Gravity family: Difference between revisions

The '''gravity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[graviton]] ({{monzo|legend=1| -13 17 -6 }}, [[ratio]]: 129140163/128000000).  

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 [[6/1|6th harmonic]]; the [[ploidacot]] for gravity is beta-hexacot. Gravity is part of the [[syntonic–chromatic equivalence continuum]] with {{nowrap| ''n'' {{=}} 6 }}, so it equates a [[2187/2048|Pythagorean apotome]] with a stack of six [[81/80|syntonic commas]].

 

[[Subgroup]]: 2.3.5

 

 

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.1831{{c}}, ~27/20 = 516.9226{{c}}

: [[error map]]: {{val| +0.183 -0.602 +0.456 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 516.8575{{c}}

: error map: {{val| 0.000 -0.810 +0.263 }}

{{Optimal ET sequence|legend=1| 7, …, 51c, 58, 65, 137, 202, 267, 469 }}

[[Badness]] (Sintel): 2.19

Full 7-limit extensions of gravity include abergravity ({{nowrap| 58 & 65d }}), marvo ({{nowrap| 65d & 72 }}), zarvo ({{nowrap| 65 & 72 }}), gravid ({{nowrap| 58 & 65 }}), and harry ({{nowrap| 58 & 72 }}), all considered below. A notable subgroup extension is 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.

Subgrop-val mapping: {{mapping| 1 -1 -5 -3 | 0 6 17 15 }}

Gencom mapping: {{mapping| 1 -1 -5 0 -3 | 0 6 17 0 15 }}

Optimal tunings:  

* WE: ~2 = 1200.0787{{c}}, ~27/20 = 516.8677{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.8400{{c}}

{{Optimal ET sequence|legend=0| 7, …, 51ce, 58, 65, 137, 202 }}

Badness (Sintel): 0.389

Abergravity is the extension of 2.3.5.11-subgroup gravity with prime 7 by extending the streak [[121/120|S11]]~[[100/99|S10]]~[[81/80|S9]]~[[64/63|''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 [[58edo|58]] & [[65edo|65d]] temperament, also supported by their [[val]] sum of 58 + 65d = [[123edo|123df]]. A sharp edo tuning of prime 7 (and hence a flat tuning of 8/7) is also possible with the extreme tuning [[51edo|51ce-edo]], in which [[1029/1024]] ([[S-expression|S7/S8]]) vanishes. (Note that while [[65edo]] doesn't appear in any of the optimal ET sequences, it is a very viable tuning if you like a sharp 7.)

An obvious extension to the 13-limit is by noticing the 'squeeze' of equated commas (S8, S9, S10, S11) as suggesting [[144/143]] ({{S|12}}) to be tempered out, which fits the 58 & 65d join, and this is intuitively confirmed by also tempering out [[847/845]] ([[S-expression|S11/S13]]) so that the spacing is made natural, but also because it tempers out [[352/351]] and [[351/350]] in the 13-limit as a natural extension for [[176/175]] ([[S-expression|S8/S10]]), their product. Arguably the best edo tuning for making sense of this spacing is [[58edo]], a great tuning for [[15-odd-limit]] where the distinction between [[12/11]]~[[13/12]] and [[14/13]]~[[15/14]] helps solidifying each other's identity. Alternatively, [[65edo]] gives a [[marvel]] tuning (16/15~15/14), and any tuning between them, such as [[123edo]], distinguishes 14/13, 15/14 and 16/15.

Abergravity was first discovered by [[User:Godtone|Godtone]] but left unnamed until being rediscovered and named by [[User:2^67-1|2^67-1]] in 2026 as a contraction of ''aberschismic'' and ''gravity''. Its S-expression-based comma list is {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]], ([[847/845|S11/S13]],) [[144/143|S12]], [[256/255|S16]]}.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 177147/175000

{{Mapping|legend=1| 1 -1 -5 11 | 0 6 17 -19 }}

* [[WE]]: ~2 = 1198.8184{{c}}, ~27/20 = 516.6336{{c}}

: [[error map]]: {{val| -1.182 -0.972 +2.366 +2.137 }}

: error map: {{val| 0.000 +0.846 +4.956 +5.637 }}

{{Optimal ET sequence|legend=0| 7, 51c, 58, 123d, 181cd, 239ccdd }}

[[Badness]] (Sintel): 3.46

Comma list: 176/175, 243/242, 2560/2541

Mapping: {{mapping| 1 -1 -5 11 -3 | 0 6 17 -19 15 }}

Optimal tunings:

