Wizmic microtemperaments

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

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 ]]

POTE generator: ~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 ]]

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~14/13 = 128.890

Optimal ET sequence: 121, 149, 270

Badness: 0.015621

Witch

See also: Amity family

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

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]]

POTE generator: ~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]]

POTE generator: ~33/26 = 412.070

Optimal ET sequence: 99ef, 166, 265, 431c, 696cc

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 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]]

POTE generator: ~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

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~49/39 = 393.949

Optimal ET sequence: 198, 463f, 661ff

Badness: 0.059591