Gravity family: Difference between revisions

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The '''gravity family''' tempers out [[graviton]], the 5-limit comma 129140163/128000000 = {{monzo| -13 17 -6 }}. The graviton equals (81/80)4/(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-chromatic equivalence continuum]], whereby (81/80)''k'' = 25/24.

The '''gravity family''' tempers out [[graviton]], the 5-limit comma 129140163/128000000 = {{monzo| -13 17 -6 }}. The graviton equals (81/80)4/(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–chromatic equivalence continuum]], whereby (81/80)''k'' = 25/24.

== Gravity ==

[[Subgroup]]: 2.3.5

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Full 7-limit extensions of gravity include marvo (65d & 72), zarvo (65 & 72), gravid (58 & 65), and harry (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|S9/10]], [[4000/3993|S10/11]], ([[847/845|S11/S13]],) [[144/143|S12]]}.

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) ===

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}

Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}

: gencom: [2 40/27; 243/242 4000/3993]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834

{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}

Badness: 0.0125

== Marvo ==

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~~{{Multival|legend=1| 6 17 46 13 56 59 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694

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

~~{{Multival|legend=1| 6 17 46 15 13 56 3 59 -24 -117 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699

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

~~{{Multival|legend=1| 6 17 46 15 62 13 56 3 76 59 -24 81 -117 4 159 }}~~

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730

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~~{{Multival|legend=1| 6 17 -26 13 -58 -108 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702

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~~{{Multival|legend=1| 6 17 39 13 45 43 }}~~

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140

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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, 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.

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~~, and the octave part of the wedgie is {{multival| 12 34 20 30 … }}~~.

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~~, with {{multival| 12 34 20 30 52 … }} as the octave wedgie~~. 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.

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.

[[Subgroup]]: 2.3.5.7

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

~~{{Multival|legend=1| 12 34 20 26 -2 -49 }}~~

[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)

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Badness: 0.015867

~~==== 2.3.5.11 subgroup (Larry) ====~~

~~Subgroup: 2.3.5.11~~

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

~~Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}~~

~~Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}~~

~~: gencom: [2 40/27; 243/242 4000/3993]~~

~~Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834~~

~~{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}~~

~~Badness: 0.0125~~

=== 13-limit ===

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Gravity family| ]]

[[Category:Gravity| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-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-chromatic equivalence continuum]], whereby (81/80)<sup>''k''</sup> = 25/24.
+{{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.
 == Gravity ==
+{{main|Gravity}}
 [[Subgroup]]: 2.3.5
@@ Line 19: / Line 22: @@
 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.
-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|S9/10]], [[4000/3993|S10/11]], ([[847/845|S11/S13]],) [[144/143|S12]]}.
+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) ===
+Subgroup: 2.3.5.11
+Comma list: 243/242, 4000/3993
+Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}
+Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}
+: gencom: [2 40/27; 243/242 4000/3993]
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834
+{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
+Badness: 0.0125
 == Marvo ==
@@ Line 27: / Line 47: @@
 {{Mapping|legend=1| 1 5 12 29 | 0 -6 -17 -46 }}
-{{Multival|legend=1| 6 17 46 13 56 59 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694
@@ Line 42: / Line 60: @@
 Mapping: {{mapping| 1 5 12 29 12 | 0 -6 -17 -46 -15 }}
-{{Multival|legend=1| 6 17 46 15 13 56 3 59 -24 -117 }}
 Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699
@@ Line 57: / Line 73: @@
 Mapping: {{mapping| 1 5 12 29 12 39 | 0 -6 -17 -46 -15 -62 }}
-{{Multival|legend=1| 6 17 46 15 62 13 56 3 76 59 -24 81 -117 4 159 }}
 Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730
@@ Line 72: / Line 86: @@
 {{Mapping|legend=1| 1 5 12 -12 | 0 -6 -17 26 }}
-{{Multival|legend=1| 6 17 -26 13 -58 -108 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702
@@ Line 113: / Line 125: @@
 {{Mapping|legend=1| 1 5 12 25 | 0 -6 -17 -39 }}
-{{Multival|legend=1| 6 17 39 13 45 43 }}
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140
@@ Line 140: / Line 150: @@
 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.
-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, and the octave part of the wedgie is {{multival| 12 34 20 30 … }}.
+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, with {{multival| 12 34 20 30 52 … }} as the octave wedgie. 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.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 151: / Line 161: @@
 : mapping generators: ~567/400, ~21/20
-{{Multival|legend=1| 12 34 20 26 -2 -49 }}
 [[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)
@@ Line 172: / Line 180: @@
 Badness: 0.015867
-==== 2.3.5.11 subgroup (Larry) ====
-Subgroup: 2.3.5.11
-Comma list: 243/242, 4000/3993
-Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}
-Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}
-: gencom: [2 40/27; 243/242 4000/3993]
-Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834
-{{Optimal ET sequence|legend=1| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
-Badness: 0.0125
 === 13-limit ===
@@ Line 217: / Line 208: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Gravity family| ]] <!-- main article -->
 [[Category:Gravity| ]] <!-- key article -->
 [[Category:Rank 2]]