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 ==

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

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

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

{{~~Val list~~|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]]: 0.442 × 10-3

== Julius aka varda ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[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 to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]

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* [[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]]~~s (unchanged intervals)~~: 2~~, 12~~/11~~, 10~~/9

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/3

{{~~Val list~~|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]]: 0.728 × 10-3

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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 }}

~~Vals:~~ {{~~Val list~~| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}

{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}

Badness: 0.970 × 10-3

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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 }}

~~Vals:~~ {{~~Val list~~| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}

{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}

=== Julius ===

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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 }}

~~Vals:~~ {{~~Val list~~| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}

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

Badness: 1.78 × 10-3

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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 }}

~~Vals:~~ {{~~Val list~~| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}

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

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

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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 }}

~~Vals:~~ {{~~Val list~~| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}

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

== Augustus ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[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 }}]

[[Minimax tuning]]:

* [[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|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/7

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

[[Badness]]: 1.58 × 10-3

== Tiberius ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[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 }}]

[[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]]~~s (unchanged intervals)~~: 2, 9/5, 7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7

{{~~Val list~~|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}

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

[[Badness]]: 1.71 × 10-3

== Claudius ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[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 }}]

[[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]]~~s (unchanged intervals)~~: 2, 7/5, 11/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5

{{~~Val list~~|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 }}

[[Badness]]: 1.97 × 10-3

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Diaschismic rank-3 family| ]]

[[Category:Diaschismic| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-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.
+{{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.
 == Rank-3 diaschismic ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
-{{Val list|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]]: 0.442 × 10<sup>-3</sup>
 == Julius aka varda ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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 }}]
@@ Line 32: / Line 33: @@
 * [[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]]s (unchanged intervals): 2, 12/11, 10/9
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/3
-{{Val list|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]]: 0.728 × 10<sup>-3</sup>
@@ Line 49: / Line 50: @@
 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 }}
-Vals: {{Val list| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
+{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
 Badness: 0.970 × 10<sup>-3</sup>
@@ Line 60: / Line 61: @@
 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 }}
-Vals: {{Val list| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
+{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
 === Julius ===
@@ Line 69: / Line 70: @@
 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 }}
-Vals: {{Val list| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
+{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
 Badness: 1.78 × 10<sup>-3</sup>
@@ Line 80: / Line 81: @@
 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 }}
-Vals: {{Val list| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
+{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
 ===== 19-limit =====
@@ Line 89: / Line 90: @@
 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 }}
-Vals: {{Val list| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
+{{Optimal ET sequence|legend=1| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
 == Augustus ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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]]:
 * [[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|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/7
-{{Val list|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
+{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
 [[Badness]]: 1.58 × 10<sup>-3</sup>
 == Tiberius ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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]]:
 * [[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]]s (unchanged intervals): 2, 9/5, 7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7
-{{Val list|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
+{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
 [[Badness]]: 1.71 × 10<sup>-3</sup>
 == Claudius ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[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]]:
 * [[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]]s (unchanged intervals): 2, 7/5, 11/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
-{{Val list|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 }}
 [[Badness]]: 1.97 × 10<sup>-3</sup>
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Diaschismic rank-3 family| ]] <!-- main article -->
+[[Category:Diaschismic| ]] <!-- key article -->
 [[Category:Rank 3]]