Metric microtemperaments
These are rank-2 temperaments that temper out the meter, 703125/702464 = [-11 2 7 -3⟩.
Temperaments discussed elsewhere include:
- Meantone (+81/80 or 126/125) → Meantone family
- Triwell (+1029/1024) → Semicomma family
- Tertiaseptal (+2401/2400) → Breedsmic temperaments
- Quintupole (+4000/3969) → Quintaleap family
- Enneadecal (+4375/4374) → Ragismic microtemperaments
- Undim (+5120/5103) → Undim family
- Grendel (+6144/6125) → Mirkwai clan
- Marfifths (+10976/10935) → Kleismic family
- Term (+32805/32768) → Schismatic family
- Qak (+420175/419904) → Wizmic microtemperaments
- Trinity (+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]]
Optimal tuning (POTE): ~15/14 = 1\10, ~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]]
Optimal tuning (POTE): ~15/14 = 1\10, ~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]]
Optimal tuning (POTE): ~15/14 = 1\10, ~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]]
Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.435
Optimal ET sequence: 50, 90, 140, 190g, 330eg
Badness: 0.021949
Stockhausenic
The stockhausenic temperament (140 & 183, named after 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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~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]]
Optimal tuning (POTE): ~2 = 1\1, ~8000/6561 = 343.863
Optimal ET sequence: 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc
Badness: 0.048770