Wizmic microtemperaments: Difference between revisions

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== Qak ==
== Qak ==
Qak tempers out the [[meter]] in addition to the wizma, and may be described as the {{nowrap| 50 & 171 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 420175/419904, 703125/702464
[[Comma list]]: 420175/419904, 703125/702464


{{Mapping|legend=1| 1 27 11 40 | 0 -41 -14 -60 }}
{{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }}
 
: mapping generators: ~2, ~125/96
: mapping generators: ~2, ~192/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 456.144
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0566{{c}}, ~125/96 = 456.1655{{c}}
: [[error map]]: {{val| +0.057 +0.039 -0.166 -0.027 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~125/96 = 456.1450{{c}}
: error map: {{val| 0.000 -0.009 -0.283 -0.125 }}


{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}
{{Optimal ET sequence|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }}


[[Badness]]: 0.029267
[[Badness]] (Sintel): 0.741


== Tokko ==
== Tokko ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Tokko]].''
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Tokko (5-limit)]].''
 
Tokko tempers out the [[mitonisma]] in addition to the wizma, and may be described as the {{nowrap| 5 & 171 }} temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 43: Line 50:
[[Comma list]]: 420175/419904, 5250987/5242880
[[Comma list]]: 420175/419904, 5250987/5242880


{{Mapping|legend=1| 1 12 56 -2 | 0 -13 -67 6 }}
{{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }}
 
: mapping generators: ~2, ~147/128
: mapping generators: ~2, ~256/147


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 238.599
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0850{{c}}, ~147/128 = 238.6157{{c}}
: [[error map]]: {{val| +0.085 -0.036 +0.004 -0.180 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~147/128 = 238.5997{{c}}
: error map: {{val| 0.000 -0.159 -0.133 -0.424 }}


{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }}
{{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }}


[[Badness]]: 0.044417
[[Badness]] (Sintel): 1.12


== Tertiathirds ==
== Tertiathirds ==
The ''tertiathirds'' temperament (121&270) tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, as well as the wizma. This temperament splits the interval of 5/4 into three [[14/13]] generators.
Tertiathirds tempers out the quasiorwellisma, [[29360128/29296875]] in the 7-limit, and may be described as the {{nowrap| 121 & 149 }} temperament. It splits the interval of [[5/4]] into three [[14/13]] generators.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 60: Line 70:
[[Comma list]]: 420175/419904, 29360128/29296875
[[Comma list]]: 420175/419904, 29360128/29296875


{{Mapping|legend=1| 1 48 5 76 | 0 -52 -3 -82 }}
{{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }}
 
: mapping generators: ~2, ~3375/3136
: mapping generators: ~2, ~6272/3375


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 128.888
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9305{{c}}, ~3375/3136 = 128.8804{{c}}
: [[error map]]: {{val| -0.070 +0.104 +0.188 -0.216 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3375/3136 = 128.8871{{c}}
: error map: {{val| 0.000 +0.173 +0.348 -0.085 }}


{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }}
{{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }}


[[Badness]]: 0.093005
[[Badness]] (Sintel): 2.35


=== 11-limit ===
=== 11-limit ===
Line 75: Line 88:
Comma list: 3025/3024, 5632/5625, 117649/117612
Comma list: 3025/3024, 5632/5625, 117649/117612


Mapping: {{mapping| 1 48 5 76 107 | 0 -52 -3 -82 -116 }}
Mapping: {{mapping| 1 -4 2 -6 -9 | 0 52 3 82 116 }}


Optimal tuning (POTE): ~2 = 1\1, ~264/245 = 128.890
Optimal tunings:
* WE: ~2 = 1199.9251{{c}}, ~264/245 = 128.8818{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~264/245 = 128.8890{{c}}


{{Optimal ET sequence|legend=1| 121, 149, 270, 2551bc, 2821bc, 3091bc, 3361bc, 3631bbcc, 3901bbccd, 4171bbccd, 4441bbccd, 4711bbccd }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.033322
Badness (Sintel): 1.10


