Metric microtemperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
These are [[rank-2 temperament|rank-2]] [[temperament]]s that [[tempering out|temper out]] the [[meter]], 703125/702464 = {{monzo| -11 2 7 -3 }}.
This is a collection of [[rank-2 temperament|rank-2]] '''metric microtemperaments''', which [[tempering out|temper out]] the [[meter]] ({{monzo|legend=1| -11 2 7 -3 }}, [[ratio]]: 703125/702464).


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]
 
Considered below are luminal, decimetra, stockhausenic, and geb.


== Luminal ==
== Luminal ==
Luminal is the intersection of [[Meter|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.  
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
[[Subgroup]]: 2.3.5.7
Line 24: Line 26:
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8505/8192 = 65.5883
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8505/8192 = 65.5883{{c}}


{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}


[[Badness]]: 0.134822
[[Badness]] (Smith): 0.134822


=== 11-limit ===
=== 11-limit ===
Line 37: Line 39:
Mapping: {{mapping| 1 0 5 8 1 | 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 = 1200.0000{{c}}, ~80/77 = 65.5881{{c}}


{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}


Badness: 0.032816
Badness (Smith): 0.032816


=== 13-limit ===
=== 13-limit ===
Line 50: Line 52:
Mapping: {{mapping| 1 0 5 8 1 -1 | 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 = 1200.0000{{c}}, ~27/26 = 65.5883{{c}}


{{Optimal ET sequence|legend=1| 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=1| 128, 183, 311, 494, 1299, 1793 }}


Badness: 0.016849
Badness (Smith): 0.016849


=== 17-limit ===
=== 17-limit ===
Line 63: Line 65:
Mapping: {{mapping| 1 0 5 8 1 -1 6 | 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 = 1200.0000{{c}}, ~27/26 = 65.5878{{c}}


{{Optimal ET sequence|legend=1| 128, 183, 311, 494 }}
{{Optimal ET sequence|legend=1| 128, 183, 311, 494 }}


Badness: 0.014674
Badness (Smith): 0.014674


== Decimetra ==
== Decimetra ==
The ''decimetra'' temperament (50 & 140) tempers out the [[procyon comma]], 823543/820125 = {{monzo| 0 -8 -3 7 }} as well as [[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
Line 78: Line 80:
{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}
{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~5/4 = 385.495
[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~5/4 = 385.495{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}


[[Badness]]: 0.121818
[[Badness]] (Smith): 0.121818


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 10 3 20 12 41 | 0 4 1 5 -2 }}
Mapping: {{mapping| 10 3 20 12 41 | 0 4 1 5 -2 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~5/4 = 385.396
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~5/4 = 385.396{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e, 520de }}


Badness: 0.056915
Badness (Smith): 0.056915


=== 13-limit ===
=== 13-limit ===
Line 104: Line 106:
Mapping: {{mapping| 10 3 20 12 41 37 | 0 4 1 5 -2 0 }}
Mapping: {{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 tuning (POTE): ~15/14 = 120.000{{c}}, ~5/4 = 385.419{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190, 330e }}


Badness: 0.029297
Badness (Smith): 0.029297


=== 17-limit ===
=== 17-limit ===
Line 117: Line 119:
Mapping: {{mapping| 10 3 20 12 41 37 28 | 0 4 1 5 -2 0 4 }}
Mapping: {{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 tuning (POTE): ~15/14 = 120.000{{c}}, ~5/4 = 385.435{{c}}


{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}


Badness: 0.021949
Badness (Smith): 0.021949


== Stockhausenic ==
== Stockhausenic ==
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 }}.
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
Line 132: Line 134:
{{Mapping|legend=1| 1 4 0 -1 | 0 -26 25 41 }}
{{Mapping|legend=1| 1 4 0 -1 | 0 -26 25 41 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 111.453
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/15 = 111.453{{c}}


{{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 }}


[[Badness]]: 0.088753
[[Badness]] (Smith): 0.088753


=== 11-limit ===
=== 11-limit ===
Line 145: Line 147:
Mapping: {{mapping| 1 4 0 -1 -1 | 0 -26 25 41 48 }}
Mapping: {{mapping| 1 4 0 -1 -1 | 0 -26 25 41 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.460
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 111.460{{c}}


{{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 }}


Badness: 0.056043
Badness (Smith): 0.056043


=== 13-limit ===
=== 13-limit ===
Line 158: Line 160:
Mapping: {{mapping| 1 4 0 -1 -1 5 | 0 -26 25 41 48 -14 }}
Mapping: {{mapping| 1 4 0 -1 -1 5 | 0 -26 25 41 48 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 111.458
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 111.458{{c}}


{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 829def, 1152deef }}
{{Optimal ET sequence|legend=1| 43, 97e, 140, 183, 323, 829def, 1152deef }}


Badness: 0.029947
Badness (Smith): 0.029947


=== 17-limit ===
=== 17-limit ===
Line 171: Line 173:
Mapping: {{mapping| 1 4 0 -1 -1 5 0 | 0 -26 25 41 48 -14 44 }}
Mapping: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~16/15 = 111.459{{c}}


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


Badness: 0.019205
Badness (Smith): 0.019205


== Geb ==
== Geb ==
Line 188: Line 190:
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8000/6561 = 343.863
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8000/6561 = 343.863{{c}}


{{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 }}


[[Badness]]: 0.048770
[[Badness]] (Smith): 0.048770


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Metric microtemperaments| ]] <!-- main article -->
[[Category:Metric microtemperaments| ]] <!-- main article -->
[[Category:Metric| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]

Latest revision as of 15:34, 3 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 metric microtemperaments, which temper out the meter (monzo: [-11 2 7 -3⟩, ratio: 703125/702464).

Temperaments discussed elsewhere include:

Considered below are luminal, decimetra, stockhausenic, and geb.

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]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~8505/8192 = 65.5883 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~80/77 = 65.5881 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~27/26 = 65.5883 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~27/26 = 65.5878 ¢

Optimal ET sequence: 128, 183, 311, 494

Badness (Smith): 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]]

Optimal tuning (POTE): ~15/14 = 120.000 ¢, ~5/4 = 385.495 ¢

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

Badness (Smith): 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 = 120.000 ¢, ~5/4 = 385.396 ¢

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

Badness (Smith): 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 = 120.000 ¢, ~5/4 = 385.419 ¢

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

Badness (Smith): 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 = 120.000 ¢, ~5/4 = 385.435 ¢

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

Badness (Smith): 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]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/15 = 111.453 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~16/15 = 111.460 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~16/15 = 111.458 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~16/15 = 111.459 ¢

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

Badness (Smith): 0.019205

Geb

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Geb (5-limit).

The geb temperament (171 & 506, named after the earth god of ancient Egyptian religion) tempers out the meter and the euzenius comma (78125000/78121827) as well as scheme comma (14348907/14336000) in the 7-limit. Osiris (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: 2.3.5.7

Comma list: 703125/702464, 14348907/14336000

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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8000/6561 = 343.863 ¢

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

Badness (Smith): 0.048770

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