Xenharmonic Wiki

Gravity family

Revision as of 19:44, 25 May 2026 by Godtone (talk | contribs) (→Abergravity: fix mistakes and add 13-limit extension)
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; 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

Optimal tunings:

  • 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 abergravity (58 & 65d), marvo (65d & 72), zarvo (65 & 72), gravid (58 & 65), and harry (58 & 72), all considered below. A notable subgroup extension is larry.

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

Abergravity

Abergravity, first discovered by User:Godtone (but left unnamed) and rediscovered and named by User:2^67-1 later, is the extension of 2.3.5.11 gravity to prime 7 by extending the streak S11 = S10 = S9 = S8, so that the generalized comma 121/120~100/99~81/80 discussed in #2.3.5.11 subgroup (larry) is equated with a shrunk ~64/63, hence a flat-tending ~8/7 is characteristic. It is the 58 & 65d temperament, also supported by their val sum of 58 + 65d = 123df. (A sharp edo tuning of prime 7 (and hence a flat tuning of 8/7) is possible if we use the extreme tuning 51ce in which we also temper out S7/S8.) An obvious extension to the 13-limit is by noticing the 'squeeze' of equated commas (S8, S9, S10, S11) as suggesting S12 to be tempered out which fits the 58 & 65d join, and this is intuitively confirmed by also implying tempering out S11/S13 so that the spacing is made natural, but also because it implies tempering out 352/351 and 351/350 in the 13-limit as a natural extension for S8/S10 = 176/175, their product. Arguably the best edo tuning for making sense of this spacing is 58edo, a great tuning for 15-odd-limit, as there we use the distinction between 14/13 and 13/12~12/11 to have 15/14~14/13 make sense, though if you want marvel (16/15~15/14) you will want to use 65edo instead.

Its S-expression-based comma list is {S8/S9, S9/S10, S10/S11, (S11/S13,) S12}.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 177147/175000

Mapping: [⟨1 -1 -5 11], ⟨0 6 17 -19]]

Optimal tunings:

  • WE: ~2 = 1198.818 ¢, ~27/20 = 516.633 ¢
error map: ⟨-1.182 -0.972 +2.366 +2.137]
  • CWE: ~2 = 1200.0000 ¢, ~27/20 = 517.134 ¢
error map: ⟨0.000 +0.846 +4.956 +5.637]

Badness (Sintel): 3.464

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 176/175, 243/242

Mapping: [⟨1 -1 -5 11 -3], ⟨0 6 17 -19 15]]

Optimal tunings:

  • WE: ~2 = 1198.737 ¢, ~27/20 = 516.587 ¢
error map: ⟨-1.263 -1.168 +1.987 +2.120 +1.282]
  • CWE: ~2 = 1200.000 ¢, ~27/20 = 517.116 ¢
error map: ⟨0.000 +0.744 +4.666 +5.961 +5.429]

Badness (Sintel): 1.555

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 5120/5103, 176/175, 243/242, 144/143

Mapping: [⟨1 -1 -5 11 -3 5], ⟨0 6 17 -19 15 -3]]

Optimal tunings:

  • WE: ~2 = 1198.562 ¢, ~27/20 = 516.528 ¢
error map: ⟨-1.438 -1.349 +1.851 +1.327 +0.916 +2.700]
  • CWE: ~2 = 1200.000 ¢, ~27/20 = 517.135 ¢
error map: ⟨0.000 +0.853 +4.975 +5.617 +5.701 +8.069]

Badness (Sintel): 1.144

Marvo

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

Mapping: [⟨1 -1 -5 -17], ⟨0 6 17 46]]

Optimal tunings:

  • 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

Zarvo was named by Petr Pařízek in 2011, for it is similar to marvo, but with prime 7 mapped to -26 steps.[1]

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33075/32768

Mapping: [⟨1 -1 -5 14], ⟨0 6 17 -26]]

Optimal tunings:

  • 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 ¢

Optimal ET sequence: 65f, 72

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]]

Optimal tunings:

  • 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

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. 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

Optimal tunings:

  • 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

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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