Sensamagic family
- 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
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°
- 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]
- [[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
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
The second comma in the comma list defines which 11-limit family member we are looking at. Undecimal sensamagic adds 385/384, sensawer adds 441/440, octarod adds 100/99, shrusus adds 176/175. These temperaments use the same generators as sensamagic. Bisector adds 121/120 with a half-octave period.
Temperaments discussed elsewhere include supernatural (→ Keemic family) and sensigh (→ Sengic family). The rest are considered below.
Undecimal 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]]
- 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]
- [[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]]
- 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]]
- 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*
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]]
- 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
- 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