Metric 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 metric microtemperaments, which temper out the meter (monzo: [-11 2 7 -3⟩, ratio: 703125/702464).

Temperaments discussed elsewhere include:

• Meantone (+81/80) → Meantone family
• Term (+32805/32768) → Schismatic family
• Undim (+5120/5103) → Undim family
• Quintupole (+4000/3969) → Quintaleap family
• Trinity (+1600000/1594323) → Amity family
• Marfifths (+10976/10935) → Kleismic family
• Enneadecal (+4375/4374) → Ragismic microtemperaments
• Triwell (+1029/1024) → Semicomma family
• Tertiaseptal (+2401/2400) → Breedsmic temperaments
• Grendel (+6144/6125) → Porwell temperaments
• Qak (+420175/419904) → Wizmic microtemperaments

Considered below are geb, stockhausenic, decimetra, and luminal, in the order of increasing badness.

Geb

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Geb (5-limit).

Geb tempers out the scheme comma as well as the euzenius comma, and may be described as 164d & 171 temperament. It was named by Xenllium in 2021 after the earth god of ancient Egyptian religion and the father of Osiris; the latter being the name of a temperament that tempers out the same 5-limit comma as the geb (→ Breedsmic temperaments #Osiris).

Subgroup: 2.3.5.7

Comma list: 703125/702464, 14348907/14336000

Mapping: [⟨1 -3 -10 -29], ⟨0 16 43 111]]

mapping generators: ~2, ~8000/6561

Optimal tunings:

• WE: ~2 = 1200.0532 ¢, ~8000/6561 = 343.8779 ¢

error map: ⟨+0.053 -0.067 -0.094 +0.084]

• CWE: ~2 = 1200.0000 ¢, ~8000/6561 = 343.8634 ¢

error map: ⟨0.000 -0.141 -0.189 +0.009]

Optimal ET sequence: 7d, …, 164d, 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc

Badness (Sintel): 1.23

Stockhausenic

Stockhausenic may be described as the 140 & 183 temperament, and has a generator chain of ~16/15 (or ~15/8, shown below), relating it to 25ed5. It was named by Xenllium in 2021 after Karlheinz Stockhausen for his experimental use of 25ed5.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 4096000/4084101

Mapping: [⟨1 -22 25 40], ⟨0 26 -25 -41]]

mapping generators: ~2, ~15/8

Optimal tunings:

• WE: ~2 = 1199.8908 ¢, ~15/8 = 1088.4483 ¢

error map: ⟨-0.109 +0.103 -0.250 +0.427]

• CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5477 ¢

error map: ⟨0.000 +0.284 -0.005 +0.720]

Optimal ET sequence: 43, 97, 140, 183, 323, 786, 1109d, 1432d

Badness (Sintel): 2.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 5632/5625, 35937/35840

Mapping: [⟨1 -22 25 40 47], ⟨0 26 -25 -41 -48]]

Optimal tunings:

• WE: ~2 = 1199.9691 ¢, ~15/8 = 1088.5116 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5398 ¢

Optimal ET sequence: 43, 97e, 140, 183, 323, 506

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095

Mapping: [⟨1 -22 25 40 47 -9], ⟨0 26 -25 -41 -48 14]]

Optimal tunings:

• WE: ~2 = 1200.0267 ¢, ~15/8 = 1088.5666 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5423 ¢

Optimal ET sequence: 43, 97e, 140, 183, 323

Badness (Sintel): 1.24

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095

Mapping: [⟨1 -22 25 40 47 -9 44], ⟨0 26 -25 -41 -48 14 -44]]

Optimal tunings:

• WE: ~2 = 1200.0454 ¢, ~15/8 = 1088.5821 ¢
• CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5407 ¢

Optimal ET sequence: 43, 97e, 140, 183, 323, 506

Badness (Sintel): 0.978

Decimetra

Named by Xenllium in 2021, decimetra tempers out the procyon comma as well as the linus comma, and may be described as 140 & 190.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 823543/820125

