Wizmic microtemperaments: Difference between revisions
m Removing from Category:Regular temperament theory using Cat-a-lot |
Add 19-limit tertiathirds (121&149) |
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{{Multival|legend=1| 41 14 60 -73 -20 100 }} | {{Multival|legend=1| 41 14 60 -73 -20 100 }} | ||
[[POTE generator]]: ~ | [[POTE generator]]: ~125/96 = 456.144 | ||
{{Val list|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }} | {{Val list|legend=1| 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd }} | ||
| Line 48: | Line 48: | ||
{{Multival|legend=1| 13 67 -6 76 -46 -202 }} | {{Multival|legend=1| 13 67 -6 76 -46 -202 }} | ||
[[POTE generator]]: ~ | [[POTE generator]]: ~147/128 = 238.599 | ||
{{Val list|legend=1| 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }} | {{Val list|legend=1| 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd }} | ||
| Line 67: | Line 67: | ||
{{Multival|legend=1| 52 3 82 -116 -16 182 }} | {{Multival|legend=1| 52 3 82 -116 -16 182 }} | ||
[[POTE generator]]: ~ | [[POTE generator]]: ~3375/3136 = 128.888 | ||
{{Val list|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }} | {{Val list|legend=1| 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc }} | ||
| Line 80: | Line 80: | ||
Mapping: [{{val| 1 48 5 76 107 }}, {{val| 0 -52 -3 -82 -116 }}] | Mapping: [{{val| 1 48 5 76 107 }}, {{val| 0 -52 -3 -82 -116 }}] | ||
POTE generator: ~245 | POTE generator: ~264/245 = 128.890 | ||
Optimal GPV sequence: {{Val list| 121, 149, 270, 2551bc, 2821bc, 3091bc, 3361bc, 3631bbcc, 3901bbccd, 4171bbccd, 4441bbccd, 4711bbccd }} | Optimal GPV sequence: {{Val list| 121, 149, 270, 2551bc, 2821bc, 3091bc, 3361bc, 3631bbcc, 3901bbccd, 4171bbccd, 4441bbccd, 4711bbccd }} | ||
| Line 93: | Line 93: | ||
Mapping: [{{val| 1 48 5 76 107 76 }}, {{val| 0 -52 -3 -82 -116 -81 }}] | Mapping: [{{val| 1 48 5 76 107 76 }}, {{val| 0 -52 -3 -82 -116 -81 }}] | ||
POTE generator: ~13 | POTE generator: ~14/13 = 128.890 | ||
Optimal GPV sequence: {{Val list| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }} | Optimal GPV sequence: {{Val list| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }} | ||
| Line 102: | Line 102: | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 676/675, 715/714, | Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224 | ||
Mapping: [{{val| 1 48 5 76 107 76 63 }}, {{val| 0 -52 -3 -82 -116 -81 -66 }}] | Mapping: [{{val| 1 48 5 76 107 76 63 }}, {{val| 0 -52 -3 -82 -116 -81 -66 }}] | ||
POTE generator: ~13 | POTE generator: ~14/13 = 128.891 | ||
Optimal GPV sequence: {{Val list| 121, 149, 270 }} | Optimal GPV sequence: {{Val list| 121, 149, 270 }} | ||
Badness: 0.019107 | Badness: 0.019107 | ||
=== 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: [{{val|1 48 5 76 107 76 63 -2}}, {{val|0 -52 -3 -82 -116 -81 -66 7}}] | |||
POTE generator: ~14/13 = 128.890 | |||
Optimal GPV sequence: {{Val list| 121, 149, 270 }} | |||
Badness: 0.015621 | |||
== Witch == | == Witch == | ||
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{{Multival|legend=1| 25 65 14 45 -48 -150 }} | {{Multival|legend=1| 25 65 14 45 -48 -150 }} | ||
[[POTE generator]]: ~63 | [[POTE generator]]: ~80/63 = 412.098 | ||
{{Val list|legend=1| 99, 265, 364, 463, 562, 1025 }} | {{Val list|legend=1| 99, 265, 364, 463, 562, 1025 }} | ||
| Line 140: | Line 153: | ||
Mapping: [{{val| 1 18 45 12 77 }}, {{val| 0 -25 -65 -14 -112 }}] | Mapping: [{{val| 1 18 45 12 77 }}, {{val| 0 -25 -65 -14 -112 }}] | ||
POTE generator: ~63 | POTE generator: ~80/63 = 412.079 | ||
Optimal GPV sequence: {{Val list| 99e, 166, 265 }} | Optimal GPV sequence: {{Val list| 99e, 166, 265 }} | ||
| Line 153: | Line 166: | ||
Mapping: [{{val| 1 18 45 12 77 93 }}, {{val| 0 -25 -65 -14 -112 -136 }}] | Mapping: [{{val| 1 18 45 12 77 93 }}, {{val| 0 -25 -65 -14 -112 -136 }}] | ||
POTE generator: ~ | POTE generator: ~33/26 = 412.070 | ||
Optimal GPV sequence: {{Val list| 99ef, 166, 265, 431c, 696cc }} | Optimal GPV sequence: {{Val list| 99ef, 166, 265, 431c, 696cc }} | ||
Revision as of 23:38, 5 March 2022
This is a collection of microtemperaments which tempers out the wizma, [-6 -8 2 5⟩ = 420175/419904.
