Gravity: Difference between revisions

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| Title = Gravity; Larry
| Title = Gravity; Larry
| Subgroups = 2.3.5, 2.3.5.11
| Subgroups = 2.3.5, 2.3.5.11
| Comma basis = [[129140163/128000000]] (2.3.5); <br> [[243/242]], [[4000/3993]] (2.3.5.11)
| Comma basis = [[129140163/128000000]] (2.3.5); <br>[[243/242]], [[4000/3993]] (2.3.5.11)
| Edo join 1 = 7 | Edo join 2 = 58
| Edo join 1 = 7 | Edo join 2 = 58
| Generator = 27/20 | Generator tuning = 516.8 | Optimization method = POTE
| MOS scales = [[2L 5s]], [[7L 2s]], [[7L 9s]], ..., [[7L 51s]]
| Mapping = 1; 6 17 15
| Mapping = 1; 6 17 15
| Generators = 27/20 | Generators tuning = 516.8 | Optimization method = CWE
| MOS scales = [[2L 5s]], [[7L 2s]], [[7L 9s]], …, [[7L 51s]]
| Pergen = (P8, P19/6)
| Pergen = (P8, P19/6)
| Color name = Lala-tribiguti
| Color name = Lala-tribiguti
| Odd limit 1 = 5 | Mistuning 1 = 0.90 | Complexity 1 = 23
| Odd limit 1 = 5 | Mistuning 1 = 0.90 | Complexity 1 = 23
| Odd limit 2 = (2.3.5.11) 15 | Mistuning 2 = 1.48 | Complexity 2 = 30
| Odd limit 2 = 2.3.5.11 15 | Mistuning 2 = 1.48 | Complexity 2 = 30
}}
}}
'''Gravity''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[gravity family]], [[generator|generated]] by a [[27/20|classical acute fourth (27/20)]], six of which stacked reach the [[6/1|6th harmonic]] (which octave reduces to the perfect fifth, [[3/2]]), dividing the fifth in two and the fourth in three. The [[5/1|5th harmonic]] is found at three perfect fifths up and one generator down, or 17 generators in total, thereby tempering out the [[graviton]] ([[ratio]]: 129140163/128000000, {{monzo|legend=1| -13 17 -6 }}). The complement of the acute fourth generator is the grave fifth, [[40/27]], whence the temperament's name follows.


'''Gravity''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[gravity family]], [[generator|generated]] by a [[27/20|classical acute fourth (27/20)]], six of which stacked reach the interval [[6/1]] (which octave reduces to the perfect fifth, [[3/2]]), dividing the fifth in two and the fourth in three. The 5th harmonic is found at three perfect fifths up and one generator down, or 17 generators in total, thereby tempering out the [[129140163/128000000|graviton]] ([[ratio]]: 129140163/128000000, {{monzo|legend=1| -13 17 -6 }}). The complement of the acute fourth generator is the grave fifth, [[40/27]], whence the temperament's name.
Gravity is most naturally seen as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament, sometimes known as '''larry'''. Here [[8019/8000]] ([[S-expression|S9/S10]]) is tempered out, so that two intervals of 40/27 reach [[11/10]], and [[4000/3993]] ([[S-expression|S10/S11]]) is tempered out, so that three intervals of 11/10 reach 4/3. These equivalences also imply that [[243/242]] is tempered out; three 27/20 fourths reach [[11/9]], which is thus equated to [[27/22]] and acts as an exact neutral third. Gravity's generator lies close to the fifth of [[7edo]], implying that the [[mos scale]]s of gravity [[cluster temperament|cluster]] heavily around 7edo, and in this interpretation the comma reached after 7 generators simultaneously represents [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}). In fact, gravity can be completely defined by making this equivalence between three adjacent square superparticulars, being the most accurate of the 5 temperaments definable in such a way.
 
Gravity is most naturally seen as a [[2.3.5.11 subgroup]] temperament, sometimes known as '''larry'''. Here {{S|9/S10}} = [[8019/8000]] is tempered out, so that two intervals of 40/27 reach [[11/10]], and S10/S11 = [[4000/3993]] is tempered out, so that three intervals of 11/10 reach 4/3. These equivalences also imply that [[243/242]] is tempered out; three 27/20 fourths reach [[11/9]], which is thus equated to [[27/22]] and acts as an exact neutral third. Gravity's generator lies close to the fifth of [[7edo]], implying that the [[MOS scale]]s of gravity [[cluster temperament|cluster]] heavily around 7edo, and in this interpretation the comma reached after 7 generators simultaneously represents S9 = [[81/80]], S10 = [[100/99]], and S11 = [[121/120]]. In fact, gravity can be completely defined by making this equivalence between three adjacent square superparticulars, being the most accurate of the 5 temperaments definable in such a way.


