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{{Technical data page}}
{{Technical data page}}
The '''gravity family''' tempers out [[graviton]], the 5-limit comma 129140163/128000000 = {{monzo| -13 17 -6 }}. The graviton equals (81/80)<sup>4</sup>/(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&ndash;chromatic equivalence continuum]], whereby (81/80)<sup>''k''</sup> = 25/24.
The '''gravity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[graviton]] ({{monzo|legend=1| -13 17 -6 }}, [[ratio]]: 129140163/128000000).  


== Gravity ==
== Gravity ==
{{main|Gravity}}
{{Main| 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 [[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
[[Subgroup]]: 2.3.5
Line 9: Line 11:
[[Comma list]]: 129140163/128000000
[[Comma list]]: 129140163/128000000


{{Mapping|legend=1| 1 5 12 | 0 -6 -17 }}
{{Mapping|legend=1| 1 -1 -5 | 0 6 17 }}


: mapping generators: ~2, ~40/27
: mapping generators: ~2, ~27/20


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.844
[[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, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469 }}
{{Optimal ET sequence|legend=1| 7, …, 51c, 58, 65, 137, 202, 267, 469 }}


[[Badness]]: 0.093184
[[Badness]] (Sintel): 2.19


=== Overview to extensions ===
=== Overview to extensions ===
Full 7-limit extensions of gravity include marvo (65d &amp; 72), zarvo (65 &amp; 72), gravid (58 &amp; 65), and harry (58 &amp; 72), all considered below. A notable subgroup extension is larry.
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.


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 {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]], ([[847/845|S11/S13]],) [[144/143|S12]]}.
=== 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.


=== 2.3.5.11 subgroup (larry) ===
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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


Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}
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 ==
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 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.1335{{c}}
: 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:
* WE: ~2 = 1198.7370{{c}}, ~27/20 = 516.5874{{c}}
: error map: {{val| -1.263 -1.168 +1.987 +2.120 +1.282 }}
* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.1165{{c}}
: error map: {{val| 0.000 +0.744 +4.666 +5.961 +5.429 }}
 
{{Optimal ET sequence|legend=0| 7, 51ce, 58, 123d, 181cde }}


Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}
Badness (Sintel): 1.56


: gencom: [2 40/27; 243/242 4000/3993]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834
Comma list: 144/143, 176/175, 243/242, 847/845


{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
Mapping: {{mapping| 1 -1 -5 11 -3 5 | 0 6 17 -19 15 -3 }}


Badness: 0.0125
Optimal tunings:
* WE: ~2 = 1198.5623{{c}}, ~27/20 = 516.5280{{c}}
: error map: {{val| -1.438 -1.349 +1.851 +1.327 +0.916 +2.700 }}
* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.1346{{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
 
=== 17-limit ===
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}}
: 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


== Marvo ==
== Marvo ==
Line 46: Line 126:
[[Comma list]]: 225/224, 78125000/78121827
[[Comma list]]: 225/224, 78125000/78121827


{{Mapping|legend=1| 1 5 12 29 | 0 -6 -17 -46 }}
{{Mapping|legend=1| 1 -1 -5 -17 | 0 6 17 46 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694
[[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}}
: error map: {{val| 0.000 -1.676 -2.191 -0.024 }}


{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
{{Optimal ET sequence|legend=1| 65d, 72, 353c, 425bc, 497bc, 569bcc }}


[[Badness]]: 0.097627
[[Badness]] (Sintel): 2.47


=== 11-limit ===
=== 11-limit ===
Line 59: Line 143:
Comma list: 225/224, 243/242, 4000/3993
Comma list: 225/224, 243/242, 4000/3993


Mapping: {{mapping| 1 5 12 29 12 | 0 -6 -17 -46 -15 }}
Mapping: {{mapping| 1 -1 -5 -17 -3 | 0 6 17 46 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699
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=1| 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=0| 65d, 72, 281, 353c, 425bc, 497bc }}


Badness: 0.031685
Badness (Sintel): 1.05


=== 13-limit ===
=== 13-limit ===
Line 72: Line 158:
Comma list: 225/224, 243/242, 351/350, 1625/1617
Comma list: 225/224, 243/242, 351/350, 1625/1617


