[Comma]]: [[32/27]]
[[Comma list]]: [[32/27]]


[[Mapping]]: [{{val| 3 5 7 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 3 5 0 | 0 0 1 }}


[[POTE generator]]: ~10/9 = 59.935
: mapping generators: ~4/3, ~5


{{Val list|legend=1| 3, 12bcc, 15bbcc, 18bbcc }}
[[Optimal tuning]]s:
* [[WE]]: ~4/3 = 389.212{{c}}, ~5/4 = 447.530{{c}} (~10/9 = 58.318{{c}})
: [[error map]]: {{val| -32.364 +44.105 -3.512 }}
* [[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 }}


[[Badness]]: 0.078042
{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}


== Yellow (2c&3) ==
[[Badness]] (Sintel): 1.83
This temperament family is characterized by [[10/9]] being tempering out.


Subgroup: 2.3.5
== Antonian ==
{{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


[[Comma list]]: [[10/9]]
[[Comma list]]: [[10/9]]


[[Mapping]]: [{{val| 1 0 -1 }}, {{val| 0 1 2 }}]
{{Mapping|legend=1| 1 0 -1 | 0 1 2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1187.236{{c}}, ~3/2 = 767.716{{c}}
: [[error map]]: {{val| -12.764 +52.997 -63.645 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: 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 }}


[[POTE generator]]: ~3/2 = 775.970
Optimal tunings:  
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}


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


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


=== Brutus ===
=== Brutus ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 28/25
Comma list: 10/9, 28/25


[[Mapping]]: [{{val| 1 0 -1 -4 }}, {{val| 0 1 2 4 }}]
Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}


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


{{Val list|legend=1| 3, 7bc, 10bc }}
{{Optimal ET sequence|legend=0| 3, 7bc }}


[[Badness]]: 0.053389
Badness (Sintel): 1.35


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


[[Comma list]]: 10/9, 35/32
Comma list: 10/9, 35/32


[[Mapping]]: [{{val| 1 0 -1 6 }}, {{val| 0 1 2 -2 }}]
Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}


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


{{Val list|legend=1| 3, 5c, 8c, 13bcc }}
{{Optimal ET sequence|legend=0| 3, 5c, 8c }}


[[Badness]]: 0.051293
Badness (Sintel): 1.30


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


[[Comma list]]: 10/9, 49/45
Comma list: 10/9, 49/45


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


[[POTE generator]]: ~3/2 = 782.604 (or ~10/9 = 182.604)
Optimal tunings:  
* 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}})


{{Val list|legend=1| 6, 14bc, 20bb }}
{{Optimal ET sequence|legend=0| 4bcd, 6 }}


[[Badness]]: 0.056404
Badness (Sintel): 1.43


=== Nessus ===
=== Nessus ===
{{See also| Semaphoresmic clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 49/48
Comma list: 10/9, 49/48
 
Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4
 
Optimal tunings:
* WE: ~2 = 1189.201{{c}}, ~7/4 = 978.002{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/4 = 983.918{{c}}
 
{{Optimal ET sequence|legend=0| 5c, 6 }}
 
Badness (Sintel): 1.49
 
== Ternary ==
Ternary is identical to [[3edo|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|legend=1| 3 5 7 0 | 0 0 0 1 }}
 
: mapping generators: ~5/4, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 391.796{{c}}, ~7/4 = 1012.806{{c}} (~15/14 = 162.582{{c}})
: [[error map]]: {{val| -24.612 +57.026 -43.741 -5.243 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}
 
[[Badness]] (Sintel): 0.726
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 10/9, 16/15, 22/21
 
Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 3de, 6 }}
 
Badness (Sintel): 0.846


[[Mapping]]: [{{val| 1 0 -1 2 }}, {{val| 0 2 4 1 }}]
== Quad ==
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.


[[POTE generator]]: ~7/6 = 213.117
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 5c, 6, 11bc, 17bb }}
[[Comma list]]: 9/8, 25/24


[[Badness]]: 0.059070
{{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 ==
{{see also| Dicot family }}
{{See also| Dicot family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 8/7, 25/21


[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 2 1 0 }}]
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}


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


{{Val list|legend=1| 3d }}, {{val| 16 26 37 48 }}, {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}
{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}


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


== Malacoda ==
== Malacoda ==
Subgroup: 2.3.5.7
{{See also| Semaphoresmic clan }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 35/32
[[Comma list]]: 15/14, 35/32


[[Mapping]]: [{{val| 1 0 3 2 }}, {{val| 0 2 -1 1 }}]
{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}
 
