Mirkwai family: Difference between revisions

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{{Technical data page}}
The '''mirkwai family''' of [[planar temperament]]s tempers out the [[mirkwai comma]], 16875/16807.
The '''mirkwai family''' of [[planar temperament]]s tempers out the [[mirkwai comma]], 16875/16807.


= Mirkwai =
== Mirkwai ==
[[Comma]]: mirkwai 16875/16807
[[Subgroup]]: 2.3.5.7


7-limit minimax
[[Comma list]]: 16875/16807 (mirkwai)


[|1 0 0 0>, |0 4/7 -4/7 5/7>,
[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 3 3 }}, {{val| 0 0 -5 -4 }}]
|0 -3/7 3/7 5/7>, |0 0 0 1>]


[[Eigenmonzo]]s: 2, 8/7, 6/5
Mapping generators: ~2, ~3, ~7/5


9-limit minimax
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 4/7 -4/7 5/7 }}, {{monzo| 0 -3/7 3/7 5/7 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 8/7, 6/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 8/11 -4/11 5/11 }}, {{monzo| 0 -6/11 3/11 10/11 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 8/7


[|1 0 0 0>, |0 8/11 -4/11 5/11>,  
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}
|0 -6/11 3/11 10/11>, |0 0 0 1>]


[[Eigenmonzo]]s: 2, 10/9, 8/7
[[Badness]]: 0.342 × 10<sup>-3</sup>
 
Map: [&lt;1 0 0 0|, &lt;0 1 3 3|, &lt;0 0 -5 -4|]
 
[[Generator]]s: 2, 3, 7/5
 
[[EDO|Edos]]: [[72edo|72]], [[152edo|152]], [[193edo|193]], [[224edo|224]], [[296edo|296]], [[327edo|327]], [[775edo|775]]
 
Badness: 0.000342


[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5
[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5


= Indra =
== Indra ==
[[Comma]]s: 540/539, 1375/1372
[[Subgroup]]: 2.3.5.7.11


[[Minimax tuning]]
[[Comma list]]: 540/539, 1375/1372


[|1 0 0 0 0&gt;, |10 10 -3 0 -5&gt;, |0 0 1 0 0&gt;,
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 3 3 0 }}, {{val| 0 0 -5 -4 3 }}]
|6 6 -1 0 -3&gt;, |20 18 -6 0 -9&gt;]


[[Eigenmonzo]]s: 2, 11/9, 5/4
Mapping generators: ~2, ~3, ~7/5


Lattice basis: 15/14 length 0.6184 55/36 length 0.7585
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]


Angle(15/14, 55/36) = 68.1048 cents
Lattice basis:
: 15/14 length = 0.6184, 55/36 length = 0.7585
: Angle (15/14, 55/36) = 68.1048 deg


Map to lattice: [&lt;0 2 1 2 3|, &lt;0 -1 2 1 -3|]
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10 10 -3 0 -5 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 6 -1 0 -3 }}, {{monzo| 20 18 -6 0 -9 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9, 5/4


Map: [&lt;1 0 0 0 2|, &lt;0 1 3 3 0|, &lt;0 0 -5 -4 3|]
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}


[[Generator]]s: 2, 3, 7/5
[[Badness]]: 0.266 × 10<sup>-3</sup>
 
[[EDO|Edos]]: 31, 41, [[72edo|72]], [[152edo|152]], [[224edo|224]], [[296edo|296]], [[703edo|703]]
 
Badness: 0.000266


[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
Line 56: Line 53:
Scales: [[indra31]]
Scales: [[indra31]]


== Shibi ==
=== Shibi ===
Commas: 540/539, 729/728, 1375/1372
Subgroup: 2.3.5.7.11.13
 
Map: [&lt;1 0 0 0 2 -4 -3|, &lt;0 1 3 3 0 11 3|, &lt;0 0 -5 -4 3 4|]


EDOs: 41, 72, 111, 183, 224, 703
Comma list: 540/539, 729/728, 1375/1372


Badness: 0.000721
Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}]


== Yajna ==
{{Optimal ET sequence|legend=1| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}
[[Comma]]s: 540/539, 625/624, 1375/1372


13-limit minimax
Badness: 0.721 × 10<sup>-3</sup>


[|1 0 0 0 0 0&gt;, |4/5 6/5 -4/5 0 0 1/5&gt;,
=== Yajna ===
|8/5 2/5 -3/5 0 0 2/5&gt;,
Subgroup: 2.3.5.7.11.13
|44/25 26/25 -24/25 0 0 11/25&gt;,
|62/25 48/25 -27/25 0 0 3/25&gt;,
|8/5 2/5 -8/5 0 0 7/5&gt;]


[[Eigenmonzo]]s: 2, 10/9, 13/10
Comma list: 540/539, 625/624, 1375/1372


15-limit minimax
Mapping: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 3 3 0 11 }}, {{val| 0 0 -5 -4 3 -20 }}]


[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,
Mapping generators: ~2, ~3, ~7/5
|5/4 1/2 0 -5/16 0 5/16&gt;,
|1 1 0 -1/4 0 1/4&gt;,  
|5/4 3/2 0 3/16 0 -3/16&gt;,  
|1 1 0 -5/4 0 5/4&gt;]


