Didymus rank-3 family: Difference between revisions

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== Didymus  ==
== Didymus  ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80
[[Comma list]]: 81/80


[[Mapping]]: [{{val| 1 0 -4 0 }}, {{val| 0 1 4 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}


[[POTE generator]]s: ~3/2 = 696.2387, ~7/4 = 964.9090
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2387, ~7/4 = 964.9090


{{Optimal ET sequence|legend=1| 12, 19, 31, 81 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81 }}
Line 15: Line 15:


== Euterpe  ==
== Euterpe  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 99/98
[[Comma list]]: 81/80, 99/98


[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]
{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}


[[POTE generator]]s: ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 34: Line 33:


== Calliope  ==
== Calliope  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 45/44, 81/80
[[Comma list]]: 45/44, 81/80


[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}


[[POTE generator]]s: ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/3, 11
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3.11


{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
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[[Badness]]: 0.530 × 10<sup>-3</sup>
[[Badness]]: 0.530 × 10<sup>-3</sup>


== Erato ==
== Erato ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 126/125
[[Comma list]]: 81/80, 126/125


[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}


[[POTE generator]]s: ~3/2 = 696.4949, ~11/8 = 547.0252
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.4949, ~11/8 = 547.0252


{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }}
Line 67: Line 64:


=== 13-limit  ===
=== 13-limit  ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 105/104, 126/125
Comma list: 81/80, 105/104, 126/125


Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}


POTE generators: ~3/2 = 695.9883, ~11/8 = 545.6817
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.9883, ~11/8 = 545.6817


{{Optimal ET sequence|legend=1| 12f, 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 12f, 19, 31, 50, 81 }}


== Clio  ==
== Clio  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 176/175
[[Comma list]]: 81/80, 176/175


Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]
{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}


[[POTE generator]]s: ~3/2 = 697.2502, ~7/4 = 968.6295
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 697.2502, ~7/4 = 968.6295


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 99: Line 94:
== Polyhymnia  ==
== Polyhymnia  ==


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 81/80, 385/384
[[Comma list]]: 81/80, 385/384


[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]
{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}


[[POTE generator]]s: ~3/2 = 696.2305, ~7/4 = 964.8695
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2305, ~7/4 = 964.8695


{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }}
{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }}
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== Thalia  ==
== Thalia  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 33/32, 55/54
[[Comma list]]: 33/32, 55/54


[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}


[[POTE generator]]s: ~3/2 = 692.0796, ~7/4 = 950.2565
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 692.0796, ~7/4 = 950.2565


{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }}
{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }}


== Melpomene  ==
== Melpomene  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 56/55
[[Comma list]]: 81/80, 56/55


[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}


[[POTE generator]]s: ~3/2 = 699.2230, ~7/4 = 964.2363
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.2230, ~7/4 = 964.2363


{{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}
{{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}


== Urania  ==
== Urania  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 121/120
[[Comma list]]: 81/80, 121/120


[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 1 0 0 2 | 0 2 8 0 5 | 0 0 0 1 0 }}


[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
Line 149: Line 141:
: Angle (11/9, 8/7) = 90 degrees
: Angle (11/9, 8/7) = 90 degrees


[[POTE generator]]s: ~11/9 = 348.0938, ~7/4 = 963.6042
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 348.0938, ~7/4 = 963.6042


{{Optimal ET sequence|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
{{Optimal ET sequence|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
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== Terpsichore  ==
== Terpsichore  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 540/539
[[Comma list]]: 81/80, 540/539


[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}


[[POTE generator]]s: ~3/2 = 696.2358, ~7/4 = 964.0006
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2358, ~7/4 = 964.0006


{{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }}
{{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }}

Revision as of 05:25, 5 August 2023

The didymus rank-3 family are rank-3 temperaments tempering out the didymus comma, 81/80. 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.

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

Mapping: [⟨1 0 -4 0], ⟨0 1 4 0], ⟨0 0 0 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2387, ~7/4 = 964.9090

Optimal ET sequence: 12, 19, 31, 81

Badness: 0.095 × 10-3

Euterpe

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]
Eigenmonzos (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 12, 17c, 19e, 26, 31, 88

Badness: 0.536 × 10-3

Calliope

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80

Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]
Eigenmonzo (unchanged-interval) basis: 2.7/3.11

Optimal ET sequence: 7d, 12, 19, 26, 45

Badness: 0.530 × 10-3

Erato

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.4949, ~11/8 = 547.0252

Optimal ET sequence: 12, 19, 31, 50, 81

Badness: 0.558 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125

Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.9883, ~11/8 = 545.6817

Optimal ET sequence: 12f, 19, 31, 50, 81

Clio

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175

Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.2502, ~7/4 = 968.6295

Minimax tuning:

[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]
Eigenmonzos (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 7, 12, 19e, 24, 31, 105, 129

Badness: 0.738 × 10-3

Polyhymnia

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384

Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2305, ~7/4 = 964.8695

Optimal ET sequence: 7, 12e, 19, 24, 26, 31

Scales: polyhymnia12

Thalia

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54

Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 692.0796, ~7/4 = 950.2565

Optimal ET sequence: 5, 7, 12e, 14c, 19e

Melpomene

Subgroup: 2.3.5.7.11

Comma list: 81/80, 56/55

Mapping: [⟨1 0 -4 0 7], ⟨0 1 4 0 -4], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.2230, ~7/4 = 964.2363

Optimal ET sequence: 7d, 12, 17c, 19, 24, 31e, 36

Urania

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807
Angle (11/9, 8/7) = 90 degrees

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.0938, ~7/4 = 963.6042

Optimal ET sequence: 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde

Badness: 0.842 × 10-3

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scales: urania24

Terpsichore

Subgroup: 2.3.5.7.11

Comma list: 81/80, 540/539

Mapping: [⟨1 0 -4 0 -2], ⟨0 1 4 0 7], ⟨0 0 0 1 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.2358, ~7/4 = 964.0006

Optimal ET sequence: 14c, 17c, 19, 31, 81, 112b

Badness: 0.850 × 10-3

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11

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