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Sensamagic family

Revision as of 04:50, 14 February 2026 by Overthink (talk | contribs) (→Octarod: asterisk, caps)
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 the sensamagic comma, 245/243.

For a list of rank-2 temperaments, see Sensamagic clan.

Sensamagic

Main article: Sensamagic

Sensamagic is generated by a perfect fifth and a wide supermajor third of ~9/7, two of which make ~5/3. Among the good edo tunings are 87edo and 128edo, as well as the optimal patent val 283edo.

Another notable tuning is given by TE, CTE and POTE, all coinciding at 703.7424 ¢, 440.9020 ¢ with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from CWE is practically unnoticeable.

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 tunings:

  • WE: ~2 = 1199.9983 ¢, ~3/2 = 703.7414 ¢, ~9/7 = 440.9014 ¢
error map: ⟨-0.002 +1.785 -0.771 -2.248]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7411 ¢, ~9/7 = 440.9017 ¢
error map: ⟨0.000 +1.786 -0.769 -2.245]

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 (Sintel): 0.570

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

Undecimal sensamagic tempers out not only 385/384, but 896/891, making itself a strong extension of parapyth.

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 tunings:

  • WE: ~2 = 1199.9667 ¢, ~3/2 = 703.7809 ¢, ~9/7 = 440.9056 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7948 ¢, ~9/7 = 440.9180 ¢
error map: ⟨0.000 +1.840 -0.683 -2.154 +0.175]

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

Badness (Sintel): 0.868

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 tunings:

  • WE: ~2 = 1199.9905 ¢, ~3/2 = 703.7325 ¢, ~9/7 = 440.9149 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7381 ¢, ~9/7 = 440.9184 ¢

Optimal ET sequence: 17, 19f, 22, 41, 46, 63, 87, 237, 283

Badness (Sintel): 1.12

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 tunings:

  • WE: ~2 = 1200.1654 ¢, ~3/2 = 703.2870 ¢, ~9/7 = 441.1967 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2917 ¢, ~9/7 = 441.1849 ¢
error map: ⟨0.000 +1.840 -0.683 -2.1554 +0.175]

Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87

Badness (Sintel): 0.957

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 tunings:

  • WE: ~2 = 1199.9800 ¢, ~3/2 = 703.4468 ¢, ~9/7 = 441.3705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4494 ¢, ~9/7 = 441.3758 ¢

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

Badness (Sintel): 0.868

Octarod

Octarod tempers out 100/99 and the interval class of 11 is found as a stack of four ~9/7's. The name octarod was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from octacot and rodan; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

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 tunings:

  • WE: ~2 = 1199.2854 ¢, ~3/2 = 704.6266 ¢, ~9/7 = 439.2433 ¢
error map: ⟨-0.715 +1.957 -3.915 -0.245 +4.226]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5246 ¢, ~9/7 = 439.2798 ¢
error map: ⟨0.000 +2.570 -3.230 +0.944 +5.801]

Optimal ET sequence: 14c, 19, 22, 27e, 41, 90e, 131e*

*Optimal patent val: 104

Badness (Sintel): 0.698

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 tunings:

  • WE: ~2 = 1198.9114 ¢, ~3/2 = 705.7294 ¢, ~9/7 = 441.7137 ¢
error map: ⟨-1.089 +2.686 +1.754 -1.258 -3.259]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.8402 ¢, ~9/7 = 442.1064 ¢
error map: ⟨0.000 +3.885 +3.739 +0.748 -1.638]

Optimal ET sequence: 19e, 22, 27e, 46, 68, 95, 141bc, 163bc

Badness (Sintel): 1.05

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 tunings:

  • WE: ~2 = 1199.7256 ¢, ~3/2 = 704.9071 ¢, ~9/7 = 443.1303 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9572 ¢, ~9/7 = 443.2018 ¢

Optimal ET sequence: 19e, 22, 27e, 46

Badness (Sintel): 1.05

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 tunings:

  • WE: ~2 = 600.3096 ¢, ~3/2 = 703.4512 ¢, ~9/7 = 441.3336 ¢
error map: ⟨+0.619 +2.115 +0.424 -2.019 -4.985]
  • CWE: ~2 = 600.0000 ¢, ~3/2 = 703.5671 ¢, ~9/7 = 441.2436 ¢
error map: ⟨0.000 +1.612 -0.259 -2.935 -6.507]

Optimal ET sequence: 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e

Badness (Sintel): 1.31

Sensigh

Sensigh uses the same mapping as 7-limit sensi with an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/243

Mapping: [⟨1 -1 -1 -2 3], ⟨0 7 9 13 0], ⟨0 0 0 0 1]]

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

Optimal tunings:

  • WE: ~2 = 1199.7081 ¢, ~9/7 = 443.2748 ¢, ~11/8 = 552.1736 ¢
error map: ⟨-0.2919 +1.2608 +3.4518 -5.6691 -0.0202]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3493 ¢, ~11/8 = 551.8069 ¢
error map: ⟨0.0000 +1.4899 +3.8297 -5.2854 +0.4890]

Optimal ET sequence: 27e, 38df, 46, 111d

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -1 -2 3 0], ⟨0 7 9 13 0 10], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~9/7 = 443.4379 ¢, ~11/8 = 550.3462 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3581 ¢, ~11/8 = 550.7092 ¢

Optimal ET sequence: 27e, 38df, 46, 111df

Badness (Sintel): 0.878

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 -1 -2 3 0 4], ⟨0 7 9 13 0 10 -1], ⟨0 0 0 0 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1200.2286 ¢, ~9/7 = 443.4291 ¢, ~11/8 = 549.2790 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3707 ¢, ~11/8 = 549.5775 ¢

Optimal ET sequence: 27eg, 38df, 46

Badness (Sintel): 0.917

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