Mapping: [⟨10 3 20 12], ⟨0 4 1 5]]

mapping generators: ~15/14, ~5/4

Optimal tunings:

• WE: ~15/14 = 120.0204 ¢, ~5/4 = 385.5608 ¢

error map: {[val| +0.204 +0.349 -0.345 -0.777 }}

• CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.5321 ¢

error map: {[val| 0.000 +0.173 -0.782 -1.166 }}

Optimal ET sequence: 50, 90, 140, 330, 470, 800cd

Badness (Sintel): 3.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800, 391314/390625

Mapping: [⟨10 3 20 12 41], ⟨0 4 1 5 -2]]

Optimal tunings:

• WE: ~15/14 = 120.0379 ¢, ~5/4 = 385.5181 ¢
• CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4329 ¢

Optimal ET sequence: 50, 90, 140, 190, 330e, 520de

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 847/845, 4459/4455

Mapping: [⟨10 3 20 12 41 37], ⟨0 4 1 5 -2 0]]

Optimal tunings:

• WE: ~15/14 = 120.0324 ¢, ~5/4 = 385.5231 ¢
• CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4445 ¢

Optimal ET sequence: 50, 90, 140, 190, 330e, 520de

Badness (Sintel): 1.21

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 385/384, 561/560, 625/624, 847/845

Mapping: [⟨10 3 20 12 41 37 28], ⟨0 4 1 5 -2 0 4]]

Optimal tunings:

• WE: ~15/14 = 120.0380 ¢, ~5/4 = 385.5574 ¢
• CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4670 ¢

Optimal ET sequence: 50, 90, 140, 190g, 330eg

Badness (Sintel): 1.12

Luminal

Luminal may be described as the 128 & 183 temperament. In the 11-limit it extends via skadi, the metric extension tempering out 3025/3024, the lehmerisma, and joins with lux so that it also tempers out 131072/130977, the olympia. It was named by Flora Canou in 2021 following the naming of lux, whose name is the SI unit for illuminance, and lx⋅m² is lumen, the SI unit for luminous flux.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 26843545600/26795786661

Mapping: [⟨1 0 5 8], ⟨0 29 -49 -95]]

mapping generators: ~2, ~8505/8192

Optimal tunings:

• WE: ~2 = 1199.9728 ¢, ~8505/8192 = 65.5868 ¢

error map: ⟨-0.027 +0.062 -0.202 +0.212]

• CWE: ~2 = 1200.0000 ¢, ~8505/8192 = 65.5884 ¢

error map: ⟨0.000 +0.108 -0.145 +0.277]

Optimal ET sequence: 55d, 128, 183, 311, 494, 1299, 1793

Badness (Sintel): 3.41

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 131072/130977, 703125/702464

Mapping: [⟨1 0 5 8 1], ⟨0 29 -49 -95 45]]

Optimal tunings:

• WE: ~2 = 1199.9685 ¢, ~80/77 = 65.5864 ¢
• CWE: ~2 = 1200.0000 ¢, ~80/77 = 65.5882 ¢

Optimal ET sequence: 55d, 128, 183, 311, 494, 1299, 1793, 2287d

Badness (Sintel): 1.08

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213

Mapping: [⟨1 0 5 8 1 -1], ⟨0 29 -49 -95 45 86]]

Optimal tunings:

• WE: ~2 = 1199.9715 ¢, ~27/26 = 65.5867 ¢
• CWE: ~2 = 1200.0000 ¢, ~27/26 = 65.5883 ¢

Optimal ET sequence: 128, 183, 311, 494, 1299, 1793

Badness (Sintel): 0.696

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095

Mapping: [⟨1 0 5 8 1 -1 6], ⟨0 29 -49 -95 45 86 -35]]

Optimal tunings:

• WE: ~2 = 1199.9941 ¢, ~27/26 = 65.5875 ¢
• CWE: ~2 = 1200.0000 ¢, ~27/26 = 65.5878 ¢

Optimal ET sequence: 128, 183, 311, 494

Badness (Sintel): 0.748