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

Main article: 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; its ploidacot is beta-hexacot. It 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 5 12], ⟨0 -6 -17]]

mapping generators: ~2, ~40/27

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.844 ¢

Optimal ET sequence: 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469

Badness (Smith): 0.093184

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 176/175 to extend larry to include prime 7 and tempering 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)

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Subgrop-val mapping: [⟨1 5 12 12], ⟨0 -6 -17 -15]]

Gencom mapping: [⟨1 5 12 0 12], ⟨0 -6 -17 0 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.834 ¢

Optimal ET sequence: 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202

Badness (Smith): 0.0125

Marvo

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

Mapping: [⟨1 5 12 29], ⟨0 -6 -17 -46]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.694 ¢

Optimal ET sequence: 65d, 72, 137, 209, 281, 569bcc

Badness (Smith): 0.097627

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4000/3993

Mapping: [⟨1 5 12 29 12], ⟨0 -6 -17 -46 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.699 ¢

Optimal ET sequence: 65d, 72, 281, 353c, 425bc, 497bc

Badness (Smith): 0.031685

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 1625/1617

Mapping: [⟨1 5 12 29 12 39], ⟨0 -6 -17 -46 -15 -62]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.730 ¢

Optimal ET sequence: 65d, 72, 137, 209, 281f, 490bcf

Badness (Smith): 0.026882

Zarvo

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33075/32768

Mapping: [⟨1 5 12 -12], ⟨0 -6 -17 26]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.702 ¢

Optimal ET sequence: 65, 72, 281d, 353cd, 425bcdd, 497bcdd

Badness (Smith): 0.096840

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 4000/3993

Mapping: [⟨1 5 12 -12 12], ⟨0 -6 -17 26 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.691 ¢

Optimal ET sequence: 65, 72, 353cd

Badness (Smith): 0.034773

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 385/384

Mapping: [⟨1 5 12 -12 12 -2], ⟨0 -6 -17 26 -15 10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 516.667 ¢

Optimal ET sequence: 65f, 72

Badness (Smith): 0.027584

Gravid

Subgroup: 2.3.5.7

Comma list: 126/125, 1605632/1594323

Mapping: [⟨1 5 12 25], ⟨0 -6 -17 -39]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 517.140 ¢

Optimal ET sequence: 58, 123, 181c

Badness (Smith): 0.131153

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 896/891

Mapping: [⟨1 5 12 25 12], ⟨0 -6 -17 -39 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/20 = 517.155 ¢

Optimal ET sequence: 58, 123, 181ce

Badness (Smith): 0.047283

Harry

Main article: 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, with generator tunings of 9\130 or 14\202 being 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

Optimal tuning (POTE): ~567/400 = 600.000 ¢, ~27/20 = 516.844 ¢ (~21/20 = 83.156 ¢)

Optimal ET sequence: 14c, 58, 72, 130, 202, 534, 736b, 938b

Badness (Smith): 0.034077

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 tuning (POTE): ~99/70 = 600.000 ¢, ~27/20 = 516.833 ¢ (~21/20 = 83.167 ¢)

Optimal ET sequence: 14c, 58, 72, 130, 202

Badness (Smith): 0.015867

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 tuning (POTE): ~55/39 = 600.000 ¢, ~27/20 = 516.884 ¢ (~21/20 = 83.116 ¢)

Optimal ET sequence: 14cf, 58, 72, 130, 332f, 462ef

Badness (Smith): 0.013046

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 tuning (POTE): ~17/12 = 600.000 ¢, ~27/20 = 516.832 ¢ (~21/20 = 83.168 ¢)

Optimal ET sequence: 14cf, 58, 72, 130, 202g

Badness (Smith): 0.012657