=== 13-limit ===
=== 13-limit ===
Line 88: Line 103:
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224


Mapping: {{mapping| 1 48 5 76 107 76 | 0 -52 -3 -82 -116 -81 }}
Mapping: {{mapping| 1 -4 2 -6 -9 -5 | 0 52 3 82 116 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
Optimal tunings:
* WE: ~2 = 1199.9510{{c}}, ~14/13 = 128.8849{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


{{Optimal ET sequence|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.019494
Badness (Sintel): 0.806


=== 17-limit ===
=== 17-limit ===
Line 101: Line 118:
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224


Mapping: {{mapping| 1 48 5 76 107 76 63 | 0 -52 -3 -82 -116 -81 -66 }}
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.891
Optimal tunings:
* WE: ~2 = 1199.8754{{c}}, ~14/13 = 128.8778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8898{{c}}


{{Optimal ET sequence|legend=1| 121, 149, 270 }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.019107
Badness (Sintel): 0.973


=== 19-limit ===
=== 19-limit ===
Line 114: Line 133:
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125
Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125


Mapping: {{mapping| 1 48 5 76 107 76 63 -2 | 0 -52 -3 -82 -116 -81 -66 7 }}
Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
Optimal tunings:
* WE: ~2 = 1199.8964{{c}}, ~14/13 = 128.8793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 128.8896{{c}}


{{Optimal ET sequence|legend=1| 121, 149, 270 }}
{{Optimal ET sequence|legend=0| 121, 149, 270 }}


Badness: 0.015621
Badness (Sintel): 0.950


== Witcher ==
== Witcher ==
{{See also| Amity family }}
{{See also| Amity family }}


The witcher temperament (99&166) tempers out wizma (420175/419904) and the amity comma (1600000/1594323) as well as the ''hemfiness'' comma (4096000/4084101, saquinru-atriyo).
Witcher tempers out the wizma (420175/419904) and the amity comma (1600000/1594323). It may be described as the {{nowrap| 99 & 166 }} temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 131: Line 152:
[[Comma list]]: 420175/419904, 1600000/1594323
[[Comma list]]: 420175/419904, 1600000/1594323


{{Mapping|legend=1| 1 18 45 12 | 0 -25 -65 -14 }}
{{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }}
 
: mapping generators: ~2, ~80/63
: mapping generators: ~2, ~63/40


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~80/63 = 412.098
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8698{{c}}, ~80/63 = 412.0531{{c}}
: [[error map]]: {{val| -0.130 +0.283 -0.260 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/63 = 412.0956{{c}}
: error map: {{val| 0.000 +0.435 -0.100 +0.512 }}


{{Optimal ET sequence|legend=1| 99, 265, 364, 463, 562, 1025 }}
{{Optimal ET sequence|legend=1| 32c, 67c, 99, 265, 364, 463, 562, 1025 }}


[[Badness]]: 0.080608
[[Badness]] (Sintel): 2.04


=== 11-limit ===
=== 11-limit ===
Line 146: Line 170:
Comma list: 540/539, 4375/4356, 1240029/1239040
Comma list: 540/539, 4375/4356, 1240029/1239040


Mapping: {{mapping| 1 18 45 12 77 | 0 -25 -65 -14 -112 }}
Mapping: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }}


Optimal tuning (POTE): ~2 = 1\1, ~80/63 = 412.079
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~80/63 = 412.0758{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/63 = 412.0788{{c}}


{{Optimal ET sequence|legend=1| 99e, 166, 265 }}
{{Optimal ET sequence|legend=0| 99e, 166, 265 }}


Badness: 0.082607
Badness (Sintel): 2.73


==== 13-limit ====
==== 13-limit ====
Line 159: Line 185:
Comma list: 540/539, 729/728, 4375/4356, 10648/10647
Comma list: 540/539, 729/728, 4375/4356, 10648/10647


