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== Abergravity ==
== 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 [[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 possible if we use the extreme tuning [[51edo|51ce]] in which we also temper out [[1029/1024|S7/S8]].) 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 implying tempering out [[847/845|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 [[176/175|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.
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.)


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]]}.
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
[[Subgroup]]: 2.3.5.7
Line 56: Line 58:
[[Comma list]]: 5120/5103, 177147/175000
[[Comma list]]: 5120/5103, 177147/175000


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.818{{c}}, ~27/20 = 516.633{{c}}
* [[WE]]: ~2 = 1198.8184{{c}}, ~27/20 = 516.6336{{c}}
: [[error map]]: {{val| -1.182 -0.972 +2.366 +2.137 }}  
: [[error map]]: {{val| -1.182 -0.972 +2.366 +2.137 }}  
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.134{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.1335{{c}}
: error map: {{val| 0.000 +0.846 +4.956 +5.637 }}
: error map: {{val| 0.000 +0.846 +4.956 +5.637 }}


[[Badness]] (Sintel): 3.464
{{Optimal ET sequence|legend=0| 7, 51c, 58, 123d, 181cd, 239ccdd }}
 
[[Badness]] (Sintel): 3.46


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 5120/5103, 176/175, 243/242
Comma list: 176/175, 243/242, 2560/2541


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


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1198.737{{c}}, ~27/20 = 516.587{{c}}
* WE: ~2 = 1198.7370{{c}}, ~27/20 = 516.5874{{c}}
: error map: {{val| -1.263 -1.168 +1.987 +2.120 +1.282 }}
: error map: {{val| -1.263 -1.168 +1.987 +2.120 +1.282 }}
* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.116{{c}}
* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.1165{{c}}
: error map: {{val| 0.000 +0.744 +4.666 +5.961 +5.429 }}
: error map: {{val| 0.000 +0.744 +4.666 +5.961 +5.429 }}


Badness (Sintel): 1.555
{{Optimal ET sequence|legend=0| 7, 51ce, 58, 123d, 181cde }}
 
Badness (Sintel): 1.56


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 5120/5103, 176/175, 243/242, 144/143
Comma list: 144/143, 176/175, 243/242, 847/845


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


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1198.562{{c}}, ~27/20 = 516.528{{c}}
* 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 }}
: error map: {{val| -1.438 -1.349 +1.851 +1.327 +0.916 +2.700 }}
* CWE: ~2 = 1200.000{{c}}, ~27/20 = 517.135{{c}}
* 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 }}
: error map: {{val| 0.000 +0.853 +4.975 +5.617 +5.701 +8.069 }}


Badness (Sintel): 1.144
{{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 ==
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