Metric microtemperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,025 edits
Undo revision 70659 and revise information
Tag: Undo
Sort by badness
 
(29 intermediate revisions by 8 users not shown)
Line 1: Line 1:
These are rank-two temperaments that temper out the [[meter comma]], 703125/702464 = {{monzo|-11 2 7 -3}}.
{{Technical data page}}
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 not dicussed here include:  
Temperaments discussed elsewhere include:  
* [[Meantone family #Septimal meantone|Septimal meantone]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Semicomma family #Triwell|Triwell]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* [[Breedsmic temperaments #Tertiaseptal|Tertiaseptal]]
* ''[[Undim]]'' (+5120/5103) → [[Undim family #Septimal undim|Undim family]]
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]]
* ''[[Quintupole]]'' (+4000/3969) → [[Quintaleap family #Quintupole|Quintaleap family]]
* [[Hemifamity temperaments #Undim|Undim]]
* ''[[Trinity]]'' (+1600000/1594323) → [[Amity family #Trinity|Amity family]]
* [[Mirkwai clan #Grendel|Grendel]]
* ''[[Marfifths]]'' (+10976/10935) → [[Kleismic family #Marfifths|Kleismic family]]
* [[Kleismic family #Marfifths|Marfifths]]
* ''[[Enneadecal]]'' (+4375/4374) → [[Ragismic microtemperaments #Enneadecal|Ragismic microtemperaments]]
* [[Schismatic family #Term|Term]]
* ''[[Triwell]]'' (+1029/1024) → [[Semicomma family #Triwell|Semicomma family]]
* [[Wizmic microtemperaments #Qak|Qak]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Grendel]]'' (+6144/6125) → [[Porwell temperaments #Grendel|Porwell temperaments]]
* ''[[Qak]]'' (+420175/419904) → [[Wizmic microtemperaments #Qak|Wizmic microtemperaments]]


= Luminal =
Considered below are geb, stockhausenic, decimetra, and luminal, in the order of increasing [[badness]].  
Luminal tempers out the [[meter comma|meter]] and the [[lux comma|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
== Geb ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Geb (5-limit)]].''


[[Comma list]]: 703125/702464, 26843545600/26795786661
Geb tempers out the [[scheme comma]] as well as the [[euzenius comma]], and may be described as {{nowrap| 164d & 171 }} temperament. It was named by [[Xenllium]] in 2021 after the earth god of ancient Egyptian religion and the father of Osiris; the latter being the name of a temperament that tempers out the same 5-limit comma as the geb (→ [[Breedsmic temperaments #Osiris]]).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 703125/702464, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -10 -29 | 0 16 43 111 }}
: mapping generators: ~2, ~8000/6561
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0532{{c}}, ~8000/6561 = 343.8779{{c}}
: [[error map]]: {{val| +0.053 -0.067 -0.094 +0.084 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8000/6561 = 343.8634{{c}}
: error map: {{val| 0.000 -0.141 -0.189 +0.009 }}
 
{{Optimal ET sequence|legend=1| 7d, …, 164d, 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
 
[[Badness]] (Sintel): 1.23
 
== Stockhausenic ==
Stockhausenic may be described as the {{nowrap| 140 & 183 }} temperament, and has a generator chain of ~16/15 (or ~15/8, shown below), relating it to [[25ed5]]. It was named by Xenllium in 2021 after {{w|Karlheinz Stockhausen}} for his experimental use of 25ed5.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 5 8 }}, {{val| 0 29 -49 -95 }}]
[[Comma list]]: 703125/702464, 4096000/4084101


{{Multival|legend=1| 29 -49 -95 -145 -232 -82 }}
{{Mapping|legend=1| 1 -22 25 40 | 0 26 -25 -41 }}
: mapping generators: ~2, ~15/8


[[POTE generator]]: ~8505/8192 = 65.5883
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8908{{c}}, ~15/8 = 1088.4483{{c}}
: [[error map]]: {{val| -0.109 +0.103 -0.250 +0.427 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5477{{c}}
: error map: {{val| 0.000 +0.284 -0.005 +0.720 }}