Mapping: {{mapping| 1 18 45 12 77 93 | 0 -25 -65 -14 -112 -136 }}
Mapping: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }}


Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 412.070
Optimal tunings:
* WE: ~2 = 1200.0660{{c}}, ~33/26 = 412.0928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/26 = 412.0709{{c}}


{{Optimal ET sequence|legend=1| 99ef, 166, 265, 431c, 696cc }}
{{Optimal ET sequence|legend=0| 99ef, 166, 265 }}


Badness: 0.057980
Badness (Sintel): 2.40


=== Semiwitcher ===
=== Semiwitcher ===
Line 172: Line 200:
Comma list: 9801/9800, 41503/41472, 585640/583443
Comma list: 9801/9800, 41503/41472, 585640/583443


Mapping: {{mapping| 2 11 25 10 16 | 0 -25 -65 -14 -29 }}
Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }}
 
: mapping generators: ~99/70, ~80/63
: mapping generators: ~99/70, ~49/44


Optimal tuning (POTE): ~99/70 = 1\2, ~49/44 = 187.904
Optimal tunings:
* WE: ~99/70 = 599.9717{{c}}, ~80/63 = 412.0766{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~80/63 = 412.0951{{c}}


{{Optimal ET sequence|legend=1| 166, 198, 364, 562, 926, 1488bd, 2414bdd }}
{{Optimal ET sequence|legend=0| 32c, 166, 198, 364, 562, 926 }}


Badness: 0.088189
Badness (Sintel): 2.92


==== 13-limit ====
==== 13-limit ====
Line 187: Line 216:
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280
Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280


Mapping: {{mapping| 2 11 25 10 16 24 | 0 -25 -65 -14 -29 -53 }}
Mapping: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~39/35 = 187.904
Optimal tunings:
* WE: ~99/70 = 599.9666{{c}}, ~33/26 = 412.0726{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/26 = 412.0944{{c}}


{{Optimal ET sequence|legend=1| 166, 198, 364, 562, 926 }}
{{Optimal ET sequence|legend=0| 32cf, 166, 198, 364, 562, 926 }}


Badness: 0.038009
Badness (Sintel): 1.57


=== Hitch ===
=== Hitch ===
Line 200: Line 231:
Comma list: 3025/3024, 117649/117612, 1600000/1594323
Comma list: 3025/3024, 117649/117612, 1600000/1594323


Mapping: {{mapping| 1 18 45 12 -10 | 0 -50 -130 -28 41 }}
Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }}
: mapping generators: ~2, ~891/560


: mapping generators: ~2, ~1120/891
Optimal tunings:  
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.950
{{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }}


{{Optimal ET sequence|legend=1| 198, 463, 661, 1124bde }}
Badness (Sintel): 3.09
 
Badness: 0.093482


==== 13-limit ====
==== 13-limit ====
Line 215: Line 247:
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323


Mapping: {{mapping| 1 18 45 12 -10 71 | 0 -50 -130 -28 41 -205 }}
Mapping: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }}


Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.948
Optimal tunings:
* WE: ~2 = 1199.8415{{c}}, ~891/560 = 805.9451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0508{{c}}


{{Optimal ET sequence|legend=1| 198, 463, 661 }}
{{Optimal ET sequence|legend=0| 198, 463, 661 }}


Badness: 0.060716
Badness (Sintel): 2.51


==== Hitcher ====
==== Hitcher ====
Line 228: Line 262:
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561


Mapping: {{mapping| 1 18 45 12 -10 6 | 0 -50 -130 -28 41 -7 }}
Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/39 = 393.949
Optimal tunings:
* WE: ~2 = 1199.7690{{c}}, ~78/49 = 805.8958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~78/49 = 806.0500{{c}}


{{Optimal ET sequence|legend=1| 198, 463f, 661ff }}
{{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }}


