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

[[Comma list]]: 129140163/128000000

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

: mapping generators: ~2, ~27/20

[[Optimal tuning]]s:  

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

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

{{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.

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

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 }}

[[Optimal tuning]]s:  

* [[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

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

: error map: {{val| -1.263 -1.168 +1.987 +2.120 +1.282 }}

: 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

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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}}

* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.1346{{c}}

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

Badness (Sintel): 1.14

Subgroup: 2.3.5.7.11.13.17

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}}

* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.099{{c}}

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

Badness (Sintel): 1.25

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:  

: 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

Subgroup: 2.3.5.7.11

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

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Gravity family| ]] <!-- main article -->

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]