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: 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: 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> | ||
| 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 | ||
[[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: 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> | ||
| 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 | Mapping: [{{val| 1 0 0 0 2 -3 }}, {{val| 0 1 3 3 0 3 }}, {{val| 0 0 -5 -4 4 }}] | ||
{{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: 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: 2, 14/13, 4/3 | : [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 4/3 | ||
{{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 }}] | ||
{{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 }}] | ||
{{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: | [[Category:Temperament families]] | ||
[[Category: | [[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
- [[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