Didymus rank-3 family: Difference between revisions

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These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth.
__FORCETOC__
=Euterpe=
Commas: 81/80, 99/98


Minimax tuning
= Euterpe =
[[Comma list]]: 81/80, 99/98


[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>,  
[[Minimax tuning]]:
|0 0 0 1 0>, |-1 0 -1/2 2 0>]
* [[11-odd-limit]]:
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [[Eigenmonzo]]s: 2, 5, 7


Eigenmonzos: 2, 5, 7
[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]


Map: [<1 0 -4 0 1|, <0 1 4 0 -2|, <0 0 0 1 2|]
{{Val list|legend=1| 5, 12, 26, 31, 88 }}


EDOs: 5, 12, 26, 31, 88
[[Badness]]: 0.536 × 10<sup>-3</sup>


Badness: 0.000536
= Calliope =
[[Comma list]]: 81/80, 45/44


=Calliope=
[[Minimax tuning]]:
Commas: 81/80, 45/44
* [[11-odd-limit]]:
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo]]s: 2, 7/3, 11


Minimax tuning
[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]


[|1 0 0 0 0&gt;, |1 0 0 0 1/6&gt;, |0 0 0 0 2/3&gt;,  
{{Val list|legend=1| 7, 12, 19, 26, 45 }}
|1 -1 0 1 1/6&gt;, |0 0 0 0 1&gt;]


Eigenmonzos: 2, 7/3, 11
[[Badness]]: 0.530 × 10<sup>-3</sup>


Map: [&lt;1 0 -4 0 -6|, &lt;0 1 4 0 6|, &lt;0 0 0 1 0|]
= Erato =
[[Comma list]]: 81/80, 126/125


EDOs: 7, 12, 19, 26, 45
[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]


Badness: 0.000530
{{Val list|legend=1| 12, 19, 31, 50, 81 }}


=Erato=
[[Badness]]: 0.558 × 10<sup>-3</sup>
Commas: 81/80, 126/125


Map: [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|]
== 13-limit ==
Comma list: 81/80, 105/104, 126/125


EDOs: 12, 19, 31, 50, 81
Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]


Badness: 0.000558
{{Val list|legend=1| 12f, 19, 31, 50, 81 }}


==13-limit==
= Clio =
Commas: 81/80, 105/104, 126/125
[[Comma list]]: 81/80, 176/175


Map: [&lt;1 0 -4 -13 0 -20|, &lt;0 1 4 10 0 15|, &lt;0 0 0 0 1 0|]
[[Minimax tuning]]
* [[11-odd-limit]]:  
: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
: [[Eigenmonzo]]s: 2, 5, 7


EDOs: 12f, 19, 31, 50, 81
Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]


=Clio=
{{Val list|legend=1| 7, 12, 24, 31, 105, 129 }}
Commas: 81/80, 176/175


Minimax tuning
Badness: 0.738 × 10<sup>-3</sup>
||1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
|0 0 0 1 0&gt;, |-4 0 2 1 0&gt;]


Eigenmonzos: 2, 5, 7
= Polyhymnia =
[[Comma list]]: 81/80, 385/384


Map: [&lt;1 0 -4 0 -12|, &lt;0 1 4 0 8|, &lt;0 0 0 1 1|]
[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]


EDOs: 7, 12, 24, 31, 105, 129
{{Val list|legend=1| 7, 19, 24, 26, 31, 38, 45 }}


Badness: 0.000738
= Thalia =
[[Comma list]]: 81/80, 55/54


=Polyhymnia=
[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
Commas: 81/80, 385/384


Map: [&lt;1 0 -4 0 11|, &lt;0 1 4 0 -3|, &lt;0 0 0 1 -1|]
{{Val list|legend=1| 5, 7, 12e, 19e }}


EDOs: 7, 19, 24, 26, 31, 38, 45
= Melpomene =
[[Comma list]]: 81/80, 56/55


=Thalia=
[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
Commas: 81/80, 55/54


Map: [&lt;1 0 -4 0 5|, &lt;0 1 4 0 -1|, &lt;0 0 0 1 0|]
{{Val list|legend=1| 5, 12, 19, 24, 36 }}


EDOs: 5, 7, 12e, 19e
= Urania =
[[Comma list]]: 81/80, 121/120


=Melpomene=
[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
Commas: 81/80, 56/55


