Hemimage temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
Line 27: Line 27:
: mapping generators: ~63/50, ~28/27
: mapping generators: ~63/50, ~28/27


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~28/27 = 60.528{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}


{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]] (Smith): 0.057499
[[Badness]] (Sintel): 1.46


=== 11-limit ===
=== 11-limit ===
Line 40: Line 44:
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.430{{c}}
Optimal tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}


Badness (Smith): 0.050379
Badness (Sintel): 1.67


==== 13-limit ====
==== 13-limit ====
Line 53: Line 59:
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.428{{c}}
Optimal tunings:
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}


{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}


Badness (Smith): 0.046006
Badness (Sintel): 1.90


===== 17-limit =====
===== 17-limit =====
Line 66: Line 74:
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.438{{c}}
Optimal tunings:
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}


Badness (Smith): 0.031678
Badness (Sintel): 1.61


==== Catachrome ====
==== Catachrome ====
Line 79: Line 89:
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.378{{c}}
Optimal tunings:
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}


{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness (Smith): 0.043844
Badness (Sintel): 1.81


===== 17-limit =====
===== 17-limit =====
Line 92: Line 104:
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.377{{c}}
Optimal tunings:
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}


{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness (Smith): 0.030218
Badness (Sintel): 1.54


==== Chromic ====
==== Chromic ====
Line 105: Line 119:
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~27/26 = 60.456{{c}}
Optimal tunings:
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}


{{Optimal ET sequence|legend=0| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness (Smith): 0.049857
Badness (Sintel): 2.06


===== 17-limit =====
===== 17-limit =====
Line 118: Line 134:
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~27/26 = 60.459{{c}}
Optimal tunings:
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}


{{Optimal ET sequence|legend=0| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness (Smith): 0.031043
Badness (Sintel): 1.58


=== Hemichromat ===
=== Hemichromat ===
Line 131: Line 149:
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~55/54 = 30.2511{{c}}
Optimal tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}


{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness (Smith): 0.067173
Badness (Sintel): 2.22


==== 13-limit ====
==== 13-limit ====
Line 144: Line 164:
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~55/54 = 30.2527{{c}}
Optimal tunings:
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}


{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}


Badness (Smith): 0.033420
Badness (Sintel): 1.38


== Bisupermajor ==
== Bisupermajor ==
Line 160: Line 182:
: mapping generators: ~1225/864, ~192/175
: mapping generators: ~1225/864, ~192/175


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000{{c}}, ~192/175 = 162.806{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}
: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}


{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]] (Smith): 0.065492
[[Badness]] (Sintel): 1.66


=== 11-limit ===
=== 11-limit ===
Line 173: Line 199:
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~11/10 = 162.773{{c}}
Optimal tunings:
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}


{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}


Badness (Smith): 0.032080
Badness (Sintel): 1.06


== Bicommatic ==
== Bicommatic ==
Line 189: Line 217:
: mapping generators: ~567/400, ~81/80
: mapping generators: ~567/400, ~81/80


[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~81/80 = 20.377{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}
: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}
: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}


{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


[[Badness]] (Smith): 0.084317
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Line 202: Line 234:
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~81/80 = 20.390{{c}}
Optimal tunings:
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}


{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}


Badness (Smith): 0.030461
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 215: Line 249:
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~66/65 = 20.427{{c}}
Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}


{{Optimal ET sequence|legend=0| 58, 118, 176f }}
{{Optimal ET sequence|legend=0| 58, 118, 176f }}


Badness (Smith): 0.026336
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 228: Line 264:
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~66/65 = 20.378{{c}}
Optimal tunings:
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


{{Optimal ET sequence|legend=0| 58, 118, 294ffg, 412dffgg }}
{{Optimal ET sequence|legend=0| 58, 118 }}


Badness (Smith): 0.022396
Badness (Sintel): 1.14


== Degrees ==
== Degrees ==
Line 253: Line 291:
: mapping generators: ~28/27, ~3
: mapping generators: ~28/27, ~3


[[Optimal tuning]] ([[POTE]]): ~28/27 = 60.000{{c}}, ~3/2 = 703.015{{c}} (~126/125 = 16.985{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}


{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


[[Badness]] (Sintel): 2.69
[[Badness]] (Sintel): 2.69
Line 266: Line 308:
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


Optimal tuning (POTE): ~28/27 = 60.000{{c}}, ~3/2 = 703.231{{c}} (~100/99 = 16.769{{c}})
Optimal tunings:
* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})


