Hemifamity temperaments: Difference between revisions

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The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal diesis, {{monzo| 2 2 -1 -1 }} = [[36/35]] into two equal steps.  
The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same circle of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  


Belonging to it and considered below are buzzard, undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:
Belonging to it and considered below are undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:
* ''[[Dominant]]'', {36/35, 64/63} → [[Meantone family #Dominant|Meantone family]]
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]]
* ''[[Diaschismic]]'', {126/125, 2048/2025} → [[Diaschismic family #Diaschismic|Diaschismic family]]
: +36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.
* [[Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan|Gamelismic clan]]
: +225/224, the 41 & 53 temperament, generated by the fifth with 5/4 mapped to the d4.
* ''[[Monkey]]'', {875/864, 5120/5103} → [[Tetracot family #Monkey|Tetracot family]]
* ''[[Kwai]]'' → [[Mirkwai clan #Kwai|Mirkwai clan]]
* ''[[Buzzard]]'', {1728/1715, 5120/5103} → [[Vulture family #Buzzard|Vulture family]]
: +16875/16807, the 41 & 70 temperament, generated by the fifth with 5/4 mapped to the 4A7 aka m3 + 3 Pyth. commas.
* [[Hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Diaschismic]]'' → [[Diaschismic family #Diaschismic|Diaschismic family]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Misty|Misty family]]
: +126/125, the 46 & 58 temperament, generated by the fifth and using a semioctave period.
* [[Amity]], {4375/4374, 5120/5103} → [[Ragismic microtemperaments #Amity|Ragismic microtemperaments]] and [[Amity family #Septimal amity|Amity family]]
* [[Hemififths]] → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Countercata]]'', {5120/5103, 15625/15552} → [[Kleismic family #Countercata|Kleismic family]]
: +2401/2400, the 41 & 58 temperament, fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma.
* ''[[Kwai]]'', {5120/5103, 16875/16807} → [[Mirkwai clan #Kwai|Mirkwai clan]]
* [[Rodan]] → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Warrior]]'', {5120/5103, 78732/78125} → [[Sensipent family #Warrior|Sensipent family]]
: +245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.
* ''[[Supers]]'', {5120/5103, 118098/117649} → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Monkey]]'' → [[Tetracot family #Monkey|Tetracot family]]
* ''[[Undim]]'', {5120/5103, 390625/388962} → [[Undim family #Septimal undim|Undim family]]
: +875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.
* ''[[Quinticosiennic]]'', {5120/5103, 395136/390625} → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* [[Buzzard]] → [[Vulture family #Buzzard|Vulture family]]
* ''[[Quintakwai]]'', {5120/5103, 9765625/9680832} → [[Quindromeda family #Quintakwai|Quindromeda family]]
: +1728/1715, the 53 & 58 temperament, twelfth sliced into four as does vulture.
* ''[[Alphaquarter]]'', {5120/5103, 29360128/29296875} → [[Escapade family #Alphaquarter|Escapade family]]
* [[Misty]] → [[Misty family #Misty|Misty family]]
: +3136/3125, the 12 & 99 temperament, generated by the fifth and using a 1/3-octave period.
* ''[[Supers]]'' → [[Stearnsmic clan #Supers|Stearnsmic clan]]
: +118098/117649, the 36c & 58 temperament, fifth sliced into three and using a semioctave period.
* ''[[Undim]]'' → [[Undim family #Septimal undim|Undim family]]
: +390625/388962, the 12 & 140 temperament, generated by the fifth and using a 1/4-octave period.
* ''[[Quinticosiennic]]'' → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
: +395136/390625, the 12 & 145 temperament, fourth sliced into five.
* ''[[Quintakwai]]'' → [[Quindromeda family #Quintakwai|Quindromeda family]]
: +9765625/9680832, the 12 & 181 temperament, fourth sliced into five.
* [[Amity]] → [[Amity family #Septimal amity|Amity family]]
: +4375/4374, the 46 & 53 temperament, eleventh sliced into five.
* ''[[Countercata]]'' → [[Kleismic family #Countercata|Kleismic family]]
: +15625/15552, the 34 & 53 temperament, twelfth sliced into six as does hanson.
* ''[[Warrior]]'' → [[Sensipent family #Warrior|Sensipent family]]
: +78732/78125, the 46 & 65d temperament, 6th harmonic sliced into seven as does sensi.
* ''[[Alphaquarter]]'' → [[Escapade family #Alphaquarter|Escapade family]]
: +29360128/29296875, the 65d & 87 temperament, fourth sliced into nine as does escapade.


