Metric microtemperaments

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

Temperaments not dicussed here include:

Luminal

Luminal tempers out the meter and the lux in the 7-limit, 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]]

POTE generator: ~8505/8192 = 65.5883

Vals55d, 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]]

POTE generator: ~80/77 = 65.5881

Vals: 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]]

POTE generator: ~27/26 = 65.5883

Vals: 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]]

POTE generator: ~27/26 = 65.5878

Vals: 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 15 23 27], 0 4 1 5]]

Wedgie⟨⟨40 10 50 -77 -33 88]]

POTE generator: ~5/4 = 385.495

Vals50, 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 15 23 27 35], 0 4 1 5 -2]]

POTE generator: ~5/4 = 385.396

Vals: 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 15 23 27 35 37], 0 4 1 5 -2 0]]

POTE generator: ~5/4 = 385.419

Vals: 50, 90, 140, 190, 330e

Badness: 0.029297

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

Vals43, 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

Vals: 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

Vals: 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

Vals: 43, 97e, 140, 183, 323, 506, 829defg

Badness: 0.019205

Geb

The geb temperaent (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: [-31 43 -16

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

POTE generator: ~8000/6561 = 343.868

Vals7, 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

Vals171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc

Badness: 0.048770