Dicot family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus
Line 14: Line 14:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 354.664
* [[WE]]: ~2 = 1206.283{{c}}, ~6/5 = 350.420{{c}}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
: error map: {{val| 0.000 +0.216 -35.228 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 354.664{{c}}
: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
: [[error map]]: {{val| 0.000 +7.374 -31.649 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 348.594
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 348.594{{c}}
: error map: {{val| 0.000 -4.766 -37.719 }}
: error map: {{val| 0.000 -4.766 -37.719 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 25: Line 29:
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


[[Badness]] (Smith): 0.013028
[[Badness]] (Sintel): 0.306


=== Overview to extensions ===
=== Overview to extensions ===
Line 44: Line 48:
Comma list: 25/24, 45/44
Comma list: 25/24, 45/44


Sval mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}


Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
Line 51: Line 55:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 352.287
* WE: ~2 = 1206.750{{c}}, ~6/5 = 348.684{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 346.734
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 348.954{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.287{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 346.734{{c}} -->


{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c, 38cc, 45cce }}
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}


RMS error: 5.621 cents
Badness (Sintel): 0.370


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Line 63: Line 69:
Comma list: 25/24, 40/39, 45/44
Comma list: 25/24, 40/39, 45/44


Sval mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}


Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}
Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}
Line 70: Line 76:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 352.420
* WE: ~2 = 1202.433{{c}}, ~6/5 = 351.237{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 350.526
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 350.978{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 352.420{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 350.526{{c}} -->


{{Optimal ET sequence|legend=0| 3e, 7, 17, 24c }}
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}


RMS error: 5.916 cents
Badness (Sintel): 0.536


== Septimal dicot ==
== Septimal dicot ==
Line 87: Line 95:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 342.257
* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}
: [[error map]]: {{val| +5.532 -20.561 -37.319 +56.032 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 342.257{{c}}
: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}
: [[error map]]: {{val| 0.000 -17.441 -44.056 +57.946 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 336.381
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 336.381{{c}}
: error map: {{val| 0.000 -29.193 -49.933 +40.316 }}
: error map: {{val| 0.000 -29.193 -49.933 +40.316 }} -->


{{Optimal ET sequence|legend=1| 3d, 4, 7 }}
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}


[[Badness]] (Smith): 0.019935
[[Badness]] (Sintel): 0.504


=== 11-limit ===
=== 11-limit ===
Line 104: Line 116:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 345.596
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 342.125
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 345.596{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 342.125{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}


Badness (Smith): 0.019854
Badness (Sintel): 0.656


=== Eudicot ===
=== Eudicot ===
Line 119: Line 133:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 340.417
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 336.051
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.417{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 336.051{{c}} -->


{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}


Badness (Smith): 0.027114
Badness (Sintel): 0.896


==== 13-limit ====
==== 13-limit ====
Line 134: Line 150:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 340.835
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 338.846
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 340.835{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 338.846{{c}} -->


{{Optimal ET sequence|legend=0| 3d, 4, 7 }}
{{Optimal ET sequence|legend=0| 3d, 4, 7 }}


Badness (Smith): 0.023828
Badness (Sintel): 0.985


== Flattie ==
== Flattie ==
Line 151: Line 169:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 346.438
* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}
: [[error map]]: {{val| +20.466 -6.335 -7.804 -45.004 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 346.438{{c}}
: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}
: [[error map]]: {{val| 0.000 -9.080 -39.876 -115.264 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 331.916
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 331.916{{c}}
: error map: {{val| 0.000 -38.123 -54.398 -100.742 }}
: error map: {{val| 0.000 -38.123 -54.398 -100.742 }} -->


{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}


[[Badness]] (Smith): 0.025381
[[Badness]] (Sintel): 0.642


=== 11-limit ===
=== 11-limit ===
Line 168: Line 190:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 343.139
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 337.532
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.139{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 337.532{{c}} -->


