Hemimage family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The '''hemimage family''' of | The '''hemimage family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] tempers out the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). | ||
== Hemimage == | == Hemimage == | ||
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{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }} | {{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }} | ||
: mapping generators: ~2, ~3, ~27/14 | : mapping generators: ~2, ~3, ~27/14 | ||
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* [[7-odd-limit]] | * [[7-odd-limit]] | ||
: {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }} | : {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7 | ||
* [[9-odd-limit]] | * [[9-odd-limit]] | ||
: {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }} | : {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5 | ||
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }} | {{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }} | ||
[[Badness]]: 0.199 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.199 × 10<sup>-3</sup> | ||
[[Projection pair]]: 5 10976/2187 to 2.3.7 | [[Projection pair]]: <code>5 10976/2187</code> to 2.3.7 | ||
== Uni == | == Uni == | ||
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* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}] | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9 | ||
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }} | {{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }} | ||
[[Badness]]: 0.702 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.702 × 10<sup>-3</sup> | ||
=== 13-limit === | === 13-limit === | ||
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Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }} | Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }} | ||
Badness: 1.06 × 10<sup>-3</sup> | Badness (Smith): 1.06 × 10<sup>-3</sup> | ||
=== Tin === | === Tin === | ||
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Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }} | Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }} | ||
Badness: 1.07 × 10<sup>-3</sup> | Badness (Smith): 1.07 × 10<sup>-3</sup> | ||
== Turan == | == Turan == | ||
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* [[11-odd-limit]] | * [[11-odd-limit]] | ||
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}] | : [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}] | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9 | ||
{{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }} | {{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }} | ||
[[Badness]]: 0.936 × 10<sup>-3</sup> | [[Badness]] (Smith): 0.936 × 10<sup>-3</sup> | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Hemimage family| ]] <!-- main article --> | [[Category:Hemimage family| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
Latest revision as of 18:49, 28 April 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 hemimage family of rank-3 temperaments tempers out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935).
Hemimage
Subgroup: 2.3.5.7
Comma list: 10976/10935
Mapping: [⟨1 0 2 -1], ⟨0 1 2 3], ⟨0 0 -3 -1]]
- mapping generators: ~2, ~3, ~27/14
- [[1 0 0 0⟩, [5/8 1/8 -1/8 3/8⟩, [5/8 -7/8 7/8 3/8⟩, [0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.5/3.7
- [[1 0 0 0⟩, [5/9 2/9 -1/9 1/3⟩, [10/9 -14/9 7/9 2/3⟩, [0 0 0 1⟩]
- unchanged-interval (eigenmonzo) basis: 2.7.9/5
Optimal ET sequence: 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695
Badness (Smith): 0.199 × 10-3
Projection pair: 5 10976/2187 to 2.3.7
Uni
Subgroup: 2.3.5.7.11
Comma list: 540/539, 896/891
Mapping: [⟨1 0 2 -1 6], ⟨0 1 2 3 -1], ⟨0 0 -3 -1 -1]]
- [[1 0 0 0 0⟩, [16/11 6/11 1/11 0 -3/11⟩, [0 0 1 0 0⟩, [19/11 14/11 6/11 0 -7/11⟩, [32/11 -10/11 2/11 0 5/11⟩]
- unchanged-interval (eigenmonzo) basis: 2.5.11/9
Optimal ET sequence: 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee
Badness (Smith): 0.702 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 540/539
Mapping: [⟨1 0 2 -1 6 11], ⟨0 1 2 3 -1 -4], ⟨0 0 -3 -1 -1 -1]]
Optimal ET sequence: 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef
Badness (Smith): 1.06 × 10-3
Tin
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 729/728
Mapping: [⟨1 0 2 -1 6 -2], ⟨0 1 2 3 -1 3], ⟨0 0 -3 -1 -1 1]]
Optimal ET sequence: 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff
Badness (Smith): 1.07 × 10-3
Turan
Subgroup: 2.3.5.7.11
Comma list: 441/440, 10976/10935
Mapping: [⟨1 0 2 -1 -7], ⟨0 1 2 3 6], ⟨0 0 -3 -1 1]]
- [[1 0 0 0 0⟩, [19/12 -2/3 1/12 0 1/4⟩, [19/6 -10/3 7/6 0 1/2⟩, [37/12 -8/3 7/12 0 3/4⟩, [19/6 -10/3 1/6 0 3/2⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5.11/9
Optimal ET sequence: 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335
Badness (Smith): 0.936 × 10-3