Gravity family: Difference between revisions

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=== Overview to extensions ===

Full 7-limit extensions of gravity include 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.

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 {{nowrap| 58 & 65d }} extension by tempering out [[176/175]] to extend larry to include prime 7 and tempering out [[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-~~expression]]s~~ 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~~]] in~~ six generators of 27/20.

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

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Comma list: 243/242, 4000/3993

Subgrop-val 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 5 12 0 12 | 0 -6 -17 0 -15 }}

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

Optimal tunings:

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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

[[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 }}

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}}

: 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 }}

Badness (Sintel): 1.25

== Marvo ==

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

[[Optimal tuning]]s:

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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 tunings:

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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 tunings:

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== 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

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

[[Optimal tuning]]s:

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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 tunings:

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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 tunings:

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[[Comma list]]: 126/125, 1605632/1594323

[[Optimal tuning]]s:

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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 tunings:

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Badness (Sintel): 0.645

== References ==

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Gravity family| ]]

~~[[Category:Gravity| ]] ~~

[[Category:Rank 2]]

@@ Line 26: / Line 26: @@
 === Overview to extensions ===
-Full 7-limit extensions of gravity include 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.
+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 {{nowrap| 58 & 65d }} extension by tempering out [[176/175]] to extend larry to include prime 7 and tempering out [[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-expression]]s 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]] in six generators of 27/20.
+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
@@ Line 37: / Line 35: @@
 Comma list: 243/242, 4000/3993
-Subgrop-val 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 5 12 0 12 | 0 -6 -17 0 -15 }}
+Gencom mapping: {{mapping| 1 -1 -5 0 -3 | 0 6 17 0 15 }}
 Optimal tunings:
@@ Line 48: / Line 46: @@
 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 }}
+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}}
+: 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 ==
@@ Line 54: / Line 126: @@
 [[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]]s:
@@ Line 71: / Line 143: @@
 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 tunings:
@@ Line 86: / Line 158: @@
 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 tunings:
@@ Line 97: / Line 169: @@
 == 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
 [[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]]s:
@@ Line 118: / Line 192: @@
 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 tunings:
@@ Line 133: / Line 207: @@
 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 tunings:
@@ Line 148: / Line 222: @@
 [[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]]s:
@@ Line 165: / Line 239: @@
 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 tunings:
@@ Line 246: / Line 320: @@
 Badness (Sintel): 0.645
+== References ==
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Gravity family| ]] <!-- main article -->
-[[Category:Gravity| ]] <!-- key article -->
 [[Category:Rank 2]]