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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.
-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 ==
+{{main|Gravity}}
-== Gravity ==
+[[Subgroup]]: 2.3.5
-Subgroup: 2.3.5
 [[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
-[[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 }}
+{{Optimal ET sequence|legend=1| 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469 }}
 [[Badness]]: 0.093184
-=== Larry ===
+=== 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.
+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: [{{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]
-POTE generator: ~40/27 = 683.166
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.834
-Vals: {{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
-== 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|legend=1| 1 5 12 29 | 0 -6 -17 -46 }}
-[[Comma list]]: 2401/2400, 19683/19600
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.694
-[[Mapping]]: [{{val| 2 4 7 7 }}, {{val| 0 -6 -17 -10 }}]
+{{Optimal ET sequence|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
-{{Multival|legend=1| 12 34 20 26 -2 -49 }}
+[[Badness]]: 0.097627
-[[POTE generator]]: ~21/20 = 83.156
-{{Val list|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
-[[Badness]]: 0.034077
 === 11-limit ===
 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: {{mapping| 1 5 12 29 12 | 0 -6 -17 -46 -15 }}
-POTE generator: ~21/20 = 83.167
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.699
-Vals: {{Val list| 14c, 58, 72, 130, 202 }}
+{{Optimal ET sequence|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
-Badness: 0.015867
+Badness: 0.031685
 === 13-limit ===
 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: {{mapping| 1 5 12 29 12 39 | 0 -6 -17 -46 -15 -62 }}
-POTE generator: ~21/20 = 83.116
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.730
-Vals: {{Val list| 14cf, 58, 72, 130, 332f, 462ef }}
+{{Optimal ET sequence|legend=1| 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|legend=1| 1 5 12 -12 | 0 -6 -17 26 }}
-POTE generator: ~21/20 = 83.168
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 516.702
-Vals: {{Val list| 14cf, 58, 72, 130, 202g }}
+{{Optimal ET sequence|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}
-Badness: 0.012657
+[[Badness]]: 0.096840
-== Gravid ==
+=== 11-limit ===
-Subgroup: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 126/125, 1605632/1594323
+Comma list: 243/242, 385/384, 4000/3993
-[[Mapping]]: [{{val|1 -1 -5 -14}}, {{val|0 6 17 39}}]
+Mapping: {{mapping| 1 5 12 -12 12 | 0 -6 -17 26 -15 }}
-{{Multival|legend=1| 6 17 39 13 45 43 }}
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.691
-[[POTE generator]]: ~27/20 = 517.140
+{{Optimal ET sequence|legend=1| 65, 72, 353cd }}
-{{Val list|legend=1| 58, 123, 181c }}
+Badness: 0.034773
-[[Badness]]: 0.131153
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-=== 11-limit ===
+Comma list: 169/168, 243/242, 325/324, 385/384
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 243/242, 896/891
+Mapping: {{mapping| 1 5 12 -12 12 -2 | 0 -6 -17 26 -15 10 }}
-Mapping: [{{val|1 -1 -5 -14 -3}}, {{val|0 6 17 39 15}}]
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.667
-POTE generator: ~27/20 = 517.155
+{{Optimal ET sequence|legend=1| 65f, 72 }}
-Vals: {{Val list| 58, 123, 181ce }}
+Badness: 0.027584
-Badness: 0.047283
+== Gravid ==
+[[Subgroup]]: 2.3.5.7
-== Marvo ==
+[[Comma list]]: 126/125, 1605632/1594323
-Subgroup: 2.3.5.7
-[[Comma list]]: 225/224, 78125000/78121827
+{{Mapping|legend=1| 1 5 12 25 | 0 -6 -17 -39 }}
-[[Mapping]]: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/20 = 517.140
-{{Multival|legend=1| 6 17 46 13 56 59 }}
+{{Optimal ET sequence|legend=1| 58, 123, 181c }}
-[[POTE generator]]: ~27/20 = 516.694
+[[Badness]]: 0.131153
-{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
-[[Badness]]: 0.097627
 === 11-limit ===
 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: {{mapping| 1 5 12 25 12 | 0 -6 -17 -39 -15 }}
-POTE generator: ~27/20 = 516.699
+Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 517.155
-Vals: {{Val list| 65d, 72, 281, 353c, 425bc, 497bc }}
+{{Optimal ET sequence|legend=1| 58, 123, 181ce }}
-Badness: 0.031685
+Badness: 0.047283
-=== 13-limit ===
+== Harry ==
-Subgroup: 2.3.5.7.11.13
+{{Main| Harry }}
-Comma list: 225/224, 243/242, 351/350, 1625/1617
+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.
-Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
+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.
-POTE generator: ~27/20 = 516.730
+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.
-Vals: {{Val list| 65d, 72, 137, 209, 281f, 490bcf }}
+[[Subgroup]]: 2.3.5.7
-Badness: 0.026882
+[[Comma list]]: 2401/2400, 19683/19600
-== Zarvo ==
+{{Mapping|legend=1| 2 4 7 7 | 0 -6 -17 -10 }}
-Subgroup: 2.3.5.7
-[[Comma list]]: 4375/4374, 33075/32768
+: mapping generators: ~567/400, ~21/20
-[[Mapping]]: [{{val|1 -1 -5 14}}, {{val|0 6 17 -26}}]
+[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~27/20 = 516.844 (~21/20 = 83.156)
-{{Multival|legend=1| 6 17 -26 13 -58 -108 }}
+{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202, 534, 736b, 938b }}
-[[POTE generator]]: ~27/20 = 516.702
+[[Badness]]: 0.034077
-{{Val list|legend=1| 65, 72, 281d, 353cd, 425bcdd, 497bcdd }}
-[[Badness]]: 0.096840
 === 11-limit ===
 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: {{mapping| 2 4 7 7 9 | 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)
-Vals: {{Val list| 65, 72, 353cd }}
+{{Optimal ET sequence|legend=1| 14c, 58, 72, 130, 202 }}
-Badness: 0.034773
+Badness: 0.015867
 === 13-limit ===
 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: {{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 ET sequence|legend=1| 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: {{mapping| 2 4 7 7 9 11 9 | 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)
-Vals: {{Val list| 65f, 72 }}
+{{Optimal ET sequence|legend=1| 14cf, 58, 72, 130, 202g }}
-Badness: 0.027584
+Badness: 0.012657
-[[Category:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Pages with mostly numerical content]]
-[[Category:Gravity]]
 [[Category:Gravity family| ]] <!-- main article -->
+[[Category:Gravity| ]] <!-- key article -->
 [[Category:Rank 2]]

Gravity family: Difference between revisions