Hemimage family: Difference between revisions

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===Vital statistics===
{{Technical data page}}
[[Comma|Comma]]: hemimage 10976/10935
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).


7-limit minimax
== Hemimage  ==
[[Subgroup]]: 2.3.5.7


[|1 0 0 0>, |5/8 1/8 -1/8 3/8>,
[[Comma list]]: 10976/10935
|5/8 -7/8 7/8 3/8>, |0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5
{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }}
: mapping generators: ~2, ~3, ~27/14


9-limit minimax
[[Minimax tuning]]:
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5


[|1 0 0 0>, |5/9 2/9 -1/9 1/3>,  
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
|10/9 -14/9 7/9 2/3>, |0 0 0 1>]


[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7
[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>


Map: [&lt;1 0 2 -1|, &lt;0 1 2 3|, &lt;0 0 -3 -1|]
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


[[generator|Generators]]: 2, 3, 27/14
== Uni ==
[[Subgroup]]: 2.3.5.7.11


[[EDO|Edos]]: 14, 19, 41, 58, 77, 80, [[99edo|99]], [[118edo|118]], [[140edo|140]], [[217edo|217]], [[316edo|316]], [[338edo|338]], [[357edo|357]], [[415edo|415]], [[456edo|456]], [[654edo|654]], [[695edo|695]]
[[Comma list]]: 540/539, 896/891


Badness: 0.000199
{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}


[[Projection_pair|Projection pair]]: 5 10976/2187 to 2.3.7
[[Minimax tuning]]:
* [[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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9


==Uni==
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
[[Comma|Commas]]: 540/539, 896/891


[[Minimax_tuning|Minimax tuning]]
[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;,
=== 13-limit  ===
|0 0 1 0 0&gt;, |19/11 14/11 6/11 0 -7/11&gt;,
Subgroup: 2.3.5.7.11.13
|32/11 -10/11 2/11 0 5/11&gt;]


[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
Comma list: 352/351, 364/363, 540/539


Map: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}


[[generator|Generators]]: 2, 3, 27/14
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}


[[EDO|Edos]]: [[19edo|19]], [[22edo|22]], [[41edo|41]], [[58edo|58]], [[80edo|80]], [[121edo|121]]
Badness (Smith): 1.06 × 10<sup>-3</sup>


Badness: 0.000702
=== Tin ===
Subgroup: 2.3.5.7.11.13


===13-limit===
Comma list: 144/143, 196/195, 729/728
Commas: 352/351, 364/363, 540/539


Map: [&lt;1 0 2 -1 6 11|, &lt;0 1 2 3 -1 -4|, &lt;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 }}


EDOs: 22, 41, 58, 80, 121
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}


Badness: 0.00106
Badness (Smith): 1.07 × 10<sup>-3</sup>


===Tin===
== Turan ==
Commas: 144/143, 196/195, 729/728
[[Subgroup]]: 2.3.5.7.11


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 1 2 3 -1 3|, &lt;0 0 -3 -1 -1 1|]
[[Comma list]]: 441/440, 10976/10935


EDOs: 19, 36, 41, 58
{{Mapping|legend=1| 1 0 2 -1 -7 | 0 1 2 3 6 | 0 0 -3 -1 1 }}


Badness: 0.00107
[[Minimax tuning]]:  
* [[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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


==Turan==
{{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}
[[Comma|Commas]]: 10976/10935, 441/440


[[Minimax_tuning|Minimax tuning]]
[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |19/12 -2/3 1/12 0 1/4&gt;,
[[Category:Temperament families]]
|19/6 -10/3 7/6 0 1/2&gt;,
[[Category:Hemimage family| ]] <!-- main article -->
|37/12 -8/3 7/12 0 3/4&gt;,
[[Category:Rank 3]]
|19/6 -10/3 1/6 0 3/2&gt;]
 
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9
 
Map: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
 
[[generator|Generators]]: 2, 3, 27/14
 
[[EDO|Edos]]: [[41edo|41]], [[58edo|58]], [[118edo|118]], [[159edo|159]], [[217edo|217]], [[335edo|335]]
 
Badness: 0.000936
 
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Hemimage]]

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

Minimax tuning:

[[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]]

Minimax tuning:

[[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]]

Minimax tuning:

[[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

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