{{Val list|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}


[[Badness]]: 0.134822
[[Badness]] (Sintel): 2.25


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 131072/130977, 703125/702464
Comma list: 1375/1372, 5632/5625, 35937/35840


Mapping: [{{val| 1 0 5 8 1 }}, {{val| 0 29 -49 -95 45 }}]
Mapping: {{mapping| 1 -22 25 40 47 | 0 26 -25 -41 -48 }}


POTE generator: ~80/77 = 65.5881
Optimal tunings:  
* WE: ~2 = 1199.9691{{c}}, ~15/8 = 1088.5116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5398{{c}}


Vals: {{val list| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness: 0.032816
Badness (Sintel): 1.85


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213
Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095


Mapping: [{{val| 1 0 5 8 1 -1 }}, {{val| 0 29 -49 -95 45 86 }}]
Mapping: {{mapping| 1 -22 25 40 47 -9 | 0 26 -25 -41 -48 14 }}


POTE generator: ~27/26 = 65.5883
Optimal tunings:  
* WE: ~2 = 1200.0267{{c}}, ~15/8 = 1088.5666{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5423{{c}}


Vals: {{val list| 128, 183, 311, 494, 1299, 1793 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323 }}


Badness: 0.016849
Badness (Sintel): 1.24


== 17-limit ==
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095
Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095


Mapping: [{{val| 1 0 5 8 1 -1 6 }}, {{val| 0 29 -49 -95 45 86 -35 }}]
Mapping: {{mapping| 1 -22 25 40 47 -9 44 | 0 26 -25 -41 -48 14 -44 }}


POTE generator: ~27/26 = 65.5878
Optimal tunings:  
* WE: ~2 = 1200.0454{{c}}, ~15/8 = 1088.5821{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1088.5407{{c}}


Vals: {{val list| 128, 183, 311, 494 }}
{{Optimal ET sequence|legend=0| 43, 97e, 140, 183, 323, 506 }}


Badness: 0.014674
Badness (Sintel): 0.978


= Decimetra =
== Decimetra ==
The ''decimetra'' temperament (50&140, named by [[User:Xenllium|Xenllium]]) tempers out the [[Procyon clan|Procyon comma]], 823543/820125 = {{monzo|0 -8 -3 7}} as well as [[15/14ths equal temperament|linus comma]].
Named by [[Xenllium]] in 2021, decimetra tempers out the [[procyon comma]] as well as the [[linus comma]], and may be described as {{nowrap| 140 & 190 }}.


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 15 23 27}}, {{val|0 4 1 5}}]
{{Mapping|legend=1| 10 3 20 12 | 0 4 1 5 }}
: mapping generators: ~15/14, ~5/4


{{Multival|legend=1| 40 10 50 -77 -33 88 }}
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 120.0204{{c}}, ~5/4 = 385.5608{{c}}
: [[error map]]: {[val| +0.204 +0.349 -0.345 -0.777 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~5/4 = 385.5321{{c}}
: error map: {[val| 0.000 +0.173 -0.782 -1.166 }}


[[POTE generator]]: ~5/4 = 385.495
{{Optimal ET sequence|legend=1| 50, 90, 140, 330, 470, 800cd }}


{{Val list|legend=1| 50, 90, 140, 330, 470, 800cd }}
[[Badness]] (Sintel): 3.08


[[Badness]]: 0.121818
=== 11-limit ===
 
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 9801/9800, 391314/390625
Comma list: 385/384, 9801/9800, 391314/390625


Mapping: [{{val|10 15 23 27 35}}, {{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 tunings:  
* WE: ~15/14 = 120.0379{{c}}, ~5/4 = 385.5181{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4329{{c}}


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


Badness: 0.056915
Badness (Sintel): 1.88


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 385/384, 625/624, 847/845, 4459/4455
Comma list: 385/384, 625/624, 847/845, 4459/4455


