Hemimage family: Difference between revisions

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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).

The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

== Hemimage ==

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[[Comma list]]: 10976/10935

: mapping generators: ~2, ~3, ~27~~/14~~

: mapping generators: ~2, ~3, ~28/27

[[Minimax tuning]]:

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* [[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

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7

{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}

[[Badness]] (~~Smith~~): 0.~~199 × 10-3~~

[[Badness]] (Sintel): 0.878

[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7

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[[Comma list]]: 540/539, 896/891

[[Minimax tuning]]:

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{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}

[[Badness]] (~~Smith~~): 0.~~702 × 10-3~~

[[Badness]] (Sintel): 0.844

=== 13-limit ===

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Comma list: 352/351, 364/363, 540/539

Mapping: {{mapping| 1 0 2 -1 ~~6 11~~ | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}

Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}

{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}

Badness (~~Smith~~): 1.~~06 × 10-3~~

Badness (Sintel): 0.993

=== Tin ===

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Comma list: 144/143, 196/195, 729/728

Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}

Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}

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

Badness (~~Smith~~): 1.~~07 × 10-3~~

Badness (Sintel): 1.00

== Turan ==

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[[Comma list]]: 441/440, 10976/10935

[[Minimax tuning]]:

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: [[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, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}

[[Badness]] (~~Smith~~): 0.~~936 × 10-3~~

[[Badness]] (Sintel): 1.13

[[Category:Temperament families]]

[[Category:Hemimage family| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-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).
+The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).
 == Hemimage  ==
@@ Line 7: / Line 7: @@
 [[Comma list]]: 10976/10935
-{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }}
+{{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
-: mapping generators: ~2, ~3, ~27/14
+: mapping generators: ~2, ~3, ~28/27
 [[Minimax tuning]]:
@@ Line 16: / Line 16: @@
 * [[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
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7
 {{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
-[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.878
 [[Projection pair]]: <code>5 10976/2187</code> to 2.3.7
@@ Line 29: / Line 29: @@
 [[Comma list]]: 540/539, 896/891
-{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}
+{{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}
 [[Minimax tuning]]:
@@ Line 38: / Line 38: @@
 {{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
-[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.844
 === 13-limit  ===
@@ Line 45: / Line 45: @@
 Comma list: 352/351, 364/363, 540/539
-Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}
+Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}
 {{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
-Badness (Smith): 1.06 × 10<sup>-3</sup>
+Badness (Sintel): 0.993
 === Tin ===
@@ Line 56: / Line 56: @@
 Comma list: 144/143, 196/195, 729/728
-Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}
+Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}
 {{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
-Badness (Smith): 1.07 × 10<sup>-3</sup>
+Badness (Sintel): 1.00
 == Turan ==
@@ Line 67: / Line 67: @@
 [[Comma list]]: 441/440, 10976/10935
-{{Mapping|legend=1| 1 0 2 -1 -7 | 0 1 2 3 6 | 0 0 -3 -1 1 }}
+{{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}
 [[Minimax tuning]]:
@@ Line 74: / Line 74: @@
 : [[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, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}
-[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.13
 [[Category:Temperament families]]
 [[Category:Hemimage family| ]] <!-- main article -->
 [[Category:Rank 3]]