Very low accuracy temperaments: Difference between revisions

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{{Todo|intro|cleanup|comment=what are the criteria for inclusion here?|inline=1}}
{{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.


Below are listed some very low accuracy temperaments.
== 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|legend=1| 2 3 0 | 0 0 1 }}
 
: 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 }}
 
{{Optimal ET sequence|legend=1| 2, 4 }}
 
[[Badness]] (Sintel): 0.508
 
; Music
* [[:File:Antitonic(8).mp3|''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: {{mapping| 2 3 0 1 | 0 0 1 1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 2, 4 }}
 
Badness (Sintel): 0.490
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9/8, 12/11, 15/14
 
Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}
 
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 ET sequence|legend=0| 2, 4 }}
 
Badness (Sintel): 0.721
 
==== Antietam ====
Subgroup: 2.3.5.7.11
 
Comma list: 9/8, 11/10, 15/14
 
Mapping: {{mapping| 2 3 0 1 2 | 0 0 1 1 1 }}
 
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 ET sequence|legend=0| 2, 4e }}
 
Badness (Sintel): 0.628
 
=== Antaeus ===
Subgroup: 2.3.5.7
 
Comma list: 9/8, 35/32
 
Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 2d, 4 }}
 
Badness (Sintel): 0.950
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 9/8, 12/11, 35/32
 
Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}
 
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 ET sequence|legend=0| 2d, 4 }}
 
Badness (Sintel): 1.12


== Alteraugment ==
== Alteraugment ==
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.


Like [[augmented]], but the generator provides 5/4 instead of 3/2.
[[Subgroup]]: 2.3.5


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


POTE generator: ~10/9 = 59.9349 cents
{{Mapping|legend=1| 3 5 0 | 0 0 1 }}


Mapping: [{{val| 3 5 7 }}, {{val| 0 0 1 }}]
: mapping generators: ~4/3, ~5


{{Val list|legend=1| 3, 12bcc }}
[[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 }}


== Antitonic ==
{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}
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.  
 
[[Badness]] (Sintel): 1.83
 
== 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]]
 
{{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: 9/8
Comma list: 10/9, 15/14


POTE generator: ~5/4 = 313.6898 or ~6/5 = 286.3102
Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}


Mapping: [{{val| 2 3 0 }}, {{val| 0 0 1 }}]
Optimal tunings:  
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}


Mapping generators: ~3/2, ~5
{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}


{{Val list|legend=1| 2, 4 }}
Badness (Sintel): 0.606


Badness:
=== Antonym ===
Subgroup: 2.3.5.7


=== 7-limit ===
Comma list: 7/6, 10/9
Comma list: 9/8, 15/14


POTE generator: ~5/4 = 301.9970 or ~6/5 = 298.0030
Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}


Mapping: [{{val| 2 3 0 1 }}, {{val| 0 0 1 1 }}]
Optimal tunings:  
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}


Mapping generators: ~3/2, ~5
{{Optimal ET sequence|legend=0| 2cd, 3 }}


{{Val list|legend=1| 2, 4 }}
Badness (Sintel): 0.568


Badness:
=== Antony ===
Subgroup: 2.3.5.7


=== 11-limit ===
Comma list: 8/7, 10/9
Comma list: 9/8, 12/11, 21/20


POTE generator: ~5/4 = 340.2928 or ~6/5 = 259.7072
Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}


Mapping: [{{val| 2 3 0 1 7 }}, {{val| 0 0 1 1 0 }}]
Optimal tunings:  
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}


Mapping generators: ~3/2, ~5
{{Optimal ET sequence|legend=0| 1c, 3d }}


{{Val list|legend=1| 2, 4 }}
Badness (Sintel): 0.720


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


== Brutus ==
Comma list: 10/9, 28/25
Comma list: 10/9, 28/25


POTE generator: ~3/2 = 848.221
Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}
 
Optimal tunings:
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7bc }}
 
Badness (Sintel): 1.35
 
=== Phlegyas ===
{{See also| Archytas clan }}
 
Subgroup: 2.3.5.7
 
Comma list: 10/9, 35/32
 
Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}
 
{{Optimal ET sequence|legend=0| 3, 5c, 8c }}
 
Badness (Sintel): 1.30
 
=== Charon ===
{{See also| Jubilismic clan }}
 
Subgroup: 2.3.5.7
 
Comma list: 10/9, 49/45
 
Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3
 
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}})
 
{{Optimal ET sequence|legend=0| 4bcd, 6 }}
 
Badness (Sintel): 1.43
 
=== Nessus ===
{{See also| Semaphoresmic clan }}
 
Subgroup: 2.3.5.7
 
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: [{{val| 1 0 -1 -4 }}, {{val| 0 1 2 4 }}]
{{Mapping|legend=1| 3 5 7 0 | 0 0 0 1 }}


{{Multival|legend=1| 1 2 4 1 4 4 }}
: mapping generators: ~5/4, ~7


{{Val list|legend=1| 3, 10bc }}
[[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 }}


Badness: 0.0534
{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}


== Geryon ==
[[Badness]] (Sintel): 0.726
Comma list: 8/7, 25/21
 
=== 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
 
== 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 }}


POTE generator: ~5/4 = 385.440
{{Optimal ET sequence|legend=1| 4 }}


Mapping: [{{val| 1 1 2 3 }}, {{val| 0 2 1 0 }}]
[[Badness]] (Sintel): 1.16


{{Multival|legend=1| 2 1 0 -3 -6 -3 }}
== 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.