Badness: 0.059591
Badness (Sintel): 2.46


== Gariwizmic ==
== Gariwizmic ==
Gariwizmic (94&176) tempers out the wizma (420175/419904) and the garischisma (33554432/33480783). It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. Unlike schismic, it finds 5/4 at the ''trans''diminished fourth (+45 fifths) minus a half pythagorean comma, or equivalently, at the ''trans''diminished octave (+39 fifths) minus a semioctave.  
Gariwizmic tempers out the wizma (420175/419904) and the garischisma ([[33554432/33480783]]), and may be described as the {{nowrap| 94 & 176 }} temperament. It assumes a [[semioctave]] period and a [[3/2|perfect fifth]] generator that is slightly sharp of just. It finds [[5/4]] 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup.  
 
It is bad in the 5-limit but extends extremely well to other subgroups, notably the 2.3.5.7.11.13.19 subgroup, as it also indirectly tempers out the [[tredekisma]], the smallest superparticular ratio in the 19- and 23-limit.


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


Comma list: 420175/419904, [[33554432/33480783]]
[[Comma list]]: 420175/419904, 33554432/33480783
 
{{Mapping|legend=1| 2 2 -41 22|0 1 39 -14}}
 
* mapping generators: ~46305/32768, ~3
[[Optimal tuning|Optimal tunings]]:


* [[WE]]: ~46305/32768 = 599.996{{c}}, ~3/2 = 702.177{{c}}
{{Mapping|legend=1| 2 0 -119 50 | 0 1 39 -14 }}
: mapping generators: ~46305/32768, ~3


: [[error map]]: {{val|-0.069, 0.153, -0.021, -0.053}}
[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9657{{c}}, ~3/2 = 702.1765{{c}}
: [[error map]]: {{val|-0.069 +0.153 -0.021 -0.053 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~3/2 = 702.2161{{c}}
: error map: {{val| 0.000 +0.261 +0.114 +0.149 }}


* [[CWE]]: ~46305/32768 = 600.000{{c}}, ~3/2 = 702.216{{c}}
{{Optimal ET sequence|legend=1| 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd }}


: error map: {{val|0.000 0.261 0.114 0.149}}
[[Badness]] (Sintel): 2.22
 
{{Optimal ET sequence|legend=1|94, 176, 270, 364, 446, 634, 904}}
 
[[Badness]]: 2.225 (Sintel)


==== 11-limit ====
==== 11-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: [[9801/9800]], [[19712/19683]], [[41503/41472]]
Comma list: 9801/9800, 19712/19683, 41503/41472


{{Mapping|legend=1| 2 2 -41 22 -20|0 1 39 -14 23}}
Mapping: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }}


* mapping generators: ~99/70, ~3
Optimal tunings:
* WE: ~99/70 = 599.9790{{c}}, ~3/2 = 702.1938{{c}}
: error map: {{val| -0.042 +0.197 +0.106 -0.001 -0.440 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2179{{c}}
: error map: {{val| 0.000 +0.263 +0.185 +0.123 -0.306 }}


* WE: ~99/70 = 599.979, ~3/2 = 702.194
{{Optimal ET sequence|legend=0| 94, 176, 270, 1174, 1444, 1714, 1984e }}
: error map: {{val|-0.042 0.197 0.106 -0.001 -0.440}}
* CWE: ~99/70 = 1\2, ~3/2 = 702.218
: error map: {{val|0.000 0.263 0.185 0.123 -0.306}}


{{Optimal ET sequence|legend=0|94, 176, 270, 364, 446, 634, 904}}
Badness (Sintel): 1.01
 
[[Badness]]: 1.008 (Sintel)


==== 13-limit ====
==== 13-limit ====
Comma list: [[2080/2079]], [[1716/1715]], [[4096/4095]], [[35035/34992]]
Subgroup: 2.3.5.7.11.13


{{Mapping|legend=1| 2 2 -41 22 -20 39|0 1 39 -14 23 -27}}
Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683


