Archytas clan: Difference between revisions

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Remove mother cuz it's too simple. Moved to father family and septisemi temperaments
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The '''archytas clan''' tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]]. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.  
The '''archytas clan''' tempers out the [[64/63|Archytas comma]], 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to [[81/80]] in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of [[22edo]]. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.  


Discussed under subgroup temperaments is the 2.3.7 [[Subgroup temperaments #Archy|archy]]. Under their respective 5-limit families are [[Diaschismic family #Pajara|pajara]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[porcupine family|porcupine]], [[Tetracot family #Modus|modus]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
Discussed under subgroup temperaments is the 2.3.7 [[Subgroup temperaments #Archy|archy]]. Under their respective 5-limit families are [[Father family #Mother]], [[Diaschismic family #Pajara|pajara]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[porcupine family|porcupine]], [[Tetracot family #Modus|modus]], and [[Immunity family #Immunized|immunized]]. The rest are considered below.
 
= Mother =
{{see also| Father family }}
 
Commas: 16/15, 21/20
 
[[POTE generator]]: 721.569
 
Map: [<1 0 4 6|, <0 1 -1 -2|]
 
Wedgie: <<1 -1 -2 -4 -6 -2||
 
{{EDOs|legend=1| 5, 148, 153 }}
 
Badness: 0.0242


= Blacksmith =
= Blacksmith =

Revision as of 14:49, 10 February 2021

The archytas clan tempers out the Archytas comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3s and 7s quite sharp, such as those of 22edo. Adding 50/49 to the list of commas gives pajara, 36/35 gives dominant, 16/15 gives mother, 126/125 gives augene, 28/27 gives blacksmith, 245/243 gives superpyth, 250/243 gives porcupine, 686/675 gives beatles, 360/343 gives schism, 3125/3087 gives passion, 2430/2401 gives quasisuper, and 4375/4374 gives modus.

Discussed under subgroup temperaments is the 2.3.7 archy. Under their respective 5-limit families are Father family #Mother, pajara, dominant, augene, porcupine, modus, and immunized. The rest are considered below.

Blacksmith

5-limit (blackwood)

Comma: 256/243

POTE generator: 399.594

Map: [<5 8 0|, <0 0 1|]

EDOs: 5, 10, 15

Badness: 0.0638

7-limit

Commas: 28/27, 49/48

POTE generator: ~5/4 = 392.767

Map: [<5 8 0 14|, <0 0 1 0|]

Wedgie: <<0 5 0 8 0 -14||

EDOs: 5, 10, 15, 40b, 55b

Badness: 0.0256

11-limit

Commas: 28/27, 49/48, 55/54

POTE generator: ~5/4 = 394.948

Map: [<5 8 0 14 29|, <0 0 1 0 -1|]

EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde

Badness: 0.0246

13-limit

Commas: 28/27, 40/39, 49/48, 55/54

POTE generator: ~5/4 = 391.0367

Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|]

EDOs: 5, 10, 15, 25e, 40bef

Badness: 0.0205

Farrier

Commas: 28/27, 49/48, 77/75

POTE generator: ~5/4 = 398.070

Map: [<5 8 0 14 -6|, <0 0 1 0 2|]

EDOs: 5e, 15

Badness: 0.0292

13-limit

Commas: 28/27, 40/39, 49/48, 66/65

POTE generator: ~5/4 = 396.812

Map: [<5 8 0 14 -6 7|, <0 0 1 0 2 1|]

EDOs: 5e, 10e, 15

Badness: 0.0223

Ferrum

Commas: 28/27, 35/33, 49/48

POTE generator: ~5/4 = 374.763

Map: [<5 8 0 14 6|, <0 0 1 0 1|]

EDOs: 10

Badness: 0.0309

blacksmith10.jpg

Superpyth

Main article: Superpyth

Commas: 64/63, 245/243

POTE generator: 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]

Wedgie: <<1 9 -2 12 -6 -30||

EDOs: 5, 17, 22, 27, 49

Badness: 0.0323

11-limit

Commas: 64/63, 100/99, 245/243

POTE generator: 710.175

Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]

EDOs: 22, 49

Badness: 0.0250

13-limit

Commas: 64/63, 78/77, 91/90, 100/99

POTE generator: ~3/2 = 710.479

Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]

EDOs: 22, 27e, 49, 76bcde

Badness: 0.0247

Suprapyth

Commas: 55/54, 64/63, 99/98

POTE generator: ~3/2 = 709.495

Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]

EDOs: 5, 7, 12, 17, 22

Badness: 0.0328

13-limit

Commas: 55/54, 64/63, 65/63, 364/363

POTE generator: ~3/2 = 708.703

Map: [<1 0 -12 6 13 18|, <0 1 9 -2 -6 -9|]

EDOs: 17, 22, 83cdf

Badness: 0.0363

Beatles

5-limit

Comma: 524288/492075

POTE generator: ~512/405 = 355.930

Map: [<1 1 5|,<0 2 -9|]

EDOs: 10, 17c, 27, 64b, 91bc, 118bc

Badness: 0.3585

7-limit

Commas: 64/63, 686/675

POTE generator: ~49/40 = 355.904

Map: [<1 1 5 4|,<0 2 -9 -4|]

Wedgie: <<2 -9 -4 -19 -12 16||

EDOs: 10, 17c, 27, 64b, 91bcd, 118bcd

Badness: 0.0459

Music: Beatles Improv by Herman Miller

11-limit

Commas: 64/63, 100/99, 686/675

POTE generator: ~49/40 = 356.140

Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|]

EDOs: 27e, 37, 64be, 91bcde

Badness: 0.0456

13-limit

Commas: 64/63, 91/90, 100/99, 169/168

POTE generator: ~16/13 = 356.229

Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|]

