Archytas clan: Difference between revisions

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Line 51: Line 51:
POTE generator: ~3/2 = 707.192
POTE generator: ~3/2 = 707.192


Vals: {{Val list| 5, 12, 17, 39d, 56d }}
Optimal GPV sequence: {{Val list| 5, 12, 17, 39d, 56d }}


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 70: Line 70:
POTE generator: ~3/2 = 706.137
POTE generator: ~3/2 = 706.137


Vals: {{Val list| 12f, 17 }}
Optimal GPV sequence: {{Val list| 12f, 17 }}


Scales: [[supra7]], [[supra12]]
Scales: [[supra7]], [[supra12]]
Line 89: Line 89:
POTE generator: ~11/9 = 353.958
POTE generator: ~11/9 = 353.958


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
Optimal GPV sequence: {{Val list| 7, 10, 17, 44e, 61de }}


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Line 108: Line 108:
POTE generator: ~11/9 = 353.775
POTE generator: ~11/9 = 353.775


Vals: {{Val list| 7, 10, 17, 44e, 61de }}
Optimal GPV sequence: {{Val list| 7, 10, 17, 44e, 61de }}


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Line 115: Line 115:
{{main| Superpyth }}
{{main| Superpyth }}


In the 5-limit, superpyth tempers out 20480/19683 (sayo comma). This temperament has a fifth generator of ~3/2 = ~710¢ and 5/4 is equated to nine generators minus 5 octaves. A weak extension [[Jubilismic clan #Bipyth|bipyth]] (10cd&22) tempers out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
Subgroup: 2.3.5
[[Comma list]]: 20480/19683
[[Mapping]]: [{{val| 1 0 -12 }}, {{val| 0 1 9 }}]
[[POTE generator]]: ~3/2 = 710.078
{{Val list|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
[[Badness]]: 0.135141
=== 7-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 138: Line 153:
POTE generator: ~3/2 = 710.175
POTE generator: ~3/2 = 710.175


Vals: {{Val list| 22, 27e, 49 }}
Optimal GPV sequence: {{Val list| 22, 27e, 49 }}


Badness: 0.024976
Badness: 0.024976
Line 151: Line 166:
POTE generator: ~3/2 = 710.479
POTE generator: ~3/2 = 710.479


Vals: {{Val list| 22, 27e, 49, 76bcde }}
Optimal GPV sequence: {{Val list| 22, 27e, 49, 76bcde }}


Badness: 0.024673
Badness: 0.024673
Line 164: Line 179:
POTE generator: ~3/2 = 709.495
POTE generator: ~3/2 = 709.495


Vals: {{Val list| 5, 12c, 17, 22 }}
Optimal GPV sequence: {{Val list| 5, 12c, 17, 22 }}


Badness: 0.032768
Badness: 0.032768
Line 177: Line 192:
POTE generator: ~3/2 = 708.703
POTE generator: ~3/2 = 708.703


Vals: {{Val list| 17, 22, 83cdf }}
Optimal GPV sequence: {{Val list| 17, 22, 83cdf }}


Badness: 0.036336
Badness: 0.036336
Line 207: Line 222:
POTE generator: ~3/2 = 708.205
POTE generator: ~3/2 = 708.205


Vals: {{Val list| 17c, 22, 39d, 61d }}
Optimal GPV sequence: {{Val list| 17c, 22, 39d, 61d }}


Badness: 0.032203
Badness: 0.032203
Line 220: Line 235:
POTE generator: ~3/2 = 708.004
POTE generator: ~3/2 = 708.004


Vals: {{Val list| 17c, 22, 39d, 61df, 100bcdf }}
Optimal GPV sequence: {{Val list| 17c, 22, 39d, 61df, 100bcdf }}


Badness: 0.030219
Badness: 0.030219
Line 233: Line 248:
POTE generator: ~3/2 = 709.021
POTE generator: ~3/2 = 709.021


Vals: {{Val list| 5ce, 17ce, 22 }}
Optimal GPV sequence: {{Val list| 5ce, 17ce, 22 }}


