Mirkwai family: Difference between revisions

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


=Vital statistics=
== Mirkwai ==
[[Comma|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|Eigenmonzos]]: 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|Eigenmonzos]]: 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|]
[[Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5


[[generator|Generators]]: 2, 3, 7/5
== Indra ==
[[Subgroup]]: 2.3.5.7.11


[[EDO|Edos]]: [[72edo|72]], [[152edo|152]], [[193edo|193]], [[224edo|224]], [[296edo|296]], [[327edo|327]], [[775edo|775]]
[[Comma list]]: 540/539, 1375/1372


Badness: 0.000342
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 3 3 0 }}, {{val| 0 0 -5 -4 3 }}]


[[Projection_pair|Projection pair]]s: 5 84375/16807 7 16875/2401 to 2.3.7/5
Mapping generators: ~2, ~3, ~7/5


=Indra=
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]
[[Comma|Commas]]: 540/539, 1375/1372


[[Minimax_tuning|Minimax tuning]]
Lattice basis:
: 15/14 length = 0.6184, 55/36 length = 0.7585
: Angle (15/14, 55/36) = 68.1048 deg


[|1 0 0 0 0&gt;, |10 10 -3 0 -5&gt;, |0 0 1 0 0&gt;,  
[[Minimax tuning]]:
|6 6 -1 0 -3&gt;, |20 18 -6 0 -9&gt;]
* [[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


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}


Lattice basis: 15/14 length 0.6184 55/36 length 0.7585
[[Badness]]: 0.266 × 10<sup>-3</sup>


Angle(15/14, 55/36) = 68.1048 cents
[[Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5


Map to lattice: [&lt;0 2 1 2 3|, &lt;0 -1 2 1 -3|]
Scales: [[indra31]]


Map: [&lt;1 0 0 0 2|, &lt;0 1 3 3 0|, &lt;0 0 -5 -4 3|]
=== Shibi ===
Subgroup: 2.3.5.7.11.13


[[generator|Generators]]: 2, 3, 7/5
Comma list: 540/539, 729/728, 1375/1372


[[EDO|Edos]]: 31, 41, [[72edo|72]], [[152edo|152]], [[224edo|224]], [[296edo|296]], [[703edo|703]]
Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}]


Badness: 0.000266
{{Optimal ET sequence|legend=1| 31f, 39df, 41, 72, 111, 152f, 183, 224, 631d, 855d }}


[[Projection_pair|Projection pair]]s: 5 84375/16807 7 16875/2401 11 1372/125 to 2.3.7/5
Badness: 0.721 × 10<sup>-3</sup>


Scales: [[indra31|indra31]]
=== Yajna ===
Subgroup: 2.3.5.7.11.13


==Shibi==
Comma list: 540/539, 625/624, 1375/1372
Commas: 540/539, 729/728, 1375/1372


Map: [&lt;1 0 0 0 2 -4 -3|, &lt;0 1 3 3 0 11 3|, &lt;0 0 -5 -4 3 4|]
Mapping: [{{val| 1 0 0 0 2 -4 }}, {{val| 0 1 3 3 0 11 }}, {{val| 0 0 -5 -4 3 -20 }}]


EDOs: 41, 72, 111, 183, 224, 703
Mapping generators: ~2, ~3, ~7/5


Badness: 0.000721
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


==Yajna==
{{Optimal ET sequence|legend=1| 31, 49f, 72, 103, 121, 152f, 193, 224 }}
[[Comma|Commas]]: 540/539, 625/624, 1375/1372


13-limit minimax
Badness: 0.700 × 10<sup>-3</sup>


[|1 0 0 0 0 0&gt;, |4/5 6/5 -4/5 0 0 1/5&gt;,
=== Vibhu ===
|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|Eigenmonzos]]: 2, 10/9, 13/10
Comma list: 351/350, 540/539, 1375/1372


15-limit minimax
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]


[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,  
{{Optimal ET sequence|legend=1| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}
|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|Eigenmonzos]]: 2, 14/13, 4/3
Badness: 0.826 × 10<sup>-3</sup>


Map: [&lt;1 0 0 0 2 -4|, &lt;0 1 3 3 0 11|, &lt;0 0 -5 -4 3 -20|]
=== Vish ===
Subgroup: 2.3.5.7.11.13


[[generator|Generators]]: 2, 3, 7/5
Comma list: 325/324, 364/363, 540/539


[[EDO|Edos]]: [[31edo|31]], [[72edo|72]], [[103edo|103]], [[193edo|193]], 224, 296, 520, 703
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 3 3 0 -2 }}, {{val| 0 0 -5 -4 3 10 }}]


Badness: 0.000700
{{Optimal ET sequence|legend=1| 31f, 41, 72, 121, 193 }}


==Vibhu==
Badness: 0.930 × 10<sup>-3</sup>
[[Comma|Commas]]: 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|]
[[Category:Temperament families]]
 
[[Category:Pages with mostly numerical content]]
EDOs: 31, 72, 103, 111, 152, 183
[[Category:Mirkwai family]] <!-- main article -->
 
[[Category:Rank 3]]
Badness: 0.000826
 
==Vish==
[[Comma|Commas]]: 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|]
 
EDOs: 41, 72, 121, 193
 
Badness: 0.000930
[[Category:family]]
[[Category:mirkwai]]
[[Category:theory]]

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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