Mapping: {{mapping| 1 5 12 29 12 39 | 0 -6 -17 -46 -15 -62 }}
Mapping: {{mapping| 1 -1 -5 -17 -3 -23 | 0 6 17 46 15 62 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730
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=1| 65d, 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=0| 65d, 72, 137, 209, 281f }}


Badness: 0.026882
Badness (Sintel): 1.10


== Zarvo ==
== 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
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 5 12 -12 | 0 -6 -17 26 }}
{{Mapping|legend=1| 1 -1 -5 14 | 0 6 17 -26 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702
[[Optimal tuning]]s:
* [[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 }}
{{Optimal ET sequence|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}


[[Badness]]: 0.096840
[[Badness]] (Sintel): 2.45


=== 11-limit ===
=== 11-limit ===
Line 98: Line 192:
Comma list: 243/242, 385/384, 4000/3993
Comma list: 243/242, 385/384, 4000/3993


Mapping: {{mapping| 1 5 12 -12 12 | 0 -6 -17 26 -15 }}
Mapping: {{mapping| 1 -1 -5 14 -3 | 0 6 17 -26 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.691
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=1| 65, 72, 353cd }}
{{Optimal ET sequence|legend=0| 65, 72, 353cd }}


Badness: 0.034773
Badness (Sintel): 1.15


=== 13-limit ===
=== 13-limit ===
Line 111: Line 207:
Comma list: 169/168, 243/242, 325/324, 385/384
Comma list: 169/168, 243/242, 325/324, 385/384


Mapping: {{mapping| 1 5 12 -12 12 -2 | 0 -6 -17 26 -15 10 }}
Mapping: {{mapping| 1 -1 -5 14 -3 8 | 0 6 17 -26 15 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.667
Optimal tunings:
* WE: ~2 = 1200.9333{{c}}, ~27/20 = 517.0690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.6698{{c}}


{{Optimal ET sequence|legend=1| 65f, 72 }}
{{Optimal ET sequence|legend=0| 65f, 72 }}


Badness: 0.027584
Badness (Sintel): 1.14


== Gravid ==
== Gravid ==
Line 124: Line 222:
[[Comma list]]: 126/125, 1605632/1594323
[[Comma list]]: 126/125, 1605632/1594323


{{Mapping|legend=1| 1 5 12 25 | 0 -6 -17 -39 }}
{{Mapping|legend=1| 1 -1 -5 -14 | 0 6 17 39 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3413{{c}}, ~27/20 = 516.8566{{c}}
: [[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 }}
{{Optimal ET sequence|legend=1| 58, 123, 181c }}


[[Badness]]: 0.131153
[[Badness]] (Sintel): 3.32


=== 11-limit ===
=== 11-limit ===
Line 137: Line 239:
Comma list: 126/125, 243/242, 896/891
Comma list: 126/125, 243/242, 896/891


Mapping: {{mapping| 1 5 12 25 12 | 0 -6 -17 -39 -15 }}
Mapping: {{mapping| 1 -1 -5 -14 -3 | 0 6 17 39 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.155
Optimal tunings:
* WE: ~2 = 1199.0523{{c}}, ~27/20 = 516.7466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 517.1210{{c}}


{{Optimal ET sequence|legend=1| 58, 123, 181ce }}
{{Optimal ET sequence|legend=0| 58, 123, 181ce }}


Badness: 0.047283
Badness (Sintel): 1.56


== Harry ==
== Harry ==
{{Main| Harry }}
{{Main| 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 &amp; 72 temperament. The [[period]] is half an [[octave]], and the generator ~21/20, with generator tunings of [[130edo|9\130]] or [[202edo|14\202]] being good choices. [[Mos]] of size 14, 16, 30, 44 or 58 are among the scale choices.
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.
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.
Line 162: Line 266:
: mapping generators: ~567/400, ~21/20
: mapping generators: ~567/400, ~21/20


[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)
[[Optimal tuning]]s:
* [[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 }}
{{Optimal ET sequence|legend=1| 14c, …, 58, 72, 130, 202, 534, 736b, 938b }}