Strong extensions with prime 7 include [[gravid]] (58 & 65), 58 & 65d, [[marvo]] (65d & 72), and [[zarvo]] (65 & 72). However, the most notable extension of gravity is [[harry]] (58 & 72), which splits the octave in two and extends well to the 13- and [[17-limit]].
Strong extensions with prime 7 include [[gravid]] (58 & 65), 58 & 65d, [[marvo]] (65d & 72), and [[zarvo]] (65 & 72). However, the most notable extension of gravity is [[harry]] (58 & 72), which splits the octave in two and extends well to the 13- and [[17-limit]].


{{tdlink|Gravity family #Gravity}}
For technical data, see [[Gravity family #Gravity]].
 
 
[[File:Gravity_construction.png|alt=Gravity construction.png|960x320px]]
 
A pictorial representation of the process of constructing the heptatonic MOS of 2.3.5.11 gravity. Splitting 3/2 in two and splitting 4/3 in three are equivalent to splitting [[6/1]] in six, and Gravity[7] is equivalent to the scale obtained by octave-reducing [[6ed6]].


== Intervals ==
== Intervals ==
[[File:Gravity construction.png|thumb|right|alt=Gravity construction.png|A pictorial representation of the process of constructing the heptatonic mos of 2.3.5.11 gravity. Splitting 3/2 in two and splitting 4/3 in three are equivalent to splitting 6/1 in six, and Gravity[7] is equivalent to the scale obtained by octave-reducing [[6ed6]].|960x320px]]


=== Interval chain ===
=== Interval chain ===
In the following table, odd harmonics 1–15 are labeled in '''bold'''.  
In the following table, [[odd harmonic]]s 1–15 are labeled in '''bold'''.  


{| class="wikitable sortable center-1 right-2"
{| class="wikitable sortable center-1 right-2"
! &#35;
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios
! class="unsortable" | Approximate ratios
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| '''15/8'''
| '''15/8'''
|}
|}
<nowiki />* In 2.3.5.11-subgroup [[CTE tuning]]
<nowiki/>* In 2.3.5.11-subgroup [[CTE tuning]]


=== As a detempering of 7et ===
=== As a detempering of 7et ===
{{todo|add detempering info|inline=1}}
{{Todo|inline=1|complete section|comment = Add detempering info.}}


== Tunings ==
== Tunings ==
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=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! EDO<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
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| [[11/8]]
| [[11/8]]
| 516.7545
| 516.7545
| (2.3.5.11) 11-odd-limit minimax tuning
| 2.3.5.11 11-odd-limit minimax tuning
|-
|-
|  
|  
| [[5/3]]
| [[5/3]]
| 516.7599
| 516.7599
| 2/11-comma, (2.3.5.11) 15-odd-limit minimax tuning
| 2/11-comma, 2.3.5.11 15-odd-limit minimax tuning
|-
|-
| [[137edo|59\137]]
| [[137edo|59\137]]
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|  
|  
| '''520.0000'''
| '''520.0000'''
| 30bccee val, '''upper bound of (2.3.5.11) 11- and 15-odd-limit diamond monotone'''
| 30bccee val, '''upper bound of 2.3.5.11 11- and 15-odd-limit diamond monotone'''
|-
|-
| '''[[23edo|10\23]]'''
| '''[[23edo|10\23]]'''
|  
|  
| '''521.7391'''
| '''521.7391'''
| 23bcccee val, '''upper bound of (2.3.5) 5- and 9-odd-limit diamond monotone'''
| 23bcccee val, '''upper bound of 2.3.5 5- and 9-odd-limit diamond monotone'''
|}
|}
<nowiki/>* Besides the octave
<nowiki/>* Besides the octave

Latest revision as of 00:31, 29 March 2026

Gravity; Larry
Subgroups 2.3.5, 2.3.5.11
Comma basis 129140163/128000000 (2.3.5);
243/242, 4000/3993 (2.3.5.11)
Reduced mapping ⟨1; 6 17 15]
ET join 7 & 58
Generators (CWE) ~27/20 = 516.8 ¢
MOS scales 2L 5s, 7L 2s, 7L 9s, …, 7L 51s
Ploidacot beta-hexacot
Pergen (P8, P19/6)
Color name Lala-tribiguti
Minimax error 5-odd-limit: 0.90 ¢;
2.3.5.11 15-odd-limit: 1.48 ¢
Target scale size 5-odd-limit: 23 notes;
2.3.5.11 15-odd-limit: 30 notes

