Horwell family: Difference between revisions

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The '''horwell family''' of rank-3 temperaments tempers out the [[horwell comma]], 65625/65536.
The '''horwell family''' of rank three temperaments tempers out the [[horwell comma]], 65625/65536.


== Horwell ==
== Horwell ==
Comma: 65625/65536
Subgroup: 2.3.5.7


7-limit minimax
[[Comma list]]: 65625/65536


[|1 0 0 0>, |16/7 6/7 -5/7 -1/7>,  
[[Mapping]]: [{{val| 1 0 0 16 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -5 }}]
|16/7 -1/7 2/7 -1/7>, |16/7 -1/7 -5/7 6/7>]


Eigenmonzos: 2, 7/6, 6/5
Mapping generators: ~2, ~3, ~5


9-limit minimax
Map to lattice: [{{val| 0 -1 1 -4 }}, {{val| 0 -1 0 -1 }}]


[|1 0 0 0>, |16/13 12/13 -5/13 -1/13>,
Lattice basis:
|32/13 -2/13 3/13 -2/13>, |32/13 -2/13 -10/13 11/13>]
: 5/4 length = 0.6213, 16/15 length = 1.3863
: Angle (5/4, 16/15) = 64.6379


Eigenmonzos: 2, 9/7, 7/5
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 16/7 6/7 -5/7 -1/7 }}, {{monzo| 16/7 -1/7 2/7 -1/7 }}, {{monzo| 16/7 -1/7 -5/7 6/7 }}]
: [[Eigenmonzo]]s: 2, 7/6, 6/5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 16/13 12/13 -5/13 -1/13 }}, {{monzo| 32/13 -2/13 3/13 -2/13 }}, {{monzo| 32/13 -2/13 -10/13 11/13 }}]
: [[Eigenmonzo]]s: 2, 9/7, 7/5


Lattice basis: 5/4 length=0.6213 16/15 length=1.3863
{{Val list|legend=1| 9, 22, 25, 28, 31, 34, 40, 53, 56, 65, 75, 84, 87, 118, 140, 171, 824, 995 }}


Angle(5/4, 16/15) = 64.6379
[[Badness]]: 0.173 × 10<sup>-3</sup>


Map to lattice: [&lt;0 -1 1 -4|, &lt;0 -1 0 -1|]
[[Projection pair]]: 7 65536/9375


Map: [&lt;1 0 0 16|, &lt;0 1 0 -1|, &lt;0 0 1 -5|]
== Horwellic  ==
 
Subgroup: 2.3.5.7.11
Generators: 2, 3, 5


Edos: 9, 22, 25, 28, 31, 34, 40, 53, 56, 65, 75, 84, 87, 118, 140, 171, 824, 995
[[Comma list]]: 3025/3024, 65625/65536


Badness: 0.000173
[[Mapping]]: [{{val| 1 0 0 16 10 }}, {{val| 0 1 0 -1 1 }}, {{val| 0 0 2 -10 -7 }}]
 
[[Projection pair]]: 7 65536/9375
 
== Horwellic  ==
Commas: 3025/3024, 65625/65536


Map: [&lt;1 0 0 16 10|, &lt;0 1 0 -1 1|, &lt;0 0 2 -10 -7|]
{{Val list|legend=1| 31, 87, 118, 193, 224, 311, 342 }}


EDOs: 31, 87, 118, 193, 224, 311, 342
[[Badness]]: 0.505 × 10<sup>-3</sup>


Badness: 0.000505
== Zelda ==
Subgroup: 2.3.5.7.11


== Zelda  ==
[[Comma list]]: 385/384, 3388/3375
Commas: 385/384, 3388/3375


Map: [&lt;1 0 0 16 -9|, &lt;0 1 0 -1 2|, &lt;0 0 1 -5 4|]
[[Mapping]]: [{{val| 1 0 0 16 -9 }}, {{val| 0 1 0 -1 2 }}, {{val| 0 0 1 -5 4 }}]


EDOs: 22, 31, 53, 84, 87, 118, 258e, 376de, 547de, 665dee
{{Val list|legend=1| 22, 31, 53, 84, 87, 118, 258e, 376de, 547de, 665dee }}


Badness: 0.000642
[[Badness]]: 0.642 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Horwell]]
[[Category:Horwell family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 18:17, 16 October 2021

The horwell family of rank-3 temperaments tempers out the horwell comma, 65625/65536.

Horwell

Subgroup: 2.3.5.7

Comma list: 65625/65536

Mapping: [⟨1 0 0 16], ⟨0 1 0 -1], ⟨0 0 1 -5]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [⟨0 -1 1 -4], ⟨0 -1 0 -1]]

Lattice basis:

5/4 length = 0.6213, 16/15 length = 1.3863
Angle (5/4, 16/15) = 64.6379

Minimax tuning:

[[1 0 0 0⟩, [16/7 6/7 -5/7 -1/7⟩, [16/7 -1/7 2/7 -1/7⟩, [16/7 -1/7 -5/7 6/7⟩]
Eigenmonzos: 2, 7/6, 6/5
[[1 0 0 0⟩, [16/13 12/13 -5/13 -1/13⟩, [32/13 -2/13 3/13 -2/13⟩, [32/13 -2/13 -10/13 11/13⟩]
Eigenmonzos: 2, 9/7, 7/5

Template:Val list

Badness: 0.173 × 10-3

Projection pair: 7 65536/9375

Horwellic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 65625/65536

Mapping: [⟨1 0 0 16 10], ⟨0 1 0 -1 1], ⟨0 0 2 -10 -7]]

Template:Val list

Badness: 0.505 × 10-3

Zelda

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375

Mapping: [⟨1 0 0 16 -9], ⟨0 1 0 -1 2], ⟨0 0 1 -5 4]]

Template:Val list

Badness: 0.642 × 10-3

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