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&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 ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 129140163/128000000
[[Comma list]]: 129140163/128000000
Line 12: Line 10:
Mapping generators: ~2, ~40/27
Mapping generators: ~2, ~40/27


[[POTE generator]]: ~40/27 = 683.156
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.844


{{Val list|legend=1| 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469 }}
{{Val list|legend=1| 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469 }}


[[Badness]]: 0.093184
[[Badness]]: 0.093184
=== Overview to extensions ===
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.


=== Larry ===
=== Larry ===
Line 29: Line 30:
Gencom: [2 40/27; 243/242 4000/3993]
Gencom: [2 40/27; 243/242 4000/3993]


POTE generator: ~40/27 = 683.166
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834


Optimal GPV sequence: {{Val list| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
Optimal GPV sequence: {{Val list| 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202 }}
Line 35: Line 36:
Badness: 0.0125
Badness: 0.0125


== Harry ==
== Marvo ==
{{main| Harry }}
[[Subgroup]]: 2.3.5.7
{{see also| Breedsmic temperaments #Harry }}


Harry temperament can be described as 58&amp;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.
[[Comma list]]: 225/224, 78125000/78121827


Subgroup: 2.3.5.7
[[Mapping]]: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]


[[Comma list]]: 2401/2400, 19683/19600
{{Multival|legend=1| 6 17 46 13 56 59 }}


[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694


{{Multival|legend=1| 12 34 20 26 -2 -49 }}
{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}


[[POTE generator]]: ~21/20 = 83.156
[[Badness]]: 0.097627
 
{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
 
[[Badness]]: 0.034077


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 4000/3993
Comma list: 225/224, 243/242, 4000/3993


Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]
Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]


POTE generator: ~21/20 = 83.167
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699


Optimal GPV sequence: {{Val list| 14c, 58, 72, 130, 202 }}
Optimal GPV sequence: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}


Badness: 0.015867
Badness: 0.031685


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 364/363, 441/440
Comma list: 225/224, 243/242, 351/350, 1625/1617


Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]
Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]


POTE generator: ~21/20 = 83.116
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730


Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}
Optimal GPV sequence: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}


Badness: 0.013046
Badness: 0.026882


=== 17-limit ===
== Zarvo ==
Subgroup: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7


Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
[[Comma list]]: 4375/4374, 33075/32768


Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]
[[Mapping]]: [{{val| 1 5 12 -12 }}, {{val| 0 -6 -17 26 }}]


POTE generator: ~21/20 = 83.168
{{Multival|legend=1| 6 17 -26 13 -58 -108 }}


Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 202g }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702


Badness: 0.012657
{{Val list|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}


== Gravid ==
[[Badness]]: 0.096840
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 1605632/1594323
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|1 -1 -5 -14}}, {{val|0 6 17 39}}]
Comma list: 243/242, 385/384, 4000/3993


{{Multival|legend=1| 6 17 39 13 45 43 }}
Mapping: [{{val| 1 5 12 -12 12 }}, {{val| 0 -6 -17 26 -15 }}]


[[POTE generator]]: ~27/20 = 517.140
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.691


{{Val list|legend=1| 58, 123, 181c }}
Optimal GPV sequence: {{Val list| 65, 72, 353cd }}


[[Badness]]: 0.131153
Badness: 0.034773


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 243/242, 896/891
Comma list: 169/168, 243/242, 325/324, 385/384


Mapping: [{{val|1 -1 -5 -14 -3}}, {{val|0 6 17 39 15}}]
Mapping: [{{val| 1 5 12 -12 12 -2 }}, {{val| 0 -6 -17 26 -15 10 }}]


POTE generator: ~27/20 = 517.155
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.667


Optimal GPV sequence: {{Val list| 58, 123, 181ce }}
Optimal GPV sequence: {{Val list| 65f, 72 }}


Badness: 0.047283
Badness: 0.027584


== Marvo ==
== Gravid ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 78125000/78121827
[[Comma list]]: 126/125, 1605632/1594323


[[Mapping]]: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
[[Mapping]]: [{{val| 1 5 12 25 }}, {{val| 0 -6 -17 -39 }}]


{{Multival|legend=1| 6 17 46 13 56 59 }}
{{Multival|legend=1| 6 17 39 13 45 43 }}


[[POTE generator]]: ~27/20 = 516.694
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
{{Val list|legend=1| 58, 123, 181c }}


[[Badness]]: 0.097627
[[Badness]]: 0.131153


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 243/242, 4000/3993
Comma list: 126/125, 243/242, 896/891


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Mapping: [{{val| 1 5 12 25 12 }}, {{val| 0 -6 -17 -39 -15 }}]


POTE generator: ~27/20 = 516.699
Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.155


Optimal GPV sequence: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
Optimal GPV sequence: {{Val list| 58, 123, 181ce }}


Badness: 0.031685
Badness: 0.047283


=== 13-limit ===
== Harry ==
Subgroup: 2.3.5.7.11.13
{{Main| Harry }}
{{See also| Breedsmic temperaments #Harry }}


Comma list: 225/224, 243/242, 351/350, 1625/1617
Harry can be described as 58 &amp; 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 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.