Gravity is a rank-2 temperament and parent of the gravity family, generated by a classical acute fourth (27/20), six of which stacked reach the 6th harmonic (which octave reduces to the perfect fifth, 3/2), dividing the fifth in two and the fourth in three. The 5th harmonic is found at three perfect fifths up and one generator down, or 17 generators in total, thereby tempering out the graviton (ratio: 129140163/128000000, monzo: [-13 17 -6⟩). The complement of the acute fourth generator is the grave fifth, 40/27, whence the temperament's name follows.

Gravity is most naturally seen as a 2.3.5.11-subgroup temperament, sometimes known as larry. Here 8019/8000 (S9/S10) is tempered out, so that two intervals of 40/27 reach 11/10, and 4000/3993 (S10/S11) is tempered out, so that three intervals of 11/10 reach 4/3. These equivalences also imply that 243/242 is tempered out; three 27/20 fourths reach 11/9, which is thus equated to 27/22 and acts as an exact neutral third. Gravity's generator lies close to the fifth of 7edo, implying that the mos scales of gravity cluster heavily around 7edo, and in this interpretation the comma reached after 7 generators simultaneously represents 81/80 (S9), 100/99 (S10), and 121/120 (S11). In fact, gravity can be completely defined by making this equivalence between three adjacent square superparticulars, being the most accurate of the 5 temperaments definable in such a way.

Strong extensions with prime 7 include gravid (58 & 65), 58 & 65d, marvo (65d & 72), and zarvo (65 & 72). However, the most notable extension of gravity is harry (58 & 72), which splits the octave in two and extends well to the 13- and 17-limit.

For technical data, see Gravity family #Gravity.

Intervals

Gravity construction.png
A pictorial representation of the process of constructing the heptatonic mos of 2.3.5.11 gravity. Splitting 3/2 in two and splitting 4/3 in three are equivalent to splitting 6/1 in six, and Gravity[7] is equivalent to the scale obtained by octave-reducing 6ed6.

Interval chain

In the following table, odd harmonics 1–15 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 516.8 27/20, 121/90
2 1033.7 20/11
3 350.5 11/9, 27/22
4 867.4 33/20, 200/121
5 184.2 10/9, 135/121
6 701.1 3/2
7 17.9 81/80, 100/99, 121/120
8 534.8 15/11, 110/81
9 1051.6 11/6, 81/44
10 368.5 99/80, 100/81, 150/121
11 885.3 5/3
12 202.2 9/8, 121/108
13 719.0 50/33, 121/80
14 35.9 45/44, 55/54
15 552.7 11/8
16 1069.6 50/27
17 386.4 5/4
18 903.3 27/16, 121/72
19 220.1 25/22
20 737.0 55/36
21 53.8 33/32, 125/121
22 570.7 25/18
23 1087.5 15/8

* In 2.3.5.11-subgroup CTE tuning

As a detempering of 7et

Todo: complete section

Add detempering info.

Tunings

Gravity.png
A chart of the tuning spectrum of gravity, showing the offsets of prime harmonics 3, 5, and 11, and odds 9 and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Comments
3\7 514.2857 Lower bound of 5- to 15-odd-limit diamond monotone
11/9 515.8026
34\79 516.4557
10/9 516.4807 1/5-comma
11/6 516.5959
31\72 516.6667
90\209 516.7464
11/8 516.7545 2.3.5.11 11-odd-limit minimax tuning
5/3 516.7599 2/11-comma, 2.3.5.11 15-odd-limit minimax tuning
59\137 516.7883
25/24 516.8097 5/28-comma
87\202 516.8317
5/4 516.8420 3/17-comma, 5- and 9-odd-limit minimax tuning
115\267 516.8539
15/8 516.8812 4/23-comma
28\65 516.9231
3/2 516.9925 1/6-comma
53\123 517.0732
15/11 517.1188
25\58 517.2414
20/11 517.4979
22\51 517.6471 51ce val
27/20 519.5513 Untempered tuning
13\30 520.0000 30bccee val, upper bound of 2.3.5.11 11- and 15-odd-limit diamond monotone
10\23 521.7391 23bcccee val, upper bound of 2.3.5 5- and 9-odd-limit diamond monotone

* Besides the octave

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