Hemimage temperaments: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). | This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). | ||
Temperaments discussed elsewhere are: | |||
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]] | * ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]] | ||
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]] | * ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]] | ||
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* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]] | * ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]] | ||
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]] | * ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]] | ||
Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]]. | |||
== Chromat == | == Chromat == | ||
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== Squarschmidt == | == Squarschmidt == | ||
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].'' | |||
[[ | Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]]. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Badness (Sintel): 1.26 | Badness (Sintel): 1.26 | ||
== Leapmonth == | |||
Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 10976/10935, 51200/50421 | |||
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}} | |||
: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}} | |||
: error map: {{val| 0.000 +2.977 +1.093 +5.150 }} | |||
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }} | |||
[[Badness]] (Sintel): 4.79 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 540/539, 896/891, 1331/1323 | |||
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }} | |||
Badness (Sintel): 1.88 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 169/168, 352/351, 364/363, 540/539 | |||
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}} | |||
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }} | |||
Badness (Sintel): 1.53 | |||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Hemimage temperaments| ]] <!-- main article --> | [[Category:Hemimage temperaments| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||