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~27/20 = 516.730
[[Comma list]]: 2401/2400, 19683/19600


Optimal GPV sequence: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]


Badness: 0.026882
Mapping generators: ~567/400, ~21/20


== Zarvo ==
{{Multival|legend=1| 12 34 20 26 -2 -49 }}
Subgroup: 2.3.5.7


[[Comma list]]: 4375/4374, 33075/32768
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)


[[Mapping]]: [{{val|1 -1 -5 14}}, {{val|0 6 17 -26}}]
{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}


{{Multival|legend=1| 6 17 -26 13 -58 -108 }}
[[Badness]]: 0.034077
 
[[POTE generator]]: ~27/20 = 516.702
 
{{Val list|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}
 
[[Badness]]: 0.096840


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 243/242, 385/384, 4000/3993
Comma list: 243/242, 441/440, 4000/3993


Mapping: [{{val|1 -1 -5 14 -3}}, {{val|0 6 17 -26 15}}]
Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]


POTE generator: ~27/20 = 516.691
Optimal tuning (POTE): ~99/70 = 1\2, ~27/20 = 516.833 (~21/20 = 83.167)


Optimal GPV sequence: {{Val list| 65, 72, 353cd }}
Optimal GPV sequence: {{Val list| 14c, 58, 72, 130, 202 }}


Badness: 0.034773
Badness: 0.015867


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 243/242, 325/324, 385/384
Comma list: 243/242, 351/350, 364/363, 441/440
 
Mapping: [{{val| 2 4 7 7 9 11 }}, {{val| 0 -6 -17 -10 -15 -26 }}]
 
Optimal tuning (POTE): ~55/39 = 1\2, ~27/20 = 516.884 (~21/20 = 83.116)
 
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}
 
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: [{{val|1 -1 -5 14 -3 8}}, {{val|0 6 17 -26 15 -10}}]
Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]


POTE generator: ~27/20 = 516.667
Optimal tuning (POTE): ~17/12 = 1\2, ~27/20 = 516.832 (~21/20 = 83.168)


Optimal GPV sequence: {{Val list| 65f, 72 }}
Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 202g }}


Badness: 0.027584
Badness: 0.012657


[[Category:Temperament families]]
[[Category:Temperament families]]

Revision as of 11:50, 24 January 2023

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.

Gravity

Subgroup: 2.3.5

Comma list: 129140163/128000000

Mapping: [⟨1 5 12], ⟨0 -6 -17]]

Mapping generators: ~2, ~40/27

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.844

Template:Val list

Badness: 0.093184

Overview to extensions

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.

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]

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

Optimal GPV sequence: Template:Val list

Badness: 0.0125

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 ]]

Optimal tuning (POTE): ~2 = 1\1, ~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]]

Optimal tuning (POTE): ~2 = 1\1, ~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]]

Optimal tuning (POTE): ~2 = 1\1, ~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 5 12 -12], ⟨0 -6 -17 26]]

Wedgie: ⟨⟨ 6 17 -26 13 -58 -108 ]]

Optimal tuning (POTE): ~2 = 1\1, ~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 5 12 -12 12], ⟨0 -6 -17 26 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~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 5 12 -12 12 -2], ⟨0 -6 -17 26 -15 10]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.667

Optimal GPV sequence: Template:Val list

Badness: 0.027584

Gravid

Subgroup: 2.3.5.7

Comma list: 126/125, 1605632/1594323

Mapping: [⟨1 5 12 25], ⟨0 -6 -17 -39]]

Wedgie: ⟨⟨ 6 17 39 13 45 43 ]]

Optimal tuning (POTE): ~2 = 1\1, ~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 5 12 25 12], ⟨0 -6 -17 -39 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.155

Optimal GPV sequence: Template:Val list

Badness: 0.047283

Harry

Main article: Harry
See also: Breedsmic temperaments § Harry

Harry 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 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]]

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

Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]

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

Optimal tuning (POTE): ~99/70 = 1\2, ~27/20 = 516.833 (~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]]

Optimal tuning (POTE): ~55/39 = 1\2, ~27/20 = 516.884 (~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]]

Optimal tuning (POTE): ~17/12 = 1\2, ~27/20 = 516.832 (~21/20 = 83.168)

Optimal GPV sequence: Template:Val list

Badness: 0.012657

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