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).  

{{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}}

{{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.

 

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

 

 

 

 

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.

 

 

 

 

 

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.1335{{c}}

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

 

 

 

 

 

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

 

 

 

 

 

 

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

 

 

 

 

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| -1.242 -1.316 +1.521 +2.755 +0.901 +3.564 -3.365 }}

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.6303{{c}}, ~27/20 = 516.9658{{c}}

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

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

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

[[Badness]] (Sintel): 2.47

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

Optimal tunings:

* WE: ~2 = 1200.5247{{c}}, ~27/20 = 516.9253{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.7142{{c}}

{{Optimal ET sequence|legend=0| 65d, 72, 281, 353c, 425bc, 497bc }}

Badness (Sintel): 1.05

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

Optimal tunings:

* WE: ~2 = 1200.4175{{c}}, ~27/20 = 516.9102{{c}}

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

{{Optimal ET sequence|legend=0| 65d, 72, 137, 209, 281f }}

Badness (Sintel): 1.10

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

* [[WE]]: ~2 = 1200.8048{{c}}, ~27/20 = 517.0487{{c}}

: [[error map]]: {{val| +0.805 -0.468 -0.510 -0.825 }}

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

: error map: {{val| 0.000 -1.730 -2.344 -3.133 }}

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

[[Badness]] (Sintel): 2.45

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

Optimal tunings:

* WE: ~2 = 1200.7023{{c}}, ~27/20 = 516.9937{{c}}

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

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

Badness (Sintel): 1.15

Mapping: {{mapping| 1 -1 -5 14 -3 8 | 0 6 17 -26 15 -10 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.6698{{c}}

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

Badness (Sintel): 1.14

{{Mapping|legend=1| 1 -1 -5 -14 | 0 6 17 39 }}

: [[error map]]: {{val| -0.659 -0.157 +3.542 -2.196 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.1162{{c}}

: error map: {{val| 0.000 +0.742 +4.662 -1.292 }}

{{Optimal ET sequence|legend=1| 58, 123, 181c }}

[[Badness]] (Sintel): 3.32

Mapping: {{mapping| 1 -1 -5 -14 -3 | 0 6 17 39 15 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 517.1210{{c}}

{{Optimal ET sequence|legend=0| 58, 123, 181ce }}

Badness (Sintel): 1.56

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.

 

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 [[cent]]s. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

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

: mapping generators: ~567/400, ~21/20

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

: [[error map]]: {{val| +0.171 -0.620 +0.431 +0.094 }}

* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~21/20 = 83.1427{{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

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

Optimal tunings:

* WE: ~99/70 = 600.0504{{c}}, ~21/20 = 83.1740{{c}}

* 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

Mapping: {{mapping| 2 4 7 7 9 11 | 0 -6 -17 -10 -15 -26 }}

Optimal tunings:

* WE: ~55/39 = 599.9967{{c}}, ~21/20 = 83.1160{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~21/20 = 83.1169{{c}}

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

Badness (Sintel): 0.539

Mapping: {{mapping| 2 4 7 7 9 11 9 | 0 -6 -17 -10 -15 -26 -6 }}

 

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

Badness (Sintel): 0.645