Metric microtemperaments: Difference between revisions
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These are rank-2 temperaments that temper out the [[meter comma]], 703125/702464 = {{monzo| -11 2 7 -3 }}. | These are rank-2 temperaments that temper out the [[meter comma]], 703125/702464 = {{monzo| -11 2 7 -3 }}. | ||
Temperaments | Temperaments discussed elsewhere include: | ||
* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]] | * [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]] | ||
* ''[[Triwell]]'', {1029/1024, 235298/234375} → [[Semicomma family #Triwell|Semicomma family]] | * ''[[Triwell]]'', {1029/1024, 235298/234375} → [[Semicomma family #Triwell|Semicomma family]] | ||
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POTE generator: ~80/77 = 65.5881 | POTE generator: ~80/77 = 65.5881 | ||
Optimal GPV sequence: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }} | |||
Badness: 0.032816 | Badness: 0.032816 | ||
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POTE generator: ~27/26 = 65.5883 | POTE generator: ~27/26 = 65.5883 | ||
Optimal GPV sequence: {{val list| 128, 183, 311, 494, 1299, 1793 }} | |||
Badness: 0.016849 | Badness: 0.016849 | ||
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POTE generator: ~27/26 = 65.5878 | POTE generator: ~27/26 = 65.5878 | ||
Optimal GPV sequence: {{val list| 128, 183, 311, 494 }} | |||
Badness: 0.014674 | Badness: 0.014674 | ||
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POTE generator: ~5/4 = 385.396 | POTE generator: ~5/4 = 385.396 | ||
Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e, 520de }} | |||
Badness: 0.056915 | Badness: 0.056915 | ||
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POTE generator: ~5/4 = 385.419 | POTE generator: ~5/4 = 385.419 | ||
Optimal GPV sequence: {{Val list| 50, 90, 140, 190, 330e }} | |||
Badness: 0.029297 | Badness: 0.029297 | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 385/384, 561/560, 595/594, 625/624, 847/845 | |||
Mapping: [{{val| 10 3 20 12 41 37 28 }}, {{val| 0 4 1 5 -2 0 4 }}] | |||
POTE generator: ~5/4 = 385.435 | |||
Optimal GPV sequence: {{Val list| 50, 90, 140, 190g, 330eg }} | |||
Badness: 0.021949 | |||
== Stockhausenic == | == Stockhausenic == | ||
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POTE generator: ~16/15 = 111.460 | POTE generator: ~16/15 = 111.460 | ||
Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }} | |||
Badness: 0.056043 | Badness: 0.056043 | ||
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POTE generator: ~16/15 = 111.458 | POTE generator: ~16/15 = 111.458 | ||
Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }} | |||
Badness: 0.029947 | Badness: 0.029947 | ||
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POTE generator: ~16/15 = 111.459 | POTE generator: ~16/15 = 111.459 | ||
Optimal GPV sequence: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }} | |||
Badness: 0.019205 | Badness: 0.019205 | ||
== Geb == | == Geb == | ||
The ''geb'' | The ''geb'' temperament (171&506, named after the earth god of ancient Egyptian religion) tempers out {{monzo|-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. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&670, named after the eldest son of Geb) tempers out the same 5-limit comma as the geb, but with the [[2401/2400|breedsma]] rather than the meter tempered out. | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
Revision as of 21:58, 29 March 2022
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 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
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
Optimal GPV sequence: Template:Val list
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
Optimal GPV sequence: Template:Val list
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
Optimal GPV sequence: Template:Val list
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
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
Badness: 0.029297
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 385/384, 561/560, 595/594, 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 GPV sequence: Template:Val list
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
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
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 GPV sequence: Template:Val list
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
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
Badness: 0.048770