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. | ||
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* [[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 | ||
{{ | {{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }} | ||
[[Badness]]: 0.342 × 10<sup>-3</sup> | [[Badness]]: 0.342 × 10<sup>-3</sup> | ||
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* [[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 | ||
{{ | {{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> | ||
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Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}] | Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}] | ||
Optimal | {{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> | ||
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* 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 | ||
Optimal | {{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> | ||
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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 }}] | ||
Optimal | {{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> | ||
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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 }}] | ||
Optimal | {{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:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Mirkwai family]] <!-- main article --> | [[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
- [[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
- [[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