Mapping: [{{val|10 15 23 27 35 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 tunings:  
* WE: ~15/14 = 120.0324{{c}}, ~5/4 = 385.5231{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4445{{c}}


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


Badness: 0.029297
Badness (Sintel): 1.21


= Stockhausenic =
=== 17-limit ===
The ''stockhausenic'' temperament (140&183, named by [[User:Xenllium|Xenllium]]) is related to [[25ed5|25ED5]] tuning, and tempers out the ''hemfiness'' comma (saquadru-atriyo, named by Xenllium), 4096000/4084101 = {{monzo|15 -5 3 -5}}.
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 289/288, 385/384, 561/560, 625/624, 847/845


[[Comma list]]: 703125/702464, 4096000/4084101
Mapping: {{mapping| 10 3 20 12 41 37 28 | 0 4 1 5 -2 0 4 }}


[[Mapping]]: [{{val|1 4 0 -1}}, {{val|0 -26 25 41}}]
Optimal tunings:  
* WE: ~15/14 = 120.0380{{c}}, ~5/4 = 385.5574{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~5/4 = 385.4670{{c}}


{{Multival|legend=1|26 -25 -41 -100 -138 -25}}
{{Optimal ET sequence|legend=1| 50, 90, 140, 190g, 330eg }}


[[POTE generator]]: ~16/15 = 111.453
Badness (Sintel): 1.12


{{Val list|legend=1| 43, 97, 140, 183, 323, 786, 1109d, 1432d }}
== Luminal ==
Luminal may be described as the {{nowrap| 128 & 183 }} temperament. In the [[11-limit]] it extends via [[skadi]], the metric extension tempering out 3025/3024, the [[lehmerisma]], and joins with [[lux]] so that it also tempers out 131072/130977, the [[olympia]]. It was named by [[Flora Canou]] in 2021 following the naming of lux, whose name is the SI unit for illuminance, and lx⋅m<sup>2</sup> is lumen, the SI unit for luminous flux.


[[Badness]]: 0.088753
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 703125/702464, 26843545600/26795786661
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 5632/5625, 35937/35840
{{Mapping|legend=1| 1 0 5 8 | 0 29 -49 -95 }}
: mapping generators: ~2, ~8505/8192


Mapping: [{{val|1 4 0 -1 -1}}, {{val|0 -26 25 41 48}}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9728{{c}}, ~8505/8192 = 65.5868{{c}}
: [[error map]]: {{val| -0.027 +0.062 -0.202 +0.212 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8505/8192 = 65.5884{{c}}
: error map: {{val| 0.000 +0.108 -0.145 +0.277 }}


POTE generator: ~16/15 = 111.460
{{Optimal ET sequence|legend=1| 55d, 128, 183, 311, 494, 1299, 1793 }}


Vals: {{Val list| 43, 97e, 140, 183, 323, 506, 829de, 1335ddee }}
[[Badness]] (Sintel): 3.41


Badness: 0.056043
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 3025/3024, 131072/130977, 703125/702464
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095
Mapping: {{mapping| 1 0 5 8 1 | 0 29 -49 -95 45 }}


Mapping: [{{val|1 4 0 -1 -1 5}}, {{val|0 -26 25 41 48 -14}}]
Optimal tunings:  
* WE: ~2 = 1199.9685{{c}}, ~80/77 = 65.5864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/77 = 65.5882{{c}}


POTE generator: ~16/15 = 111.458
{{Optimal ET sequence|legend=0| 55d, 128, 183, 311, 494, 1299, 1793, 2287d }}


Vals: {{Val list| 43, 97e, 140, 183, 323, 829def, 1152deef }}
Badness (Sintel): 1.08


Badness: 0.029947
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 17-limit ==
Comma list: 2080/2079, 3025/3024, 4096/4095, 31250/31213
Subgroup: 2.3.5.7.11.13.17


Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095
Mapping: {{mapping| 1 0 5 8 1 -1 | 0 29 -49 -95 45 86 }}


