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 page collects rank-2 wizmic microtemperaments, which temper out the wizma (monzo: [-6 -8 2 5⟩, ratio: 420175/419904). For the rank-3 temperament, see Miscellaneous 7-limit temperaments #Wizmic.

Temperaments discussed elsewhere are:

• Superpyth (+64/63 or 245/243) → Archytas clan
• Gariwizmic (+33554432/33480783) → Septischismic clan
• Cloudtone (+81/80) → Meantone family
• Tsaharuk (+32805/32768) → Schismatic family
• Hemiseven (+1029/1024) → Gamelismic clan
• Septiquarter (+5120/5103) → Aberschismic temperaments
• Hendecatonic (+6144/6125 or 10976/10935) → Porwell temperaments
• Wizard (+225/224) → Stearnsmic clan
• Witcher (+1600000/1594323) → Amity family
• Oolong (+126/125) → Starling temperaments
• Ennealimmal (+2401/2400 or 4375/4374) → Septiennealimmal clan
• Fifthplus (+65625/65536) → Horwell temperaments
• Octowerck (+321489/320000) → Varunismic temperaments
• Sengagen (+3136/3125) → Hemimean clan
• Sqrtphi (+15625/15552 or 16875/16807) → Kleismic family
• Quinwell (+2109375/2097152) → Semicomma family
• Decavish (+2202927104/2197265625) → Vishnu family

Considered below are qak, tokko, and tertiathirds, in the order of increasing badness.

Qak

Qak tempers out the meter in addition to the wizma, and may be described as the 50 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 703125/702464

Mapping: [⟨1 -14 -3 -20], ⟨0 41 14 60]]

mapping generators: ~2, ~125/96

Optimal tunings:

• WE: ~2 = 1200.0566 ¢, ~125/96 = 456.1655 ¢

error map: ⟨+0.057 +0.039 -0.166 -0.027]

• CWE: ~2 = 1200.0000 ¢, ~125/96 = 456.1450 ¢

error map: ⟨0.000 -0.009 -0.283 -0.125]

Optimal ET sequence: 50, 121, 171, 976, 1147, 1318, 1489, 1660, 1831, 2002c, 3833cd

Badness (Sintel): 0.741

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 5 & 171 temperament.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 5250987/5242880

Mapping: [⟨1 -1 -11 4], ⟨0 13 67 -6]]

mapping generators: ~2, ~147/128

Optimal tunings:

• WE: ~2 = 1200.0850 ¢, ~147/128 = 238.6157 ¢

error map: ⟨+0.085 -0.036 +0.004 -0.180]

• CWE: ~2 = 1200.0000 ¢, ~147/128 = 238.5997 ¢

error map: ⟨0.000 -0.159 -0.133 -0.424]

Optimal ET sequence: 5, 161c, 166, 171, 1544d, 1715d, …, 2912dd, 3083cdd

Badness (Sintel): 1.12

Tertiathirds

Tertiathirds tempers out the quasiorwellisma, 29360128/29296875 in the 7-limit, and may be described as the 121 & 149 temperament. It splits the interval of 5/4 into three 14/13 generators.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 29360128/29296875

Mapping: [⟨1 -4 2 -6], ⟨0 52 3 82]]

mapping generators: ~2, ~3375/3136

Optimal tunings:

• WE: ~2 = 1199.9305 ¢, ~3375/3136 = 128.8804 ¢

error map: ⟨-0.070 +0.104 +0.188 -0.216]

• CWE: ~2 = 1200.0000 ¢, ~3375/3136 = 128.8871 ¢

error map: ⟨0.000 +0.173 +0.348 -0.085]

Optimal ET sequence: 121, 149, 270, 2309c, 2579c

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 117649/117612

Mapping: [⟨1 -4 2 -6 -9], ⟨0 52 3 82 116]]

Optimal tunings:

• WE: ~2 = 1199.9251 ¢, ~264/245 = 128.8818 ¢
• CWE: ~2 = 1200.0000 ¢, ~264/245 = 128.8890 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 1.10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5], ⟨0 52 3 82 116 81]]

Optimal tunings:

• WE: ~2 = 1199.9510 ¢, ~14/13 = 128.8849 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.806

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1225/1224, 4225/4224

Mapping: [⟨1 -4 2 -6 -9 -5 -3], ⟨0 52 3 82 116 81 66]]

Optimal tunings:

• WE: ~2 = 1199.8754 ¢, ~14/13 = 128.8778 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8898 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.973

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 -4 2 -6 -9 -5 -3 5], ⟨0 52 3 82 116 81 66 -7]]

Optimal tunings:

• WE: ~2 = 1199.8964 ¢, ~14/13 = 128.8793 ¢
• CWE: ~2 = 1200.0000 ¢, ~14/13 = 128.8896 ¢

Optimal ET sequence: 121, 149, 270

Badness (Sintel): 0.950