EDOs: 27e, 37, 64be

Badness: 0.0302

Ringo

Commas: 56/55, 64/63, 540/539

POTE generator: ~11/9 = 355.419

Map: [<1 1 5 4 2|,<0 2 -9 -4 5|]

EDOs: 10, 17c, 27e

Badness: 0.0329

13-limit

Commas: 56/55, 64/63, 78/77, 91/90

POTE generator: ~11/9 = 355.456

Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|]

EDOs: 10, 17c, 27e

Badness: 0.0226

Schism

See also: Schismatic family § Schism

Commas: 64/63, 360/343

POTE generator: ~3/2 = 701.556

Map: [<1 0 15 6|, <0 1 -8 -2|]

Wedgie: <<1 -8 -2 -15 -6 18||

EDOs: 12, 41d, 53d

Badness: 0.0566

11-limit

Commas: 45/44, 64/63, 99/98

POTE generator ~3/2 = 702.136

Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]

EDOs: 12, 29de, 41de

Badness: 0.0375

Passion

5-limit

Comma: 262144/253125

POTE generator: ~16/15 = 98.670

Map: [<1 2 2|, <0 -5 4|]

EDOs: 11, 12, 49, 61, 73

Badness: 0.1686

Passive

Commas: 225/224, 256/245

POTE generator: ~16/15 = 98.809

Map: [<1 2 2 3|, <0 -5 4 -2|]

EDOs: 1, 11, 12, 49d

Badness: 0.0751

7-limit

Commas: 64/63, 3125/3087

POTE generator: ~16/15 = 98.153

Map: [<1 2 2 2|, <0 -5 4 10|]

Wedgie: <<5 -4 -10 -18 -30 -12||

Generators: 2, 16/15

EDOs: 12, 37, 49, 110bcd

Badness: 0.0623

11-limit

Commas: 64/63, 100/99, 1375/1372

POTE generator: ~16/15 = 98.019

Map: [<1 2 2 2 2|, <0 -5 4 10 18|]

EDOs: 12, 37, 49

Badness: 0.0408

13-limit

Commas: 64/63, 100/99, 196/195, 275/273

POTE generator: ~16/15 = 97.910

Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|]

EDOs: 12f, 37, 49f

Badness: 0.0309

Fervor

5-limit

Comma: 67108864/61509375

POTE generator: ~64/45 = 577.705

Map: [<1 4 -2|, <0 -5 9|]

EDOs: 25, 27

Badness: 0.8526

7-limit

Commas: 64/63, 9604/9375

POTE generator: ~7/5 = 577.777

Map: [<1 4 -2 -2|, <0 -5 9 10|]

Wedgie: <<5 -9 -10 -26 -30 2||

EDOs: 25, 27

Badness: 0.1085

11-limit

Commas: 56/55, 64/63, 1350/1331

POTE generator: ~7/5 = 577.850

Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|]

EDOs: 25e, 27e

Badness: 0.0521

13-limit

Commas: 56/55, 64/63, 78/77, 507/500

POTE generator: ~7/5 = 578.060

Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|]

EDOs: 27e

Badness: 0.0397

Quasisuper

Commas: 64/63, 2430/2401

POTE generator: 708.328

Map: [<1 0 23 6|, <0 1 -13 -2|]

Wedgie: <<1 -13 -2 -23 -2 -6 32||

EDOs: 22, 61

Badness: 0.0638

Quasisupra

Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in (rather than the superpyth mapping of 5, which results in suprapyth).

Commas: 64/63, 99/98, 121/120

POTE generator: ~3/2 = 708.205

Map: [<1 2 -3 2 1|, <0 -1 13 2 6|]

EDOs: 17c, 22, 27c, 39d, 61d

Badness: 0.0322

13-limit

Commas: 64/63, 78/77, 91/90, 121/120

POTE generator: ~3/2 = 708.004

Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|]

EDOs: 17c, 22, 39d, 61df, 100bcdf

Badness: 0.0302

Quasisoup

Commas: 55/54, 64/63, 2430/2401

POTE generator: ~3/2 = 709.021

Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|]

EDOs: 22

Badness: 0.0835

Progress

5-limit

Comma: 32768/30375

POTE generator: ~64/45 = 561.264

Map: [<1 0 5|, <0 3 -5|]

EDOs: 4, 13, 15, 32c, 47bc, 62bc

Badness: 0.2461

7-limit

Commas: 64/63, 392/375

POTE generator: ~7/5 = 562.122

Map: [<1 0 5 6|, <0 3 -5 -6|]

Wedgie: <<3 -5 -6 -15 -18 0||

EDOs: 13, 15, 32c, 79bcc, 111bcc

Badness: 0.0664

11-limit

Commas: 56/55, 64/63, 77/75

POTE generator: ~7/5 = 562.085

Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|]

EDOs: 13, 15, 32c, 47bc, 79bcce

Badness: 0.0310

13-limit

Commas: 56/55, 64/63, 66/65, 77/75

POTE generator: ~7/5 = 562.365

Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|]

EDOs: 15, 17c, 32cf

Badness: 0.0262

Progressive

Commas: 26/25, 56/55, 64/63, 77/75

POTE generator: ~7/5 = 563.239

Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|]

EDOs: 15f, 17c, 32c, 49c

Badness: 0.0327

Sixix

5-limit

Comma: 3125/2916

POTE generator: ~6/5 = 338.365

Map: [<1 3 4|, <0 -5 -6|]

EDOs: 7, 25, 32

Badness: 0.1531

7-limit

Commas: 3125/2916, 64/63

POTE generator: ~6/5 = 337.4419

Map: [<1 3 4 0 |, <0 -5 -6 10|]

EDOs: 7, 25, 32

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