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 ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807 (mirkwai)
[[Comma list]]: 16875/16807 (mirkwai)
Line 13: Line 14:
* [[7-odd-limit]]
* [[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 }}]
: [{{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
: [[Eigenmonzo]]s (unchanged-intervals): 2, 8/7, 6/5
* [[9-odd-limit]]
* [[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 }}]
: [{{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
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 8/7


{{Val list|legend=1| 31, 41, 72, 152, 224 }}
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}


[[Badness]]: 0.342 × 10<sup>-3</sup>
[[Badness]]: 0.342 × 10<sup>-3</sup>
Line 24: Line 25:
[[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 ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 1375/1372
[[Comma list]]: 540/539, 1375/1372
Line 33: Line 34:
Mapping generators: ~2, ~3, ~7/5
Mapping generators: ~2, ~3, ~7/5


Map to lattice: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]
[[Mapping to lattice]]: [{{val| 0 2 1 2 3 }}, {{val| 0 -1 2 1 -3 }}]


Lattice basis:  
Lattice basis:  
Line 42: Line 43:
* [[11-odd-limit]]
* [[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 }}]
: [{{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]]s (unchanged-intervals): 2, 11/9, 5/4


{{Val list|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224, 631d, 855d }}


[[Badness]]: 0.266 × 10<sup>-3</sup>
[[Badness]]: 0.266 × 10<sup>-3</sup>
Line 52: Line 53:
Scales: [[indra31]]
Scales: [[indra31]]


=== Shibi ===
=== Shibi ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


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


Badness: 0.721 × 10<sup>-3</sup>
Badness: 0.721 × 10<sup>-3</sup>


=== Yajna ===
=== Yajna ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 75: Line 76:
* 13-odd-limit  
* 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 }}]
: [{{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
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 13/10
* 15-odd-limit minimax
* 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 }}]
: [{{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
: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 4/3


Vals: {{Val list| 31, 49f, 72, 103, 121, 152f, 193, 224 }}
{{Optimal ET sequence|legend=1| 31, 49f, 72, 103, 121, 152f, 193, 224 }}


Badness: 0.700 × 10<sup>-3</sup>
Badness: 0.700 × 10<sup>-3</sup>


=== Vibhu ===
=== Vibhu ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 91: Line 92:
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]
Mapping: [{{val| 1 0 0 0 2 1 }}, {{val| 0 1 3 3 0 6 }}, {{val| 0 0 -5 -4 3 -14 }}]


Vals: {{Val list| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}
{{Optimal ET sequence|legend=1| 31, 64cdf, 72, 103, 111, 152, 175f, 183, 469f }}


Badness: 0.826 × 10<sup>-3</sup>
Badness: 0.826 × 10<sup>-3</sup>


=== Vish ===
=== Vish ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 102: Line 103:
Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 1 3 3 0 -2 }}, {{val| 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 }}]


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


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


[[Category:Regular temperament 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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