{{Optimal ET sequence|legend=0| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness (Smith): 0.024988
Badness (Sintel): 0.826


=== 13-limit ===
=== 13-limit ===
Line 183: Line 207:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~6/5 = 343.655
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* POTE: ~2 = 1200.000, ~6/5 = 341.023
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~6/5 = 343.655{{c}}
* POTE: ~2 = 1200.000{{c}}, ~6/5 = 341.023{{c}} -->


{{Optimal ET sequence|legend=0| 3, 4, 7d }}
{{Optimal ET sequence|legend=0| 3, 4, 7d }}


Badness (Smith): 0.023420
Badness (Sintel): 0.968


== Sharpie ==
== Sharpie ==
Line 200: Line 226:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 359.564
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 359.564{{c}}
: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}
: [[error map]]: {{val| 0.000 +17.173 -26.750 -11.442 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 357.938
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 357.938{{c}}
: error map: {{val| 0.000 +13.921 -28.376 -21.198 }}
: error map: {{val| 0.000 +13.921 -28.376 -21.198 }} -->


{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}


[[Badness]] (Smith): 0.028942
[[Badness]] (Sintel): 0.732


=== 11-limit ===
=== 11-limit ===
Line 217: Line 247:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 357.261
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 356.106
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 357.261{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 356.106{{c}} -->


{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}


Badness (Smith): 0.022366
Badness (Sintel): 0.739


== Dichotic ==
== Dichotic ==
Line 234: Line 266:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~5/4 = 356.333
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
: [[error map]]: {{val| +0.802 +11.851 -28.208 +8.374 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.333{{c}}
: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}
: [[error map]]: {{val| 0.000 +10.710 -29.981 +5.844 }}
* [[POTE]]: ~2 = 1200.000, ~5/4 = 356.264
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.264{{c}}
: error map: {{val| 0.000 +10.573 -30.050 +6.119 }}
: error map: {{val| 0.000 +10.573 -30.050 +6.119 }} -->


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}


[[Badness]] (Smith): 0.037565
[[Badness]] (Sintel): 0.951


=== 11-limit ===
=== 11-limit ===
Line 251: Line 287:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 354.183
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.262
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.183{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.262{{c}} -->


{{Optimal ET sequence|legend=0| 7, 10, 17 }}
{{Optimal ET sequence|legend=0| 7, 10, 17 }}


Badness (Smith): 0.030680
Badness (Sintel): 1.01


==== 13-limit ====
==== 13-limit ====
Line 266: Line 304:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 354.247
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.365
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 354.247{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.365{{c}} -->


{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}


Badness (Smith): 0.021674
Badness (Sintel): 0.896


=== Dichotomic ===
=== Dichotomic ===
Line 281: Line 321:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 353.751
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.073
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.751{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.073{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness (Smith): 0.031719
Badness (Sintel): 1.05


==== 13-limit ====
==== 13-limit ====
Line 296: Line 338:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 353.850
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 354.313
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 353.850{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 354.313{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7, 10e }}
{{Optimal ET sequence|legend=0| 3, 7, 10e }}


Badness (Smith): 0.022741
Badness (Sintel): 0.940


=== Dichosis ===
=== Dichosis ===
Line 311: Line 355:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 361.081
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 360.659
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.081{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.659{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness (Smith): 0.041361
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 326: Line 372:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~5/4 = 361.061
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* POTE: ~2 = 1200.000, ~5/4 = 360.646
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/4 = 361.061{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 360.646{{c}} -->


{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}


Badness (Smith): 0.027938
Badness (Sintel): 1.15


== Decimal ==
== Decimal ==
Line 346: Line 394:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
: [[error map]]: {{val| +6.571 +5.318 -22.821 -2.047 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}
: [[error map]]: {{val| 0.000 +9.260 -30.706 -13.218 }}
* [[POTE]]: ~7/5 = 600.000, ~7/4 = 948.443 (~7/6 = 251.557)
* [[POTE]]: ~7/5 = 600.000{{c}}, ~7/4 = 948.443{{c}} (~7/6 = 251.557{{c}})
: error map: {{val| 0.000 -5.069 -37.871 -20.383 }}
: error map: {{val| 0.000 -5.069 -37.871 -20.383 }} -->


