Wizmic microtemperaments
This is a collection of rank-2 microtemperaments which tempers out the wizma, [-6 -8 2 5⟩ = 420175/419904.
Discussed elsewhere are:
- Superpyth (+64/63 or 245/243) → Archytas clan
- Cloudtone (+81/80) → Meantone family
- Oolong (+126/125) → Starling temperaments
- Wizard (+225/224) → Marvel temperaments
- Hemiseven (+1029/1024) → Gamelismic clan
- Ennealimmal (+2401/2400 or 4375/4374) → Ragismic microtemperaments
- Sengagen (+3136/3125) → Hemimean clan
- Septiquarter (+5120/5103) → Hemifamity temperaments
- Hendecatonic (+6144/6125 or 10976/10935) → Porwell temperaments
- Sqrtphi (+15625/15552 or 16875/16807) → Kleismic family
- Tsaharuk (+32805/32768) → Schismatic family
- Fifthplus (+5625/65536) → Horwell temperaments
- Octowerck (+321489/320000) → Varunismic temperaments
- Quinwell (+2109375/2097152) → Semicomma family
- Decavish (+2202927104/2197265625) → Vishnuzmic family
Considered below are qak, tokko, tertiathirds, and witcher.
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]]
Optimal tuning (POTE): ~2 = 1\1, ~125/96 = 456.144
Optimal ET sequence: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd
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]]
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 238.599
Optimal ET sequence: 5, 161c, 166, 171, 1544d, 1715d, 1886d, 2057d, 2228d, 2399d, 2570d, 2741d, 2912dd, 3083cdd
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]]
Optimal tuning (POTE): ~2 = 1\1, ~3375/3136 = 128.888
Optimal ET sequence: 121, 149, 270, 2309c, 2579c, 2849c, 3119bc, 3389bc, 3659bcc, 3929bcc, 4199bcc
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]]
Optimal tuning (POTE): ~2 = 1\1, ~264/245 = 128.890
Optimal ET sequence: 121, 149, 270, 2551bc, 2821bc, 3091bc, 3361bc, 3631bbcc, 3901bbccd, 4171bbccd, 4441bbccd, 4711bbccd
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]]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
Optimal ET sequence: 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff
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]]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.891
Optimal ET sequence: 121, 149, 270
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]]
Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 128.890
Optimal ET sequence: 121, 149, 270
Badness: 0.015621
Witcher
- 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).
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]]
Optimal tuning (POTE): ~2 = 1\1, ~80/63 = 412.098
Optimal ET sequence: 99, 265, 364, 463, 562, 1025
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]]
Optimal tuning (POTE): ~2 = 1\1, ~80/63 = 412.079
Optimal ET sequence: 99e, 166, 265
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]]
Optimal tuning (POTE): ~2 = 1\1, ~33/26 = 412.070
Optimal ET sequence: 99ef, 166, 265, 431c, 696cc
Badness: 0.057980
Semiwitcher
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
Optimal tuning (POTE): ~99/70 = 1\2, ~49/44 = 187.904
Optimal ET sequence: 166, 198, 364, 562, 926, 1488bd, 2414bdd
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]]
Optimal tuning (POTE): ~99/70 = 1\2, ~39/35 = 187.904
Optimal ET sequence: 166, 198, 364, 562, 926
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
Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.950
Optimal ET sequence: 198, 463, 661, 1124bde
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]]
Optimal tuning (POTE): ~2 = 1\1, ~1120/891 = 393.948
Optimal ET sequence: 198, 463, 661
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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/39 = 393.949
Optimal ET sequence: 198, 463f, 661ff
Badness: 0.059591