Gravity 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 gravity family of temperaments tempers out the graviton (monzo: [-13 17 -6⟩, ratio: 129140163/128000000).
Gravity
The generator for the gravity temperament is a grave fifth of ~40/27, and hence the name. However, the functional generator is the acute fourth of ~27/20, six of which reach the 6th harmonic; the ploidacot for gravity is beta-hexacot. Gravity is part of the syntonic–chromatic equivalence continuum with n = 6, so it equates a Pythagorean apotome with a stack of six syntonic commas.
Subgroup: 2.3.5
Comma list: 129140163/128000000
Mapping: [⟨1 -1 -5], ⟨0 6 17]]
- mapping generators: ~2, ~27/20
- WE: ~2 = 1200.1831 ¢, ~27/20 = 516.9226 ¢
- error map: ⟨+0.183 -0.602 +0.456]
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.8575 ¢
- error map: ⟨0.000 -0.810 +0.263]
Optimal ET sequence: 7, …, 51c, 58, 65, 137, 202, 267, 469
Badness (Sintel): 2.19
Overview to extensions
Full 7-limit extensions of gravity include marvo (65d & 72), zarvo (65 & 72), gravid (58 & 65), and harry (58 & 72), all considered below. A notable subgroup extension is larry.
There is also an unnamed 58 & 65d extension by tempering out 176/175 to extend larry to include prime 7 and tempering out 847/845 to extend it to the 13-limit, with an S-expression-based comma list of {S8/S9, S9/S10, S10/S11, (S11/S13,) S12}.
2.3.5.11 subgroup (larry)
Gravity is most naturally thought of as a 2.3.5.11 subgroup temperament, which in terms of S-expressions is defined by equating S9 (81/80), S10 (100/99), and S11 (121/120). By tempering out S10/S11, 4/3 is split into three intervals of 11/10, and by tempering out S9/S11, 3/2 is split into two intervals of 11/9. The overall structure therefore divides 6/1 into six generators of 27/20.
Subgroup: 2.3.5.11
Comma list: 243/242, 4000/3993
Subgrop-val mapping: [⟨1 -1 -5 -3], ⟨0 6 17 15]]
Gencom mapping: [⟨1 -1 -5 0 -3], ⟨0 6 17 0 15]]
Optimal tunings:
- WE: ~2 = 1200.0787 ¢, ~27/20 = 516.8677 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.8400 ¢
Optimal ET sequence: 7, …, 51ce, 58, 65, 137, 202
Badness (Sintel): 0.389
Marvo
Subgroup: 2.3.5.7
Comma list: 225/224, 78125000/78121827
Mapping: [⟨1 -1 -5 -17], ⟨0 6 17 46]]
- WE: ~2 = 1200.6303 ¢, ~27/20 = 516.9658 ¢
- error map: ⟨+0.630 -0.791 -1.047 +0.885]
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.7131 ¢
- error map: ⟨0.000 -1.676 -2.191 -0.024]
Optimal ET sequence: 65d, 72, 353c, 425bc, 497bc, 569bcc
Badness (Sintel): 2.47
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 4000/3993
Mapping: [⟨1 -1 -5 -17 -3], ⟨0 6 17 46 15]]
Optimal tunings:
- WE: ~2 = 1200.5247 ¢, ~27/20 = 516.9253 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.7142 ¢
Optimal ET sequence: 65d, 72, 281, 353c, 425bc, 497bc
Badness (Sintel): 1.05
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 351/350, 1625/1617
Mapping: [⟨1 -1 -5 -17 -3 -23], ⟨0 6 17 46 15 62]]
Optimal tunings:
- WE: ~2 = 1200.4175 ¢, ~27/20 = 516.9102 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.7401 ¢
Optimal ET sequence: 65d, 72, 137, 209, 281f
Badness (Sintel): 1.10
Zarvo
Subgroup: 2.3.5.7
Comma list: 4375/4374, 33075/32768
Mapping: [⟨1 -1 -5 14], ⟨0 6 17 -26]]
- WE: ~2 = 1200.8048 ¢, ~27/20 = 517.0487 ¢
- error map: ⟨+0.805 -0.468 -0.510 -0.825]