[[Eigenmonzo]]s: 2, 14/13, 4/3
Minimax tuning:
* 13-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 4/5 6/5 -4/5 0 0 1/5 }}, {{monzo| 8/5 2/5 -3/5 0 0 2/5 }}, {{monzo| 44/25 26/25 -24/25 0 0 11/25 }}, {{monzo| 62/25 48/25 -27/25 0 0 3/25 }}, {{monzo| 8/5 2/5 -8/5 0 0 7/5 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 13/10
* 15-odd-limit minimax
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 1 0 0 0 0 }}, {{monzo| 5/4 1/2 0 -5/16 0 5/16 }}, {{monzo| 1 1 0 -1/4 0 1/4 }}, {{monzo| 5/4 3/2 0 3/16 0 -3/16 }}, {{monzo| 1 1 0 -5/4 0 5/4 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 4/3


Map: [&lt;1 0 0 0 2 -4|, &lt;0 1 3 3 0 11|, &lt;0 0 -5 -4 3 -20|]
{{Optimal ET sequence|legend=1| 31, 49f, 72, 103, 121, 152f, 193, 224 }}


[[Generator]]s: 2, 3, 7/5
Badness: 0.700 × 10<sup>-3</sup>


[[EDO|Edos]]: [[31edo|31]], [[72edo|72]], [[103edo|103]], [[193edo|193]], 224, 296, 520, 703
=== Vibhu ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000700
Comma list: 351/350, 540/539, 1375/1372


== Vibhu ==
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]
[[Comma]]s: 540/539, 1375/1372, 351/350


Map: [&lt;1 0 0 0 2 1|, &lt;0 1 3 3 0 6|, &lt;0 0 -5 -4 3 -14|]
{{Optimal ET sequence|legend=1| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}


EDOs: 31, 72, 103, 111, 152, 183
Badness: 0.826 × 10<sup>-3</sup>


Badness: 0.000826
=== Vish ===
Subgroup: 2.3.5.7.11.13


== Vish ==
Comma list: 325/324, 364/363, 540/539
[[Comma]]s: 325/324, 364/363, 540/539


Map: [&lt;1 0 0 0 2 2], &lt;0 1 3 3 0 -2], &lt;0 0 -5 -4 3 10|]
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 3 3 0 -2 }}, {{val| 0 0 -5 -4 3 10 }}]


EDOs: 41, 72, 121, 193
{{Optimal ET sequence|legend=1| 31f, 41, 72, 121, 193 }}


Badness: 0.000930
Badness: 0.930 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Mirkwai]]
[[Category:Mirkwai family]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:28, 24 June 2025

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.

The mirkwai family of planar temperaments tempers out the mirkwai comma, 16875/16807.

Mirkwai

Subgroup: 2.3.5.7

Comma list: 16875/16807 (mirkwai)

Mapping: [⟨1 0 0 0], ⟨0 1 3 3], ⟨0 0 -5 -4]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

[[1 0 0 0⟩, [0 4/7 -4/7 5/7⟩, [0 -3/7 3/7 5/7⟩, [0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 8/7, 6/5
[[1 0 0 0⟩, [0 8/11 -4/11 5/11⟩, [0 -6/11 3/11 10/11⟩, [0 0 0 1⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9, 8/7

Optimal ET sequence: 31, 41, 72, 152, 224

Badness: 0.342 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 to 2.3.7/5

Indra

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372

Mapping: [⟨1 0 0 0 2], ⟨0 1 3 3 0], ⟨0 0 -5 -4 3]]

Mapping generators: ~2, ~3, ~7/5

Mapping to lattice: [⟨0 2 1 2 3], ⟨0 -1 2 1 -3]]

Lattice basis:

15/14 length = 0.6184, 55/36 length = 0.7585
Angle (15/14, 55/36) = 68.1048 deg

Minimax tuning:

[[1 0 0 0 0⟩, [10 10 -3 0 -5⟩, [0 0 1 0 0⟩, [6 6 -1 0 -3⟩, [20 18 -6 0 -9⟩]
Eigenmonzos (unchanged-intervals): 2, 11/9, 5/4

Optimal ET sequence: 31, 41, 72, 152, 224, 631d, 855d

Badness: 0.266 × 10-3

Projection pairs: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5

Scales: indra31

Shibi

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 0 0 2 -3], ⟨0 1 3 3 0 3], ⟨0 0 -5 -4 4]]

Optimal ET sequence: 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d

Badness: 0.721 × 10-3

Yajna

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 1375/1372

Mapping: [⟨1 0 0 0 2 -4], ⟨0 1 3 3 0 11], ⟨0 0 -5 -4 3 -20]]

Mapping generators: ~2, ~3, ~7/5

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0⟩, [4/5 6/5 -4/5 0 0 1/5⟩, [8/5 2/5 -3/5 0 0 2/5⟩, [44/25 26/25 -24/25 0 0 11/25⟩, [62/25 48/25 -27/25 0 0 3/25⟩, [8/5 2/5 -8/5 0 0 7/5⟩]
Eigenmonzos (unchanged-intervals): 2, 10/9, 13/10
  • 15-odd-limit minimax
[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [5/4 1/2 0 -5/16 0 5/16⟩, [1 1 0 -1/4 0 1/4⟩, [5/4 3/2 0 3/16 0 -3/16⟩, [1 1 0 -5/4 0 5/4⟩]
Eigenmonzos (unchanged-intervals): 2, 14/13, 4/3

Optimal ET sequence: 31, 49f, 72, 103, 121, 152f, 193, 224

Badness: 0.700 × 10-3

Vibhu

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1375/1372

Mapping: [⟨1 0 0 0 2 1], ⟨0 1 3 3 0 6], ⟨0 0 -5 -4 3 -14]]

Optimal ET sequence: 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f

Badness: 0.826 × 10-3

Vish

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 540/539

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 3 3 0 -2], ⟨0 0 -5 -4 3 10]]

Optimal ET sequence: 31f, 41, 72, 121, 193

Badness: 0.930 × 10-3

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