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Diaschismic rank three family: Difference between revisions

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These are the rank three 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.
{{Technical data page}}
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.


==Julius==
== Rank-3 diaschismic ==
Commas: 176/175, 896/891
[[Subgroup]]: 2.3.5.7


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


[|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>,
{{Mapping|legend=1| 2 0 11 0 | 0 1 -2 0 | 0 0 0 1 }}
|-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]


Eigenmonzos: 2, 12/11, 10/9
: mapping generators: ~45/32, ~3, ~7


Lattice basis: 3/2, 9/7
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982, ~7/4 = 972.1330


Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}


Angle(3/2, 9/7) = 82.289 degrees
[[Badness]]: 0.442 × 10<sup>-3</sup>


Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1 |]
== Julius aka varda ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 58, 68, 80, 126
[[Comma list]]: 176/175, 896/891


Badness: 0.000728
{{Mapping|legend=1| 2 0 11 0 14 | 0 1 -2 0 -4 | 0 0 0 1 1 }}


Julius[22] [[Hobbits|hobbit]] [[Transversal|transversal]]:
[[Mapping to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]


28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2
Lattice basis:
: 3/2, 9/7
: Angle (3/2, 9/7) = 82.289 degrees
 
[[Minimax tuning]]:
* [[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 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/3
 
{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}
 
[[Badness]]: 0.728 × 10<sup>-3</sup>
 
Julius[22] [[hobbit]] [[transversal]]:
: 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2


Julius[24] hobbit transversal
Julius[24] hobbit transversal
: 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
=== Varda ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 364/363
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 }}
Badness: 0.970 × 10<sup>-3</sup>
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 256/255, 351/350
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 }}
=== Julius ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 896/891
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 }}
Badness: 1.78 × 10<sup>-3</sup>
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
Comma list: 136/135, 176/175, 256/255, 325/324


===13-limit===
Mapping: {{mapping| 2 0 11 0 14 -18 5 | 0 1 -2 0 -4 8 1 | 0 0 0 1 1 0 0 }}
Commas: 176/175, 325/324, 896/891


Map: [&lt;2 0 11 0 14 -18|, &lt;0 1 -2 0 -4 8|, &lt;0 0 0 1 1 0|]
{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}


EDOs: 12, 46, 68, 80
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.00178
Comma list: 136/135, 176/175, 190/189, 256/255, 286/285


==Augustus==
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 }}
Commas: 385/384, 1232/1215


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


[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
== Augustus ==
|45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
[[Subgroup]]: 2.3.5.7.11


Eigenmonzos: 2, 10/9, 14/11
[[Comma list]]: 385/384, 1232/1215


Map: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]
{{Mapping|legend=1| 2 0 11 0 3 | 0 1 -2 0 3 | 0 0 0 1 -1 }}


EDOs: 46, 68, 114, 124
[[Minimax tuning]]:  
* [[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| 45/16 3/4 -3/8 1/2 -1/2 }}, {{monzo| 45/16 3/4 -3/8 -1/2 1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/7


Badness: 0.00158
{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}


==Tiberius==
[[Badness]]: 1.58 × 10<sup>-3</sup>
Commas: 2048/2025, 441/440


Minimax tuning
== Tiberius ==
[[Subgroup]]: 2.3.5.7.11


[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
[[Comma list]]: 441/440, 2048/2025
|0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]


Eigenmonzos: 2, 9/5, 7
{{Mapping|legend=1| 2 0 11 0 -17 | 0 1 -2 0 4 | 0 0 0 1 2 }}


Map: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]
[[Minimax tuning]]:  
* [[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 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


EDOs: 58, 148, 160
{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}


Badness: 0.00171
[[Badness]]: 1.71 × 10<sup>-3</sup>


==Claudius==
== Claudius ==
Commas: 2048/2025, 540/539
[[Subgroup]]: 2.3.5.7.11


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


[|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;,
{{Mapping|legend=1| 2 0 11 0 15 | 0 1 -2 0 1 | 0 0 0 1 -2 }}
|41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]


Eigenmonzos: 2, 7/5, 11/5
[[Minimax tuning]]:
* [[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 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


Map: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
{{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}


EDOs: 58, 80
[[Badness]]: 1.97 × 10<sup>-3</sup>


Badness: 0.00197
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Diaschismic rank-3 family| ]] <!-- main article -->
[[Category:Diaschismic| ]] <!-- key article -->
[[Category:Rank 3]]
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