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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 148: Line 147:
== Harry ==
== Harry ==
{{Main| Harry }}
{{Main| Harry }}
{{See also| Stearnsmic clan #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 156: 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 173: 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 186: 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 199: 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]]

Latest revision as of 00:36, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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

Main article: 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

Optimal ET sequence: 7, 37cc, 44c, 51c, 58, 65, 137, 202, 267, 469

Badness: 0.093184

Overview to extensions

Full 7-limit extensions of gravity include marvo (65d & 72), zarvo (65 & 72), gravid (58 & 65), and harry (58 & 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 {S8/S9, S9/S10, S10/S11, (S11/S13,) S12}.

2.3.5.11 subgroup (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 ET sequence: 7, 37ccee, 44ce, 51ce, 58, 65, 137, 202

Badness: 0.0125

Marvo

Subgroup: 2.3.5.7

Comma list: 225/224, 78125000/78121827

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

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

Optimal ET sequence: 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

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

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

Optimal ET sequence: 65d, 72, 281, 353c, 425bc, 497bc

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 ET sequence: 65d, 72, 137, 209, 281f, 490bcf

Badness: 0.026882

Zarvo

Subgroup: 2.3.5.7

Comma list: 4375/4374, 33075/32768

Mapping: [⟨1 5 12 -12], ⟨0 -6 -17 26]]

Optimal tuning (POTE): ~2 = 1\1, ~27/20 = 516.702

Optimal ET sequence: 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

Mapping: [⟨1 5 12 -12 12], ⟨0 -6 -17 26 -15]]

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

Optimal ET sequence: 65, 72, 353cd

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 ET sequence: 65f, 72

Badness: 0.027584

Gravid

Subgroup: 2.3.5.7

Comma list: 126/125, 1605632/1594323

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

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

Optimal ET sequence: 58, 123, 181c

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 ET sequence: 58, 123, 181ce

Badness: 0.047283

Harry

Main article: Harry

Harry adds the breedsma, 2401/2400, and the cataharry comma, 19683/19600, to the set of commas, and may be described as the 58 & 72 temperament. The period is half an octave, and the generator ~21/20, with generator tunings of 9\130 or 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 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.

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

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

Optimal ET sequence: 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

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 ET sequence: 14c, 58, 72, 130, 202

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 ET sequence: 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: [⟨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 ET sequence: 14cf, 58, 72, 130, 202g

Badness: 0.012657

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