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

Optimal tunings:

• 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:

• Septimal passion adds 64/63;
• Passionate adds 126/125;
• Passive adds 225/224.

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]]

Optimal tunings:

• 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]]

Optimal tunings:

• 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]]

Optimal tunings:

• 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

Optimal tunings:

• 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