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Diaschismic rank three family: Difference between revisions

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These are the rank-3 temperaments tempering out the [[diaschisma]], 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
The '''diaschismic rank-3 family''' of [[temperament]]s [[tempering out|tempers out]] the diaschisma, [[2048/2025]]. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.


== Rank-3 diaschismic ==
== Rank-3 diaschismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[2048/2025]]
[[Comma list]]: [[2048/2025]]


[[Mapping]]: [{{val| 2 0 11 0 }}, {{val| 0 1 -2 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 2 0 11 0 | 0 1 -2 0 | 0 0 0 1 }}


Mapping generators: ~45/32, ~3, ~7
: mapping generators: ~45/32, ~3, ~7


[[POTE generator]]s: ~3/2 = 704.8982, ~7/4 = 972.1330
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982, ~7/4 = 972.1330


{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}
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== Julius aka varda ==
== Julius aka varda ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 896/891
[[Comma list]]: 176/175, 896/891


[[Mapping]]: [{{val| 2 0 11 0 14 }}, {{val| 0 1 -2 0 -4 }}, {{val| 0 0 0 1 1 }}]
{{Mapping|legend=1| 2 0 11 0 14 | 0 1 -2 0 -4 | 0 0 0 1 1 }}


[[Mapping to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]
[[Mapping to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]
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* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| -1/8 3/2 -5/4 0 1 }}, {{monzo| 11/8 -1/2 -1/4 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| -1/8 3/2 -5/4 0 1 }}, {{monzo| 11/8 -1/2 -1/4 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 12/11, 10/9
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/3


{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}
{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}
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Comma list: 176/175, 351/350, 364/363
Comma list: 176/175, 351/350, 364/363


Mapping: [{{val| 2 0 11 0 14 24 }}, {{val| 0 1 -2 0 -4 -7 }}, {{val| 0 0 0 1 1 1 }}]
Mapping: {{mapping| 2 0 11 0 14 24 | 0 1 -2 0 -4 -7 | 0 0 0 1 1 1 }}


{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
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Comma list: 136/135, 176/175, 256/255, 351/350
Comma list: 136/135, 176/175, 256/255, 351/350


Mapping: [{{val| 2 0 11 0 14 24 5}}, {{val| 0 1 -2 0 -4 -7 1}}, {{val| 0 0 0 1 1 1 0}}]
Mapping: {{mapping| 2 0 11 0 14 24 5 | 0 1 -2 0 -4 -7 1 | 0 0 0 1 1 1 0 }}


{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
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Comma list: 176/175, 325/324, 896/891
Comma list: 176/175, 325/324, 896/891


Mapping: [{{val| 2 0 11 0 14 -18 }}, {{val| 0 1 -2 0 -4 8 }}, {{val| 0 0 0 1 1 0 }}]
Mapping: {{mapping| 2 0 11 0 14 -18 | 0 1 -2 0 -4 8 | 0 0 0 1 1 0 }}


{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
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Comma list: 136/135, 176/175, 256/255, 325/324
Comma list: 136/135, 176/175, 256/255, 325/324


Mapping: [{{val| 2 0 11 0 14 -18 5}}, {{val| 0 1 -2 0 -4 8 1}}, {{val| 0 0 0 1 1 0 0}}]
Mapping: {{mapping| 2 0 11 0 14 -18 5 | 0 1 -2 0 -4 8 1 | 0 0 0 1 1 0 0 }}


{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
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Comma list: 136/135, 176/175, 190/189, 256/255, 286/285
Comma list: 136/135, 176/175, 190/189, 256/255, 286/285


Mapping: [{{val| 2 0 11 0 14 -18 5 -13}}, {{val| 0 1 -2 0 -4 8 1 5}}, {{val| 0 0 0 1 1 0 0 1}}]
Mapping: {{mapping| 2 0 11 0 14 -18 5 -13 | 0 1 -2 0 -4 8 1 5 | 0 0 0 1 1 0 0 1 }}


{{Optimal ET sequence|legend=1| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
{{Optimal ET sequence|legend=1| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}


== Augustus ==
== Augustus ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 1232/1215
[[Comma list]]: 385/384, 1232/1215


[[Mapping]]: [{{val| 2 0 11 0 3 }}, {{val| 0 1 -2 0 3 }}, {{val| 0 0 0 1 -1 }}]
{{Mapping|legend=1| 2 0 11 0 3 | 0 1 -2 0 3 | 0 0 0 1 -1 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|1 0 0 0 0 }}, |11/8 1/2 -1/4 0 0 }}, |11/4 -1 1/2 0 0 }}, |45/16 3/4 -3/8 1/2 -1/2 }}, |45/16 3/4 -3/8 -1/2 1/2 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| 45/16 3/4 -3/8 1/2 -1/2 }}, {{monzo| 45/16 3/4 -3/8 -1/2 1/2 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 14/11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/7


{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
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== Tiberius ==
== Tiberius ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 2048/2025
[[Comma list]]: 441/440, 2048/2025


[[Mapping]]: [{{val| 2 0 11 0 -17 }}, {{val| 0 1 -2 0 4 }}, {{val| 0 0 0 1 2 }}]
{{Mapping|legend=1| 2 0 11 0 -17 | 0 1 -2 0 4 | 0 0 0 1 2 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -3 2 -1 2 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -3 2 -1 2 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/5, 7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.7


{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
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== Claudius ==
== Claudius ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 2048/2025
[[Comma list]]: 540/539, 2048/2025


[[Mapping]]: [{{val| 2 0 11 0 15 }}, {{val| 0 1 -2 0 1 }}, {{val| 0 0 0 1 -2 }}]
{{Mapping|legend=1| 2 0 11 0 15 | 0 1 -2 0 1 | 0 0 0 1 -2 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 9/7 0 -3/7 2/7 1/7 }}, {{monzo| 41/14 0 6/7 -4/7 -2/7 }}, {{monzo| 41/14 0 -1/7 3/7 -2/7 }}, {{monzo| 41/14 0 -1/7 -4/7 5/7 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 9/7 0 -3/7 2/7 1/7 }}, {{monzo| 41/14 0 6/7 -4/7 -2/7 }}, {{monzo| 41/14 0 -1/7 3/7 -2/7 }}, {{monzo| 41/14 0 -1/7 -4/7 5/7 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/5, 11/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/5


{{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}
{{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Diaschismic rank-3 family| ]] <!-- main article -->
[[Category:Diaschismic rank-3 family| ]] <!-- main article -->
[[Category:Diaschismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
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