{{Optimal ET sequence|legend=0| 20cd, 60e, 80, 140, 360 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness (Sintel): 1.55
Badness (Sintel): 1.55
Line 279: Line 323:
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


Optimal tuning (POTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.080{{c}} (~100/99 = 16.920{{c}})
Optimal tunings:
* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})


{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Sintel): 1.35
Badness (Sintel): 1.35
Line 292: Line 338:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.107{{c}} (~100/99 = 16.893{{c}})
Optimal tunings:
* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})


{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Sintel): 1.17
Badness (Sintel): 1.17
Line 305: Line 353:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.107{{c}} (~100/99 = 16.893{{c}})
Optimal tunings:
* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})


{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Sintel): 1.27
Badness (Sintel): 1.27
Line 318: Line 368:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.169{{c}} (~100/99 = 16.831{{c}})
Optimal tunings:
* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})


{{Optimal ET sequence|legend=0| 20cdei, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Sintel): 1.21
Badness (Sintel): 1.21
Line 331: Line 383:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.171{{c}} (~100/99 = 16.829{{c}})
Optimal tunings:
* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})


{{Optimal ET sequence|legend=0| 20cdeij, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Sintel): 1.13
Badness (Sintel): 1.13


=== no-31's 37-limit ===
=== 2.3.5.7.11.13.17.19.23.29.37 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


Line 344: Line 398:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}


Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.222{{c}} (~100/99 = 16.778{{c}})
Optimal tunings:
* WE: ~29/28 = 60.0001{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})


{{Optimal ET sequence|legend=0| 20cdeijl, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Badness (Sintel): 1.13
Badness (Sintel): 1.13


=== no-31's 41-limit ===
=== 2.3.5.7.11.13.17.19.23.29.37.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


Line 357: Line 413:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}


Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.207{{c}}
Optimal tunings:
* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})


{{Optimal ET sequence|legend=0| 20cdeijlm, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Badness (Sintel): 1.10
Badness (Sintel): 1.10


== Squarschmidt ==
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as {{nowrap| 118 & 239 }} temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as {{nowrap| 118 & 239 }} temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 373: Line 431:
: mapping generators: ~2, ~98304/78125
: mapping generators: ~2, ~98304/78125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98304/78125 = 396.621{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9653{{c}}, ~98304/78125 = 396.6094{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98304/78125 = 396.6201{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc }}


[[Badness]] (Smith): 0.218314
[[Badness]] (Sintel): 5.12


=== 7-limit ===
=== 7-limit ===
Line 386: Line 448:
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1125/896 = 396.643{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


[[Badness]] (Smith): 0.132821
[[Badness]] (Sintel): 3.36


=== 11-limit ===
=== 11-limit ===
Line 399: Line 465:
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~44/35 = 396.644{{c}}
Optimal tunings:
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}


{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}


Badness (Smith): 0.038186
Badness (Sintel): 1.26


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 11:36, 28 April 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.

This is a collection of rank-2 temperaments tempering out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:

Chromat

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an amity extension with third-octave period.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

mapping generators: ~63/50, ~28/27

Optimal tunings:

  • WE: ~63/50 = 399.9549 ¢, ~28/27 = 60.5216 ¢
error map: ⟨-0.135 +0.473 +0.241 -0.751]
  • CWE: ~63/50 = 400.0000 ¢, ~28/27 = 60.5162 ¢
error map: ⟨0.000 +0.626 +0.397 -0.567]

Optimal ET sequence: 39d, 60, 99, 258, 357, 456

Badness (Sintel): 1.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 10976/10935

Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

Optimal tunings:

  • WE: ~44/35 = 400.0359 ¢, ~28/27 = 60.4357 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

Optimal ET sequence: 60e, 99e, 159, 258

Badness (Sintel): 1.67

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 10976/10935

Mapping: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]

Optimal tunings:

  • WE: ~44/35 = 400.0382 ¢, ~28/27 = 60.4342 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4331 ¢

Optimal ET sequence: 60eff, 99ef, 159, 258, 417d

Badness (Sintel): 1.90

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757

Mapping: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]

Optimal tunings:

  • WE: ~44/35 = 399.9982 ¢, ~28/27 = 60.4374 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

Optimal ET sequence: 99ef, 159, 258, 417dg

Badness (Sintel): 1.61

Catachrome

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 1001/1000, 10976/10935

Mapping: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]

Optimal tunings:

  • WE: ~44/35 = 400.1386 ¢, ~28/27 = 60.3986 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3929 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Sintel): 1.81

