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.
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, S9/10, S10/11, (S11/S13,) S12}.
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]]
Wedgie: ⟨⟨ 6 17 46 15 13 56 3 59 -24 -117 ]]
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]]
Wedgie: ⟨⟨ 6 17 46 15 62 13 56 3 76 59 -24 81 -117 4 159 ]]
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
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
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