* mapping generators: ~99/70, ~3
Mapping: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }}


[[Optimal tuning|Optimal tunings]]:
Optimal tunings:
* WE: ~99/70 = 599.996, ~3/2 = 702.210
* WE: ~99/70 = 599.9958{{c}}, ~3/2 = 702.2096{{c}}
: error map: {{val|-0.008 0.246 0.035 0.146 -0.412 -0.353}}
: error map: {{val| -0.008 +0.246 +0.035 +0.146 -0.412 -0.353 }}
* CWE: ~99/70 = 1\2, ~3/2 = 702.215
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2145{{c}}
: error map: {{val|-0.000 0.260 0.054 0.170 -0.383 -0.321}}
: error map: {{val| 0.000 +0.260 +0.054 +0.170 -0.383 -0.321 }}


{{Optimal ET sequence|legend=0|94, 176, 270, 364, 446, 634, 904}}
{{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }}


[[Badness]]: 0.822 (Sintel)
Badness (Sintel): 0.822


==== Subgroup: 2.3.5.7.11.13.19 ====
==== 2.3.5.7.11.13.19 subgroup ====
Comma list: [[2080/2079]], [[1540/1539]], [[1216/1215]], [[1716/1715]], [[1729/1728]]
Subgroup: 2.3.5.7.11.13.19


{{Mapping|legend=1| 2 2 -41 22 -20 39 -43|0 1 39 -14 23 -27 44}}
Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079


* mapping generators: ~99/70, ~3
Mapping: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }}


[[Optimal tuning|Optimal tunings]]:
Optimal tunings:
* WE: ~99/70 = 599.996, ~3/2 = 702.210
* WE: ~99/70 = 599.9969{{c}}, ~3/2 = 702.2114{{c}}
: error map: {{val|-0.006 0.250 0.057 0.147 -0.394 -0.355 -0.079}}
: error map: {{val| -0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079 }}
* CWE: ~99/70 = 1\2, ~3/2 = 702.215
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2150{{c}}
: error map: {{val|-0.000 0.260 0.070 0.165 -0.374 -0.332 -0.055}}
: error map: {{val| 0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055 }}


{{Optimal ET sequence|legend=0|94, 176, 270, 364, 446, 634, 904}}
{{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }}


[[Badness]]: 0.655 (Sintel)
Badness (Sintel): 0.655


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Revision as of 19:09, 11 May 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 microtemperaments which tempers out the wizma (monzo: [-6 -8 2 5⟩, ratio: 420175/419904).

Discussed elsewhere are:

Considered below are qak, tokko, tertiathirds, witcher, and gariwizmic.

Qak

Qak tempers out the meter in addition to the wizma, and may be described as the 50 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 703125/702464

Mapping: [⟨1 -14 -3 -20], ⟨0 41 14 60]]

mapping generators: ~2, ~125/96

Optimal tunings:

  • WE: ~2 = 1200.0566 ¢, ~125/96 = 456.1655 ¢
error map: ⟨+0.057 +0.039 -0.166 -0.027]
  • CWE: ~2 = 1200.0000 ¢, ~125/96 = 456.1450 ¢
error map: ⟨0.000 -0.009 -0.283 -0.125]

Optimal ET sequence: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd

Badness (Sintel): 0.741

Tokko

For the 5-limit version, see Syntonic–diatonic equivalence continuum #Tokko (5-limit).