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}


[[Badness]] (Smith): 0.028334
[[Badness]] (Sintel): 0.717


=== 11-limit ===
=== 11-limit ===
Line 363: Line 415:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 952.812 (~8/7 = 247.188)
* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 946.507 (~7/6 = 253.493)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 952.812{{c}} (~8/7 = 247.188{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 946.507{{c}} (~7/6 = 253.493{{c}}) -->


{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}


Badness (Smith): 0.026712
Badness (Sintel): 0.883


==== 13-limit ====
==== 13-limit ====
Line 378: Line 432:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 954.469 (~8/7 = 245.531)
* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 947.955 (~7/6 = 252.045)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 954.469{{c}} (~8/7 = 245.531{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 947.955{{c}} (~7/6 = 252.045{{c}}) -->


{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}


Badness (Smith): 0.021326
Badness (Sintel): 0.881


=== Decimated ===
=== Decimated ===
Line 393: Line 449:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 950.940 (~7/6 = 249.060)
* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 944.934 (~7/6 = 255.066)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 950.940{{c}} (~7/6 = 249.060{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 944.934{{c}} (~7/6 = 255.066{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}


Badness (Smith): 0.031456
Badness (Sintel): 1.04


=== Decibel ===
=== Decibel ===
Line 408: Line 466:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
* POTE: ~7/5 = 600.000, ~7/4 = 956.507 (~8/7 = 243.493)
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})
<!-- * CTE: ~7/5 = 600.000{{c}}, ~7/4 = 955.608{{c}} (~8/7 = 244.392{{c}})
* POTE: ~7/5 = 600.000{{c}}, ~7/4 = 956.507{{c}} (~8/7 = 243.493{{c}}) -->


{{Optimal ET sequence|legend=0| 4, 6, 10 }}
{{Optimal ET sequence|legend=0| 4, 6, 10 }}


Badness (Smith): 0.032385
Badness (Sintel): 1.07


== Sidi ==
== Sidi ==
Line 420: Line 480:
[[Comma list]]: 25/24, 245/243
[[Comma list]]: 25/24, 245/243


{{Mapping|legend=1| 1 3 3 6 | 0 -4 -2 -9 }}
{{Mapping|legend=1| 1 -1 1 -3 | 0 4 2 9 }}


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 424.452
* [[WE]]: ~2 = 1207.178{{c}}, ~14/9 = 777.414{{c}}
: [[error map]]: {{val| +7.178 +0.523 -24.308 +6.367 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 775.548{{c}}
: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}
: [[error map]]: {{val| 0.000 +0.238 -35.217 +11.108 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 427.208
* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 772.792{{c}}
: error map: {{val| 0.000 -10.789 -40.731 -13.702 }}
: error map: {{val| 0.000 -10.789 -40.731 -13.702 }} -->


{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}


[[Badness]] (Smith): 0.056586
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 439: Line 503:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 424.587
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 427.273
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/7 = 775.413{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/7 = 772.727{{c}} -->


{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}
{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}


Badness (Smith): 0.032957
Badness (Sintel): 1.09


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 13:46, 17 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.

The dicot family of temperaments tempers out 25/24, the classical chromatic semitone. The head of this family, dicot, is generated by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, (5/4)2 = (3/2)(25/24).

Possible tunings for dicot are 7edo, 10edo, 17edo, 24edo using the val ⟨24 38 55] (24c), and 31edo using the val ⟨31 49 71] (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an exotemperament.

Dicot

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1206.283 ¢, ~6/5 = 350.420 ¢
error map: ⟨+6.283 +5.167 -23.328]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 351.086 ¢
error map: ⟨0.000 +0.216 -35.228]

Tuning ranges:

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness (Sintel): 0.306

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot adds 36/35, sharpie adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.

