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 temperament|rank-2]] [[temperament]]s that [[Tempering out|temper out]] the [[meter comma]], 703125/702464 = {{monzo| -11 2 7 -3 }}.


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Meantone]]'', {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Meantone]]'' (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Triwell]]'', {1029/1024, 235298/234375} → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Tertiaseptal]]'', {2401/2400, 65625/65536} → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Tertiaseptal]]'' (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Quintupole]]'', {4000/3969, 458752/455625} → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Enneadecal]]'', {4375/4374, 703125/702464} → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Undim]]'', {5120/5103, 390625/388962} → [[Undim family #Septimal undim|Undim family]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* ''[[Grendel]]'', {6144/6125, 16875/16807} → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Grendel]]'' (+6144/6125) → [[Mirkwai clan #Grendel|Mirkwai clan]]
* ''[[Marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Term]]'', {32805/32768, 250047/250000} → [[Schismatic family #Term|Schismatic family]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Qak]]'', {420175/419904, 703125/702464} → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Trinity]]'', {703125/702464, 1600000/1594323} → [[Amity family #Trinity|Amity family]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]


== Luminal ==
== Luminal ==
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[[Comma list]]: 703125/702464, 26843545600/26795786661
[[Comma list]]: 703125/702464, 26843545600/26795786661


[[Mapping]]: [{{val| 1 0 5 8 }}, {{val| 0 29 -49 -95 }}]
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}


{{Multival|legend=1| 29 -49 -95 -145 -232 -82 }}
{{Multival|legend=1| 29 -49 -95 -145 -232 -82 }}
Line 36: Line 36:
Comma list: 3025/3024, 131072/130977, 703125/702464
Comma list: 3025/3024, 131072/130977, 703125/702464


Mapping: [{{val| 1 0 5 8 1 }}, {{val| 0 29 -49 -95 45 }}]
Mapping: {{mapping| 1 0 5 8 1 | 0 29 -49 -95 45 }}


Optimal tuning (POTE): ~2 = 1\1, ~80/77 = 65.5881
Optimal tuning (POTE): ~2 = 1\1, ~80/77 = 65.5881
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Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213
Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213


Mapping: [{{val| 1 0 5 8 1 -1 }}, {{val| 0 29 -49 -95 45 86 }}]
Mapping: {{mapping| 1 0 5 8 1 -1 | 0 29 -49 -95 45 86 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 65.5883
Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 65.5883
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Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095
Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095


Mapping: [{{val| 1 0 5 8 1 -1 6 }}, {{val| 0 29 -49 -95 45 86 -35 }}]
Mapping: {{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 tuning (POTE): ~2 = 1\1, ~27/26 = 65.5878
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== Decimetra ==
== Decimetra ==
The ''decimetra'' temperament (50&140) tempers out the [[Procyon clan|Procyon comma]], 823543/820125 = {{monzo|0 -8 -3 7}} as well as [[15/14ths equal temperament|linus comma]].
The ''decimetra'' temperament (50 & 140) tempers out the [[procyon comma]], 823543/820125 = {{monzo| 0 -8 -3 7 }} as well as [[linus comma]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 703125/702464, 823543/820125
[[Comma list]]: 703125/702464, 823543/820125


[[Mapping]]: [{{val| 10 3 20 12 }}, {{val| 0 4 1 5 }}]
{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}


{{Multival|legend=1| 40 10 50 -77 -33 88 }}
{{Multival|legend=1| 40 10 50 -77 -33 88 }}


[[POTE generator]]: ~5/4 = 385.495
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~5/4 = 385.495


{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}
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Comma list: 385/384, 9801/9800, 391314/390625
Comma list: 385/384, 9801/9800, 391314/390625


Mapping: [{{val| 10 3 20 12 41 }}, {{val| 0 4 1 5 -2 }}]
Mapping: {{mapping| 10 3 20 12 41 | 0 4 1 5 -2 }}


POTE generator: ~5/4 = 385.396
Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.396


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}
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Comma list: 385/384, 625/624, 847/845, 4459/4455
Comma list: 385/384, 625/624, 847/845, 4459/4455


Mapping: [{{val| 10 3 20 12 41 37 }}, {{val| 0 4 1 5 -2 0 }}]
Mapping: {{mapping| 10 3 20 12 41 37 | 0 4 1 5 -2 0 }}


POTE generator: ~5/4 = 385.419
Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.419


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e }}
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Comma list: 289/288, 385/384, 561/560, 625/624, 847/845
Comma list: 289/288, 385/384, 561/560, 625/624, 847/845