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913

Mapping: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]

Optimal tunings:

  • WE: ~44/35 = 400.1115 ¢, ~28/27 = 60.3935 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3893 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Sintel): 1.54

Chromic

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1875/1859

Mapping: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]

Optimal tunings:

  • WE: ~44/35 = 399.9082 ¢, ~28/27 = 60.4425 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4380 ¢

Optimal ET sequence: 60e, 99ef, 159f

Badness (Sintel): 2.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 352/351, 375/374, 595/594

Mapping: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]

Optimal tunings:

  • WE: ~44/35 = 399.8948 ¢, ~28/27 = 60.4435 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4385 ¢

Optimal ET sequence: 60e, 99ef, 159f

Badness (Sintel): 1.58

Hemichromat

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 10976/10935, 102487/102400

Mapping: [⟨3 4 5 6 10], ⟨0 10 26 32 5]]

Optimal tunings:

  • WE: ~63/50 = 399.9750 ¢, ~55/54 = 30.2568 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2561 ¢

Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b

Badness (Sintel): 2.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935

Mapping: [⟨3 4 5 6 10 8], ⟨0 10 26 32 5 41]]

Optimal tunings:

  • WE: ~63/50 = 399.9741 ¢, ~55/54 = 30.2584 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2577 ¢

Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b

Badness (Sintel): 1.38

Bisupermajor

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [⟨2 1 6 1], ⟨0 8 -5 17]]

mapping generators: ~1225/864, ~192/175

Optimal tunings:

  • WE: ~1225/864 = 600.0294 ¢, ~192/175 = 162.8141 ¢
error map: ⟨+0.059 +0.587 -0.208 -0.957]
  • CWE: ~1225/864 = 600.0000 ¢, ~192/175 = 162.8082 ¢
error map: ⟨0.000 +0.510 -0.355 -1.087]

Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d

Badness (Sintel): 1.66

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [⟨2 1 6 1 8], ⟨0 8 -5 17 -4]]

Optimal tunings:

  • WE: ~99/70 = 600.1224 ¢, ~11/10 = 162.8065 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~11/10 = 162.7788 ¢

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de, 634dee

Badness (Sintel): 1.06

Bicommatic

Used to be known simply as the commatic temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

mapping generators: ~567/400, ~81/80

Optimal tunings:

  • WE: ~567/400 = 600.0497 ¢, ~81/80 = 20.3790 ¢
error map: ⟨+0.099 +0.089 +1.085 -1.756]
  • CWE: ~567/400 = 600.0000 ¢, ~81/80 = 20.3837 ¢
error map: ⟨0.000 -0.037 +0.976 -1.920]

Optimal ET sequence: 58, 118, 294, 412d

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

Optimal tunings:

  • WE: ~99/70 = 600.0401 ¢, ~81/80 = 20.3913 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~81/80 = 20.3948 ¢

Optimal ET sequence: 58, 118, 294, 412d

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1001/1000

Mapping: [⟨2 3 4 5 6 7], ⟨0 5 19 18 27 12]]

Optimal tunings:

  • WE: ~99/70 = 599.8514 ¢, ~66/65 = 20.4215 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/65 = 20.4093 ¢

Optimal ET sequence: 58, 118, 176f

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

Mapping: [⟨2 3 4 5 6 7 8], ⟨0 5 19 18 27 12 5]]

Optimal tunings:

  • WE: ~17/12 = 600.0257 ¢, ~66/65 = 20.3789 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 20.3804 ¢

Optimal ET sequence: 58, 118

Badness (Sintel): 1.14

Degrees

This page is about the regular temperament. For scale degrees, see degree.
See also: 20th-octave temperaments

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with 23/20, 6\20 = 3\10 with 69/56, 7\20 with 23/18, etc. By observing that 1\20 works as 30/29~29/28~28/27, with 29/28 being especially accurate, and by equating 29/22 with 2\5 = 240 ¢, we get a uniquely elegant extension to the 29-limit which tempers out (33/25)/(29/22) = 726/725, S28 = 784/783 and S29 = 841/840. An edo as large as 220 supports it by patent val, though it does not appear in the optimal ET sequence, and 80edo and 140edo are both much more recommendable tunings.

By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.