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.7041 ¢
- error map: ⟨0.000 -1.730 -2.344 -3.133]
Optimal ET sequence: 65, 72, 281d, 353cd, 425bcdd, 497bcdd
Badness (Sintel): 2.45
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 385/384, 4000/3993
Mapping: [⟨1 -1 -5 14 -3], ⟨0 6 17 -26 15]]
Optimal tunings:
- WE: ~2 = 1200.7023 ¢, ~27/20 = 516.9937 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.6957 ¢
Optimal ET sequence: 65, 72, 353cd
Badness (Sintel): 1.15
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 243/242, 325/324, 385/384
Mapping: [⟨1 -1 -5 14 -3 8], ⟨0 6 17 -26 15 -10]]
Optimal tunings:
- WE: ~2 = 1200.9333 ¢, ~27/20 = 517.0690 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 516.6698 ¢
Badness (Sintel): 1.14
Gravid
Subgroup: 2.3.5.7
Comma list: 126/125, 1605632/1594323
Mapping: [⟨1 -1 -5 -14], ⟨0 6 17 39]]
- WE: ~2 = 1199.3413 ¢, ~27/20 = 516.8566 ¢
- error map: ⟨-0.659 -0.157 +3.542 -2.196]
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 517.1162 ¢
- error map: ⟨0.000 +0.742 +4.662 -1.292]
Optimal ET sequence: 58, 123, 181c
Badness (Sintel): 3.32
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242, 896/891
Mapping: [⟨1 -1 -5 -14 -3], ⟨0 6 17 39 15]]
Optimal tunings:
- WE: ~2 = 1199.0523 ¢, ~27/20 = 516.7466 ¢
- CWE: ~2 = 1200.0000 ¢, ~27/20 = 517.1210 ¢
Optimal ET sequence: 58, 123, 181ce
Badness (Sintel): 1.56
Harry
Harry adds the breedsma, 2401/2400, and the cataharry comma, 19683/19600, to the set of commas, and may be described as the 58 & 72 temperament. The period is half an octave, and the generator ~21/20. The ploidacot for harry is diploid delta-hexacot. Generator tunings of 9\130 or 14\202 are good choices. Mos of size 14, 16, 30, 44 or 58 are among the scale choices.
It becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 9\130 and especially 14\202 still make for good tuning choices.
Similar comments apply to the 13-limit, where we can add 351/350, 364/363, and 729/728 to the commas. 130edo is again a good tuning choice, but even better might be tuning the harmonic 7 justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 19683/19600
Mapping: [⟨2 4 7 7], ⟨0 -6 -17 -10]]
- mapping generators: ~567/400, ~21/20
- WE: ~567/400 = 600.0856 ¢, ~21/20 = 83.1679 ¢
- error map: ⟨+0.171 -0.620 +0.431 +0.094]
- CWE: ~567/400 = 1200.0000 ¢, ~21/20 = 83.1427 ¢
- error map: ⟨0.000 -0.811 +0.261 -0.253]
Optimal ET sequence: 14c, …, 58, 72, 130, 202, 534, 736b, 938b
Badness (Sintel): 0.862
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4000/3993
Mapping: [⟨2 4 7 7 9], ⟨0 -6 -17 -10 -15]]
Optimal tunings:
- WE: ~99/70 = 600.0504 ¢, ~21/20 = 83.1740 ¢
- CWE: ~99/70 = 600.0000 ¢, ~21/20 = 83.1589 ¢
Optimal ET sequence: 14c, …, 58, 72, 130, 202
Badness (Sintel): 0.525
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 364/363, 441/440
Mapping: [⟨2 4 7 7 9 11], ⟨0 -6 -17 -10 -15 -26]]
Optimal tunings:
- WE: ~55/39 = 599.9967 ¢, ~21/20 = 83.1160 ¢
- CWE: ~55/39 = 600.0000 ¢, ~21/20 = 83.1169 ¢
Optimal ET sequence: 14cf, …, 58, 72, 130
Badness (Sintel): 0.539
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
Mapping: [⟨2 4 7 7 9 11 9], ⟨0 -6 -17 -10 -15 -26 -6]]
Optimal tunings:
- WE: ~17/12 = 600.1620 ¢, ~21/20 = 83.1904 ¢
- CWE: ~17/12 = 600.0000 ¢, ~21/20 = 83.1482 ¢
Optimal ET sequence: 14cf, 58, 72, 130, 202g
Badness (Sintel): 0.645