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.
{{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 ==
== Rank-3 diaschismic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[2048/2025]]
[[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>
[[Badness]]: 0.442 × 10<sup>-3</sup>


== Julius aka varda ==
== Julius aka varda ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 896/891
[[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 }}]
[[Mapping to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]
Line 32: Line 33:
* [[11-odd-limit]]
* [[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 }}]
: [{{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>
[[Badness]]: 0.728 × 10<sup>-3</sup>
Line 47: Line 48:
Subgroup: 2.3.5.7.11.13
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, 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>
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 ===
=== Julius ===
Line 60: Line 70:
Comma list: 176/175, 325/324, 896/891
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| 12f, 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>
Badness: 1.78 × 10<sup>-3</sup>
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 256/255, 325/324
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 }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 136/135, 176/175, 190/189, 256/255, 286/285
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 }}


== Augustus ==
== Augustus ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 1232/1215
[[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]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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>
[[Badness]]: 1.58 × 10<sup>-3</sup>


== Tiberius ==
== Tiberius ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 2048/2025
[[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]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[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 }}]
: [{{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>
[[Badness]]: 1.71 × 10<sup>-3</sup>


== Claudius ==
== Claudius ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 2048/2025
[[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]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[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 }}]
: [{{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>
[[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 rank-3 family| ]] <!-- main article -->
[[Category:Diaschismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

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:

[[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:

[[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:

[[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:

[[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

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