[[Val]]s: {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0510
[[Comma list]]: 16/15, 27/25


== Antaeus ==
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
Comma list: 9/8, 35/32


POTE generator: ~5/4 = 315.962
: Mapping generators: ~6/5, ~7


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


{{Multival|legend=1| 0 2 -2 3 -3 -10 }}
{{Optimal ET sequence|legend=1| 5, 15ccd }}


{{Val list|legend=1| 4, 10bd, 12bcd }}
[[Badness]] (Sintel): 1.22


Badness: 0.0375
== Geryon ==
{{See also| Dicot family }}


== Phlegyas ==
[[Subgroup]]: 2.3.5.7
Comma list: 10/9, 35/32


POTE generator: ~3/2 = 743.135
[[Comma list]]: 8/7, 25/21


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


{{Multival|legend=1| 1 2 -2 1 -6 -10 }}
[[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| 3, 10c, 11bcd, 13c }}
{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}


Badness: 0.0513
[[Badness]] (Sintel): 1.29


== Malacoda ==
== Malacoda ==
Comma list: 15/14, 35/32
{{See also| Semaphoresmic clan }}


POTE generator: ~8/7 = 276.283
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 3 2 }}, {{val| 0 2 -1 1 }}]
[[Comma list]]: 15/14, 35/32


{{Multival|legend=1| 2 -1 1 -6 -4 5 }}
{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}


{{Val list|legend=1| 1, 4, 13bc }}
: mapping generators: ~2, ~7/4


Badness: 0.0372
[[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 }}
 
{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}
 
[[Badness]] (Sintel): 0.942


== Ugolino ==
== Ugolino ==
Comma list: 15/14, 27/25
{{See also| Bug family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15/14, 27/25
 
{{Mapping|legend=1| 1 0 0 -1 | 0 2 3 5 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}
 
[[Badness]] (Sintel): 1.11
 
== Medusa ==
{{See also| Archytas clan | Mavila family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15/14, 64/63
 
{{Mapping|legend=1| 1 0 7 6 | 0 1 -3 -2 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
 
[[Badness]] (Sintel): 1.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 278.360
Comma list: 15/14, 22/21, 33/32


Mapping: [{{val| 1 0 0 -1 }}, {{val| 0 2 3 5 }}]
Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}


{{Multival|legend=1| 2 3 5 0 2 3 }}
Optimal tunings:
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}


{{Val list|legend=1| 4, 13bd }}
{{Optimal ET sequence|legend=0| 2, 5c, 7 }}


Badness: 0.0438
Badness (Sintel): 0.887


== Charon ==
== Wallaby ==
Comma list: 10/9, 49/45
{{See also| Trienstonic clan | Mavila family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 28/27, 35/32
 
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}
 
[[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 ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15/14, 81/80
 
{{Mapping|legend=1| 1 0 -4 -5 | 0 1 4 5 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
 
[[Badness]] (Sintel): 1.14
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 15/14, 22/21, 81/80
 
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}}
 
{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}
 
Badness (Sintel): 1.08
 
== 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.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: [[45/32]]
 
{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}
 
: mapping generators: ~2, ~3
 
[[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 ===
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


POTE generator: ~3/2 = 782.604
[[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 }}


Mapping: [{{val| 2 0 -2 -1 }}, {{val| 0 1 2 2 }}]
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]


{{Multival|legend=1| 2 4 4 2 1 -2 }}
[[Badness]] (Sintel): 0.751


{{Val list|legend=1| 6, 13bd }}
== 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.


Badness: 0.0564
[[Subgroup]]: 2.3.5.7


== Nessus ==
[[Comma list]]: [[27/25]], [[50/49]]
Comma list: 10/9, 49/48


POTE generator: ~8/7 = 213.117
{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}


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


{{Multival|legend=1| 2 4 1 2 -4 -9 }}
{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}


{{Val list|legend=1| 6, 11bc, 17b }}
[[Badness]] (Sintel): 1.95


Badness: 0.0591
== 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]].


== Medusa ==
[[Subgroup]]: 2.3.5.7
Comma list: 15/14, 64/63


POTE generator: ~3/2 = 685.632
[[Comma list]]: 21/20, 175/162


Mapping: [{{val| 1 0 7 6 }}, {{val| 0 1 -3 -2 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}


{{Multival|legend=1| 1 -3 -2 -7 -6 4 }}
[[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 }}


{{Val list|legend=1| 2, 7, 12c, 14cd }}
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*


Badness: 0.0427
<nowiki/>*[[Optimal patent val]]: [[8edo|8]]


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

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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