Passion 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 passion family of temperaments tempers out the passion comma, 262144/253125 ([18 -4 -5⟩) in the 5-limit, which equates five instances of 16/15 with 4/3.
Passion
Passion is generated by a ~16/15 semitone, five of which reach a ~4/3 perfect fourth; its ploidacot is omega-pentacot. It follows that both 3 and 5 should be tuned sharp. 73edo does this and may be recommended as a tuning.
Subgroup: 2.3.5
Comma list: 262144/253125
Mapping: [⟨1 2 2], ⟨0 -5 4]]
- mapping generators: ~2, ~16/15
- WE: ~2 = 1197.8074 ¢, ~16/15 = 98.4898 ¢
- error map: ⟨-2.193 +1.211 +3.260]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7335 ¢
- error map: ⟨0.000 +4.378 +8.620]
Optimal ET sequence: 12, 49, 61, 73, 231bccc
Badness (Sintel): 3.95
Overview to extensions
The second comma of the comma list defines which 7-limit family member we are looking at:
Those all use the same nominal generator as passion.
The only weak extension we are considering is freivald, which adds 6144/6125 to the comma list and splits the generator in two.
Septimal passion
Septimal passion tempers out 64/63, the archytas comma, connecting itself to the archytas clan. It may be described as the 12 & 49 temperament. The interval class of 7 is found by exactly 10 generator steps, which calls for a flatter tuning of the fourth, and 49edo may be recommended as a tuning.
Subgroup: 2.3.5.7
Comma list: 64/63, 3125/3087
Mapping: [⟨1 2 2 2], ⟨0 -5 4 10]]
- WE: ~2 = 1197.0350 ¢, ~21/20 = 97.9106 ¢
- error map: ⟨-2.965 +2.562 -0.601 +4.351]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.1019 ¢
- error map: ⟨0.000 +7.535 +6.094 +12.193]
Optimal ET sequence: 12, 37, 49, 110bcd
Badness (Sintel): 1.58
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 1375/1372
Mapping: [⟨1 2 2 2 2], ⟨0 -5 4 10 18]]
Optimal tunings:
- WE: ~2 = 1196.9304 ¢, ~21/20 = 97.7686 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 97.9333 ¢
Optimal ET sequence: 12, 37, 49
Badness (Sintel): 1.35
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 196/195, 275/273
Mapping: [⟨1 2 2 2 2 2], ⟨0 -5 4 10 18 21]]
Optimal tunings:
- WE: ~2 = 1196.8712 ¢, ~21/20 = 97.6548 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 97.8141 ¢
Optimal ET sequence: 12f, 25f, 37, 49f
Badness (Sintel): 1.28
Passionate
Passionate tempers out 126/125 and may be described as 12 & 73. The optimum of this temperament is closer to its 5-limit counterpart's, but prime 7 is much more complex, at +22 generator steps away. 73edo and 85edo in the 85c val are among the possible tunings.
Subgroup: 2.3.5.7
Comma list: 126/125, 131072/127575
Mapping: [⟨1 2 2 1], ⟨0 -5 4 22]]
- WE: ~2 = 1197.9486 ¢, ~16/15 = 98.6479 ¢
- error map: ⟨-2.051 +0.703 +4.175 -0.623]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7770 ¢
- error map: ⟨0.000 +4.160 +8.794 +4.269]
Optimal ET sequence: 12, 49d, 61, 73, 85c
Badness (Sintel): 3.07
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 8192/8019
Mapping: [⟨1 2 2 1 1], ⟨0 -5 4 22 30]]
Optimal tunings:
- WE: ~2 = 1197.8835 ¢, ~16/15 = 98.5469 ¢
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.6733 ¢
Optimal ET sequence: 12, 49de, 61, 73, 158bccee
Badness (Sintel): 2.26
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 343/338, 364/363
Mapping: [⟨1 2 2 1 1 1], ⟨0 -5 4 22 30 33]]
Optimal tunings:
- WE: ~2 = 1197.8309 ¢, ~16/15 = 98.4485 ¢
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.5760 ¢
Optimal ET sequence: 12f, 49de, 61, 73f
Badness (Sintel): 2.16
Pash
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 1352/1331
Mapping: [⟨1 2 2 1 1 0], ⟨0 -5 4 22 30 45]]
Optimal tunings:
- WE: ~2 = 1197.8306 ¢, ~16/15 = 98.6124 ¢
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.7468 ¢
Optimal ET sequence: 12f, 73, 85ce
Badness (Sintel): 2.33
Passive
This low-complexity extension tempers out 225/224 as well as 256/245. Two generator steps represent ~8/7. As one might expect, 12edo is about as accurate as it can be tuned.
Subgroup: 2.3.5.7
Comma list: 225/224, 256/245
Mapping: [⟨1 2 2 3], ⟨0 -5 4 -2]]
- WE: ~2 = 1195.3551 ¢, ~16/15 = 98.4266 ¢
- error map: ⟨-4.645 -3.378 -1.897 +20.386]
- CWE: ~2 = 1200.0000 ¢, ~16/15 = 98.9690 ¢
- error map: ⟨0.000 +3.200 +9.562 +33.236]
Optimal ET sequence: 1, 11, 12
Badness (Sintel): 1.90
Freivald
Freivald splits the octave complement of the semitone generator of passion in two, each of which is used for ~11/8. It may be described as 24 & 37; its ploidacot is delta-decacot. 61edo may be recommended as a tuning.
Subgroup: 2.3.5.7
Comma list: 6144/6125, 6272/6075
Mapping: [⟨1 -3 6 -5], ⟨0 10 -8 17]]
- mapping generator: ~2, ~48/35
- WE: ~2 = 1198.4021 ¢, ~48/35 = 550.2038 ¢
- error map: ⟨-1.598 +4.877 +2.468 -7.371]
- CWE: ~2 = 1200.0000 ¢, ~48/35 = 550.8980 ¢
- error map: ⟨0.000 +7.025 +6.502 -3.559]
Optimal ET sequence: 13d, 24, 37, 61, 98b
Badness (Sintel): 5.29
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 2744/2673
Mapping: [⟨1 -3 6 -5 3], ⟨0 10 -8 17 1]]
Optimal tunings:
- WE: ~2 = 1198.8053 ¢, ~11/8 = 550.3522 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/8 = 550.8760 ¢
Optimal ET sequence: 13d, 24, 37, 61
Badness (Sintel): 2.56
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 121/120, 176/175, 512/507
Mapping: [⟨1 7 -2 12 4 1], ⟨0 -10 8 -17 -1 5]]
Optimal tunings:
- WE: ~2 = 1198.7706 ¢, ~11/8 = 550.3418 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/8 = 550.8892 ¢
Optimal ET sequence: 13d, 24, 37, 61
Badness (Sintel): 1.90