Very low accuracy temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (2/2). Complete missing data
- CTE & POTE tunings
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 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 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 154: Line 136:
* [[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 }}
Line 173: Line 151:
* 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 190: Line 166:
* 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 207: Line 181:
* 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 228: Line 200:
* WE: ~2 = 1189.201{{c}}, ~3/2 = 978.002{{c}}
* WE: ~2 = 1189.201{{c}}, ~3/2 = 978.002{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 983.918{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 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 221:
* [[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 236:
* 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 }}
Line 291: Line 255:
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 354.664{{c}}
: [[error map]]: {{val| 0.000 +7.373 -31.649 +231.174 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 385.440{{c}}
: error map: {{val| 0.000 +68.926 -0.873 +231.174 }} -->


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
Line 316: Line 276:
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~7/4 = 938.796{{c}}
: [[error map]]: {{val| 0.000 -24.362 -125.110 -30.029 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.717{{c}}
: error map: {{val| 0.000 -54.521 -110.031 -45.109 }} -->


{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}
{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}
Line 339: Line 295:
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/3 = 926.868{{c}}
: [[error map]]: {{val| 0.000 -48.218 -5.708 +65.517 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/3 = 921.640{{c}}
: error map: {{val| 0.000 -58.675 -21.394 +39.374 }} -->


{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}
{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}
Line 362: Line 314:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.511{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.511{{c}}
: error map: {{val| 0.000 -16.443 -42.849 +60.150 }}
: error map: {{val| 0.000 -16.443 -42.849 +60.150 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 684.722{{c}}
: [[error map]]: {{val| 0.000 -17.233 -40.480 +61.730 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.632{{c}}
: error map: {{val| 0.000 -16.323 -43.209 +59.911 }} -->


{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
Line 381: Line 329:
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 683.589{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 685.809{{c}} -->


{{Optimal ET sequence|legend=0| 2, 5c, 7 }}
{{Optimal ET sequence|legend=0| 2, 5c, 7 }}
Line 402: Line 348:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}
<!-- * [[CTE]]: ~2 = 1200.000, ~3/2 = 693.440{{c}}
: [[error map]]: {{val| 0.000 -8.515 -66.635 -88.505 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 691.351{{c}}
: 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 }}
Line 425: Line 367:
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 683.935{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 683.935{{c}}
: error map: {{val| 0.000 -18.020 -50.573 +50.850 }}
: error map: {{val| 0.000 -18.020 -50.573 +50.850 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.837{{c}}
: [[error map]]: {{val| 0.000 -16.118 -42.965 +60.359 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 682.895{{c}}
: error map: {{val| 0.000 -19.060 -54.734 +45.649 }} -->


{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
Line 444: Line 382:
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 687.743{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 685.234{{c}} -->


{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}
{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}
Line 463: Line 399:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 509.892{{c}}
: [[error map]]: {{val| 0.000 -192.063 -206.097 }} -->
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: error map: {{val| +65.405 -84.203 -94.789 }}
: error map: {{val| +65.405 -84.203 -94.789 }}
Line 482: Line 416:


Optimal tunings:  
Optimal tunings:  
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 509.892{{c}} -->
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
Line 502: Line 435:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
<!-- * [[CTE]]: ~3 = 1901.955{{c}}, ~5/3 = 701.863{{c}}
: [[error map]]: {{val| 0.000 -182.496 -266.779 }} -->
* [[WE]]: ~3 = 1988.549{{c}}, ~3/2 = 719.679{{c}}
* [[WE]]: ~3 = 1988.549{{c}}, ~3/2 = 719.679{{c}}
: error map: {{val| +86.594 -78.085 -111.407 }}
: error map: {{val| +86.594 -78.085 -111.407 }}
Line 525: Line 456:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~5/3 = 935.289{{c}}
: [[error map]]: {{val| 0.000 -31.377 +19.554 +37.041 }} -->
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: error map: {{val| -2.821 -31.998 +18.621 +34.699 }}
: error map: {{val| -2.821 -31.998 +18.621 +34.699 }}
Line 550: Line 479:
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 161.341{{c}}
: [[error map]]: {{val| 0.000 +14.023 +6.982 -91.507 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 169.112{{c}}
: error map: {{val| 0.000 -9.291 -31.873 -107.050 }} -->


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

Revision as of 14:31, 10 August 2025

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. 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 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 x31eq. 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.

7-limit

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 is like augmented, but the generator provides 5/4 instead of 3/2. 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

Yo (2c & 3)

This temperament family is characterized by the yo 2nd 10/9 being tempering out.

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

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

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

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]

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 ¢, ~3/2 = 978.002 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 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

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

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 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 ¢, ~3/2 = 719.679 ¢
error map: ⟨+86.594 -78.085 -111.407]
  • CWE: ~3 = 1901.995 ¢, ~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

This temperament 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.

Subgroup: 2.3.5.7

Comma list: 27/25, 54/49

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

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

Badness (Sintel): 1.52

Bixby

Main article: Bixby

Subgroup: 2.3.5

Comma list: 4/3

Mapping: [⟨1 2 0], ⟨0 0 1]]

Optimal tunings:

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

Optimal ET sequence: 1c, 2b, 3bbcc

Badness (Sintel): 0.424

Archon

Subgroup: 2.3.5

Comma list: 5/4

Mapping: [⟨1 0 2], ⟨0 1 0]]

Optimal tunings:

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

Optimal ET sequence: 2c

Badness (Sintel): 0.474

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