Mapping: [{{val| 10 3 20 12 41 37 28 }}, {{val| 0 4 1 5 -2 0 4 }}]
Mapping: {{mapping| 10 3 20 12 41 37 28 | 0 4 1 5 -2 0 4 }}


POTE generator: ~5/4 = 385.435
Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.435


{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}
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== Stockhausenic ==
== Stockhausenic ==
The ''stockhausenic'' temperament (140&183, named after a German composer [[Wikipedia:Karlheinz Stockhausen|Karlheinz Stockhausen]]) is related to [[25ed5|25ED5]] tuning, and tempers out the ''hemfiness'' comma (saquadru-atriyo), 4096000/4084101 = {{monzo|15 -5 3 -5}}.
The stockhausenic temperament (140 & 183, named after [[Wikipedia:Karlheinz Stockhausen|Karlheinz Stockhausen]]) is related to [[25ed5]] tuning, and tempers out the hemfiness comma (saquadru-atriyo), 4096000/4084101 = {{monzo| 15 -5 3 -5 }}.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 703125/702464, 4096000/4084101
[[Comma list]]: 703125/702464, 4096000/4084101


[[Mapping]]: [{{val| 1 4 0 -1 }}, {{val| 0 -26 25 41 }}]
{{Mapping|legend=1| 1 4 0 -1 | 0 -26 25 41 }}


{{Multival|legend=1|26 -25 -41 -100 -138 -25}}
{{Multival|legend=1| 26 -25 -41 -100 -138 -25 }}


[[POTE generator]]: ~16/15 = 111.453
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 111.453


{{Optimal ET sequence|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}
{{Optimal ET sequence|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}
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Comma list: 1375/1372, 5632/5625, 35937/35840
Comma list: 1375/1372, 5632/5625, 35937/35840


Mapping: [{{val| 1 4 0 -1 -1 }}, {{val| 0 -26 25 41 48 }}]
Mapping: {{mapping| 1 4 0 -1 -1 | 0 -26 25 41 48 }}


POTE generator: ~16/15 = 111.460
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.460


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
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Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095
Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095


Mapping: [{{val| 1 4 0 -1 -1 5 }}, {{val| 0 -26 25 41 48 -14 }}]
Mapping: {{mapping| 1 4 0 -1 -1 5 | 0 -26 25 41 48 -14 }}


POTE generator: ~16/15 = 111.458
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.458


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 829def, 1152deef }}
{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 829def, 1152deef }}
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Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095
Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095


Mapping: [{{val| 1 4 0 -1 -1 5 0 }}, {{val| 0 -26 25 41 48 -14 44 }}]
Mapping: {{mapping| 1 4 0 -1 -1 5 0 | 0 -26 25 41 48 -14 44 }}


POTE generator: ~16/15 = 111.459
Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.459


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829defg }}
{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 506, 829defg }}
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== Geb ==
== 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.
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


[[Comma list]]: {{monzo| -31 43 -16 }}
[[Comma list]]: {{monzo| -31 43 -16 }}


[[Mapping]]: [{{val| 1 -3 -10 }}, {{val| 0 16 43 }}]
{{Mapping|legend=1| 1 -3 -10 | 0 16 43 }}


[[POTE generator]]: ~8000/6561 = 343.868
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8000/6561 = 343.868


{{Optimal ET sequence|legend=1| 7, 143c, 150c, 157, 164, 171, 506, 677, 848 }}
{{Optimal ET sequence|legend=1| 7, 143c, 150c, 157, 164, 171, 506, 677, 848 }}
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 703125/702464, 14348907/14336000
[[Comma list]]: 703125/702464, 14348907/14336000


[[Mapping]]: [{{val| 1 -3 -10 -29 }}, {{val| 0 16 43 111 }}]
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}


{{Multival|legend=1| 16 43 111 31 131 137 }}
{{Multival|legend=1| 16 43 111 31 131 137 }}


[[POTE generator]]: ~8000/6561 = 343.863
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8000/6561 = 343.863


{{Optimal ET sequence|legend=1| 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
{{Optimal ET sequence|legend=1| 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Metric microtemperaments| ]] <!-- main article -->
[[Category:Metric| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Metric microtemperaments| ]] <!-- main article -->
[[Category:Microtemperaments]]
[[Category:Microtemperaments]]

Revision as of 14:19, 15 September 2023

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

Temperaments discussed elsewhere include:

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

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