Badness: 0.083490
Badness: 0.083490
Line 262: Line 277:
Mapping: [{{val| 1 0 -20 6 -25 }}, {{val| 0 1 14 -2 18 }}]
Mapping: [{{val| 1 0 -20 6 -25 }}, {{val| 0 1 14 -2 18 }}]


POTE generator: ~3/2 = 713.7453
POTE generator: ~3/2 = 713.745


Vals: {{Val list| 5, 32, 37, 79bc }}
Optimal GPV sequence: {{Val list| 5, 32, 37, 79bc }}


Badness: 0.0547
Badness: 0.0547
Line 277: Line 292:
POTE generator: ~3/2 = 713.395
POTE generator: ~3/2 = 713.395


Vals: {{Val list| 5, 32, 37 }}
Optimal GPV sequence: {{Val list| 5, 32, 37 }}


Badness: 0.068238
Badness: 0.068238
Line 290: Line 305:
POTE generator: ~3/2 = 713.500
POTE generator: ~3/2 = 713.500


Vals: {{Val list| 5, 32, 37 }}
Optimal GPV sequence: {{Val list| 5, 32, 37 }}


Badness: 0.049172
Badness: 0.049172
Line 303: Line 318:
POTE generator: ~3/2 = 713.791
POTE generator: ~3/2 = 713.791


Vals: {{Val list| 5e, 32e, 37, 79bce, 116bbce }}
Optimal GPV sequence: {{Val list| 5e, 32e, 37, 79bce, 116bbce }}


Badness: 0.078068
Badness: 0.078068
Line 316: Line 331:
POTE generator: ~3/2 = 713.811
POTE generator: ~3/2 = 713.811


Vals: {{Val list| 5e, 32e, 37, 79bcef, 116bbcef }}
Optimal GPV sequence: {{Val list| 5e, 32e, 37, 79bcef, 116bbcef }}


Badness: 0.045653
Badness: 0.045653
Line 346: Line 361:
POTE generator: ~3/2 = 702.136
POTE generator: ~3/2 = 702.136


Vals: {{Val list| 12, 29de, 41de }}
Optimal GPV sequence: {{Val list| 12, 29de, 41de }}


Badness: 0.037482
Badness: 0.037482
Line 379: Line 394:
POTE generator: ~49/40 = 356.140
POTE generator: ~49/40 = 356.140


Vals: {{Val list| 27e, 37, 64be, 91bcde }}
Optimal GPV sequence: {{Val list| 27e, 37, 64be, 91bcde }}


Badness: 0.045639
Badness: 0.045639
Line 392: Line 407:
POTE generator: ~16/13 = 356.229
POTE generator: ~16/13 = 356.229


Vals: {{Val list| 27e, 37, 64be }}
Optimal GPV sequence: {{Val list| 27e, 37, 64be }}


Badness: 0.030161
Badness: 0.030161
Line 405: Line 420:
POTE generator: ~11/9 = 355.419
POTE generator: ~11/9 = 355.419


Vals: {{Val list| 10, 17c, 27e }}
Optimal GPV sequence: {{Val list| 10, 17c, 27e }}


Badness: 0.032863
Badness: 0.032863
Line 418: Line 433:
POTE generator: ~11/9 = 355.456
POTE generator: ~11/9 = 355.456


Vals: {{Val list| 10, 17c, 27e }}
Optimal GPV sequence: {{Val list| 10, 17c, 27e }}


Badness: 0.022619
Badness: 0.022619
Line 431: Line 446:
POTE generator: ~49/40 = 356.710
POTE generator: ~49/40 = 356.710


Vals: {{Val list| 10, 27, 37 }}
Optimal GPV sequence: {{Val list| 10, 27, 37 }}


Badness: 0.058084
Badness: 0.058084
Line 444: Line 459:
POTE generator: ~16/13 = 356.701
POTE generator: ~16/13 = 356.701


Vals: {{Val list| 10, 27, 37 }}
Optimal GPV sequence: {{Val list| 10, 27, 37 }}


Badness: 0.033971
Badness: 0.033971
Line 474: Line 489:
POTE generator: ~7/5 = 577.850
POTE generator: ~7/5 = 577.850