* WE: ~2 = 1198.7370{{c}}, ~27/20 = 516.5874{{c}}

: error map: {{val| 0.000 +0.744 +4.666 +5.961 +5.429 }}

{{Optimal ET sequence|legend=0| 7, 51ce, 58, 123d, 181cde }}

Badness (Sintel): 1.56

Comma list: 144/143, 176/175, 243/242, 847/845

Mapping: {{mapping| 1 -1 -5 11 -3 5 | 0 6 17 -19 15 -3 }}

Optimal tunings:

* WE: ~2 = 1198.5623{{c}}, ~27/20 = 516.5280{{c}}

: error map: {{val| 0.000 +0.853 +4.975 +5.617 +5.701 +8.069 }}

{{Optimal ET sequence|legend=0| 7, 51ce, 58, 123df, 181cdeff, 239ccddeefff }}

Badness (Sintel): 1.14

Comma list: 144/143, 170/169, 176/175, 243/242, 847/845

Mapping: {{mapping| 1 -1 -5 11 -3 5 14 | 0 6 17 -19 15 -3 -23 }}

Optimal tunings:

* WE: ~2 = 1198.758{{c}}, ~27/20 = 516.566{{c}}

: error map: {{val| 0.000 +0.640 +4.373 +6.289 +5.171 +8.175 +1.762 }}

{{Optimal ET sequence|legend=0| 7, 58, 123df }}

Badness (Sintel): 1.25

== Marvo ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 78125000/78121827

{{Mapping|legend=1| 1 -1 -5 -17 | 0 6 17 46 }}

: [[error map]]: {{val| +0.630 -0.791 -1.047 +0.885 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 516.7131{{c}}

: error map: {{val| 0.000 -1.676 -2.191 -0.024 }}

{{Optimal ET sequence|legend=1| 65d, 72, 353c, 425bc, 497bc, 569bcc }}

[[Badness]] (Sintel): 2.47

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

 

 

 

 

 

 

Mapping: {{mapping| 1 -1 -5 -17 -3 -23 | 0 6 17 46 15 62 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.7401{{c}}

{{Optimal ET sequence|legend=0| 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.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 33075/32768

{{Mapping|legend=1| 1 -1 -5 14 | 0 6 17 -26 }}

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 516.7041{{c}}

{{Optimal ET sequence|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}

[[Badness]] (Sintel): 2.45

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

Mapping: {{mapping| 1 -1 -5 14 -3 | 0 6 17 -26 15 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.6957{{c}}

{{Optimal ET sequence|legend=0| 65, 72, 353cd }}

Badness (Sintel): 1.15

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 {{nowrap| 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 [[130edo|9\130]] or [[202edo|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.

 

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 19683/19600

{{Mapping|legend=1| 2 4 7 7 | 0 -6 -17 -10 }}

[[Optimal tuning]]s:

* [[WE]]: ~567/400 = 600.0856{{c}}, ~21/20 = 83.1679{{c}}

: error map: {{val| 0.000 -0.811 +0.261 -0.253 }}

{{Optimal ET sequence|legend=1| 14c, …, 58, 72, 130, 202, 534, 736b, 938b }}

[[Badness]] (Sintel): 0.862

Comma list: 243/242, 441/440, 4000/3993

Mapping: {{mapping| 2 4 7 7 9 | 0 -6 -17 -10 -15 }}

Optimal tunings:  

* CWE: ~99/70 = 600.0000{{c}}, ~21/20 = 83.1589{{c}}

{{Optimal ET sequence|legend=0| 14c, …, 58, 72, 130, 202 }}

Badness (Sintel): 0.525

Comma list: 243/242, 351/350, 364/363, 441/440

 

 

 

 

 

 

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

 

Optimal tunings:  

* WE: ~17/12 = 600.1620{{c}}, ~21/20 = 83.1904{{c}}

* CWE: ~17/12 = 600.0000{{c}}, ~21/20 = 83.1482{{c}}

{{Optimal ET sequence|legend=0| 14cf, 58, 72, 130, 202g }}

Badness (Sintel): 0.645