Orwellismic family: Difference between revisions

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__FORCETOC__
The '''orwellismic family''' of rank-3 temperaments tempers out the orwellisma, [[1728/1715]].
The '''orwellismic family''' of rank three temperaments tempers out the orwellisma, [[1728/1715]].


= Orwellismic =
= Orwellismic =
[[Comma]]: 1728/1715
Subgroup: 2.3.5.7


[[7-odd-limit|7-]] and [[9-odd-limit|9-limit]] [[minimax tuning|minimax]]
[[Comma list]]: 1728/1715


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


<3/2 3/4 -1/4 1/4|]
Mapping generators: ~2, ~3, ~12/7


[[Eigenmonzo]]s: 2, 4/3, 7/5
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 3/2 3/4 3/4 -3/4 }}, {{monzo| 3/2 3/4 -1/4 1/4 }}]
: [[Eigenmonzo]]s: 2, 4/3, 7/5


Lattice basis: 7/6 0.7526 3/2 1.3931
Map to lattice: [{{val| 0 0 -3 1 }}, {{val| 0 1 0 1 }}]


Angle(7/6, 3/2) = 97.5081
Lattice basis:
: 7/6 length = 0.7526, 3/2 length = 1.3931
: Angle (7/6, 3/2) = 97.5081


Map to lattice: [<0 0 -3 1|, <0 1 0 1|]
{{Val list|legend=1| 9, 14, 17, 18, 22, 26, 27, 31, 49, 53, 58, 80, 84, 111, 142, 164d, 306d, 417cd }}


[[Map]]: [<1 0 0 2|, <0 1 0 1|, <0 0 3 -1|]
[[Badness]]: 0.187 × 10<sup>-3</sup>
 
[[Generator]]s: 2, 3, 12/7
 
{{EDOs|legend=1| 9, 14, 17, 18, 22, 26, 27, 31, 49, 53, 58, 80, 84, 111, 142, 164d, 306d, 417cd }}
 
[[Badness]]: 0.000187


[[Projection pair]]: 5 1728/343 to 2.3.7
[[Projection pair]]: 5 1728/343 to 2.3.7


[[Scale]]s: [[orwellismic9]]
Scales: [[orwellismic9]]


= Orwellian =
= Orwellian =
Commas: 99/98, 385/384
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 0 2 5|, &lt;0 1 0 1 0| &lt;0 0 3 -1 -2|]
[[Comma list]]: 99/98, 385/384


{{EDOs|legend=1| 22, 31, 53, 84e, 163e, 194e, 247cde, 278bdee }}
[[Mapping]]: [{{val| 1 0 0 2 5 }}, {{val| 0 1 0 1 0 }}, {{val| 0 0 3 -1 -2 }}]


Badness: 0.000715
{{Val list|legend=1| 22, 31, 53, 84e, 163e, 194e, 247cde, 278bdee }}
 
[[Badness]]: 0.715 × 10<sup>-3</sup>


= Guanyin =
= Guanyin =
[[Comma]]s: 176/175, 540/539
[[Comma list]]: 176/175, 540/539


[[11-limit]] [[minimax tuning|minimax]]
[[Mapping]]: [{{val| 1 0 0 2 -2 }}, {{val| 0 1 0 1 1 }}, {{val| 0 0 3 -1 5 }}]


[&lt;1 0 0 0 0|, &lt;-2/9 0 -5/9 5/9 4/9|, &lt;4/3 0 1/3 -1/3 1/3|,  
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| -2/9 0 -5/9 5/9 4/9 }}, {{monzo| 4/3 0 1/3 -1/3 1/3 }}, {{monzo| 4/3 0 -2/3 2/3 1/3 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo]]s: 2, 11/8, 7/5


&lt;4/3 0 -2/3 2/3 1/3|, &lt;0 0 0 0 1|]
Map to lattice: [{{val| 0 0 -3 1 -5 }}, {{val| 0 1 0 1 1 }}]


[[Eigenmonzo]]s: 2, 11/8, 7/5
Lattice basis:  
: 7/6 length = 0.5167, 3/2 length = 1.3055
: Angle (7/6, 3/2) = 79.0940


Lattice basis: 7/6 0.5167 3/2 1.3055
{{Val list|legend=1| 9, 22, 31, 53, 58, 80, 111, 280cd, 391cd }}


Angle(7/6, 3/2) = 79.0940
[[Badness]]: 0.411 × 10<sup>-3</sup>
 
Map to lattice: [&lt;0 0 -3 1 -5|, &lt;0 1 0 1 1|]
 
[[Map]]: [&lt;1 0 0 2 -2|, &lt;0 1 0 1 1|, &lt;0 0 3 -1 5|]
 
[[Generator]]s: 2, 3, 12/7
 
{{EDOs|legend=1| 9, 22, 31, 53, 58, 80, 111, 280cd, 391cd }}
 
[[Badness]]: 0.000411


[[Projection pair]]s: 5 1728/343 11 186624/16807 to 2.3.7
[[Projection pair]]s: 5 1728/343 11 186624/16807 to 2.3.7


