Wizmic microtemperaments

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:

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) → Septiennealimmal clan
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, witcher, and gariwizmic.

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

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

For the 5-limit version, see Syntonic–diatonic equivalence continuum #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

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

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

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

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 transdiminished fourth (+45 fifths) minus a half pythagorean comma, or equivalently, at the transdiminished octave (+39 fifths) minus a semioctave.

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

Comma list: 420175/419904, 33554432/33480783

Mapping: [⟨2 2 -41 22], ⟨0 1 39 -14]]