Vals: {{Val list| 2, 25e, 27e }}
Optimal GPV sequence: {{Val list| 2, 25e, 27e }}


Badness: 0.052054
Badness: 0.052054
Line 487: Line 502:
POTE generator: ~7/5 = 578.060
POTE generator: ~7/5 = 578.060


Vals: {{Val list| 2f, 25ef, 27e }}
Optimal GPV sequence: {{Val list| 2f, 25ef, 27e }}


Badness: 0.039705
Badness: 0.039705
Line 517: Line 532:
POTE generator: ~7/5 = 562.085
POTE generator: ~7/5 = 562.085


Vals: {{Val list| 2, 13, 15, 32c, 47bc, 79bcce }}
Optimal GPV sequence: {{Val list| 2, 13, 15, 32c, 47bc, 79bcce }}


Badness: 0.031036
Badness: 0.031036
Line 530: Line 545:
POTE generator: ~7/5 = 562.365
POTE generator: ~7/5 = 562.365


Vals: {{Val list| 15, 17c, 32cf }}
Optimal GPV sequence: {{Val list| 15, 17c, 32cf }}


Badness: 0.026214
Badness: 0.026214
Line 543: Line 558:
POTE generator: ~7/5 = 563.239
POTE generator: ~7/5 = 563.239


Vals: {{Val list| 15f, 17c, 32c, 49c }}
Optimal GPV sequence: {{Val list| 15f, 17c, 32c, 49c }}


Badness: 0.032721
Badness: 0.032721

Revision as of 01:43, 8 January 2022

The archytas clan (or archy family) 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 their respective 5-limit families are:

The rest are considered below.

Archy

Subgroup: 2.3.7

Comma list: 64/63

Sval mapping: [1 0 6], 0 1 -2]]

Sval mapping generators: ~2, ~3

Gencom mapping: [1 1 0 4], 0 1 0 -2]]

Gencom: [2 3/2; 64/63]

POTE generator: ~3/2 = 709.321

Template:Val list

Scales: archy5, archy7, archy12

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Sval mapping: [1 0 6 13], 0 1 -2 -6]]

Sval mapping generators: ~2, ~3

Gencom mapping: [1 1 0 4 7], 0 1 0 -2 -6]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~3/2 = 707.192

Optimal GPV sequence: Template:Val list

Scales: supra7, supra12

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Sval mapping: [1 0 6 13 18], 0 1 -2 -6 -9]]

Sval mapping generators: ~2, ~3

Gencom mapping: [1 1 0 4 7 9], 0 1 0 -2 -6 -9]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~3/2 = 706.137

Optimal GPV sequence: Template:Val list

Scales: supra7, supra12

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Sval mapping: [1 1 4 2], 0 2 -4 5]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [1 1 0 4 2], 0 1 0 -4 5]]

Gencom: [2 3/2; 64/63 99/98]

POTE generator: ~11/9 = 353.958

Optimal GPV sequence: Template:Val list

Scales: suhajira7, suhajira10, suhajira17

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Sval mapping: [1 1 4 2 4], 0 2 -4 5 -1]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [1 1 0 4 2 4], 0 1 0 -4 5 -1]]

Gencom: [2 3/2; 64/63 78/77 99/98]

POTE generator: ~11/9 = 353.775

Optimal GPV sequence: Template:Val list

Scales: suhajira7, suhajira10, suhajira17

Superpyth

In the 5-limit, superpyth tempers out 20480/19683 (sayo comma). This temperament has a fifth generator of ~3/2 = ~710¢ and 5/4 is equated to nine generators minus 5 octaves. A weak extension bipyth (10cd&22) tempers out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.

