Didymus rank-3 family: Difference between revisions

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= Euterpe =
= Euterpe =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.1982, ~7/4 = 968.4280
EDO generators: [[12edo|(7, 10)\12]], [[14edo|(8, 11)\14]], [[31edo|(18, 25)\31]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 20: Line 8:


[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]
[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]
[[POTE generator]]s: ~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 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [[Eigenmonzo]]s: 2, 5, 7
: [[Eigenmonzo]]s: 2, 5, 7
Line 29: Line 19:


[[Badness]]: 0.536 × 10<sup>-3</sup>
[[Badness]]: 0.536 × 10<sup>-3</sup>
</div></div>


= Calliope =
= Calliope =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.1982, ~7/4 = 968.4280
EDO generators: [[7edo|(4, 5)\7]], [[7edo|(4, 6)\7]], [[12edo|(7, 10)\12]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 51: Line 27:


[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]
[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]
[[POTE generator]]s: ~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: 2, 7/3, 11
: [[Eigenmonzo]]s: 2, 7/3, 11
Line 60: Line 38:


[[Badness]]: 0.530 × 10<sup>-3</sup>
[[Badness]]: 0.530 × 10<sup>-3</sup>
</div></div>


= Erato =
= Erato =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.4949, ~11/8 = 547.0252
EDO generators: [[12edo|(7, 6)\12]], [[19edo|(11, 9)\19]], [[31edo|(18, 14)\31]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 82: Line 46:


[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]
[[POTE generator]]s: ~3/2 = 696.4949, ~11/8 = 547.0252


{{Val list|legend=1| 12, 19, 31, 50, 81 }}
{{Val list|legend=1| 12, 19, 31, 50, 81 }}


[[Badness]]: 0.558 × 10<sup>-3</sup>
[[Badness]]: 0.558 × 10<sup>-3</sup>
</div></div>


== 13-limit ==
== 13-limit ==
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 695.9883, ~11/8 = 545.6817
EDO generators: [[12edo|(7, 6)\12]], [[19edo|(11, 9)\19]], [[31edo|(18, 14)\31]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 108: Line 60:


Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]
POTE generators: ~3/2 = 695.9883, ~11/8 = 545.6817


{{Val list|legend=1| 12f, 19, 31, 50, 81 }}
{{Val list|legend=1| 12f, 19, 31, 50, 81 }}
</div></div>


= Clio =
= Clio =


Period: 1\1
Subgroup: 2.3.5.7.11


Optimal ([[POTE]]) generators: ~3/2 = 697.2502, ~7/4 = 968.6295
[[Comma list]]: 81/80, 176/175


EDO generators: [[7edo|(4, 6)\7]], [[12edo|(7, 10)\12]], [[31edo|(18, 25)\31]]
Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]


Scales:  
[[POTE generator]]s: ~3/2 = 697.2502, ~7/4 = 968.6295


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Minimax tuning]]:
<div style="line-height:1.6;">Technical data</div>
* [[11-odd-limit]]
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 81/80, 176/175
 
[[Minimax tuning]]
* [[11-odd-limit]]:
: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
: [[Eigenmonzo]]s: 2, 5, 7
: [[Eigenmonzo]]s: 2, 5, 7
Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]


{{Val list|legend=1| 7, 12, 19e, 24, 31, 105, 129 }}
{{Val list|legend=1| 7, 12, 19e, 24, 31, 105, 129 }}


Badness: 0.738 × 10<sup>-3</sup>
[[Badness]]: 0.738 × 10<sup>-3</sup>
 
</div></div>


= Polyhymnia =
= Polyhymnia =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.2305, ~7/4 = 964.8695
EDO generators: [[7edo|(4, 6)\7]], [[19edo|(11, 15)\19]], [[31edo|(18, 25)\31]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 163: Line 91:


[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]
[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]
[[POTE generator]]s: ~3/2 = 696.2305, ~7/4 = 964.8695


