Diaschismic rank three family: Difference between revisions

Latest revision as of 00:35, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The diaschismic rank-3 family of temperaments 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

Comma list: 2048/2025

Mapping: [⟨2 0 11 0], ⟨0 1 -2 0], ⟨0 0 0 1]]

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

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

Optimal ET sequence: 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

Comma list: 176/175, 896/891

Mapping: [⟨2 0 11 0 14], ⟨0 1 -2 0 -4], ⟨0 0 0 1 1]]

Mapping to lattice: [⟨0 1 -2 2 -2], ⟨0 0 0 -1 -1]]

Lattice basis:

3/2, 9/7

Angle (3/2, 9/7) = 82.289 degrees

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⟩, [-1/8 3/2 -5/4 0 1⟩, [11/8 -1/2 -1/4 0 1⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.11/3

Optimal ET sequence: 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde

Badness: 0.728 × 10^-3

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

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: [⟨2 0 11 0 14 24], ⟨0 1 -2 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde

Badness: 0.970 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 351/350

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: 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: [⟨2 0 11 0 14 -18], ⟨0 1 -2 0 -4 8], ⟨0 0 0 1 1 0]]

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

Badness: 1.78 × 10^-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324

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: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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: 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd

Augustus

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1232/1215

Mapping: [⟨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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.11/7

Optimal ET sequence: 10, 12e, 22, 46, 68, 114, 170b

Badness: 1.58 × 10^-3

Tiberius

Subgroup: 2.3.5.7.11

Comma list: 441/440, 2048/2025

Mapping: [⟨2 0 11 0 -17], ⟨0 1 -2 0 4], ⟨0 0 0 1 2]]

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⟩, [0 0 0 1 0⟩, [-3 2 -1 2 0⟩]

unchanged-interval (eigenmonzo) basis: 2.9/5.7

Optimal ET sequence: 46, 58, 104c, 160, 162ce, 172ce, 218ce

Badness: 1.71 × 10^-3

Claudius

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2048/2025

Mapping: [⟨2 0 11 0 15], ⟨0 1 -2 0 1], ⟨0 0 0 1 -2]]

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [9/7 0 -3/7 2/7 1/7⟩, [41/14 0 6/7 -4/7 -2/7⟩, [41/14 0 -1/7 3/7 -2/7⟩, [41/14 0 -1/7 -4/7 5/7⟩]

unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee

Badness: 1.97 × 10^-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 47: / Line 48: @@
 Subgroup: 2.3.5.7.11.13
-Comma list: 176/175, 352/351, 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 }}
-Vals: {{Val list| 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 58: / Line 59: @@
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 126/125, 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 }}
-Vals: {{Val list| 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 =====
+Subgroup: 2.3.5.7.11.13.17.19
 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:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Diaschismic rank-3 family| ]] <!-- main article -->
+[[Category:Diaschismic| ]] <!-- key article -->
 [[Category:Rank 3]]

Diaschismic rank three family: Difference between revisions

Latest revision as of 00:35, 24 June 2025

Contents

Rank-3 diaschismic

Julius aka varda

Varda

17-limit

Julius

17-limit

19-limit

Augustus

Tiberius

Claudius

Navigation menu

Diaschismic rank three family: Difference between revisions

Latest revision as of 00:35, 24 June 2025

Rank-3 diaschismic

Julius aka varda

Varda

17-limit

Julius

17-limit

19-limit

Augustus

Tiberius

Claudius

Navigation menu

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