Tokko tempers out the mitonisma in addition to the wizma, and may be described as the 5 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 5250987/5242880

Mapping: [⟨1 -1 -11 4], ⟨0 13 67 -6]]

mapping generators: ~2, ~147/128

Optimal tunings:

  • WE: ~2 = 1200.0850 ¢, ~147/128 = 238.6157 ¢
error map: ⟨+0.085 -0.036 +0.004 -0.180]
  • CWE: ~2 = 1200.0000 ¢, ~147/128 = 238.5997 ¢
error map: ⟨0.000 -0.159 -0.133 -0.424]

Optimal ET sequence: 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd

Badness (Sintel): 1.12

Tertiathirds

Tertiathirds tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, and may be described as the 121 & 149 temperament. It splits the interval of 5/4 into three 14/13 generators.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 29360128/29296875

Mapping: [⟨1 -4 2 -6], ⟨0 52 3 82]]

mapping generators: ~2, ~3375/3136

Optimal tunings:

  • WE: ~2 = 1199.9305 ¢, ~3375/3136 = 128.8804 ¢
error map: ⟨-0.070 +0.104 +0.188 -0.216]
  • CWE: ~2 = 1200.0000 ¢, ~3375/3136 = 128.8871 ¢
error map: ⟨0.000 +0.173 +0.348 -0.085]

Optimal ET sequence: 121, 149, 270, 2309c, 2579c

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 117649/117612

Mapping: [⟨1 -4 2 -6 -9], ⟨0 52 3 82 116]]

Optimal tunings:

  • WE: ~2 = 1199.9251 ¢, ~264/245 = 128.8818 ¢
  • CWE: ~2 = 1200.0000 ¢, ~264/245 = 128.8890 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 1.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5], ⟨0 52 3 82 116 81]]

Optimal tunings:

  • WE: ~2 = 1199.9510 ¢, ~14/13 = 128.8849 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.806

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5 -3], ⟨0 52 3 82 116 81 66]]

Optimal tunings:

  • WE: ~2 = 1199.8754 ¢, ~14/13 = 128.8778 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8898 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.973

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 936/935, 1225/1224, 1540/1539, 2128/2125

Mapping: [⟨1 -4 2 -6 -9 -5 -3 5], ⟨0 52 3 82 116 81 66 -7]]

Optimal tunings:

  • WE: ~2 = 1199.8964 ¢, ~14/13 = 128.8793 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.950

Witcher

See also: Amity family

Witcher tempers out the wizma (420175/419904) and the amity comma (1600000/1594323). It may be described as the 99 & 166 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 1600000/1594323

Mapping: [⟨1 -7 -20 -2], ⟨0 25 65 14]]

mapping generators: ~2, ~80/63

Optimal tunings:

  • WE: ~2 = 1199.8698 ¢, ~80/63 = 412.0531 ¢
error map: ⟨-0.130 +0.283 -0.260 +0.178]
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0956 ¢
error map: ⟨0.000 +0.435 -0.100 +0.512]

Optimal ET sequence: 32c, 67c, 99, 265, 364, 463, 562, 1025

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 1240029/1239040

Mapping: [⟨1 -7 -20 -2 -35], ⟨0 25 65 14 112]]

Optimal tunings:

  • WE: ~2 = 1199.9909 ¢, ~80/63 = 412.0758 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/63 = 412.0788 ¢

Optimal ET sequence: 99e, 166, 265

Badness (Sintel): 2.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4375/4356, 10648/10647

Mapping: [⟨1 -7 -20 -2 -35 -43], ⟨0 25 65 14 112 136]]

Optimal tunings:

  • WE: ~2 = 1200.0660 ¢, ~33/26 = 412.0928 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/26 = 412.0709 ¢

Optimal ET sequence: 99ef, 166, 265

Badness (Sintel): 2.40

Semiwitcher

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472, 585640/583443

Mapping: [⟨2 -14 -40 -4 -13], ⟨0 25 65 14 29]]

mapping generators: ~99/70, ~80/63

Optimal tunings:

  • WE: ~99/70 = 599.9717 ¢, ~80/63 = 412.0766 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~80/63 = 412.0951 ¢

Optimal ET sequence: 32c, 166, 198, 364, 562, 926

Badness (Sintel): 2.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 17303/17280

Mapping: [⟨2 11 25 10 16 24], ⟨0 25 65 14 29 53]]