[[Badness]]: 0.034077
[[Badness]] (Sintel): 0.862


=== 11-limit ===
=== 11-limit ===
Line 175: Line 283:
Mapping: {{mapping| 2 4 7 7 9 | 0 -6 -17 -10 -15 }}
Mapping: {{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 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=1| 14c, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=0| 14c, …, 58, 72, 130, 202 }}


Badness: 0.015867
Badness (Sintel): 0.525


=== 13-limit ===
=== 13-limit ===
Line 188: Line 298:
Mapping: {{mapping| 2 4 7 7 9 11 | 0 -6 -17 -10 -15 -26 }}
Mapping: {{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 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=1| 14cf, 58, 72, 130, 332f, 462ef }}
{{Optimal ET sequence|legend=0| 14cf, …, 58, 72, 130 }}


Badness: 0.013046
Badness (Sintel): 0.539


=== 17-limit ===
=== 17-limit ===
Line 201: Line 313:
Mapping: {{mapping| 2 4 7 7 9 11 9 | 0 -6 -17 -10 -15 -26 -6 }}
Mapping: {{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 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 }}


{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 202g }}
Badness (Sintel): 0.645


Badness: 0.012657
== References ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Gravity family| ]] <!-- main article -->
[[Category:Gravity family| ]] <!-- main article -->
[[Category:Gravity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 19:26, 26 May 2026

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 is the extension of 2.3.5.11-subgroup gravity with 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 also possible with the extreme tuning 51ce-edo, in which 1029/1024 (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 (S12) to be tempered out, which fits the 58 & 65d join, and this is intuitively confirmed by also tempering out 847/845 (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 (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 Godtone but left unnamed until being rediscovered and named by 2^67-1 in 2026 as a contraction of aberschismic and gravity. Its S-expression-based comma list is {S8/S9, S9/S10, S10/S11, (S11/S13,) S12, S16}.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 177147/175000

Mapping: [⟨1 -1 -5 11], ⟨0 6 17 -19]]

Optimal tunings:

  • WE: ~2 = 1198.8184 ¢, ~27/20 = 516.6336 ¢
error map: ⟨-1.182 -0.972 +2.366 +2.137]
  • CWE: ~2 = 1200.0000 ¢, ~27/20 = 517.1335 ¢
error map: ⟨0.000 +0.846 +4.956 +5.637]

Optimal ET sequence: 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: [⟨1 -1 -5 11 -3], ⟨0 6 17 -19 15]]

Optimal tunings:

  • WE: ~2 = 1198.7370 ¢, ~27/20 = 516.5874 ¢
error map: ⟨-1.263 -1.168 +1.987 +2.120 +1.282]
  • CWE: ~2 = 1200.000 ¢, ~27/20 = 517.1165 ¢
error map: ⟨0.000 +0.744 +4.666 +5.961 +5.429]

Optimal ET sequence: 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: [⟨1 -1 -5 11 -3 5], ⟨0 6 17 -19 15 -3]]

Optimal tunings:

  • WE: ~2 = 1198.5623 ¢, ~27/20 = 516.5280 ¢
error map: ⟨-1.438 -1.349 +1.851 +1.327 +0.916 +2.700]
  • CWE: ~2 = 1200.000 ¢, ~27/20 = 517.1346 ¢
error map: ⟨0.000 +0.853 +4.975 +5.617 +5.701 +8.069]

Optimal ET sequence: 7, 51ce, 58, 123df, 181cdeff, 239ccddeefff

Badness (Sintel): 1.14

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 -5 11 -3 5 14], ⟨0 6 17 -19 15 -3 -23]]

Optimal tunings:

  • WE: ~2 = 1198.758 ¢, ~27/20 = 516.566 ¢
error map: ⟨-1.242 -1.316 +1.521 +2.755 +0.901 +3.564 -3.365]
  • CWE: ~2 = 1200.000 ¢, ~27/20 = 517.099 ¢
error map: ⟨0.000 +0.640 +4.373 +6.289 +5.171 +8.175 +1.762]

Optimal ET sequence: 7, 58, 123df

Badness (Sintel): 1.25

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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