Diaschismic rank-3 family: Difference between revisions

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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.

The '''diaschismic rank-3 family''' of [[rank-3 temperament|rank-3]] [[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 ==

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

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

[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982{{c}}, ~7/4 = 972.1330{{c}}

{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}

[[Badness]]: 0.442 × 10-3

[[Badness]] (Smith): 0.442 × 10-3

== Julius ~~aka~~ varda ==

== Julius a.k.a. varda ==

[[Subgroup]]: 2.3.5.7.11

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{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}

[[Badness]]: 0.728 × 10-3

[[Badness]] (Smith): 0.728 × 10-3

Julius[22] [[hobbit]] [[transversal]]:

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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=0| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}

Badness: 0.970 × 10-3

Badness (Smith): 0.970 × 10-3

===== 17-limit =====

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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=0| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}

=== Julius ===

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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=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}

Badness: 1.78 × 10-3

Badness (Smith): 1.78 × 10-3

===== 17-limit =====

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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=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}

===== 19-limit =====

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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=0| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}

== Augustus ==

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

[[Badness]]: 1.58 × 10-3

[[Badness]] (Smith): 1.58 × 10-3

== Tiberius ==

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

[[Badness]]: 1.71 × 10-3

[[Badness]] (Smith): 1.71 × 10-3

== Claudius ==

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{{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}

[[Badness]]: 1.97 × 10-3

[[Badness]] (Smith): 1.97 × 10-3

[[Category:Temperament families]]

@@ Line 1: / Line 1: @@
 {{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.
+The '''diaschismic rank-3 family''' of [[rank-3 temperament|rank-3]] [[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 ==
@@ Line 11: / Line 11: @@
 : mapping generators: ~45/32, ~3, ~7
-[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982, ~7/4 = 972.1330
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982{{c}}, ~7/4 = 972.1330{{c}}
 {{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}
-[[Badness]]: 0.442 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.442 × 10<sup>-3</sup>
-== Julius aka varda ==
+== Julius a.k.a. varda ==
 [[Subgroup]]: 2.3.5.7.11
@@ Line 37: / Line 37: @@
 {{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}
-[[Badness]]: 0.728 × 10<sup>-3</sup>
+[[Badness]] (Smith): 0.728 × 10<sup>-3</sup>
 Julius[22] [[hobbit]] [[transversal]]:
@@ Line 52: / Line 52: @@
 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=0| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
-Badness: 0.970 × 10<sup>-3</sup>
+Badness (Smith): 0.970 × 10<sup>-3</sup>
 ===== 17-limit =====
@@ Line 63: / Line 63: @@
 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=0| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
 === Julius ===
@@ Line 72: / Line 72: @@
 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=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
-Badness: 1.78 × 10<sup>-3</sup>
+Badness (Smith): 1.78 × 10<sup>-3</sup>
 ===== 17-limit =====
@@ Line 83: / Line 83: @@
 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=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
 ===== 19-limit =====
@@ Line 92: / Line 92: @@
 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=0| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
 == Augustus ==
@@ Line 108: / Line 108: @@
 {{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
-[[Badness]]: 1.58 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.58 × 10<sup>-3</sup>
 == Tiberius ==
@@ Line 124: / Line 124: @@
 {{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
-[[Badness]]: 1.71 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.71 × 10<sup>-3</sup>
 == Claudius ==
@@ Line 140: / Line 140: @@
 {{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}
-[[Badness]]: 1.97 × 10<sup>-3</sup>
+[[Badness]] (Smith): 1.97 × 10<sup>-3</sup>
 [[Category:Temperament families]]