Map: [&lt;1 0 -4 0 7|, &lt;0 1 4 0 -4|, &lt;0 0 0 1 1|]
Lattice basis:  
: 11/9 length = 0.2536, 8/7 length = 2.807
: Angle (11/9, 8/7) = 90 degrees


EDOs: 5, 12, 19, 24, 36
Map to lattice: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]


=Urania=
{{Val list|legend=1| 7, 14c, 24, 31, 100de, 131bde, 162bde }}
Commas: 81/80, 121/120


Lattice basis: 11/9 length 0.2536 8/7 length 2.807
[[Badness]]: 0.842 × 10<sup>-3</sup>


Angle(11/9, 8/7) = 90 degrees
[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
 
Map to lattice: [&lt;0 2 8 0 5|, &lt;0 0 0 -1 0|]
 
Map: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|, &lt;0 0 0 1 0|]
 
EDOs: 7, 14c, 24, 31, 100de, 131bde, 162bde
 
Badness: 0.000842
 
[[Spectrum_of_a_temperament|Spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5


Scale: [[urania24]]
Scale: [[urania24]]


=Terpsichore=
= Terpsichore =
Commas: 81/80, 540/539
[[Comma list]]: 81/80, 540/539


Map: [&lt;1 0 -4 0 -2|, &lt;0 1 4 0 7|, &lt;0 0 0 1 -2|]
[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]


EDOs: 19, 31, 81, 112b
{{Val list|legend=1| 19, 31, 81, 112b }}


Badness: 0.000850
[[Badness]]: 0.850 × 10<sup>-3</sup>


Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11


[[Category:Theory]]
[[Category:Theory]]

Revision as of 07:42, 20 March 2021

The didymus rank-3 family are rank-3 temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Euterpe

Comma list: 81/80, 99/98

Minimax tuning:

[[1 0 0 0 0, [1 0 1/4 0 0, [0 0 1 0 0, [0 0 0 1 0, [-1 0 -1/2 2 0]
Eigenmonzos: 2, 5, 7

Mapping: [1 0 -4 0 1], 0 1 4 0 -2], 0 0 0 1 2]]

Template:Val list

Badness: 0.536 × 10-3

Calliope

Comma list: 81/80, 45/44

Minimax tuning:

[[1 0 0 0 0, [1 0 0 0 1/6, [0 0 0 0 2/3, [1 -1 0 1 1/6, [0 0 0 0 1]
Eigenmonzos: 2, 7/3, 11

Mapping: [1 0 -4 0 -6], 0 1 4 0 6], 0 0 0 1 0]]

Template:Val list

Badness: 0.530 × 10-3

Erato

Comma list: 81/80, 126/125

Mapping: [1 0 -4 -13 0], 0 1 4 10 0], 0 0 0 0 1]]

Template:Val list

Badness: 0.558 × 10-3

13-limit

Comma list: 81/80, 105/104, 126/125

Mapping: [1 0 -4 -13 0 -20], 0 1 4 10 0 15], 0 0 0 0 1 0]]

Template:Val list

Clio

Comma list: 81/80, 176/175

Minimax tuning

[1 0 0 0 0, [1 0 1/4 0 0, [0 0 1 0 0, [0 0 0 1 0, [-4 0 2 1 0]
Eigenmonzos: 2, 5, 7

Mapping: [1 0 -4 0 -12], 0 1 4 0 8], 0 0 0 1 1]]

Template:Val list

Badness: 0.738 × 10-3

Polyhymnia

Comma list: 81/80, 385/384

Mapping: [1 0 -4 0 11], 0 1 4 0 -3], 0 0 0 1 -1]]

Template:Val list

Thalia

Comma list: 81/80, 55/54

Mapping: [1 0 -4 0 5], 0 1 4 0 -1], 0 0 0 1 0]]

Template:Val list

Melpomene

Comma list: 81/80, 56/55

Mapping: [1 0 -4 0 7], 0 1 4 0 -4], 0 0 0 1 1]]

Template:Val list

Urania

Comma list: 81/80, 121/120

Mapping: [1 1 0 0 2], 0 2 8 0 5], 0 0 0 1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807
Angle (11/9, 8/7) = 90 degrees

Map to lattice: [0 2 8 0 5], 0 0 0 -1 0]]

Template:Val list

Badness: 0.842 × 10-3

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scale: urania24

Terpsichore

Comma list: 81/80, 540/539

Mapping: [1 0 -4 0 -2], 0 1 4 0 7], 0 0 0 1 -2]]

Template:Val list

Badness: 0.850 × 10-3

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11