[[Scale]]s: [[guanyin22]]
Scales: [[guanyin22]]


Music:  
Music:  
Line 74: Line 69:


== 13-limit ==
== 13-limit ==
[[Comma]]s: 176/175, 351/350, 540/539
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 540/539


[[Map]]: [&lt;1 0 0 2 -2 3|, &lt;0 1 0 1 1 -2|, &lt;0 0 3 -1 5 5|]
Mapping: [{{val| 1 0 0 2 -2 3 }}, {{val| 0 1 0 1 1 -2 }}, {{val| 0 0 3 -1 5 5 }}]


{{EDOs|legend=1| 22, 31, 53, 58, 80, 111, 280cdf, 391cdf }}
Vals: {{Val list| 22, 31, 53, 58, 80, 111, 280cdf, 391cdf }}


[[Badness]]: 0.000758
Badness: 0.758 × 10<sup>-3</sup>


Projection pairs: 5 1728/343 11 186624/16807 221184/16897 to 2.3.7
Projection pairs: 5 1728/343 11 186624/16807 221184/16897 to 2.3.7


== Georgian ==
== Georgian ==
[[Comma]]s: 144/143, 176/175, 196/195
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 196/195


[[Map]]: [&lt;1 0 0 2 -2 6|, &lt;0 1 0 1 1 1|, &lt;0 0 3 -1 5 -5|]
Mapping: [{{val| 1 0 0 2 -2 6 }}, {{val| 0 1 0 1 1 1 }}, {{val| 0 0 3 -1 5 -5 }}]


{{EDOs|legend=1| 9, 31, 46, 48, 58 }}
Vals: {{Val list| 9, 31, 46, 48, 58 }}


[[Badness]]: 0.000904
Badness: 0.904 × 10<sup>-3</sup>


Projection pairs: 5 1728/343 11 186624/16807 16807/1296 to 2.3.7
Projection pairs: 5 1728/343 11 186624/16807 16807/1296 to 2.3.7


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Orwellismic]]
[[Category:Orwellismic]]
[[Category:Rank 3]]
[[Category:Guanyin]]
[[Category:Guanyin]]
[[Category:Rank 3]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 15:28, 25 April 2021

The orwellismic family of rank-3 temperaments tempers out the orwellisma, 1728/1715.

Orwellismic

Subgroup: 2.3.5.7

Comma list: 1728/1715

Mapping: [⟨1 0 0 2], ⟨0 1 0 1], ⟨0 0 3 -1]]

Mapping generators: ~2, ~3, ~12/7

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [3/2 3/4 3/4 -3/4⟩, [3/2 3/4 -1/4 1/4⟩]
Eigenmonzos: 2, 4/3, 7/5

Map to lattice: [⟨0 0 -3 1], ⟨0 1 0 1]]

Lattice basis:

7/6 length = 0.7526, 3/2 length = 1.3931
Angle (7/6, 3/2) = 97.5081

Template:Val list

Badness: 0.187 × 10-3

Projection pair: 5 1728/343 to 2.3.7

Scales: orwellismic9

Orwellian

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384

Mapping: [⟨1 0 0 2 5], ⟨0 1 0 1 0], ⟨0 0 3 -1 -2]]

Template:Val list

Badness: 0.715 × 10-3

Guanyin

Comma list: 176/175, 540/539

Mapping: [⟨1 0 0 2 -2], ⟨0 1 0 1 1], ⟨0 0 3 -1 5]]

Minimax tuning:

[[1 0 0 0 0⟩, [-2/9 0 -5/9 5/9 4/9⟩, [4/3 0 1/3 -1/3 1/3⟩, [4/3 0 -2/3 2/3 1/3⟩, [0 0 0 0 1⟩]
Eigenmonzos: 2, 11/8, 7/5

Map to lattice: [⟨0 0 -3 1 -5], ⟨0 1 0 1 1]]

Lattice basis:

7/6 length = 0.5167, 3/2 length = 1.3055
Angle (7/6, 3/2) = 79.0940

Template:Val list

Badness: 0.411 × 10-3

Projection pairs: 5 1728/343 11 186624/16807 to 2.3.7

Scales: guanyin22

Music:

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539

Mapping: [⟨1 0 0 2 -2 3], ⟨0 1 0 1 1 -2], ⟨0 0 3 -1 5 5]]

Vals: Template:Val list

Badness: 0.758 × 10-3

Projection pairs: 5 1728/343 11 186624/16807 221184/16897 to 2.3.7

Georgian

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195

Mapping: [⟨1 0 0 2 -2 6], ⟨0 1 0 1 1 1], ⟨0 0 3 -1 5 -5]]

Vals: Template:Val list

Badness: 0.904 × 10-3

Projection pairs: 5 1728/343 11 186624/16807 16807/1296 to 2.3.7

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