Mapping: [{{val|1 4 0 -1 -1 5 0}}, {{val|0 -26 25 41 48 -14 44}}
Optimal tunings:  
* WE: ~2 = 1199.9715{{c}}, ~27/26 = 65.5867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5883{{c}}


POTE generator: ~16/15 = 111.459
{{Optimal ET sequence|legend=0| 128, 183, 311, 494, 1299, 1793 }}


Vals: {{Val list| 43, 97e, 140, 183, 323, 506, 829defg }}
Badness (Sintel): 0.696


Badness: 0.019205
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


= Geb =
Comma list: 1156/1155, 1275/1274, 2080/2079, 2431/2430, 4096/4095
The ''geb'' temperaent (171&amp;506, named after the earth god of ancient Egyptian religion by [[User: Xenllium|Xenllium]]) tempers out {{monzo|-31 43 -16}} (quadla-quadquadgu) in the 5-limit; the meter and the euzenius comma, 78125000/78121827 as well as 14348907/14336000 (lalarutrigu) in the 7-limit. [[Breedsmic temperaments #Osiris|Osiris temperament]] (171&amp;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
 
[[Comma]]: {{monzo|-31 43 -16}}
 
[[Mapping]]: [{{val|1 -3 -10}}, {{val|0 16 43}}]
 
[[POTE generator]]: ~8000/6561 = 343.868
 
{{Val list|legend=1| 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]]: [{{val|1 -3 -10 -29}}, {{val|0 16 43 111}}]


{{Multival|legend=1|16 43 111 31 131 137}}
Mapping: {{mapping| 1 0 5 8 1 -1 6 | 0 29 -49 -95 45 86 -35 }}


[[POTE generator]]: ~8000/6561 = 343.863
Optimal tunings:  
* WE: ~2 = 1199.9941{{c}}, ~27/26 = 65.5875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 65.5878{{c}}


{{Val list|legend=1| 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc }}
{{Optimal ET sequence|legend=0| 128, 183, 311, 494 }}


[[Badness]]: 0.048770
Badness (Sintel): 0.748


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

Latest revision as of 05:58, 9 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 geb, stockhausenic, decimetra, and luminal, in the order of increasing badness.

Geb

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

Geb tempers out the scheme comma as well as the euzenius comma, and may be described as 164d & 171 temperament. It was named by Xenllium in 2021 after the earth god of ancient Egyptian religion and the father of Osiris; the latter being the name of a temperament that tempers out the same 5-limit comma as the geb (→ Breedsmic temperaments #Osiris).

Subgroup: 2.3.5.7

Comma list: 703125/702464, 14348907/14336000

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

mapping generators: ~2, ~8000/6561

Optimal tunings:

  • WE: ~2 = 1200.0532 ¢, ~8000/6561 = 343.8779 ¢
error map: ⟨+0.053 -0.067 -0.094 +0.084]
  • CWE: ~2 = 1200.0000 ¢, ~8000/6561 = 343.8634 ¢
error map: ⟨0.000 -0.141 -0.189 +0.009]

Optimal ET sequence: 7d, …, 164d, 171, 1190, 1361, 1532, 1703, 1874, 2045, 2216, 4603bc

Badness (Sintel): 1.23

Stockhausenic

Stockhausenic may be described as the 140 & 183 temperament, and has a generator chain of ~16/15 (or ~15/8, shown below), relating it to 25ed5. It was named by Xenllium in 2021 after Karlheinz Stockhausen for his experimental use of 25ed5.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 4096000/4084101

Mapping: [⟨1 -22 25 40], ⟨0 26 -25 -41]]

mapping generators: ~2, ~15/8

Optimal tunings:

  • WE: ~2 = 1199.8908 ¢, ~15/8 = 1088.4483 ¢
error map: ⟨-0.109 +0.103 -0.250 +0.427]
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5477 ¢
error map: ⟨0.000 +0.284 -0.005 +0.720]

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

Badness (Sintel): 2.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 5632/5625, 35937/35840