== Undecental ==
== Undecental ==

Revision as of 08:52, 11 March 2023

The hemifamity temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103, dividing an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same circle of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth and 50/49 by the Pythagorean comma.

Belonging to it and considered below are undecental, leapday, mystery, quanic and ketchup. Other hemifamity temperaments are:

+36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3.
+225/224, the 41 & 53 temperament, generated by the fifth with 5/4 mapped to the d4.
+16875/16807, the 41 & 70 temperament, generated by the fifth with 5/4 mapped to the 4A7 aka m3 + 3 Pyth. commas.
+126/125, the 46 & 58 temperament, generated by the fifth and using a semioctave period.
+2401/2400, the 41 & 58 temperament, fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma.
+245/243, the 41 & 46 temperament, fifth sliced into three as does slendric.
+875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot.
+1728/1715, the 53 & 58 temperament, twelfth sliced into four as does vulture.
+3136/3125, the 12 & 99 temperament, generated by the fifth and using a 1/3-octave period.
+118098/117649, the 36c & 58 temperament, fifth sliced into three and using a semioctave period.
+390625/388962, the 12 & 140 temperament, generated by the fifth and using a 1/4-octave period.
+395136/390625, the 12 & 145 temperament, fourth sliced into five.
+9765625/9680832, the 12 & 181 temperament, fourth sliced into five.
+4375/4374, the 46 & 53 temperament, eleventh sliced into five.
+15625/15552, the 34 & 53 temperament, twelfth sliced into six as does hanson.
+78732/78125, the 46 & 65d temperament, 6th harmonic sliced into seven as does sensi.
+29360128/29296875, the 65d & 87 temperament, fourth sliced into nine as does escapade.

Undecental

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

Wedgie: ⟨⟨ 1 -37 -43 -61 -71 4 ]]

POTE generator: ~3/2 = 703.039

Template:Val list

Badness: 0.094603

Leapday

Main article: Leapday
For the 5-limit version of this temperament, see High badness temperaments #Leapday.

Leapday tempers out [31 -21 1⟩ (trisayo) in the 5-limit. This temperament can be described as the 29 & 46 temperament, which tempers out the hemifamity and 686/675 (senga). The alternative extension polypyth (46 & 121) tempers out the same 5-limit comma as the leapday, but with the porwell (6144/6125) rather than the hemifamity tempered out.

Subgroup: 2.3.5.7

Comma list: 686/675, 5120/5103

Mapping: [⟨1 0 -31 -21], ⟨0 1 21 15]]

Wedgie: ⟨⟨ 1 21 15 31 21 -24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.263

Template:Val list

Badness: 0.096123

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: [⟨1 0 -31 -21 -14], ⟨0 1 21 15 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.250

Optimal GPV sequence: Template:Val list

Badness: 0.038624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -31 -21 -14 -9], ⟨0 1 21 15 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.214

Optimal GPV sequence: Template:Val list

Badness: 0.024732

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34], ⟨0 1 21 15 11 8 24]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.229

Optimal GPV sequence: Template:Val list

Badness: 0.017863

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9], ⟨0 1 21 15 11 8 24 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.135

Optimal GPV sequence: Template:Val list

Badness: 0.017356

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 9 -5], ⟨0 1 21 15 11 8 24 -3 6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.141

Optimal GPV sequence: Template:Val list

Badness: 0.014065

Leapling

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37], ⟨0 1 21 15 11 8 24 26]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.123

Optimal GPV sequence: Template:Val list

Badness: 0.019065

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160

Mapping: [⟨1 0 -31 -21 -14 -9 -34 -37 -5], ⟨0 1 21 15 11 8 24 26 6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.114

Optimal GPV sequence: Template:Val list

Badness: 0.016067

Mystery

Main article: Mystery
For the 5-limit version of this temperament, see High badness temperaments #Mystery.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

Wedgie: ⟨⟨ 0 29 29 46 46 -14 ]]

