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)<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.
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.


Full 7-limit extensions of gravity include gravid (58 & 65), marvo (65d & 72), zarvo (65 & 72), and harry (58 & 72), all considered below. A notable subgroup extension is larry.  
Full 7-limit extensions of gravity include gravid (58&amp;65), marvo (65d&amp;72), zarvo (65&amp;72), and harry (58&amp;72), all considered below. A notable subgroup extension is larry.  


== Gravity ==
== Gravity ==
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POTE generator: ~40/27 = 683.166
POTE generator: ~40/27 = 683.166


Vals: {{Val list| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
Optimal GPV sequence: {{Val list| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}


Badness: 0.0125
Badness: 0.0125
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POTE generator: ~21/20 = 83.167
POTE generator: ~21/20 = 83.167


Vals: {{Val list| 14c, 58, 72, 130, 202 }}
Optimal GPV sequence: {{Val list| 14c, 58, 72, 130, 202 }}


Badness: 0.015867
Badness: 0.015867
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POTE generator: ~21/20 = 83.116
POTE generator: ~21/20 = 83.116


Vals: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}


Badness: 0.013046
Badness: 0.013046
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POTE generator: ~21/20 = 83.168
POTE generator: ~21/20 = 83.168


Vals: {{Val list| 14cf, 58, 72, 130, 202g }}
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 202g }}


Badness: 0.012657
Badness: 0.012657
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POTE generator: ~27/20 = 517.155
POTE generator: ~27/20 = 517.155


Vals: {{Val list| 58, 123, 181ce }}
Optimal GPV sequence: {{Val list| 58, 123, 181ce }}


Badness: 0.047283
Badness: 0.047283
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POTE generator: ~27/20 = 516.699
POTE generator: ~27/20 = 516.699


Vals: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
Optimal GPV sequence: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}


Badness: 0.031685
Badness: 0.031685
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POTE generator: ~27/20 = 516.730
POTE generator: ~27/20 = 516.730


Vals: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
Optimal GPV sequence: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}


Badness: 0.026882
Badness: 0.026882
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POTE generator: ~27/20 = 516.691
POTE generator: ~27/20 = 516.691


Vals: {{Val list| 65, 72, 353cd }}
Optimal GPV sequence: {{Val list| 65, 72, 353cd }}


Badness: 0.034773
Badness: 0.034773
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POTE generator: ~27/20 = 516.667
POTE generator: ~27/20 = 516.667


Vals: {{Val list| 65f, 72 }}
Optimal GPV sequence: {{Val list| 65f, 72 }}


Badness: 0.027584
Badness: 0.027584

Revision as of 13:35, 8 January 2022

The gravity family tempers out graviton, the 5-limit comma 129140163/128000000 = [-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.

Full 7-limit extensions of gravity include gravid (58&65), marvo (65d&72), zarvo (65&72), and harry (58&72), all considered below. A notable subgroup extension is larry.

Gravity

Subgroup: 2.3.5

Comma list: 129140163/128000000

Mapping: [⟨1 5 12], ⟨0 -6 -17]]

Mapping generators: ~2, ~40/27

POTE generator: ~40/27 = 683.156

Template:Val list

Badness: 0.093184

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Sval mapping: [⟨1 5 12 12], ⟨0 -6 -17 -15]]

Gencom mapping: [⟨1 5 12 0 12], ⟨0 -6 -17 0 -15]]

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

POTE generator: ~40/27 = 683.166

Optimal GPV sequence: Template:Val list

Badness: 0.0125

Harry

Main article: Harry
See also: Breedsmic temperaments § Harry

Harry temperament can be described as 58&72 temperament, tempering out 7-limit commas 2401/2400 and 19683/19600; 11-limit commas 243/242, 441/440 and 4000/3993, leading to a period of a half octave and a generator of minor semitone tempered between 21/20 and 22/21. Harry temperament also equates 14/13 with 27/25 in the 13-limit, 17/16 with 18/17 and 13/11 with 20/17 in the 17-limit.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 19683/19600

Mapping: [⟨2 4 7 7], ⟨0 -6 -17 -10]]

Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]

POTE generator: ~21/20 = 83.156

Template:Val list

Badness: 0.034077

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4000/3993

Mapping: [⟨2 4 7 7 9], ⟨0 -6 -17 -10 -15]]

POTE generator: ~21/20 = 83.167

Optimal GPV sequence: Template:Val list

Badness: 0.015867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 441/440

Mapping: [⟨2 4 7 7 9 11], ⟨0 -6 -17 -10 -15 -26]]

POTE generator: ~21/20 = 83.116

Optimal GPV sequence: Template:Val list

Badness: 0.013046

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 289/288, 351/350, 441/440

Mapping: [⟨2 4 7 7 9 11 9], ⟨0 -6 -17 -10 -15 -26 -6]]

POTE generator: ~21/20 = 83.168

Optimal GPV sequence: Template:Val list

Badness: 0.012657

Gravid

Subgroup: 2.3.5.7

Comma list: 126/125, 1605632/1594323

Mapping: [⟨1 -1 -5 -14], ⟨0 6 17 39]]

Wedgie: ⟨⟨ 6 17 39 13 45 43 ]]

POTE generator: ~27/20 = 517.140

Template:Val list

Badness: 0.131153

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 896/891

Mapping: [⟨1 -1 -5 -14 -3], ⟨0 6 17 39 15]]

POTE generator: ~27/20 = 517.155

Optimal GPV sequence: Template:Val list

Badness: 0.047283

Marvo

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

Mapping: [⟨1 5 12 29], ⟨0 -6 -17 -46]]

Wedgie: ⟨⟨ 6 17 46 13 56 59 ]]

POTE generator: ~27/20 = 516.694

Template:Val list

Badness: 0.097627

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4000/3993

Mapping: [⟨1 5 12 29 12], ⟨0 -6 -17 -46 -15]]

POTE generator: ~27/20 = 516.699

Optimal GPV sequence: Template:Val list

Badness: 0.031685

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 1625/1617

Mapping: [⟨1 5 12 29 12 39], ⟨0 -6 -17 -46 -15 -62]]

POTE generator: ~27/20 = 516.730

Optimal GPV sequence: Template:Val list

Badness: 0.026882

Zarvo

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33075/32768

Mapping: [⟨1 -1 -5 14], ⟨0 6 17 -26]]

Wedgie: ⟨⟨ 6 17 -26 13 -58 -108 ]]

POTE generator: ~27/20 = 516.702

Template:Val list

Badness: 0.096840

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 4000/3993

Mapping: [⟨1 -1 -5 14 -3], ⟨0 6 17 -26 15]]

POTE generator: ~27/20 = 516.691

Optimal GPV sequence: Template:Val list

Badness: 0.034773

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 385/384

Mapping: [⟨1 -1 -5 14 -3 8], ⟨0 6 17 -26 15 -10]]

POTE generator: ~27/20 = 516.667

Optimal GPV sequence: Template:Val list

Badness: 0.027584

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