Mapping: [⟨1 -22 25 40 47], ⟨0 26 -25 -41 -48]]

Optimal tunings:

  • WE: ~2 = 1199.9691 ¢, ~15/8 = 1088.5116 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5398 ¢

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

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1375/1372, 4096/4095

Mapping: [⟨1 -22 25 40 47 -9], ⟨0 26 -25 -41 -48 14]]

Optimal tunings:

  • WE: ~2 = 1200.0267 ¢, ~15/8 = 1088.5666 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5423 ¢

Optimal ET sequence: 43, 97e, 140, 183, 323

Badness (Sintel): 1.24

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 561/560, 676/675, 715/714, 4096/4095

Mapping: [⟨1 -22 25 40 47 -9 44], ⟨0 26 -25 -41 -48 14 -44]]

Optimal tunings:

  • WE: ~2 = 1200.0454 ¢, ~15/8 = 1088.5821 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/8 = 1088.5407 ¢

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

Badness (Sintel): 0.978

Decimetra

Named by Xenllium in 2021, decimetra tempers out the procyon comma as well as the linus comma, and may be described as 140 & 190.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 823543/820125

Mapping: [⟨10 3 20 12], ⟨0 4 1 5]]

mapping generators: ~15/14, ~5/4

Optimal tunings:

  • WE: ~15/14 = 120.0204 ¢, ~5/4 = 385.5608 ¢
error map: {[val| +0.204 +0.349 -0.345 -0.777 }}
  • CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.5321 ¢
error map: {[val| 0.000 +0.173 -0.782 -1.166 }}

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

Badness (Sintel): 3.08

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 tunings:

  • WE: ~15/14 = 120.0379 ¢, ~5/4 = 385.5181 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4329 ¢

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

Badness (Sintel): 1.88

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 tunings:

  • WE: ~15/14 = 120.0324 ¢, ~5/4 = 385.5231 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4445 ¢

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

Badness (Sintel): 1.21

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 tunings:

  • WE: ~15/14 = 120.0380 ¢, ~5/4 = 385.5574 ¢
  • CWE: ~15/14 = 120.0000 ¢, ~5/4 = 385.4670 ¢

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

Badness (Sintel): 1.12

Luminal

Luminal may be described as the 128 & 183 temperament. In the 11-limit it extends via skadi, the metric extension tempering out 3025/3024, the lehmerisma, and joins with lux so that it also tempers out 131072/130977, the olympia. It was named by Flora Canou in 2021 following the naming of lux, whose name is the SI unit for illuminance, and lx⋅m2 is lumen, the SI unit for luminous flux.

Subgroup: 2.3.5.7

Comma list: 703125/702464, 26843545600/26795786661

Mapping: [⟨1 0 5 8], ⟨0 29 -49 -95]]

mapping generators: ~2, ~8505/8192

Optimal tunings:

  • WE: ~2 = 1199.9728 ¢, ~8505/8192 = 65.5868 ¢
error map: ⟨-0.027 +0.062 -0.202 +0.212]
  • CWE: ~2 = 1200.0000 ¢, ~8505/8192 = 65.5884 ¢
error map: ⟨0.000 +0.108 -0.145 +0.277]

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

Badness (Sintel): 3.41

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 tunings:

  • WE: ~2 = 1199.9685 ¢, ~80/77 = 65.5864 ¢
  • CWE: ~2 = 1200.0000 ¢, ~80/77 = 65.5882 ¢

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

Badness (Sintel): 1.08

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 tunings:

  • WE: ~2 = 1199.9715 ¢, ~27/26 = 65.5867 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 65.5883 ¢

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

Badness (Sintel): 0.696

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 tunings:

  • WE: ~2 = 1199.9941 ¢, ~27/26 = 65.5875 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/26 = 65.5878 ¢

Optimal ET sequence: 128, 183, 311, 494

Badness (Sintel): 0.748

Retrieved from "https://en.xen.wiki/index.php?title=Metric_microtemperaments&oldid=232075"