Canopic family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Cleanup (2/2)
Lériendil (talk | contribs)
8,044 edits
m Lériendil moved page Mirkwai family to Canopus family
(15 intermediate revisions by 6 users not shown)
Line 1: Line 1:
The '''mirkwai family''' of [[planar temperament]]s tempers out the [[mirkwai comma] }}, 16875/16807.
{{Technical data page}}
The '''canopus family''' of [[planar temperament]]s tempers out the '''canopus''' or '''[[mirkwai comma]]''' ({{monzo|legend=1| 0 3 4 -5 }}, [[ratio]]: 16875/16807), a no-twos comma.  


= Mirkwai =
== Canopic aka Mirkwai ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16875/16807 (mirkwai)
[[Comma list]]: 16875/16807 (canopus)


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


{{Val list|legend=1| 72, 152, 193, 224, 296, 327, 775 }}
{{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


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


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: 2, 11/9, 5/4
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9, 5/4


{{Val list|legend=1| 31, 41, 72, 152, 224, 296, 703 }}
{{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 }}]


EDOs: 41, 72, 111, 183, 224, 703
{{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


Vals: {{Val list| 31, 72, 103, 193, 224, 296, 520, 703 }}
{{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, 72, 103, 111, 152, 183 }}
{{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| 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:Mirkwai family]] <!-- main article -->
[[Category:Mirkwai]]
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 16:39, 5 June 2026

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 canopus family of planar temperaments tempers out the canopus or mirkwai comma (monzo: [0 3 4 -5⟩, ratio: 16875/16807), a no-twos comma.

Canopic aka Mirkwai

Subgroup: 2.3.5.7

Comma list: 16875/16807 (canopus)

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

Retrieved from "https://en.xen.wiki/index.php?title=Canopic_family&oldid=231589"