Hemimage family: Difference between revisions
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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). | 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 == | ||
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. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 0 -1 -2 | 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, ~28/27 | : 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]]: | [[Minimax tuning]]: | ||
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{{Mapping|legend=1| 1 0 -1 -2 5 | 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]]: | [[Minimax tuning]]: | ||
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Mapping: {{mapping| 1 0 -1 -2 5 10 | 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 }} | {{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }} | ||
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Mapping: {{mapping| 1 0 -1 -2 5 -1 | 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 }} | {{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }} | ||
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{{Mapping|legend=1| 1 0 -1 -2 -6 | 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]]: | [[Minimax tuning]]: | ||
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[[Badness]] (Sintel): 1.13 | [[Badness]] (Sintel): 1.13 | ||
[[Category:Hemimage family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-3 temperaments]] | ||