{{Val list|legend=1| 7, 12e, 19, 24, 26, 31 }}
{{Val list|legend=1| 7, 12e, 19, 24, 26, 31 }}
</div></div>


= Thalia =
= Thalia =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 692.0796, ~7/4 = 950.2565
EDO generators: [[5edo|(3, 4)\5]], [[7edo|(4, 5)\7]], [[7edo|(4, 6)\7]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 187: Line 103:


[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
[[POTE generator]]s: ~3/2 = 692.0796, ~7/4 = 950.2565


{{Val list|legend=1| 5, 7, 12e, 14c, 19e }}
{{Val list|legend=1| 5, 7, 12e, 14c, 19e }}
</div></div>


= Melpomene =
= Melpomene =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 699.2230, ~7/4 = 964.2363
EDO generators: [[5edo|(3, 4)\5]], [[7edo|(4, 5)\7]], [[12edo|(7, 10)\12]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 211: Line 115:


[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
[[POTE generator]]s: ~3/2 = 699.2230, ~7/4 = 964.2363


{{Val list|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}
{{Val list|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}
</div></div>


= Urania =
= Urania =
Period: 1\1
Optimal ([[POTE]]) generators: ~11/9 = 348.0938, ~7/4 = 963.6042
EDO generators: [[7edo|(2, 5)\7]], [[14edo|(4, 11)\14]], [[31edo|(9, 25)\31]]
Scales: [[urania24]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 235: Line 127:


[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
Map to lattice: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]


Lattice basis:  
Lattice basis:  
Line 240: Line 134:
: Angle (11/9, 8/7) = 90 degrees
: Angle (11/9, 8/7) = 90 degrees


Map to lattice: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
[[POTE generator]]s: ~11/9 = 348.0938, ~7/4 = 963.6042


{{Val list|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
{{Val list|legend=1| 7, 14c, 17c, 24, 31, 100de, 131bde, 162bde }}
Line 248: Line 142:
[[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
[[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


</div></div>
Scales: [[urania24]]


= Terpsichore =
= Terpsichore =
Period: 1\1
Optimal ([[POTE]]) generators: ~3/2 = 696.2358, ~7/4 = 964.0006
EDO generators: [[14edo|(8, 11)\14]], [[19edo|(11, 8)\7]], [[31edo|(18, 25)\31]]
Scales:
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 269: Line 151:


[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]
[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]
[[POTE generator]]s: ~3/2 = 696.2358, ~7/4 = 964.0006


{{Val list|legend=1| 14c, 17c, 19, 31, 81, 112b }}
{{Val list|legend=1| 14c, 17c, 19, 31, 81, 112b }}
Line 275: Line 159:


[[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
[[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
</div></div>


[[Category:Theory]]
[[Category:Theory]]

Revision as of 09:38, 29 March 2021

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.

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]]

POTE generators: ~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: 2, 5, 7

Template:Val list

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]]

POTE generators: ~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⟩]
Eigenmonzos: 2, 7/3, 11

Template:Val list

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]]

POTE generators: ~3/2 = 696.4949, ~11/8 = 547.0252

Template:Val list

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]]

POTE generators: ~3/2 = 695.9883, ~11/8 = 545.6817

Template:Val list

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]]

POTE generators: ~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: 2, 5, 7

Template:Val list

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]]

POTE generators: ~3/2 = 696.2305, ~7/4 = 964.8695

Template:Val list

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]]

POTE generators: ~3/2 = 692.0796, ~7/4 = 950.2565

Template:Val list

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]]

POTE generators: ~3/2 = 699.2230, ~7/4 = 964.2363

Template:Val list

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]]

Map 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

POTE generators: ~11/9 = 348.0938, ~7/4 = 963.6042

Template:Val list

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]]

POTE generators: ~3/2 = 696.2358, ~7/4 = 964.0006

Template:Val list

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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