Optimal tunings:

  • WE: ~99/70 = 599.9666 ¢, ~33/26 = 412.0726 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/26 = 412.0944 ¢

Optimal ET sequence: 32cf, 166, 198, 364, 562, 926

Badness (Sintel): 1.57

Hitch

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 117649/117612, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31], ⟨0 50 130 28 -41]]

mapping generators: ~2, ~891/560

Optimal tunings:

  • WE: ~2 = 1199.8455 ¢, ~891/560 = 805.9457 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0488 ¢

Optimal ET sequence: 67c, 198, 463, 661, 1124bde

Badness (Sintel): 3.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323

Mapping: [⟨1 -32 -85 -16 31 -134], ⟨0 50 130 28 -41 205]]

Optimal tunings:

  • WE: ~2 = 1199.8415 ¢, ~891/560 = 805.9451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~891/560 = 806.0508 ¢

Optimal ET sequence: 198, 463, 661

Badness (Sintel): 2.51

Hitcher

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561

Mapping: [⟨1 -32 -85 -16 31 -1], ⟨0 50 130 28 -41 7]]

Optimal tunings:

  • WE: ~2 = 1199.7690 ¢, ~78/49 = 805.8958 ¢
  • CWE: ~2 = 1200.0000 ¢, ~78/49 = 806.0500 ¢

Optimal ET sequence: 67c, 198, 463f, 661ff

Badness (Sintel): 2.46

Gariwizmic

Gariwizmic tempers out the wizma (420175/419904) and the garischisma (33554432/33480783), and may be described as the 94 & 176 temperament. It assumes a semioctave period and a perfect fifth generator that is slightly sharp of just. It finds 5/4 39 fifths away, shifted by a semioctave. It extends extremely well to the 2.3.5.7.11.13.19 subgroup.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 33554432/33480783

Mapping: [⟨2 0 -119 50], ⟨0 1 39 -14]]

mapping generators: ~46305/32768, ~3

Optimal tunings:

  • WE: ~46305/32768 = 599.9657 ¢, ~3/2 = 702.1765 ¢
error map: ⟨-0.069 +0.153 -0.021 -0.053]
  • CWE: ~46305/32768 = 600.0000 ¢, ~3/2 = 702.2161 ¢
error map: ⟨0.000 +0.261 +0.114 +0.149]

Optimal ET sequence: 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd

Badness (Sintel): 2.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 19712/19683, 41503/41472

Mapping: [⟨2 0 -119 50 -66], ⟨0 1 39 -14 23]]

Optimal tunings:

  • WE: ~99/70 = 599.9790 ¢, ~3/2 = 702.1938 ¢
error map: ⟨-0.042 +0.197 +0.106 -0.001 -0.440]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2179 ¢
error map: ⟨0.000 +0.263 +0.185 +0.123 -0.306]

Optimal ET sequence: 94, 176, 270, 1174, 1444, 1714, 1984e

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨2 0 -119 50 -66 93], ⟨0 1 39 -14 23 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9958 ¢, ~3/2 = 702.2096 ¢
error map: ⟨-0.008 +0.246 +0.035 +0.146 -0.412 -0.353]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2145 ¢
error map: ⟨0.000 +0.260 +0.054 +0.170 -0.383 -0.321]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.822

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1716/1715, 1729/1728, 2080/2079

Mapping: [⟨2 0 -119 50 -66 93 -131], ⟨0 1 39 -14 23 -27 44]]

Optimal tunings:

  • WE: ~99/70 = 599.9969 ¢, ~3/2 = 702.2114 ¢
error map: ⟨-0.006 +0.250 +0.057 +0.147 -0.394 -0.355 -0.079]
  • CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.2150 ¢
error map: ⟨0.000 +0.260 +0.070 +0.165 -0.374 -0.332 -0.055]

Optimal ET sequence: 94, 176, 270, 634, 904, 1174

Badness (Sintel): 0.655

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