Gravity family

The gravity family tempers out graviton, the 5-limit comma 129140163/128000000 = [-13 17 -6⟩. The graviton equals (81/80)⁴/(25/24), so that four 81/80 commas come to a classic chromatic semitone. The generator of gravity temperament is a grave fifth of ~40/27, and hence the name. It is part of the syntonic-chromatic equivalence continuum, whereby (81/80)^k = 25/24.

Gravity

Subgroup: 2.3.5

Comma list: 129140163/128000000

Mapping: [⟨1 5 12], ⟨0 -6 -17]]

Mapping generators: ~2, ~40/27

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.844

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

Badness: 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.

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

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

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

Gencom: [2 40/27; 243/242 4000/3993]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834

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

Badness: 0.0125

Marvo

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

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

Wedgie: ⟨⟨ 6 17 46 13 56 59 ]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.694

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

Badness: 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 = 1\1, ~27/20 = 516.699

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

Badness: 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 = 1\1, ~27/20 = 516.730

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

Badness: 0.026882

Zarvo

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33075/32768

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

Wedgie: ⟨⟨ 6 17 -26 13 -58 -108 ]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.702

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

Badness: 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 = 1\1, ~27/20 = 516.691

Optimal ET sequence: 65, 72, 353cd

Badness: 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 = 1\1, ~27/20 = 516.667

Optimal ET sequence: 65f, 72

Badness: 0.027584

Gravid

Subgroup: 2.3.5.7

Comma list: 126/125, 1605632/1594323

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

Wedgie: ⟨⟨ 6 17 39 13 45 43 ]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.140

Optimal ET sequence: 58, 123, 181c

Badness: 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 = 1\1, ~27/20 = 517.155

Optimal ET sequence: 58, 123, 181ce

Badness: 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, and the octave part of the wedgie is ⟨⟨ 12 34 20 30 … ]].

Similar comments apply to the 13-limit, where we can add 351/350, 364/363, and 729/728 to the commas, with ⟨⟨ 12 34 20 30 52 … ]] as the octave wedgie. 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

Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]

Optimal tuning (POTE): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)

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

Badness: 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 = 1\2, ~27/20 = 516.833 (~21/20 = 83.167)

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

Badness: 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 = 1\2, ~27/20 = 516.884 (~21/20 = 83.116)

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

Badness: 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 = 1\2, ~27/20 = 516.832 (~21/20 = 83.168)

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

Badness: 0.012657