Sensamagic family: Difference between revisions

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== Sensamagic  ==
== Sensamagic  ==
{{main| Sensamagic }}
{{Main| Sensamagic }}


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 34: Line 34:
[[Projection pair]]: 5 243/49 to 2.3.7
[[Projection pair]]: 5 243/49 to 2.3.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{databox|[[Minkowski blocks]]|
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">
 
{2, 3, 7} subgroup
{2, 3, 7} subgroup
* 12: 729/686, 64/63
* 12: 729/686, 64/63
Line 44: Line 41:
* 22: 64/63, 537824/531441
* 22: 64/63, 537824/531441
* 24: 64/63, 15059072/14348907
* 24: 64/63, 15059072/14348907
}}


</div></div>
== Undecimal sensamagic ==
 
== Undecimal sensamagic ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 69: Line 64:
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7


=== 13-limit ===
=== 13-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 245/243, 352/351, 385/384
Comma list: 245/243, 352/351, 364/363


Mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 1 2 -2 -5 }}, {{val| 0 0 2 -1 -1 -1 }}]
Mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 1 2 -2 -5 }}, {{val| 0 0 2 -1 -1 -1 }}]


Vals: {{Val list| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}
Optimal GPV sequence: {{Val list| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}


Badness: 1.20 × 10<sup>-3</sup>
Badness: 1.20 × 10<sup>-3</sup>


== Sensawer ==
== Sensawer ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 95: Line 88:
[[Badness]]: 0.796 × 10<sup>-3</sup>
[[Badness]]: 0.796 × 10<sup>-3</sup>


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


Line 103: Line 95:
Mapping: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 1 2 5 2 }}, {{val| 0 0 2 -1 -4 -4 }}]
Mapping: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 1 2 5 2 }}, {{val| 0 0 2 -1 -4 -4 }}]


Vals: {{Val list| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
Optimal GPV sequence: {{Val list| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}


Badness: 0.928 × 10<sup>-3</sup>
Badness: 0.928 × 10<sup>-3</sup>


== Octarod ==
== Octarod ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 123: Line 114:
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]


== Shrusus ==
== Shrusus ==
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 137: Line 127:
[[Badness]]: 0.877 × 10<sup>-3</sup>
[[Badness]]: 0.877 × 10<sup>-3</sup>


=== Shrusic ===
=== Shrusic ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 145: Line 134:
Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]
Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]


Vals: {{Val list| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
Optimal GPV sequence: {{Val list| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}


Badness: 1.125 × 10<sup>-3</sup>
Badness: 1.125 × 10<sup>-3</sup>


== Sensigh  ==
== Bisector ==
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 121/120, 245/243
 
[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]
 
Mapping generators: ~77/54, ~3, ~9/7
 
[[POTE generator]]s: ~3/2 = 703.0884, ~9/7 = 441.1060
 
{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}


[[Badness]]: 1.089 × 10<sup>-3</sup>
== Sensigh ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 157: Line 160:
Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]


Vals: {{Val list| 19, 27, 46, 111df, 157df }}
Mapping generators: ~2, ~9/7, ~11
 
Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}


Badness: 0.939 × 10<sup>-3</sup>
Badness: 0.939 × 10<sup>-3</sup>


=== 17-limit ===
=== 17-limit ===


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 169: Line 174:
Mapping: [{{val| 1 6 8 11 0 10 0 }}, {{val| 0 -7 -9 -13 0 -10 1 }}, {{val| 0 0 0 0 1 0 1 }}]
Mapping: [{{val| 1 6 8 11 0 10 0 }}, {{val| 0 -7 -9 -13 0 -10 1 }}, {{val| 0 0 0 0 1 0 1 }}]


Vals: {{Val list| 19, 27, 46 }}
Optimal GPV sequence: {{Val list| 19, 27, 46 }}
 
== Bisector  ==
 
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 121/120, 245/243
 
[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]
 
Mapping generators: ~77/54, ~3, ~9/7
 
[[POTE generator]]s: ~3/2 = 703.0884, ~9/7 = 441.1060
 
{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
 
[[Badness]]: 1.089 × 10<sup>-3</sup>


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 00:37, 15 March 2022

The sensamagic family of rank-3 temperaments tempers out 245/243.

Sensamagic

Main article: Sensamagic

Subgroup: 2.3.5.7

Comma list: 245/243

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

Mapping generators: ~2, ~3, ~9/7

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

Lattice basis:

3/2 length = 0.9644, 9/7 length = 1.0807
Angle (3/2, 9/7) = 86.5288°

POTE generators: ~3/2 = 703.7424, ~9/7 = 440.9020

Minimax tuning:

[[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]
Eigenmonzos (unchanged intervals): 2, 8/7, 5/4
[[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
Eigenmonzos (unchanged intervals): 2, 4/3, 7/5

Template:Val list

Badness: 0.129 × 10-3

Projection pair: 5 243/49 to 2.3.7

Minkowski blocks

{2, 3, 7} subgroup

  • 12: 729/686, 64/63
  • 17: 64/63, 19683/19208
  • 19: 49/48, 177147/175616
  • 22: 64/63, 537824/531441
  • 24: 64/63, 15059072/14348907

Undecimal sensamagic

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384

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

POTE generators: ~3/2 = 703.8004, ~9/7 = 440.9178

Minimax tuning:

[[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]
Eigenmonzos (unchanged intervals): 2, 11/9, 7/5

Template:Val list

Badness: 0.722 × 10-3

Projection pairs: 5 243/49 11 896/81 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]

Optimal GPV sequence: Template:Val list

Badness: 1.20 × 10-3

Sensawer

Subgroup: 2.3.5.7.11

Comma list: 245/243, 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]

POTE generators: ~3/2 = 703.1900, ~9/7 = 441.1359

Template:Val list

Badness: 0.796 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]

Optimal GPV sequence: Template:Val list

Badness: 0.928 × 10-3

Octarod

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243

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

POTE generators: ~3/2 = 705.0464, ~9/7 = 439.5050

Template:Val list

Badness: 0.581 × 10-3

Scales: octarod1, octarod2, octarod3, octarod4, octarod5

Shrusus

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243

Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]

POTE generators: ~3/2 = 706.3702, ~9/7 = 442.1147

Template:Val list

Badness: 0.877 × 10-3

Shrusic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243

Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]

Optimal GPV sequence: Template:Val list

Badness: 1.125 × 10-3

Bisector

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243

Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]

Mapping generators: ~77/54, ~3, ~9/7

POTE generators: ~3/2 = 703.0884, ~9/7 = 441.1060

Template:Val list

Badness: 1.089 × 10-3

Sensigh

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10], ⟨0 0 0 0 1 0]]

Mapping generators: ~2, ~9/7, ~11

Optimal GPV sequence: Template:Val list

Badness: 0.939 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: [⟨1 6 8 11 0 10 0], ⟨0 -7 -9 -13 0 -10 1], ⟨0 0 0 0 1 0 1]]

Optimal GPV sequence: Template:Val list

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