Subgroup: 2.3.5

Comma list: 20480/19683

Mapping: [1 0 -12], 0 1 9]]

POTE generator: ~3/2 = 710.078

Template:Val list

Badness: 0.135141

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 245/243

Mapping: [1 0 -12 6], 0 1 9 -2]]

Wedgie⟨⟨ 1 9 -2 12 -6 -30 ]]

POTE generator: ~3/2 = 710.291

Template:Val list

Badness: 0.032318

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 245/243

Mapping: [1 0 -12 6 -22], 0 1 9 -2 16]]

POTE generator: ~3/2 = 710.175

Optimal GPV sequence: Template:Val list

Badness: 0.024976

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 100/99

Mapping: [1 0 -12 6 -22 -17], 0 1 9 -2 16 13]]

POTE generator: ~3/2 = 710.479

Optimal GPV sequence: Template:Val list

Badness: 0.024673

Suprapyth

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 99/98

Mapping: [1 0 -12 6 13], 0 1 9 -2 -6]]

POTE generator: ~3/2 = 709.495

Optimal GPV sequence: Template:Val list

Badness: 0.032768

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 99/98

Mapping: [1 0 -12 6 13 18], 0 1 9 -2 -6 -9]]

POTE generator: ~3/2 = 708.703

Optimal GPV sequence: Template:Val list

Badness: 0.036336

Quasisuper

Subgroup: 2.3.5.7

Comma list: 64/63, 2430/2401

Mapping: [1 0 23 6], 0 1 -13 -2]]

Wedgie⟨⟨ 1 -13 -2 -23 -6 32 ]]

POTE generator: ~3/2 = 708.328

Template:Val list

Badness: 0.063794

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).

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 121/120

Mapping: [1 0 23 6 13], 0 1 -13 -2 -6]]

POTE generator: ~3/2 = 708.205

Optimal GPV sequence: Template:Val list

Badness: 0.032203

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 78/77, 91/90, 121/120

Mapping: [1 0 23 6 13 18], 0 1 -13 -2 -6 -9]]

POTE generator: ~3/2 = 708.004

Optimal GPV sequence: Template:Val list

Badness: 0.030219

Quasisoup

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2430/2401

Mapping: [1 0 23 6 -22], 0 1 -13 -2 16]]

POTE generator: ~3/2 = 709.021

Optimal GPV sequence: Template:Val list

Badness: 0.083490

Ultrapyth

Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 oceanfront temperament, with mapping 5th harmonic to 14 fifths.

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 6860/6561

Mapping: [1 0 -20 6], 0 1 14 -2]]

Wedgie⟨⟨ 1 14 -2 20 -6 -44 ]]

POTE generator: ~3/2 = 713.651

Template:Val list

Badness: 0.108466

2.3.5.7.13

Subgroup: 2.3.5.7.13

Comma list: 64/63, 91/90, 4394/4375

Mapping: [1 0 -20 6 -25], 0 1 14 -2 18]]

POTE generator: ~3/2 = 713.745

Optimal GPV sequence: Template:Val list

Badness: 0.0547

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 2401/2376

Mapping: [1 0 -20 6 21], 0 1 14 -2 -11]]

POTE generator: ~3/2 = 713.395

Optimal GPV sequence: Template:Val list

Badness: 0.068238

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 1573/1568

Mapping: [1 0 -20 6 21 -25], 0 1 14 -2 -11 18]]

POTE generator: ~3/2 = 713.500

Optimal GPV sequence: Template:Val list

Badness: 0.049172

Counterultrapyth

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 3773/3645

Mapping: [1 0 -20 6 -38], 0 1 14 -2 26]]

POTE generator: ~3/2 = 713.791

Optimal GPV sequence: Template:Val list

Badness: 0.078068

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 847/845

Mapping: [1 0 -20 6 -38 -25], 0 1 14 -2 26 18]]

POTE generator: ~3/2 = 713.811

Optimal GPV sequence: Template:Val list

Badness: 0.045653

Schism

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping: [1 0 15 6], 0 1 -8 -2]]

Wedgie⟨⟨ 1 -8 -2 -15 -6 18 ]]

POTE generator: ~3/2 = 701.556

Template:Val list

Badness: 0.056648

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [1 0 15 6 13], 0 1 -8 -2 -6]]

POTE generator: ~3/2 = 702.136

Optimal GPV sequence: Template:Val list

Badness: 0.037482

Beatles

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

Subgroup: 2.3.5.7

Comma list: 64/63, 686/675

Mapping: [1 1 5 4], 0 2 -9 -4]]

