Gravity family: Difference between revisions

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Badness (Sintel): 0.389

== 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.

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]]}.

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1198.818{{c}}, ~27/20 = 516.633{{c}}

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.134{{c}}

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

[[Badness]] (Sintel): 3.464

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

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

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

Badness (Sintel): 1.555

== Marvo ==

@@ Line 48: / Line 48: @@
 Badness (Sintel): 0.389
+== 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.
+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]]}.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 5120/5103, 177147/175000
+Mapping: {{mapping| 1 -1 -5 11 | 0 6 17 -19 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.818{{c}}, ~27/20 = 516.633{{c}}
+: [[error map]]: {{val| -1.182 -0.972 +2.366 +2.137 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/20 = 517.134{{c}}
+: error map: {{val| 0.000 +0.846 +4.956 +5.637 }}
+[[Badness]] (Sintel): 3.464
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 5120/5103, 176/175, 243/242
+Mapping: {{mapping| 1 -1 -5 11 -3 | 0 6 17 -19 15 }}
+Optimal tunings:
+* WE: ~2 = 1200.6303{{c}}, ~27/20 = 516.9658{{c}}
+: error map: {{val| -1.263 -1.168 +1.987 +2.120 +1.282 }}
+* CWE: ~2 = 1200.0000{{c}}, ~27/20 = 516.7131{{c}}
+: error map: {{val| 0.000 +0.744 +4.666 +5.961 +5.429 }}
+Badness (Sintel): 1.555
 == Marvo ==