Archytas clan: Difference between revisions
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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 [[ | 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. | ||
=Mother= | Discussed under their respective 5-limit families are [[Diaschismic family #Pajara|pajara]], [[Meantone family #Dominant|dominant]], [[Augmented family #Augene|augene]], [[porcupine family|porcupine]], and [[Tetracot family #Modus|modus]]. The rest are considered below. | ||
= Mother = | |||
Commas: 16/15, 21/20 | Commas: 16/15, 21/20 | ||
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Wedgie: <<1 -1 -2 -4 -6 -2|| | Wedgie: <<1 -1 -2 -4 -6 -2|| | ||
EDOs: 5, 148, 153 | EDOs: {{EDOs|5, 148, 153}} | ||
Badness: 0.0242 | Badness: 0.0242 | ||
=Blacksmith= | = Blacksmith = | ||
== 5-limit (blackwood) == | |||
Comma: 256/243 | |||
[[POTE_tuning|POTE generator]]: 399.594 | |||
Map: [<5 8 0|, <0 0 1|] | |||
EDOs: {{EDOs|5, 10, 15}} | |||
Badness: 0.0638 | |||
== 7-limit == | |||
Commas: 28/27, 49/48 | Commas: 28/27, 49/48 | ||
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Wedgie: <<0 5 0 8 0 -14|| | Wedgie: <<0 5 0 8 0 -14|| | ||
EDOs: 5, 10, 15, 40b, 55b | EDOs: {{EDOs|5, 10, 15, 40b, 55b}} | ||
Badness: 0.0256 | Badness: 0.0256 | ||
==11-limit== | == 11-limit == | ||
Commas: 28/27, 49/48, 55/54 | Commas: 28/27, 49/48, 55/54 | ||
| Line 35: | Line 49: | ||
Map: [<5 8 0 14 29|, <0 0 1 0 -1|] | Map: [<5 8 0 14 29|, <0 0 1 0 -1|] | ||
EDOs: 5, 10, 15, 40be, 55be, 70bde, 85bcde | EDOs: {{EDOs|5, 10, 15, 40be, 55be, 70bde, 85bcde}} | ||
Badness: 0.0246 | Badness: 0.0246 | ||
==13-limit== | === 13-limit === | ||
Commas: 28/27, 40/39, 49/48, 55/54 | Commas: 28/27, 40/39, 49/48, 55/54 | ||
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Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|] | Map: [<5 8 0 14 29 7|, <0 0 1 0 -1 1|] | ||
EDOs: 5, 10, 15, 25e, 40bef | EDOs: {{EDOs|5, 10, 15, 25e, 40bef}} | ||
Badness: 0.0205 | Badness: 0.0205 | ||
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[[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]] | [[File:blacksmith10.jpg|alt=blacksmith10.jpg|blacksmith10.jpg]] | ||
=Superpyth= | =Superpyth= | ||
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Wedgie: <<1 9 -2 12 -6 -30|| | Wedgie: <<1 9 -2 12 -6 -30|| | ||
EDOs: 5, 17, 22, 27, 49 | EDOs: {{EDOs|5, 17, 22, 27, 49}} | ||
Badness: 0.0323 | Badness: 0.0323 | ||
| Line 120: | Line 121: | ||
Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|] | Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|] | ||
EDOs: 22, 49 | EDOs: {{EDOs|22, 49}} | ||
Badness: 0.0250 | Badness: 0.0250 | ||
==13-limit== | === 13-limit === | ||
Commas: 64/63, 78/77, 91/90, 100/99 | Commas: 64/63, 78/77, 91/90, 100/99 | ||
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Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|] | Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|] | ||
EDOs: 22, 27e, 49, 76bcde | EDOs: {{EDOs|22, 27e, 49, 76bcde}} | ||
Badness: 0.0247 | Badness: 0.0247 | ||
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Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|] | Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|] | ||
EDOs: 5, 7, 12, 17, 22 | EDOs: {{EDOs|5, 7, 12, 17, 22}} | ||
