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Diaschismic rank-3 family: Difference between revisions

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Line 11: Line 11:
: mapping generators: ~45/32, ~3, ~7
: mapping generators: ~45/32, ~3, ~7


[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982{{c}}, ~7/4 = 972.1330{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}, ~7/4 = 971.1782{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 -0.005 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}, ~7/4 = 971.2952{{c}}
: error map: {{val| 0.000 +3.003 +3.769 +2.469 }}


{{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 }}


[[Badness]] (Smith): 0.442 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.95
 
== Varda ==
Varda tempers out 176/175, 896/891 as well as 9801/9800, and is the main extension of rank-3 diaschismic to the 11-limit. Notice the identity {{nowrap| 2048/2025 {{=}} (176/175)⋅(896/891) }}. In the natural 13-limit extension, 351/350, 352/351 and 364/363 are further tempered out, and in the 17-limit, 136/135, 256/255 and 289/288.


== Julius a.k.a. varda ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 29: Line 35:
: 3/2, 9/7
: 3/2, 9/7
: Angle (3/2, 9/7) = 82.289 degrees
: Angle (3/2, 9/7) = 82.289 degrees
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4084{{c}}, ~3/2 = 703.9071{{c}}, ~7/4 = 971.8498{{c}}
: [[error map]]: {{val| -1.183 +0.769 +1.731 +0.657 -1.013 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.6576{{c}}, ~7/4 = 971.9790{{c}}
: error map: {{val| 0.000 +2.703 +4.371 +3.153 +2.031 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 37: Line 49:
{{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 }}


[[Badness]] (Smith): 0.728 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.875


Julius[22] [[hobbit]] [[transversal]]:
; Scales
* Varda[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
: 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
 
* Varda[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
: 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 ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 52: Line 64:
Mapping: {{mapping| 2 0 11 0 14 24 | 0 1 -2 0 -4 -7 | 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=0| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
Optimal tunings:
* WE: ~45/32 = 599.4000{{c}}, ~3/2 = 703.5375{{c}}, ~7/4 = 971.9244{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.2888{{c}}, ~7/4 = 972.0576{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d, 46, 58, 80, 104c, 138cde, 242ccde }}


Badness (Smith): 0.970 × 10<sup>-3</sup>
Badness (Sintel): 0.908


===== 17-limit =====
===== 17-limit =====
Line 63: Line 79:
Mapping: {{mapping| 2 0 11 0 14 24 5 | 0 1 -2 0 -4 -7 1 | 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=0| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
Optimal tunings:
* WE: ~17/12 = 599.5734{{c}}, ~3/2 = 703.8639{{c}}, ~7/4 = 971.4469{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.3790{{c}}, ~7/4 = 971.6313{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d, 46, 58, 80, 104c, 126, 138cde }}
 
Badness (Sintel): 0.841


=== Julius ===
=== Julius ===
Line 71: Line 93:


Mapping: {{mapping| 2 0 11 0 14 -18 | 0 1 -2 0 -4 8 | 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 tunings:
* WE: ~45/32 = 599.5350{{c}}, ~3/2 = 704.6999{{c}}, ~7/4 = 972.2975{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1906{{c}}, ~7/4 = 972.3332{{c}}


{{Optimal ET sequence|legend=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
{{Optimal ET sequence|legend=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}


Badness (Smith): 1.78 × 10<sup>-3</sup>
Badness (Sintel): 1.66


===== 17-limit =====
===== 17-limit =====
Line 83: Line 109:
Mapping: {{mapping| 2 0 11 0 14 -18 5 | 0 1 -2 0 -4 8 1 | 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=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
Optimal tunings:
* WE: ~17/12 = 599.6247{{c}}, ~3/2 = 704.7816{{c}}, ~7/4 = 971.8531{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 705.1801{{c}}, ~7/4 = 971.9669{{c}}
 
{{Optimal ET sequence|legend=0| 12, 22f, 34d, 46, 68, 80, 126, 148d }}
 
Badness (Sintel): 1.18


===== 19-limit =====
===== 19-limit =====
Line 91: Line 123:


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 }}
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 tunings:
* WE: ~17/12 = 599.6179{{c}}, ~3/2 = 704.8484{{c}}, ~7/4 = 972.3437{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 705.2584{{c}}, ~7/4 = 972.4875{{c}}


{{Optimal ET sequence|legend=0| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
{{Optimal ET sequence|legend=0| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
Badness (Sintel): 1.22


== Augustus ==
== Augustus ==
Line 100: Line 138:


{{Mapping|legend=1| 2 0 11 0 3 | 0 1 -2 0 3 | 0 0 0 1 -1 }}
{{Mapping|legend=1| 2 0 11 0 3 | 0 1 -2 0 3 | 0 0 0 1 -1 }}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7497{{c}}, ~3/2 = 705.5140{{c}}, ~7/4 = 967.5010{{c}}
: [[error map]]: {{val| -0.501 +3.058 +0.906 -2.326 -3.528 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 705.7687{{c}}, ~7/4 = 967.8291{{c}}
: error map: {{val| 0.000 +3.814 +2.149 -0.997 -1.841 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 108: Line 152:
{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}


[[Badness]] (Smith): 1.58 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.90


== Tiberius ==
== Tiberius ==
Line 116: Line 160:


{{Mapping|legend=1| 2 0 11 0 -17 | 0 1 -2 0 4 | 0 0 0 1 2 }}
{{Mapping|legend=1| 2 0 11 0 -17 | 0 1 -2 0 4 | 0 0 0 1 2 }}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4432{{c}}, ~3/2 = 703.7469{{c}}, ~7/4 = 968.7525{{c}}
: [[error map]]: {{val| -0.501 +3.058 +0.906 -2.326 -3.528 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.4801{{c}}, ~7/4 = 968.9703{{c}}
: error map: {{val| 0.000 +3.814 +2.149 -0.997 -1.841 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 122: Line 172:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
{{Optimal ET sequence|legend=1| 10, 12, 22e, 34e, 36e, 44d, 46, 58, 104c, 160, 162ce, 172ce, 218ce }}


[[Badness]] (Smith): 1.71 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.05


== Claudius ==
== Claudius ==
Line 132: Line 182:


{{Mapping|legend=1| 2 0 11 0 15 | 0 1 -2 0 1 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 2 0 11 0 15 | 0 1 -2 0 1 | 0 0 0 1 -2 }}
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2988{{c}}, ~3/2 = 703.9668{{c}}, ~7/4 = 972.7468{{c}}
: [[error map]]: {{val| -1.402 +0.609 +0.845 +1.116 +0.845 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7949{{c}}, ~7/4 = 974.0012{{c}}
: error map: {{val| 0.000 +2.840 +4.097 +5.175 +5.475 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 138: Line 194:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) 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| 10, 12e, 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}


[[Badness]] (Smith): 1.97 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.37


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