Discussed elsewhere are:
- Superpyth, {64/63, 245/243} → Archytas clan
- Cloudtone, {81/80, 16875/16384} → Meantone family
- Oolong, {126/125, 117649/116640} → Starling temperaments
- Wizard, {225/224, 118098/117649} → Marvel temperaments
- Hemiseven, {1029/1024, 19683/19600} → Gamelismic clan
- Ennealimmal, {2401/2400, 4375/4374} → Ragismic microtemperaments
- Sengagen, {3136/3125, 420175/419904} → Hemimean clan
- Septiquarter, {5120/5103, 420175/419904} → Hemifamity temperaments
- Hendecatonic, {6144/6125, 10976/10935} → Porwell temperaments
- Sqrtphi, {15625/15552, 16875/16807} → Kleismic family
- Tsaharuk, {32805/32768, 420175/419904} → Schismatic family
- Fifthplus, {65625/65536, 420175/419904} → Horwell temperaments
- Octowerck, {321489/320000, 420175/419904} → Varunismic temperaments
- Quinwell, {420175/419904, 2109375/2097152} → Semicomma family
- Decavish, {420175/419904, 2202927104/2197265625} → Vishnuzmic family
Considered below are qak, tokko, tertiathirds, and witch.
Qak
Subgroup: 2.3.5.7
Comma list: 420175/419904, 703125/702464
Mapping: [⟨1 27 11 40], ⟨0 -41 -14 -60]]
Mapping generators: ~2, ~192/125
Wedgie: ⟨⟨ 41 14 60 -73 -20 100 ]]
POTE generator: ~125/96 = 456.144
Badness: 0.029267
Tokko
Subgroup: 2.3.5.7
Comma list: 420175/419904, 5250987/5242880
Mapping: [⟨1 12 56 -2], ⟨0 -13 -67 6]]
Mapping generators: ~2, ~256/147
Wedgie: ⟨⟨ 13 67 -6 76 -46 -202 ]]
POTE generator: ~147/128 = 238.599
Badness: 0.044417
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.
Subgroup: 2.3.5.7
Comma list: 420175/419904, 29360128/29296875
Mapping: [⟨1 48 5 76], ⟨0 -52 -3 -82]]
Mapping generators: ~2, ~6272/3375
Wedgie: ⟨⟨ 52 3 82 -116 -16 182 ]]
POTE generator: ~3375/3136 = 128.888
Badness: 0.093005
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 5632/5625, 117649/117612
Mapping: [⟨1 48 5 76 107], ⟨0 -52 -3 -82 -116]]
POTE generator: ~264/245 = 128.890
Optimal GPV sequence: Template:Val list
Badness: 0.033322
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224
Mapping: [⟨1 48 5 76 107 76], ⟨0 -52 -3 -82 -116 -81]]
POTE generator: ~14/13 = 128.890
Optimal GPV sequence: Template:Val list
Badness: 0.019494
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224
Mapping: [⟨1 48 5 76 107 76 63], ⟨0 -52 -3 -82 -116 -81 -66]]
POTE generator: ~14/13 = 128.891
Optimal GPV sequence: Template:Val list
Badness: 0.019107
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 48 5 76 107 76 63 -2], ⟨0 -52 -3 -82 -116 -81 -66 7]]
POTE generator: ~14/13 = 128.890
Optimal GPV sequence: Template:Val list
Badness: 0.015621
Witch
The witch temperament (99&166) tempers out wizma (420175/419904) and the amity comma (1600000/1594323) as well as the hemfiness comma (4096000/4084101, saquinru-atriyo).
Subgroup: 2.3.5.7
Comma list: 420175/419904, 1600000/1594323
Mapping: [⟨1 18 45 12], ⟨0 -25 -65 -14]]
Mapping generators: ~2, ~63/40
Wedgie: ⟨⟨ 25 65 14 45 -48 -150 ]]
POTE generator: ~80/63 = 412.098
Badness: 0.080608
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4356, 1240029/1239040
Mapping: [⟨1 18 45 12 77], ⟨0 -25 -65 -14 -112]]
POTE generator: ~80/63 = 412.079
Optimal GPV sequence: Template:Val list
Badness: 0.082607
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4375/4356, 10648/10647
Mapping: [⟨1 18 45 12 77 93], ⟨0 -25 -65 -14 -112 -136]]
POTE generator: ~33/26 = 412.070
Optimal GPV sequence: Template:Val list
Badness: 0.057980
Semiwitch
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 41503/41472, 585640/583443
Mapping: [⟨2 11 25 10 16], ⟨0 -25 -65 -14 -29]]
Mapping generators: ~99/70, ~49/44
POTE generator: ~49/44 = 187.904
Optimal GPV sequence: Template:Val list
Badness: 0.088189
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]]
POTE generator: ~39/35 = 187.904
Optimal GPV sequence: Template:Val list
Badness: 0.038009
Hitch
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 117649/117612, 1600000/1594323
Mapping: [⟨1 18 45 12 -10], ⟨0 -50 -130 -28 41]]
Mapping generators: ~2, ~1120/891
POTE generator: ~1120/891 = 393.950
Optimal GPV sequence: Template:Val list
Badness: 0.093482
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1716/1715, 3025/3024, 1600000/1594323
Mapping: [⟨1 18 45 12 -10 71], ⟨0 -50 -130 -28 41 -205]]
POTE generator: ~1120/891 = 393.948
Optimal GPV sequence: Template:Val list
Badness: 0.060716
Hitcher
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 3025/3024, 13720/13689, 28672/28561
Mapping: [⟨1 18 45 12 -10 6], ⟨0 -50 -130 -28 41 -7]]
POTE generator: ~49/39 = 393.949
Optimal GPV sequence: Template:Val list
Badness: 0.059591