: mapping generators: ~2, ~7/4


[[POTE generator]]: ~7/6 = 276.283
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1223.542{{c}}, ~7/4 = 941.838{{c}}
: [[error map]]: {{val| +23.542 -18.278 -57.528 +20.096 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}


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


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


== Ugolino ==
== Ugolino ==
{{see also| Bug family }}
{{See also| Bug family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 27/25
[[Comma list]]: 15/14, 27/25


[[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 3 5 }}]
{{Mapping|legend=1| 1 0 0 -1 | 0 2 3 5 }}


[[POTE generator]]: ~6/5 = 278.360
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.628{{c}}, ~7/4 = 926.730{{c}}
: [[error map]]: {{val| +6.628 -48.494 -6.122 +58.198 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}


{{Val list|legend=1| 4, 9d, 13bd }}
{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}


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


== Medusa ==
== Medusa ==
{{see also| Pelogic family }}
{{See also| Archytas clan | Mavila family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 64/63
[[Comma list]]: 15/14, 64/63


[[Mapping]]: [{{val| 1 0 7 6 }}, {{val| 0 1 -3 -2 }}]
{{Mapping|legend=1| 1 0 7 6 | 0 1 -3 -2 }}


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 228: Line 416:
Comma list: 15/14, 22/21, 33/32
Comma list: 15/14, 22/21, 33/32


Mapping: [{{val| 1 0 7 6 5 }}, {{val| 0 1 -3 -2 -1 }}]
Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}
 
{{Optimal ET sequence|legend=0| 2, 5c, 7 }}
 
Badness (Sintel): 0.887
 
== Wallaby ==
{{See also| Trienstonic clan | Mavila family }}
 
[[Subgroup]]: 2.3.5.7


POTE generator: ~3/2 = 685.809
[[Comma list]]: 28/27, 35/32


Optimal GPV sequence: {{Val list| 2, 5c, 7 }}
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}


Badness: 0.026828
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1216.024{{c}}, ~3/2 = 700.583{{c}}
: [[error map]]: {{val| +16.024 +14.652 -23.967 -51.053 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}
 
{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}
 
[[Badness]] (Sintel): 1.48


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 81/80
[[Comma list]]: 15/14, 81/80


[[Mapping]]: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]
{{Mapping|legend=1| 1 0 -4 -5 | 0 1 4 5 }}


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 256: Line 470:
Comma list: 15/14, 22/21, 81/80
Comma list: 15/14, 22/21, 81/80


Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}
 
Optimal tunings:
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}


POTE generator: ~3/2 = 685.234
{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}


Optimal GPV sequence: {{Val list| 2cde, 5de, 7 }}
Badness (Sintel): 1.08


Badness: 0.032521
== Involution ==
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.  


== Abomination ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: [[4/3]]
[[Comma list]]: [[45/32]]
 
{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}


[[CTE|CTE generator]]: ~[[5/3]] = 692.233
: mapping generators: ~2, ~3


[[Mapping]]: [{{val|1 2 2}}, {{val|0 0 1}}]]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}
 
{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
 
[[Badness]] (Sintel): 3.12


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
Comma list: 8/7, 45/28
Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}
Optimal tunings:
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
Optimal ET sequence: none
Badness (Sintel): 3.01
== Devolution ==
Devolution was named by [[User:Akselai|Akselai]] in 2024.
[[Subgroup]]: 3.5.7
[[Comma list]]: [[35/27]]
{{Mapping|legend=1| 1 0 3 | 0 1 -1 }}
: mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
[[Badness]] (Sintel): 0.751
== Codex ==
{{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.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4/3]], [[7/5]]
[[Comma list]]: [[27/25]], [[50/49]]


[[CTE|CTE generator]]: ~[[5/3]] = 928.851
{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}


[[Mapping]]: [{{val|1 2 2 2}}, {{val|0 0 1 1}}]]
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: [[error map]]: {{val| -2.821 -31.998 +18.621 +34.699 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}


=== 11-limit ===
{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}
[[Subgroup]]: 2.3.5.7.11
 
[[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


[[Comma list]]: [[4/3]], [[7/5]], [[12/11]]
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}


[[CTE|CTE generator]]: ~[[5/3]] = 935.119
[[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 }}


[[Mapping]]: [{{val|1 2 2 2 4}}, {{val|0 0 1 1 0}}]]
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*


<nowiki/>*[[Optimal patent val]]: [[8edo|8]]


[[Category:Temperament collections]]
[[Badness]] (Sintel): 1.52
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