Badness: 0.0328 | Badness: 0.0328 | ||
| Line 157: | Line 158: | ||
Badness: 0.0363 | Badness: 0.0363 | ||
=Beatles= | = Beatles = | ||
== 5-limit == | |||
Comma: 524288/492075 | Comma: 524288/492075 | ||
| Line 190: | Line 192: | ||
Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|] | Map: [<1 1 5 4 10|,<0 2 -9 -4 -22|] | ||
EDOs: 27e, 37, 64be, 91bcde | EDOs: {{EDOs|27e, 37, 64be, 91bcde}} | ||
Badness: 0.0456 | Badness: 0.0456 | ||
==13-limit== | === 13-limit === | ||
Commas: 64/63, 91/90, 100/99, 169/168 | Commas: 64/63, 91/90, 100/99, 169/168 | ||
| Line 201: | Line 203: | ||
Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|] | Map: [<1 1 5 4 10 4|,<0 2 -9 -4 -22 -1|] | ||
EDOs: 27e, 37, 64be | EDOs: {{EDOs|27e, 37, 64be}} | ||
Badness: 0.0302 | Badness: 0.0302 | ||
| Line 212: | Line 214: | ||
Map: [<1 1 5 4 2|,<0 2 -9 -4 5|] | Map: [<1 1 5 4 2|,<0 2 -9 -4 5|] | ||
EDOs: 10, 17c, 27e | EDOs: {{EDOs|10, 17c, 27e}} | ||
Badness: 0.0329 | Badness: 0.0329 | ||
| Line 223: | Line 225: | ||
Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|] | Map: [<1 1 5 4 2 4|,<0 2 -9 -4 5 -1|] | ||
EDOs: 10, 17c, 27e | EDOs: {{EDOs|10, 17c, 27e}} | ||
Badness: 0.0226 | Badness: 0.0226 | ||
=Schism= | = Schism = | ||
{{see also|Schismatic family #Schism}} | |||
Commas: 64/63, 360/343 | Commas: 64/63, 360/343 | ||
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Wedgie: <<1 -8 -2 -15 -6 18|| | Wedgie: <<1 -8 -2 -15 -6 18|| | ||
EDOs: 12, 41d, 53d | EDOs: {{EDOs|12, 41d, 53d}} | ||
Badness: 0.0566 | Badness: 0.0566 | ||
| Line 247: | Line 251: | ||
Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|] | Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|] | ||
EDOs: 12, 29de, 41de | EDOs: {{EDOs|12, 29de, 41de}} | ||
Badness: 0.0375 | Badness: 0.0375 | ||
=Passion= | = Passion = | ||
== 5-limit == | |||
==5-limit== | |||
Comma: 262144/253125 | Comma: 262144/253125 | ||
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Map: [<1 2 2|, <0 -5 4|] | Map: [<1 2 2|, <0 -5 4|] | ||
EDOs: 11, 12, 49, 61, 73 | EDOs: {{EDOs|11, 12, 49, 61, 73}} | ||
Badness: 0.1686 | Badness: 0.1686 | ||
===Passive=== | === Passive === | ||
Commas: 225/224, 256/245 | Commas: 225/224, 256/245 | ||
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Map: [<1 2 2 3|, <0 -5 4 -2|] | Map: [<1 2 2 3|, <0 -5 4 -2|] | ||
EDOs: 1, 11, 12, 49d | EDOs: {{EDOs|1, 11, 12, 49d}} | ||
Badness: 0.0751 | Badness: 0.0751 | ||
==7-limit== | == 7-limit == | ||
Commas: 64/63, 3125/3087 | Commas: 64/63, 3125/3087 | ||
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Generators: 2, 16/15 | Generators: 2, 16/15 | ||
EDOs: 12, 37, 49, 110bcd | EDOs: {{EDOs|12, 37, 49, 110bcd}} | ||
Badness: 0.0623 | Badness: 0.0623 | ||
| Line 297: | Line 300: | ||
Map: [<1 2 2 2 2|, <0 -5 4 10 18|] | Map: [<1 2 2 2 2|, <0 -5 4 10 18|] | ||
EDOs: 12, 37, 49 | EDOs: {{EDOs|12, 37, 49}} | ||
Badness: 0.0408 | Badness: 0.0408 | ||
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Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|] | Map: [<1 2 2 2 2 2|, <0 -5 4 10 18 21|] | ||
EDOs: 12f, 37, 49f | EDOs: {{EDOs|12f, 37, 49f}} | ||
Badness: 0.0309 | Badness: 0.0309 | ||
=Fervor= | = Fervor = | ||
== 5-limit == | |||
