Xenharmonic Wiki

Mirkwai family: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m recat
 
(15 intermediate revisions by 5 users not shown)
Line 1: Line 1:
__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:Theory]]
[[Category:Temperament family]]
[[Category:Planar temperament]]
[[Category:Mirkwai]]
Retrieved from "https://en.xen.wiki/w/Mirkwai_family"