By looking at the mapping, we observe an 80-note mos scale is ideal, so that 80edo is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of 20edo by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in 140edo.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

mapping generators: ~28/27, ~3

Optimal tunings:

  • WE: ~28/27 = 59.9922 ¢, ~3/2 = 702.9233 ¢ (~126/125 = 16.9828 ¢)
error map: ⟨-0.157 +0.812 -0.647 -0.220]
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 702.9324 ¢ (~126/125 = 17.0676 ¢)
error map: ⟨0.000 +0.977 -0.449 -0.029]

Optimal ET sequence: 60, 80, 140, 640b, 780b

Badness (Sintel): 2.69

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]

Optimal tunings:

  • WE: ~28/27 = 59.9929 ¢, ~3/2 = 703.1478 ¢ (~100/99 = 16.7666 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1556 ¢ (~100/99 = 16.8444 ¢)

Optimal ET sequence: 60e, 80, 140, 360

Badness (Sintel): 1.55

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]

Optimal tunings:

  • WE: ~28/27 = 59.9996 ¢, ~3/2 = 703.0749 ¢ (~100/99 = 16.9197 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0770 ¢ (~100/99 = 16.9230 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.35

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

Mapping: [⟨20 0 -17 -39 -26 74 50], ⟨0 1 2 3 3 0 1]]

Optimal tunings:

  • WE: ~28/27 = 60.0058 ¢, ~3/2 = 703.0364 ¢ (~100/99 = 17.0335 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0061 ¢ (~100/99 = 16.9939 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.17

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

Mapping: [⟨20 0 -17 -39 -26 74 50 85], ⟨0 1 2 3 3 0 1 0]]

Optimal tunings:

  • WE: ~28/27 = 59.9961 ¢, ~3/2 = 703.1523 ¢ (~100/99 = 16.8015 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1777 ¢ (~100/99 = 16.8223 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.27

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27], ⟨0 1 2 3 3 0 1 0 2]]

Optimal tunings:

  • WE: ~28/27 = 59.9990 ¢, ~3/2 = 703.1804 ¢ (~100/99 = 16.8074 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1870 ¢ (~100/99 = 16.8130 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.21

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2], ⟨0 1 2 3 3 0 1 0 2 3]]

Optimal tunings:

  • WE: ~29/28 = 59.9990 ¢, ~3/2 = 703.1829 ¢ (~100/99 = 16.8055 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.1891 ¢ (~100/99 = 16.8109 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.13

2.3.5.7.11.13.17.19.23.29.37 subgroup

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9], ⟨0 1 2 3 3 0 1 0 2 3 3]]

Optimal tunings:

  • WE: ~29/28 = 60.0001 ¢, ~3/2 = 703.2183 ¢ (~100/99 = 16.7827 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2178 ¢ (~100/99 = 16.7822 ¢)

Optimal ET sequence: 60el, 80, 140

Badness (Sintel): 1.13

2.3.5.7.11.13.17.19.23.29.37.41 subgroup

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9 12], ⟨0 1 2 3 3 0 1 0 2 3 3 3]]

Optimal tunings:

  • WE: ~29/28 = 59.9998 ¢, ~3/2 = 703.2088 ¢ (~100/99 = 16.7882 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2104 ¢ (~100/99 = 16.7896 ¢)

Optimal ET sequence: 60el, 80, 140

Badness (Sintel): 1.10

Squarschmidt

A generator for the squarschimidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents. The squarschimidt temperament can be described as 118 & 239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

Subgroup: 2.3.5

Comma list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

Optimal tunings:

  • WE: ~2 = 1199.9653 ¢, ~98304/78125 = 396.6094 ¢
error map: ⟨-0.099 +0.543 +0.029 -0.719]
  • CWE: ~2 = 1200.0000 ¢, ~98304/78125 = 396.6201 ¢
error map: ⟨0.000 +0.653 +0.253 -0.552]

Optimal ET sequence: 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc

Badness (Sintel): 5.12

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]

Optimal tunings:

  • WE: ~2 = 1199.9006 ¢, ~1125/896 = 396.6104 ¢
error map: ⟨-0.099 +0.543 +0.029 -0.719]
  • CWE: ~2 = 1200.0000 ¢, ~1125/896 = 396.6417 ¢
error map: ⟨0.000 +0.653 +0.253 -0.552]

Optimal ET sequence: 118, 239, 357, 596

Badness (Sintel): 3.36

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: [⟨1 -8 1 -20 -21], ⟨0 29 4 69 74]]

Optimal tunings:

  • WE: ~2 = 1199.9005 ¢, ~44/35 = 396.6107 ¢
  • CWE: ~2 = 1200.0000 ¢, ~44/35 = 396.6419 ¢

Optimal ET sequence: 118, 239, 357, 596

Badness (Sintel): 1.26

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