Decimal adds 49/48, sidi adds 245/243, and jamesbond adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

Temperaments discussed elsewhere are:

The rest are considered below.

2.3.5.11 subgroup

The 2.3.5.11-subgroup extension is related to septimal dicot, sharpie, and dichotic.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Subgroup val mapping: [⟨1 1 2 2], ⟨0 2 1 5]]

Gencom mapping: [⟨1 1 2 0 2], ⟨0 2 1 0 5]]

gencom: [2 5/4; 25/24 45/44]

Optimal tunings:

  • WE: ~2 = 1206.750 ¢, ~6/5 = 348.684 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 348.954 ¢

Optimal ET sequence: 3e, 4e, 7, 24c, 31c

Badness (Sintel): 0.370

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

Comma list: 25/24, 40/39, 45/44

Subgroup val mapping: [⟨1 1 2 2 4], ⟨0 2 1 5 -1]]

Gencom mapping: [⟨1 1 2 0 2 4], ⟨0 2 1 0 5 -1]]

gencom: [2 5/4; 25/24 40/39 45/44]

Optimal tunings:

  • WE: ~2 = 1202.433 ¢, ~6/5 = 351.237 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 350.978 ¢

Optimal ET sequence: 3e, 7, 17

Badness (Sintel): 0.536

Septimal dicot

Septimal dicot is the extension where 7/6 and 9/7 are also conflated into 5/4~6/5. Although 5/4~6/5 is a giant block already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the canonicity of this extension, despite the relatively poor accuracy.

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Optimal tunings:

  • WE: ~2 = 1205.532 ¢, ~6/5 = 337.931 ¢
error map: ⟨+5.532 -20.561 -37.319 +56.032]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.561 ¢
error map: ⟨0.000 -24.834 -47.753 +46.856]

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.504

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Optimal tunings:

  • WE: ~2 = 1203.346 ¢, ~6/5 = 343.078 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 343.260 ¢

Optimal ET sequence: 3de, 4e, 7

Badness (Sintel): 0.656

Eudicot

Subgroup: 2.3.5.7.11

Comma list: 15/14, 25/24, 33/32

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Optimal tunings:

  • WE: ~2 = 1205.828 ¢, ~6/5 = 337.683 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 336.909 ¢

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness (Sintel): 0.896

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 15/14, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.660 ¢, ~6/5 = 339.597 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 339.104 ¢

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.985

Flattie

This temperament used to be known as flat. Unlike septimal dicot where 7/6 is added to the neutral third, here 8/7 is added instead.

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Optimal tunings:

  • WE: ~2 = 1220.466 ¢, ~6/5 = 337.577 ¢
error map: ⟨+20.466 -6.335 -7.804 -45.004]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 335.391 ¢
error map: ⟨0.000 -31.173 -50.922 -104.217]

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness (Sintel): 0.642

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1216.069 ¢, ~6/5 = 342.052 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.467 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.826

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 14/13, 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4 4], ⟨0 2 1 -1 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1211.546 ¢, ~6/5 = 344.304 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 341.373 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.968

Sharpie

This temperament used to be known as sharp. This is where you find 7/6 at the major second and 7/4 at the major sixth.

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tunings:

  • WE: ~2 = 1202.488 ¢, ~5/4 = 358.680 ¢
error map: ⟨+2.488 +17.893 -22.658 -14.258]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 358.495 ¢
error map: ⟨0.000 +15.035 -27.818 -17.854]

Optimal ET sequence: 3d, 7d, 10

Badness (Sintel): 0.732

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Optimal tunings:

  • WE: ~2 = 1201.518 ¢, ~5/4 = 356.557 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.457 ¢

Optimal ET sequence: 3de, 7d, 10, 17d

Badness (Sintel): 0.739

Dichotic

In dichotic, 7/4 is found at a stack of two perfect fourths.

