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 ==

Hemimage is generated by a [[3/2|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.

~~== Hemimage ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8555{{c}}, ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}

: [[error map]]: {{val| -0.145 +0.507 +0.155 -0.683 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6137{{c}}, ~28/27 = 60.4761{{c}}

: error map: {{val| 0.000 +0.659 +0.342 -0.509 }}

[[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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.1497{{c}}, ~3/2 = 703.3014{{c}}, ~28/27 = 60.0617{{c}}

: [[error map]]: {{val| -0.859 +0.496 -0.376 +0.290 +2.041 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6475{{c}}, ~28/27 = 59.9675{{c}}

: error map: {{val| 0.000 +1.693 +0.884 +2.084 +5.002 }}

[[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 tunings:

* WE: ~2 = 1199.0867{{c}}, ~3/2 = 703.4048{{c}}, ~28/27 = 60.0034{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9457{{c}}, ~28/27 = 59.8167{{c}}

{{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 tunings:

* WE: ~2 = 1198.8801{{c}}, ~3/2 = 702.5101{{c}}, ~28/27 = 60.9480{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8952{{c}}, ~28/27 = 60.9129{{c}}

{{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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0289{{c}}, ~3/2 = 702.1366{{c}}, ~28/27 = 60.8231{{c}}

: [[error map]]: {{val| +0.029 +0.211 +0.458 -1.564 +0.679 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1290{{c}}, ~28/27 = 60.8257{{c}}

: error map: {{val| 0.000 +0.174 +0.421 -1.613 +0.630 }}

[[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 ==
+Hemimage is generated by a [[3/2|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.
-== Hemimage  ==
 [[Subgroup]]: 2.3.5.7
 [[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
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8555{{c}}, ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
+: [[error map]]: {{val| -0.145 +0.507 +0.155 -0.683 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6137{{c}}, ~28/27 = 60.4761{{c}}
+: error map: {{val| 0.000 +0.659 +0.342 -0.509 }}
 [[Minimax tuning]]:
@@ Line 16: / Line 24: @@
 * [[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 37: @@
 [[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 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.1497{{c}}, ~3/2 = 703.3014{{c}}, ~28/27 = 60.0617{{c}}
+: [[error map]]: {{val| -0.859 +0.496 -0.376 +0.290 +2.041 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6475{{c}}, ~28/27 = 59.9675{{c}}
+: error map: {{val| 0.000 +1.693 +0.884 +2.084 +5.002 }}
 [[Minimax tuning]]:
@@ Line 38: / Line 52: @@
 {{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 59: @@
 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 tunings:
+* WE: ~2 = 1199.0867{{c}}, ~3/2 = 703.4048{{c}}, ~28/27 = 60.0034{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9457{{c}}, ~28/27 = 59.8167{{c}}
 {{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 74: @@
 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 tunings:
+* WE: ~2 = 1198.8801{{c}}, ~3/2 = 702.5101{{c}}, ~28/27 = 60.9480{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8952{{c}}, ~28/27 = 60.9129{{c}}
 {{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 89: @@
 [[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 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0289{{c}}, ~3/2 = 702.1366{{c}}, ~28/27 = 60.8231{{c}}
+: [[error map]]: {{val| +0.029 +0.211 +0.458 -1.564 +0.679 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1290{{c}}, ~28/27 = 60.8257{{c}}
+: error map: {{val| 0.000 +0.174 +0.421 -1.613 +0.630 }}
 [[Minimax tuning]]:
@@ Line 74: / Line 102: @@
 : [[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]]