Metric microtemperaments

These are rank-2 temperaments that temper out the meter comma, 703125/702464 = [-11 2 7 -3⟩.

Temperaments discussed elsewhere include:

• Meantone, {81/80, 126/125} → Meantone family
• Triwell, {1029/1024, 235298/234375} → Semicomma family
• Tertiaseptal, {2401/2400, 65625/65536} → Breedsmic temperaments
• Quintupole, {4000/3969, 458752/455625} → Quintaleap family
• Enneadecal, {4375/4374, 703125/702464} → Ragismic microtemperaments
• Undim, {5120/5103, 390625/388962} → Undim family
• Grendel, {6144/6125, 16875/16807} → Mirkwai clan
• Marfifths, {10976/10935, 15625/15552} → Kleismic family
• Term, {32805/32768, 250047/250000} → Schismatic family
• Qak, {420175/419904, 703125/702464} → Wizmic microtemperaments
• Trinity, {703125/702464, 1600000/1594323} → Amity family

Luminal

Luminal is the intersection of metric and lux, tempering out 3025/3024 in the 11-limit, 2080/2079, 4096/4095 in the 13-limit, and 1156/1155 and 2601/2600 in the 17-limit.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 26843545600/26795786661

Mapping: [⟨1 0 5 8], ⟨0 29 -49 -95]]

Wedgie: ⟨⟨ 29 -49 -95 -145 -232 -82 ]]

Optimal tuning (POTE): ~2 = 1\1, ~8505/8192 = 65.5883

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

Badness: 0.134822

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 tuning (POTE): ~2 = 1\1, ~80/77 = 65.5881

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

Badness: 0.032816

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 tuning (POTE): ~2 = 1\1, ~27/26 = 65.5883

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

Badness: 0.016849

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 tuning (POTE): ~2 = 1\1, ~27/26 = 65.5878

Optimal ET sequence: 128, 183, 311, 494

Badness: 0.014674

Decimetra

The decimetra temperament (50&140) tempers out the Procyon comma, 823543/820125 = [0 -8 -3 7⟩ as well as linus comma.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 823543/820125

Mapping: [⟨10 3 20 12], ⟨0 4 1 5]]

Wedgie: ⟨⟨ 40 10 50 -77 -33 88 ]]

POTE generator: ~5/4 = 385.495

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

Badness: 0.121818

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]]

POTE generator: ~5/4 = 385.396

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

Badness: 0.056915

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]]

POTE generator: ~5/4 = 385.419

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

Badness: 0.029297

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]]

POTE generator: ~5/4 = 385.435

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

Badness: 0.021949

Stockhausenic

The stockhausenic temperament (140&183, named after a German composer Karlheinz Stockhausen) is related to 25ED5 tuning, and tempers out the hemfiness comma (saquadru-atriyo), 4096000/4084101 = [15 -5 3 -5⟩.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 4096000/4084101

Mapping: [⟨1 4 0 -1], ⟨0 -26 25 41]]

Wedgie: ⟨⟨ 26 -25 -41 -100 -138 -25 ]]

POTE generator: ~16/15 = 111.453

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

Badness: 0.088753

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 4 0 -1 -1], ⟨0 -26 25 41 48]]

POTE generator: ~16/15 = 111.460

Optimal ET sequence: 43, 97e, 140, 183, 323, 506, 829de, 1335ddee

Badness: 0.056043

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 0 -1 -1 5], ⟨0 -26 25 41 48 -14]]

POTE generator: ~16/15 = 111.458

Optimal ET sequence: 43, 97e, 140, 183, 323, 829def, 1152deef

Badness: 0.029947

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 4 0 -1 -1 5 0], ⟨0 -26 25 41 48 -14 44]]

POTE generator: ~16/15 = 111.459

Optimal ET sequence: 43, 97e, 140, 183, 323, 506, 829defg

Badness: 0.019205

Geb

The geb temperament (171&506, named after the earth god of ancient Egyptian religion) tempers out [-31 43 -16⟩ (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as skeetsma (14348907/14336000, lalarutrigu) in the 7-limit. Osiris temperament (171&670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the breedsma rather than the meter tempered out.

Subgroup: 2.3.5

Comma list: [-31 43 -16⟩

Mapping: [⟨1 -3 -10], ⟨0 16 43]]

POTE generator: ~8000/6561 = 343.868

Optimal ET sequence: 7, 143c, 150c, 157, 164, 171, 506, 677, 848

Badness: 0.118577

7-limit

Subgroup: 2.3.5.7

Comma list: 703125/702464, 14348907/14336000

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

Wedgie: ⟨⟨ 16 43 111 31 131 137 ]]

POTE generator: ~8000/6561 = 343.863

Optimal ET sequence: 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc

Badness: 0.048770