Wizmic microtemperaments: Difference between revisions
→Gariwizmic: fix errors |
Switch to Sintel's badness, WE & CWE tunings |
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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 | {{Mapping|legend=1| 1 -14 -3 -20 | 0 41 14 60 }} | ||
: mapping generators: ~2, ~125/96 | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]] | [[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. | [[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 | ||
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[[Comma list]]: 420175/419904, 5250987/5242880 | [[Comma list]]: 420175/419904, 5250987/5242880 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -1 -11 4 | 0 13 67 -6 }} | ||
: mapping generators: ~2, ~147/128 | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]] | [[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, | {{Optimal ET sequence|legend=1| 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.12 | ||
== Tertiathirds == | == Tertiathirds == | ||
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 | ||
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[[Comma list]]: 420175/419904, 29360128/29296875 | [[Comma list]]: 420175/419904, 29360128/29296875 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -4 2 -6 | 0 52 3 82 }} | ||
: mapping generators: ~2, ~3375/3136 | |||
: mapping generators: ~2, ~ | |||
[[Optimal tuning]] | [[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 | {{Optimal ET sequence|legend=1| 121, 149, 270, 2309c, 2579c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.35 | ||
=== 11-limit === | === 11-limit === | ||
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Comma list: 3025/3024, 5632/5625, 117649/117612 | Comma list: 3025/3024, 5632/5625, 117649/117612 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -4 2 -6 -9 | 0 52 3 82 116 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 121, 149, 270 }} | ||
Badness: | Badness (Sintel): 1.10 | ||
=== 13-limit === | === 13-limit === | ||
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Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224 | Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -4 2 -6 -9 -5 | 0 52 3 82 116 81 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 121, 149, 270 }} | ||
Badness: 0. | Badness (Sintel): 0.806 | ||
=== 17-limit === | === 17-limit === | ||
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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 | Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 | 0 52 3 82 116 81 66 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 121, 149, 270 }} | ||
Badness: 0. | Badness (Sintel): 0.973 | ||
=== 19-limit === | === 19-limit === | ||
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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 | Mapping: {{mapping| 1 -4 2 -6 -9 -5 -3 5 | 0 52 3 82 116 81 66 -7 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 121, 149, 270 }} | ||
Badness: 0. | Badness (Sintel): 0.950 | ||
== Witcher == | == Witcher == | ||
{{See also| Amity family }} | {{See also| Amity family }} | ||
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 | ||
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[[Comma list]]: 420175/419904, 1600000/1594323 | [[Comma list]]: 420175/419904, 1600000/1594323 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -7 -20 -2 | 0 25 65 14 }} | ||
: mapping generators: ~2, ~80/63 | |||
: mapping generators: ~2, ~63 | |||
[[Optimal tuning]] | [[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]]: | [[Badness]] (Sintel): 2.04 | ||
=== 11-limit === | === 11-limit === | ||
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Comma list: 540/539, 4375/4356, 1240029/1239040 | Comma list: 540/539, 4375/4356, 1240029/1239040 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -7 -20 -2 -35 | 0 25 65 14 112 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 99e, 166, 265 }} | ||
Badness: | Badness (Sintel): 2.73 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Comma list: 540/539, 729/728, 4375/4356, 10648/10647 | Comma list: 540/539, 729/728, 4375/4356, 10648/10647 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -7 -20 -2 -35 -43 | 0 25 65 14 112 136 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 99ef, 166, 265 }} | ||
Badness: | Badness (Sintel): 2.40 | ||
=== Semiwitcher === | === Semiwitcher === | ||
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Comma list: 9801/9800, 41503/41472, 585640/583443 | Comma list: 9801/9800, 41503/41472, 585640/583443 | ||
Mapping: {{mapping| 2 | Mapping: {{mapping| 2 -14 -40 -4 -13 | 0 25 65 14 29 }} | ||
: mapping generators: ~99/70, ~80/63 | |||
: mapping generators: ~99/70, ~ | |||
Optimal | 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= | {{Optimal ET sequence|legend=0| 32c, 166, 198, 364, 562, 926 }} | ||
Badness: | Badness (Sintel): 2.92 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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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 | Mapping: {{mapping| 2 11 25 10 16 24 | 0 25 65 14 29 53 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 32cf, 166, 198, 364, 562, 926 }} | ||
Badness: | Badness (Sintel): 1.57 | ||
=== Hitch === | === Hitch === | ||
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Comma list: 3025/3024, 117649/117612, 1600000/1594323 | Comma list: 3025/3024, 117649/117612, 1600000/1594323 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -32 -85 -16 31 | 0 50 130 28 -41 }} | ||
: mapping generators: ~2, ~891/560 | |||
: | Optimal tunings: | ||
* WE: ~2 = 1199.8455{{c}}, ~891/560 = 805.9457{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~891/560 = 806.0488{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 67c, 198, 463, 661, 1124bde }} | ||
Badness (Sintel): 3.09 | |||
Badness: | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323 | Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -32 -85 -16 31 -134 | 0 50 130 28 -41 205 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 198, 463, 661 }} | ||
Badness: | Badness (Sintel): 2.51 | ||
==== Hitcher ==== | ==== Hitcher ==== | ||
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Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561 | Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -32 -85 -16 31 -1 | 0 50 130 28 -41 7 }} | ||
Optimal | 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= | {{Optimal ET sequence|legend=0| 67c, 198, 463f, 661ff }} | ||
Badness: | Badness (Sintel): 2.46 | ||
== Gariwizmic == | == Gariwizmic == | ||
Gariwizmic | 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 | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Comma list: 420175/419904, | [[Comma list]]: 420175/419904, 33554432/33480783 | ||
{{Mapping|legend=1| 2 0 -119 50 | 0 1 39 -14 }} | |||
: mapping generators: ~46305/32768, ~3 | |||
: [[error map]]: {{val|-0.069 | [[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 }} | |||
{{Optimal ET sequence|legend=1| 94, 176, 270, 904, 1174, 1444, 1714, 3158b, 4872bbcd }} | |||
[[Badness]] (Sintel): 2.22 | |||
[[Badness]]: 2. | |||
==== 11-limit ==== | ==== 11-limit ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 9801/9800, 19712/19683, 41503/41472 | ||
{{ | Mapping: {{mapping| 2 0 -119 50 -66 | 0 1 39 -14 23 }} | ||
* | 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 }} | |||
{{Optimal ET sequence|legend=0| 94, 176, 270, 1174, 1444, 1714, 1984e }} | |||
Badness (Sintel): 1.01 | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 1716/1715, 2080/2079, 4096/4095, 19712/19683 | |||
Mapping: {{mapping| 2 0 -119 50 -66 93 | 0 1 39 -14 23 -27 }} | |||
Optimal tunings: | |||
* WE: ~99/70 = 599. | * 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 = | * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2145{{c}} | ||
: error map: {{val| | : error map: {{val| 0.000 +0.260 +0.054 +0.170 -0.383 -0.321 }} | ||
{{Optimal ET sequence|legend=0|94, 176, 270 | {{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 -119 50 -66 93 -131 | 0 1 39 -14 23 -27 44 }} | |||
Optimal tunings: | |||
* WE: ~99/70 = 599. | * 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 = | * CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 702.2150{{c}} | ||
: error map: {{val| | : 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 | {{Optimal ET sequence|legend=0| 94, 176, 270, 634, 904, 1174 }} | ||
Badness (Sintel): 0.655 | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||