Wedgie⟨⟨ 2 -9 -4 -19 -12 16 ]]

POTE generator: ~49/40 = 355.904

Template:Val list

Badness: 0.045872

Music

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 686/675

Mapping: [1 1 5 4 10], 0 2 -9 -4 -22]]

POTE generator: ~49/40 = 356.140

Optimal GPV sequence: Template:Val list

Badness: 0.045639

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 91/90, 100/99, 169/168

Mapping: [1 1 5 4 10 4], 0 2 -9 -4 -22 -1]]

POTE generator: ~16/13 = 356.229

Optimal GPV sequence: Template:Val list

Badness: 0.030161

Ringo

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 540/539

Mapping: [1 1 5 4 2], 0 2 -9 -4 5]]

POTE generator: ~11/9 = 355.419

Optimal GPV sequence: Template:Val list

Badness: 0.032863

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 91/90

Mapping: [1 1 5 4 2 4], 0 2 -9 -4 5 -1]]

POTE generator: ~11/9 = 355.456

Optimal GPV sequence: Template:Val list

Badness: 0.022619

Beetle

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 686/675

Mapping: [1 1 5 4 -1], 0 2 -9 -4 15]]

POTE generator: ~49/40 = 356.710

Optimal GPV sequence: Template:Val list

Badness: 0.058084

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 169/168

Mapping: [1 1 5 4 -1 4], 0 2 -9 -4 15 -1]]

POTE generator: ~16/13 = 356.701

Optimal GPV sequence: Template:Val list

Badness: 0.033971

Fervor

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

Subgroup: 2.3.5.7

Comma list: 64/63, 9604/9375

Mapping: [1 4 -2 -2], 0 -5 9 10]]

Wedgie⟨⟨ 5 -9 -10 -26 -30 2 ]]

POTE generator: ~7/5 = 577.776

Template:Val list

Badness: 0.108455

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 1350/1331

Mapping: [1 4 -2 -2 3], 0 -5 9 10 1]]

POTE generator: ~7/5 = 577.850

Optimal GPV sequence: Template:Val list

Badness: 0.052054

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 507/500

Mapping: [1 4 -2 -2 3 -4], 0 -5 9 10 1 16]]

POTE generator: ~7/5 = 578.060

Optimal GPV sequence: Template:Val list

Badness: 0.039705

Progress

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

Subgroup: 2.3.5.7

Comma list: 64/63, 392/375

Mapping: [1 0 5 6], 0 3 -5 -6]]

Wedgie⟨⟨ 3 -5 -6 -15 -18 0 ]]

POTE generator: ~7/5 = 562.122

Template:Val list

Badness: 0.066400

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 77/75

Mapping: [1 0 5 6 4], 0 3 -5 -6 -1]]

POTE generator: ~7/5 = 562.085

Optimal GPV sequence: Template:Val list

Badness: 0.031036

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 66/65, 77/75

Mapping: [1 0 5 6 4 0], 0 3 -5 -6 -1 7]]

POTE generator: ~7/5 = 562.365

Optimal GPV sequence: Template:Val list

Badness: 0.026214

Progressive

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 64/63, 77/75

Mapping: [1 0 5 6 4 9], 0 3 -5 -6 -1 -10]]

POTE generator: ~7/5 = 563.239

Optimal GPV sequence: Template:Val list

Badness: 0.032721

Sixix

See also: Dual-fifth temperaments#Dual-3 Sixix

Subgroup: 2.3.5

Comma: 3125/2916

Mapping: [1 3 4], 0 -5 -6]]

POTE generator: ~6/5 = 338.365

Template:Val list

Badness: 0.153088

7-limit

Subgroup: 2.3.5.7

Comma list: 64/63, 3125/2916

Mapping: [1 3 4 0], 0 -5 -6 10]]

Wedgie⟨⟨ 5 6 -10 -2 -30 -40 ]]

POTE generator: ~6/5 = 337.442

Template:Val list

Badness: 0.158903