POTE generator: ~5/4 = 388.646

Template:Val list

Badness: 0.103734

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

POTE generator: ~5/4 = 388.460

Optimal GPV sequence: Template:Val list

Badness: 0.034291

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

POTE generator: ~5/4 = 388.354

Optimal GPV sequence: Template:Val list

Badness: 0.018591

Quanic

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

POTE generator: ~160/147 = 140.493

Template:Val list

Badness: 0.179475

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1331/1323, 5120/5103

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

POTE generator: ~88/81 = 140.489

Optimal GPV sequence: Template:Val list

Badness: 0.058678

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

POTE generator: ~13/12 = 140.496

Optimal GPV sequence: Template:Val list

Badness: 0.032481

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 715/714, 847/845

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

POTE generator: ~13/12 = 140.497

Optimal GPV sequence: Template:Val list

Badness: 0.021112

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

POTE generator: ~13/12 = 140.496

Optimal GPV sequence: Template:Val list

Badness: 0.017273

Septiquarter

Subgroup: 2.3.5.7

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 3 10 2], ⟨0 -7 -38 4]]

Wedgie: ⟨⟨ 7 38 -4 44 -26 -116 ]]

POTE generator: ~147/128 = 242.453

Template:Val list

Badness: 0.053760

Semiseptiquarter

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 9801/9800, 14641/14580

Mapping: [⟨2 6 20 4 15], ⟨0 -7 -38 4 -20]]

POTE generators: ~121/105 = 242.4511

Optimal GPV sequence: Template:Val list

Badness: 0.064160

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 14641/14580

Mapping: [⟨2 6 20 4 15 7], ⟨0 -7 -38 4 -20 1]]

POTE generators: ~121/105 = 242.4448

Optimal GPV sequence: Template:Val list

Badness: 0.034834

Tricot

See also: Tricot family

The generator for tricot is the real cube root of third harmonic, 31/3, tuned between 63/44 and 13/9. Tricot can be described as 53&70 temperament (also called as "trimot", as confirmed by the data from x31eq), tempering out the tricot comma, [39 -29 3⟩ in the 5-limit, 2430/2401 (nuwell comma) and 5120/5103 in the 7-limit, 99/98 and 121/120 in the 11-limit, 169/168, 352/351, 640/637, and 729/728 in the 13-limit.

Subgroup: 2.3.5.7

Comma list: 2430/2401, 5120/5103

Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]

Wedgie: ⟨⟨ 3 29 11 39 9 -56 ]]

POTE generator: ~81/56 = 634.026

Template:Val list

Badness: 0.100127

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 5120/5103

Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]

POTE generator: ~63/44 = 634.027

Optimal GPV sequence: Template:Val list

Badness: 0.056134

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]

POTE generator: ~13/9 = 634.012

Optimal GPV sequence: Template:Val list

Badness: 0.032102

Countriton

For the 5-limit version of this temperament, see High badness temperaments #Countriton.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 7558272/7503125

Mapping: [⟨1 6 19 -7], ⟨0 -9 -34 20]]

Wedgie: ⟨⟨ 9 34 -20 33 -57 -142 ]]

POTE generator: ~1728/1225 = 588.582

Template:Val list

Badness: 0.131191

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103, 41503/41472

Mapping: [⟨1 6 19 -7 27], ⟨0 -9 -34 20 -48]]

POTE generator: ~108/77 = 588.545

Optimal GPV sequence: Template:Val list

Badness: 0.084782

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 6 19 -7 27 14], ⟨0 -9 -34 20 -48 -21]]

POTE generator: ~108/77 = 588.544

Optimal GPV sequence: Template:Val list

Badness: 0.042321

Ketchup

Subgroup: 2.3.5.7

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

Wedgie: ⟨⟨ 8 30 -18 29 -51 -126 ]]

POTE generator: ~64/63 = ~81/80 = 25.719

Template:Val list

Badness: 0.084538

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.693

Optimal GPV sequence: Template:Val list

Badness: 0.039555

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 847/845, 1331/1323

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.697

Optimal GPV sequence: Template:Val list

Badness: 0.024824

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.701

Optimal GPV sequence: Template:Val list

Badness: 0.016591

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560

Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 -12]]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.660

Optimal GPV sequence: Template:Val list

Badness: 0.018170

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384

Mapping: [⟨2 3 4 6 7 8 8 9 9], ⟨0 4 15 -9 -2 -14 4 -12 1]]

POTE generator: ~55/54 = ~64/63 = ~81/80 = 25.661

Optimal GPV sequence: Template:Val list

Badness: 0.014033

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