Xenharmonic Wiki

Very low accuracy temperaments: Difference between revisions

← Older edit

FloraC (talk | contribs)

4 user groups
27,345 edits
VisualWikitext
Move oxygen here (prime 7 has semitone-level error & cuz it doesn't fit anywhere in the spectrum of porcupine)
Delete antonym, antony, and geryon, from the precedent of deletion of {7/6}. Perhaps {9/8} should be the threshold for tempering out superparticulars
 
(47 intermediate revisions by 9 users not shown)
Line 1: Line 1:
{{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 14: Line 14:


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


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


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


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


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


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


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


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


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


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


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


Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}
{{Mapping|legend=0| 2 3 0 1 7 | 0 0 1 1 0 }}


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


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


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


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


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


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


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


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


=== Antaeus ===
=== Antaeus ===
Line 76: Line 76:
Comma list: 9/8, 35/32
Comma list: 9/8, 35/32


Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}
{{Mapping|legend=0| 2 3 0 10 | 0 0 1 -1 }}


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


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


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


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


Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}
{{Mapping|legend=0| 2 3 0 10 7 | 0 0 1 -1 0 }}


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


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


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


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 109: Line 109:


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


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


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


[[Badness]] (Smith): 0.078042
[[Badness]] (Sintel): 1.83
 
== Antonian ==
{{Main| Antonian }}


== Yo (2c & 3)  ==
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]].  
This temperament family is characterized by the [[color notation|yo 2nd]] [[10/9]] being tempering out.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 132: Line 133:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 761.310
* [[WE]]: ~2 = 1187.236{{c}}, ~3/2 = 767.716{{c}}
: [[error map]]: {{val| 0.000 +74.015 -34.374 }}
: [[error map]]: {{val| -12.764 +52.997 -63.645 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 775.970
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: error map: {{val| 0.000 +65.763 -50.878 }}
: error map: {{val| 0.000 +65.763 -50.877 }}
 
{{Optimal ET sequence|legend=1| 2c, 3 }}
 
[[Badness]] (Sintel): 0.392
 
=== Septimal antonian ===
{{See also| Trienstonic clan }}
 
Subgroup: 2.3.5.7
 
Comma list: 10/9, 15/14
 
{{Mapping|legend=0| 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=1| 3, 11bc, 14bc, 17bb }}
{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}


[[Badness]] (Smith): 0.016712
Badness (Sintel): 0.606


=== Brutus ===
=== Brutus ===
Line 146: Line 164:
Comma list: 10/9, 28/25
Comma list: 10/9, 28/25


Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}
{{Mapping|legend=0| 1 0 -1 -4 | 0 1 2 4 }}


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


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


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


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


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


Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}
{{Mapping|legend=0| 1 0 -1 6 | 0 1 2 -2 }}


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


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


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


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


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


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


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


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


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


=== Nessus ===
=== Nessus ===
Line 193: Line 216:
Comma list: 10/9, 49/48
Comma list: 10/9, 49/48


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


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


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


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


== Ternary ==
== Ternary ==
Line 215: Line 237:


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


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 233: Line 253:
Comma list: 10/9, 16/15, 22/21
Comma list: 10/9, 16/15, 22/21


Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=0| 3 5 7 0 2 | 0 0 0 1 1 }}


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


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


Badness (Smith): 0.025592
Badness (Sintel): 0.846
 
== Quad ==
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 9/8, 25/24
 
{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}}
: [[error map]]: {{val| +32.295 -53.513 -13.650 -5.150 }}
* [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}}
: error map: {{val| 0.000 -101.955 -86.314 -71.236 }}
 
{{Optimal ET sequence|legend=1| 4 }}
 
[[Badness]] (Sintel): 1.16


== Geryon ==
== Quint ==
{{See also| Dicot family }}
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 16/15, 27/25


{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 354.664
* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
: [[error map]]: {{val| 0.000 +7.373 -31.649 +231.174 }}
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 385.440
* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
: error map: {{val| 0.000 +68.926 -0.873 +231.174 }}
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }}


[[Badness]] (Smith): 0.051009
[[Badness]] (Sintel): 1.22


== Malacoda ==
== Malacoda ==
Line 270: Line 310:


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


[[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 293: Line 332:


[[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 312: Line 351:


[[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 326: Line 365:
Comma list: 15/14, 22/21, 33/32
Comma list: 15/14, 22/21, 33/32


Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}
{{Mapping|legend=0| 1 0 7 6 5 | 0 1 -3 -2 -1 }}


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 345: Line 384:
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}


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


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


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


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


Line 367: Line 405:


[[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 381: Line 419:
Comma list: 15/14, 22/21, 81/80
Comma list: 15/14, 22/21, 81/80


Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}
{{Mapping|legend=0| 1 0 -4 -5 -6 | 0 1 4 5 6 }}


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 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:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 509.892
: [[error map]]: {{val| 0.000 -192.063 -206.097 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 520.625
: error map: {{val| 0.000 -181.330 -227.565 }}
 
{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
 
[[Badness]] (Smith): 0.133021
 
=== 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:
* CTE: ~2 = 1200.000, ~3/2 = 509.892
* CWE: ~2 = 1200.000, ~3/2 = 515.099
 
Optimal ET sequence: none
 
Badness (Smith): 0.118764
 
== 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:
* [[CTE]]: ~3 = 1901.955, ~5/3 = 701.863
: [[error map]]: {{val| 0.000 -182.496 -266.779 }}
* [[CWE]]: ~3 = 1901.995, ~5/3 = 691.468
: error map: {{val| 0.000 -192.891 -256.384 }}
 
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
 
[[Badness]]:
* Smith: 0.00406
* Dirichlet: 0.751


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


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


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


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


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


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


[[Badness]] (Smith): 0.077065
[[Badness]] (Sintel): 1.95


== Oxygen ==
== Oxygen ==
{{See also| Porcupine family }}
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI 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 479: Line 458:


{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}
{{Multival|legend=1| 3 5 2 1 -5 -9 }}


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


[[Badness]] (Smith): 0.059866
<nowiki/>* [[optimal patent val]]: [[8edo|8]]


[[Category:Temperament collections]]
[[Badness]] (Sintel): 1.52
Retrieved from "https://en.xen.wiki/w/Very_low_accuracy_temperaments"