Diaschismic rank-3 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 [[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.


== Julius aka varda ==
== Rank-3 diaschismic ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: [[2048/2025]]
 
{{Mapping|legend=1| 2 0 11 0 | 0 1 -2 0 | 0 0 0 1 }}
 
: mapping generators: ~45/32, ~3, ~7
 
[[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 }}
 
[[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.
 
[[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 }}]


Lattice basis:  
Lattice basis:  
: 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]]:  
* [[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]] (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


Comma list: 176/175, 352/351, 364/363
Comma list: 176/175, 351/350, 364/363
 
Mapping: {{mapping| 2 0 11 0 14 24 | 0 1 -2 0 -4 -7 | 0 0 0 1 1 1 }}
 
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 (Sintel): 0.908
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 256/255, 351/350


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 5 | 0 1 -2 0 -4 -7 1 | 0 0 0 1 1 1 0 }}


Vals: {{Val list| 22, 46, 58, 80, 138cde, 242cde, 322bcde }}
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}}


Badness: 0.970 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12f, 22, 34d, 46, 58, 80, 104c, 126, 138cde }}
 
Badness (Sintel): 0.841


=== Julius ===
=== Julius ===
Line 45: Line 92:
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 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}}


Badness: 1.78 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
 
Badness (Sintel): 1.66
 
===== 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 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 =====
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 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 }}
 
Badness (Sintel): 1.22


== 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 }}
 
[[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]]:  
* [[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]] (Sintel): 1.90


== 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 }}
 
[[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]]:  
* [[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| 10, 12, 22e, 34e, 36e, 44d, 46, 58, 104c, 160, 162ce, 172ce, 218ce }}


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


== 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 }}
 
[[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]]:
* [[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| 10, 12e, 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}


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


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[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 12:21, 27 November 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 rank-3 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 tunings:

  • WE: ~45/32 = 599.4107 ¢, ~3/2 = 704.2059 ¢, ~7/4 = 971.1782 ¢
error map: ⟨-1.179 +1.072 +1.150 -0.005]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.9585 ¢, ~7/4 = 971.2952 ¢
error map: ⟨0.000 +3.003 +3.769 +2.469]

Optimal ET sequence: 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd

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 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.

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

Optimal tunings:

  • WE: ~45/32 = 599.4084 ¢, ~3/2 = 703.9071 ¢, ~7/4 = 971.8498 ¢
error map: ⟨-1.183 +0.769 +1.731 +0.657 -1.013]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.6576 ¢, ~7/4 = 971.9790 ¢
error map: ⟨0.000 +2.703 +4.371 +3.153 +2.031]

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 (Sintel): 0.875

Scales
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
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

13-limit

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 tunings:

  • WE: ~45/32 = 599.4000 ¢, ~3/2 = 703.5375 ¢, ~7/4 = 971.9244 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.2888 ¢, ~7/4 = 972.0576 ¢

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 104c, 138cde, 242ccde

Badness (Sintel): 0.908

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 tunings:

  • WE: ~17/12 = 599.5734 ¢, ~3/2 = 703.8639 ¢, ~7/4 = 971.4469 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 704.3790 ¢, ~7/4 = 971.6313 ¢

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 104c, 126, 138cde

Badness (Sintel): 0.841

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 tunings:

  • WE: ~45/32 = 599.5350 ¢, ~3/2 = 704.6999 ¢, ~7/4 = 972.2975 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.1906 ¢, ~7/4 = 972.3332 ¢

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

Badness (Sintel): 1.66

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 tunings:

  • WE: ~17/12 = 599.6247 ¢, ~3/2 = 704.7816 ¢, ~7/4 = 971.8531 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 705.1801 ¢, ~7/4 = 971.9669 ¢

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 126, 148d

Badness (Sintel): 1.18

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 tunings:

  • WE: ~17/12 = 599.6179 ¢, ~3/2 = 704.8484 ¢, ~7/4 = 972.3437 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~3/2 = 705.2584 ¢, ~7/4 = 972.4875 ¢

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

Badness (Sintel): 1.22

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]]

Optimal tunings:

  • WE: ~45/32 = 599.7497 ¢, ~3/2 = 705.5140 ¢, ~7/4 = 967.5010 ¢
error map: ⟨-0.501 +3.058 +0.906 -2.326 -3.528]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 705.7687 ¢, ~7/4 = 967.8291 ¢
error map: ⟨0.000 +3.814 +2.149 -0.997 -1.841]

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 (Sintel): 1.90

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]]

Optimal tunings:

  • WE: ~45/32 = 599.4432 ¢, ~3/2 = 703.7469 ¢, ~7/4 = 968.7525 ¢
error map: ⟨-0.501 +3.058 +0.906 -2.326 -3.528]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.4801 ¢, ~7/4 = 968.9703 ¢
error map: ⟨0.000 +3.814 +2.149 -0.997 -1.841]

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: 10, 12, 22e, 34e, 36e, 44d, 46, 58, 104c, 160, 162ce, 172ce, 218ce

Badness (Sintel): 2.05

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]]

Optimal tunings:

  • WE: ~45/32 = 599.2988 ¢, ~3/2 = 703.9668 ¢, ~7/4 = 972.7468 ¢
error map: ⟨-1.402 +0.609 +0.845 +1.116 +0.845]
  • CWE: ~45/32 = 600.0000 ¢, ~3/2 = 704.7949 ¢, ~7/4 = 974.0012 ¢
error map: ⟨0.000 +2.840 +4.097 +5.175 +5.475]

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: 10, 12e, 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee

Badness (Sintel): 2.37

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