Sensamagic family: Difference between revisions

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The '''sensamagic family''' of rank-3 temperaments tempers out [[245/243]].  
The '''sensamagic family''' of [[rank-3 temperament]]s tempers out [[245/243]]. For a list of rank-2 temperaments, see [[Sensamagic clan]].  


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


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243
[[Comma list]]: 245/243
Line 18: Line 18:
: Angle (3/2, 9/7) = 86.5288°
: Angle (3/2, 9/7) = 86.5288°


[[POTE generator]]s: ~3/2 = 703.7424, ~9/7 = 440.9020
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.7424, ~9/7 = 440.9020


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 42: Line 42:
* 24: 64/63, 15059072/14348907
* 24: 64/63, 15059072/14348907
}}
}}
=== Overview to extensions ===
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.


== Undecimal sensamagic ==
== Undecimal sensamagic ==
Subgroup: 2.3.5.7.11
{{Main| Sensamagic }}
 
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 245/243, 385/384
[[Comma list]]: 245/243, 385/384
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[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 1 2 -2 }}, {{val| 0 0 2 -1 -1 }}]
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 1 2 -2 }}, {{val| 0 0 2 -1 -1 }}]


[[POTE generator]]s: ~3/2 = 703.8004, ~9/7 = 440.9178
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8004, ~9/7 = 440.9178


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 76: Line 81:


== Sensawer ==
== Sensawer ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 245/243, 441/440
[[Comma list]]: 245/243, 441/440
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[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]
[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]


[[POTE generator]]s: ~3/2 = 703.1900, ~9/7 = 441.1359
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.1900, ~9/7 = 441.1359


{{Val list|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
{{Val list|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
Line 100: Line 105:


== Octarod ==
== Octarod ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 100/99, 245/243
[[Comma list]]: 100/99, 245/243
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[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 1 2 0 }}, {{val| 0 0 2 -1 4 }}]
[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 1 2 0 }}, {{val| 0 0 2 -1 4 }}]


[[POTE generator]]s: ~3/2 = 705.0464, ~9/7 = 439.5050
[[Optimal tuning]] ([[POTE]]): ~3/2 = 705.0464, ~9/7 = 439.5050


{{Val list|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
{{Val list|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
Line 115: Line 120:


== Shrusus ==
== Shrusus ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 245/243
[[Comma list]]: 176/175, 245/243
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[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]
[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]


[[POTE generator]]s: ~3/2 = 706.3702, ~9/7 = 442.1147
[[Optimal tuning]] ([[POTE]]): ~3/2 = 706.3702, ~9/7 = 442.1147


{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
Line 139: Line 144:


== Bisector ==
== Bisector ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 121/120, 245/243
[[Comma list]]: 121/120, 245/243
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Mapping generators: ~77/54, ~3, ~9/7
Mapping generators: ~77/54, ~3, ~9/7


[[POTE generator]]s: ~3/2 = 703.0884, ~9/7 = 441.1060
[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0884, ~9/7 = 441.1060


{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
Line 167: Line 172:


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Revision as of 14:02, 12 September 2022

The sensamagic family of rank-3 temperaments tempers out 245/243. For a list of rank-2 temperaments, see Sensamagic clan.

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°

Optimal tuning (POTE): ~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

Overview to extensions

Temperaments discussed elsewhere include supernatural (→ Keemic family). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.

Undecimal sensamagic

Main article: 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]]

Optimal tuning (POTE): ~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: 2.3.5.7.11.13

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]]

Optimal tuning (POTE): ~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]]

Optimal tuning (POTE): ~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]]

Optimal tuning (POTE): ~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

Optimal tuning (POTE): ~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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