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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.
{{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 ==
== Gravity ==
{{main|Gravity}}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 129140163/128000000
[[Comma list]]: 129140163/128000000


[[Mapping]]: [{{val| 1 5 12 }}, {{val| 0 -6 -17 }}]
{{Mapping|legend=1| 1 5 12 | 0 -6 -17 }}


Mapping generators: ~2, ~40/27
: mapping generators: ~2, ~40/27


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


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


[[Badness]]: 0.093184
[[Badness]]: 0.093184


=== Overview to extensions ===
=== Overview to extensions ===
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.  
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/S10]], [[4000/3993|S10/S11]], ([[847/845|S11/S13]],) [[144/143|S12]]}.


=== Larry ===
=== 2.3.5.11 subgroup (larry) ===
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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


Sval mapping: [{{val| 1 5 12 12 }}, {{val| 0 -6 -17 -15 }}]
Sval mapping: {{mapping| 1 5 12 12 | 0 -6 -17 -15 }}


Gencom mapping: [{{val| 1 5 12 0 12 }}, {{val| 0 -6 -17 0 -15 }}]
Gencom mapping: {{mapping| 1 5 12 0 12 | 0 -6 -17 0 -15 }}


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


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


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


Badness: 0.0125
Badness: 0.0125
Line 41: Line 46:
[[Comma list]]: 225/224, 78125000/78121827
[[Comma list]]: 225/224, 78125000/78121827


[[Mapping]]: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281, 569bcc }}


[[Badness]]: 0.097627
[[Badness]]: 0.097627
Line 56: Line 59:
Comma list: 225/224, 243/242, 4000/3993
Comma list: 225/224, 243/242, 4000/3993


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


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


Optimal GPV sequence: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}


Badness: 0.031685
Badness: 0.031685
Line 69: Line 72:
Comma list: 225/224, 243/242, 351/350, 1625/1617
Comma list: 225/224, 243/242, 351/350, 1625/1617


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


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


Optimal GPV sequence: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}


Badness: 0.026882
Badness: 0.026882
Line 82: Line 85:
[[Comma list]]: 4375/4374, 33075/32768
[[Comma list]]: 4375/4374, 33075/32768


[[Mapping]]: [{{val| 1 5 12 -12 }}, {{val| 0 -6 -17 26 }}]
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702


{{Val list|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}
{{Optimal ET sequence|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}


[[Badness]]: 0.096840
[[Badness]]: 0.096840
Line 97: Line 98:
Comma list: 243/242, 385/384, 4000/3993
Comma list: 243/242, 385/384, 4000/3993


Mapping: [{{val| 1 5 12 -12 12 }}, {{val| 0 -6 -17 26 -15 }}]
Mapping: {{mapping| 1 5 12 -12 12 | 0 -6 -17 26 -15 }}


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


Optimal GPV sequence: {{Val list| 65, 72, 353cd }}
{{Optimal ET sequence|legend=1| 65, 72, 353cd }}


Badness: 0.034773
Badness: 0.034773
Line 110: Line 111:
Comma list: 169/168, 243/242, 325/324, 385/384
Comma list: 169/168, 243/242, 325/324, 385/384


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


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


Optimal GPV sequence: {{Val list| 65f, 72 }}
{{Optimal ET sequence|legend=1| 65f, 72 }}


Badness: 0.027584
Badness: 0.027584
Line 123: Line 124:
[[Comma list]]: 126/125, 1605632/1594323
[[Comma list]]: 126/125, 1605632/1594323


[[Mapping]]: [{{val| 1 5 12 25 }}, {{val| 0 -6 -17 -39 }}]
{{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
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140


{{Val list|legend=1| 58, 123, 181c }}
{{Optimal ET sequence|legend=1| 58, 123, 181c }}


[[Badness]]: 0.131153
[[Badness]]: 0.131153
Line 138: Line 137:
Comma list: 126/125, 243/242, 896/891
Comma list: 126/125, 243/242, 896/891


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


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


Optimal GPV sequence: {{Val list| 58, 123, 181ce }}
{{Optimal ET sequence|legend=1| 58, 123, 181ce }}


Badness: 0.047283
Badness: 0.047283
Line 149: Line 148:
{{Main| Harry }}
{{Main| Harry }}


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.
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.
 
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
[[Subgroup]]: 2.3.5.7
Line 155: Line 158:
[[Comma list]]: 2401/2400, 19683/19600
[[Comma list]]: 2401/2400, 19683/19600


[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]
{{Mapping|legend=1| 2 4 7 7 | 0 -6 -17 -10 }}
 
Mapping generators: ~567/400, ~21/20


{{Multival|legend=1| 12 34 20 26 -2 -49 }}
: mapping generators: ~567/400, ~21/20


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


{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}


[[Badness]]: 0.034077
[[Badness]]: 0.034077
Line 172: Line 173:
Comma list: 243/242, 441/440, 4000/3993
Comma list: 243/242, 441/440, 4000/3993


Mapping: [{{val| 2 4 7 7 9 }}, {{val| 0 -6 -17 -10 -15 }}]
Mapping: {{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 tuning (POTE): ~99/70 = 1\2, ~27/20 = 516.833 (~21/20 = 83.167)


Optimal GPV sequence: {{Val list| 14c, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202 }}


Badness: 0.015867
Badness: 0.015867
Line 185: Line 186:
Comma list: 243/242, 351/350, 364/363, 441/440
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 }}]
Mapping: {{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 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 }}
{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 332f, 462ef }}


Badness: 0.013046
Badness: 0.013046
Line 198: Line 199:
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440


Mapping: [{{val| 2 4 7 7 9 11 9 }}, {{val| 0 -6 -17 -10 -15 -26 -6 }}]
Mapping: {{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 tuning (POTE): ~17/12 = 1\2, ~27/20 = 516.832 (~21/20 = 83.168)


Optimal GPV sequence: {{Val list| 14cf, 58, 72, 130, 202g }}
{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 202g }}


Badness: 0.012657
Badness: 0.012657


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gravity]]
[[Category:Pages with mostly numerical content]]
[[Category:Gravity family| ]] <!-- main article -->
[[Category:Gravity family| ]] <!-- main article -->
[[Category:Gravity| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Gravity_family"