Sensamagic family: Difference between revisions

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The '''sensamagic family''' of rank-3 temperaments tempers out [[245/243]].  
{{Technical data page}}
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


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


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


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


Lattice basis:  
Lattice basis:  
Line 18: Line 19:
: 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]]): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 0 1/5 2/5 }}, {{monzo| 0 0 1 0 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 8/7, 5/4
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 0 5/3 2/3 -2/3 }}, {{monzo| 0 5/3 -1/3 1/3 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 4/3, 7/5  
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5  


{{Val list|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}


[[Badness]]: 0.129 × 10<sup>-3</sup>
[[Badness]]: 0.129 × 10<sup>-3</sup>
Line 34: Line 35:
[[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 block]]s|
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
2.3.7 subgroup
<div class="mw-collapsible-content">
 
{2, 3, 7} subgroup
* 12: 729/686, 64/63
* 12: 729/686, 64/63
* 17: 64/63, 19683/19208
* 17: 64/63, 19683/19208
Line 44: Line 42:
* 22: 64/63, 537824/531441
* 22: 64/63, 537824/531441
* 24: 64/63, 15059072/14348907
* 24: 64/63, 15059072/14348907
}}


</div></div>
=== 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 ==
{{Main| Sensamagic }}


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 245/243, 385/384
[[Comma list]]: 245/243, 385/384


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


[[POTE generator]]s: ~3/2 = 703.8004, ~9/7 = 440.9178
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


{{Val list|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}


[[Badness]]: 0.722 × 10<sup>-3</sup>
[[Badness]]: 0.722 × 10<sup>-3</sup>
Line 69: Line 69:
[[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: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11
Comma list: 245/243, 352/351, 364/363


Comma list: 245/243, 352/351, 385/384
Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 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 }}]
{{Optimal ET sequence|legend=1| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}
 
Vals: {{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


[[Comma list]]: 245/243, 441/440
[[Comma list]]: 245/243, 441/440


[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}


[[POTE generator]]s: ~3/2 = 703.1900, ~9/7 = 441.1359
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359


{{Val list|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}


[[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


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


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


Vals: {{Val list| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
{{Optimal ET sequence|legend=1| 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


[[Comma list]]: 100/99, 245/243
[[Comma list]]: 100/99, 245/243


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


[[POTE generator]]s: ~3/2 = 705.0464, ~9/7 = 439.5050
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050


{{Val list|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
{{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}


[[Badness]]: 0.581 × 10<sup>-3</sup>
[[Badness]]: 0.581 × 10<sup>-3</sup>
Line 123: Line 119:
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


[[Comma list]]: 176/175, 245/243
[[Comma list]]: 176/175, 245/243


[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}


[[POTE generator]]s: ~3/2 = 706.3702, ~9/7 = 442.1147
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147


{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}


[[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


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


Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}


Vals: {{Val list| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
{{Optimal ET sequence|legend=1| 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


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 121/120, 245/243


Comma list: 91/90, 126/125, 169/168
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}


Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]
: mapping generators: ~77/54, ~3, ~9/7


Vals: {{Val list| 19, 27, 46, 111df, 157df }}
[[Optimal tuning]] ([[POTE]]): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060


Badness: 0.939 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}


=== 17-limit  ===
[[Badness]]: 1.089 × 10<sup>-3</sup>


Subgroup: 2.3.5.7.11.13.17
== Sensigh ==
Subgroup: 2.3.5.7.11.13


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


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: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}


Vals: {{Val list| 19, 27, 46 }}
: mapping generators: ~2, ~9/7, ~11


== Bisector  ==
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df, 157df }}


Subgroup: 2.3.5.7.11
Badness: 0.939 × 10<sup>-3</sup>


[[Comma list]]: 121/120, 245/243
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping generators: ~77/54, ~3, ~9/7


[[POTE generator]]s: ~3/2 = 703.0884, ~9/7 = 441.1060
Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}


{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
 
[[Badness]]: 1.089 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:40, 24 June 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 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): ~2 = 1\1, ~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⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7
[[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283

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): ~2 = 1\1, ~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⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/9

Optimal ET sequence: 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd

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 ET sequence: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd

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): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359

Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd

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 ET sequence: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd

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): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050

Optimal ET sequence: 8d, 14c, 19, 22, 27e, 41, 104, 131e

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): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147

Optimal ET sequence: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc

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 ET sequence: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf

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): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060

Optimal ET sequence: 22, 46, 68, 82e, 106de, 114, 128e

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 ET sequence: 19, 27, 46, 111df, 157df

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 ET sequence: 19, 27, 46

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