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.802 ¢, ~5/4 = 356.502 ¢
error map: ⟨+0.802 +11.851 -28.208 +8.374]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.275 ¢
error map: ⟨0.000 +10.595 -30.039 +6.074]

Optimal ET sequence: 3, 7, 10, 17, 27c

Badness (Sintel): 0.951

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Optimal tunings:

  • WE: ~2 = 1199.504 ¢, ~5/4 = 354.115 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.236 ¢

Optimal ET sequence: 7, 10, 17

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

Mapping: [⟨1 1 2 4 2 4], ⟨0 2 1 -4 5 -1]]

Optimal tunings:

  • WE: ~2 = 1199.289 ¢, ~5/4 = 354.156 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.340 ¢

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness (Sintel): 0.896

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1203.949 ¢, ~5/4 = 355.239 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.024 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 4 4 4], ⟨0 2 1 -4 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1202.979 ¢, ~5/4 = 355.193 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.254 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 0.940

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Optimal tunings:

  • WE: ~2 = 1197.526 ¢, ~5/4 = 359.915 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.745 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

Mapping: [⟨1 1 2 4 5 4], ⟨0 2 1 -4 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1197.922 ¢, ~5/4 = 360.021 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.722 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.15

Decimal

Main article: Decimal
See also: Jubilismic clan

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

mapping generators: ~7/5, ~7/4

Optimal tunings:

  • WE: ~7/5 = 603.286 ¢, ~7/4 = 953.637 ¢ (~7/6 = 252.935 ¢)
error map: ⟨+6.571 +5.318 -22.821 -2.047]
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.957 ¢ (~7/6 = 249.043 ¢)
error map: ⟨0.000 -0.041 -35.357 -17.869]

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd

Badness (Sintel): 0.717

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 49/48

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Optimal tunings:

  • WE: ~7/5 = 603.558 ¢, ~7/4 = 952.121 ¢ (~7/6 = 254.996 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 948.610 ¢ (~7/6 = 251.390 ¢)

Optimal ET sequence: 4e, 10, 14c, 24c

Badness (Sintel): 0.883

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 45/44, 49/48, 91/90

Mapping: [⟨2 0 3 4 -1 1], ⟨0 2 1 1 5 4]]

Optimal tunings:

  • WE: ~7/5 = 603.612 ¢, ~7/4 = 953.663 ¢ (~7/6 = 253.562 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.116 ¢ (~7/6 = 249.884 ¢)

Optimal ET sequence: 4ef, 10, 14cf, 24cf

Badness (Sintel): 0.881

Decimated

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 49/48

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Optimal tunings:

  • WE: ~7/5 = 604.535 ¢, ~7/4 = 952.076 ¢ (~7/6 = 256.994 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 946.108 ¢ (~7/6 = 253.892 ¢)

Optimal ET sequence: 4, 10e, 14c

Badness (Sintel): 1.04

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Optimal tunings:

  • WE: ~7/5 = 599.404 ¢, ~7/4 = 955.557 ¢ (~8/7 = 243.251 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 956.169 ¢ (~8/7 = 243.831 ¢)

Optimal ET sequence: 4, 6, 10

Badness (Sintel): 1.07

Sidi

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 -1 1 -3], ⟨0 4 2 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1207.178 ¢, ~14/9 = 777.414 ¢
error map: ⟨+7.178 +0.523 -24.308 +6.367]
  • CWE: ~2 = 1200.000 ¢, ~14/9 = 773.872 ¢
error map: ⟨0.000 -6.464 -38.569 -3.973]

Optimal ET sequence: 3d, …, 11cd, 14c

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

Mapping: [⟨1 -1 1 -3 -3], ⟨0 4 2 9 10]]

Optimal tunings:

  • WE: ~2 = 1207.200 ¢, ~11/7 = 777.363 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/7 = 773.777 ¢

Optimal ET sequence: 3de, …, 11cdee, 14c

Badness (Sintel): 1.09

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