Comma: 67108864/61509375 | Comma: 67108864/61509375 | ||
| Line 319: | Line 323: | ||
Map: [<1 4 -2|, <0 -5 9|] | Map: [<1 4 -2|, <0 -5 9|] | ||
EDOs: 25, 27 | EDOs: {{EDOs|25, 27}} | ||
Badness: 0.8526 | Badness: 0.8526 | ||
| Line 332: | Line 336: | ||
Wedgie: <<5 -9 -10 -26 -30 2|| | Wedgie: <<5 -9 -10 -26 -30 2|| | ||
EDOs: 25, 27 | EDOs: {{EDOs|25, 27}} | ||
Badness: 0.1085 | Badness: 0.1085 | ||
| Line 343: | Line 347: | ||
Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|] | Map: [<1 4 -2 -2 3|, <0 -5 9 10 1|] | ||
EDOs: 25e, 27e | EDOs: {{EDOs|25e, 27e}} | ||
Badness: 0.0521 | Badness: 0.0521 | ||
| Line 354: | Line 358: | ||
Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|] | Map: [<1 4 -2 -2 3 -4|, <0 -5 9 10 1 16|] | ||
EDOs: 27e | EDOs: {{EDOs|27e}} | ||
Badness: 0.0397 | Badness: 0.0397 | ||
| Line 367: | Line 371: | ||
Wedgie: <<1 -13 -2 -23 -2 -6 32|| | Wedgie: <<1 -13 -2 -23 -2 -6 32|| | ||
EDOs: 22, 61 | EDOs: {{EDOs|22, 61}} | ||
Badness: 0.0638 | Badness: 0.0638 | ||
==Quasisupra== | ==Quasisupra== | ||
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament [[Supra|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 | Commas: 64/63, 99/98, 121/120 | ||
POTE generator: ~3/2 = 708.205 | POTE generator: ~3/2 = 708.205 | ||
| Line 380: | Line 384: | ||
Map: [<1 2 -3 2 1|, <0 -1 13 2 6|] | Map: [<1 2 -3 2 1|, <0 -1 13 2 6|] | ||
EDOs: 17c, 22, 27c, 39d, 61d | EDOs: {{EDOs|17c, 22, 27c, 39d, 61d}} | ||
Badness: 0.0322 | Badness: 0.0322 | ||
| Line 391: | Line 395: | ||
Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|] | Map: [<1 0 23 6 13 18|, <0 1 -13 -2 -6 -9|] | ||
EDOs: 17c, 22, 39d, 61df, 100bcdf | EDOs: {{EDOs|17c, 22, 39d, 61df, 100bcdf}} | ||
Badness: 0.0302 | Badness: 0.0302 | ||
| Line 402: | Line 406: | ||
Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|] | Map: [<1 0 23 6 -22|, <0 1 -13 -2 16|] | ||
EDOs: 22 | EDOs: {{EDOs|22}} | ||
Badness: 0.0835 | Badness: 0.0835 | ||
=Progress= | = Progress = | ||
== 5-limit == | |||
Comma: 32768/30375 | Comma: 32768/30375 | ||
| Line 413: | Line 418: | ||
Map: [<1 0 5|, <0 3 -5|] | Map: [<1 0 5|, <0 3 -5|] | ||
EDOs: 4, 13, 15, 32c, 47bc, 62bc | EDOs: {{EDOs|4, 13, 15, 32c, 47bc, 62bc}} | ||
Badness: 0.2461 | Badness: 0.2461 | ||
| Line 426: | Line 431: | ||
Wedgie: <<3 -5 -6 -15 -18 0|| | Wedgie: <<3 -5 -6 -15 -18 0|| | ||
EDOs: 13, 15, 32c, 79bcc, 111bcc | EDOs: {{EDOs|13, 15, 32c, 79bcc, 111bcc}} | ||
Badness: 0.0664 | Badness: 0.0664 | ||
| Line 437: | Line 442: | ||
Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|] | Map: [<1 0 5 6 4|, <0 3 -5 -6 -1|] | ||
EDOs: 13, 15, 32c, 47bc, 79bcce | EDOs: {{EDOs|13, 15, 32c, 47bc, 79bcce}} | ||
Badness: 0.0310 | Badness: 0.0310 | ||
==13-limit== | === 13-limit === | ||
Commas: 56/55, 64/63, 66/65, 77/75 | Commas: 56/55, 64/63, 66/65, 77/75 | ||
| Line 448: | Line 453: | ||
Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|] | Map: [<1 0 5 6 4 0|, <0 3 -5 -6 -1 7|] | ||
EDOs: 15, 17c, 32cf | EDOs: {{EDOs|15, 17c, 32cf}} | ||
Badness: 0.0262 | Badness: 0.0262 | ||
==Progressive== | === Progressive === | ||
Commas: 26/25, 56/55, 64/63, 77/75 | Commas: 26/25, 56/55, 64/63, 77/75 | ||
| Line 459: | Line 464: | ||
Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|] | Map: [<1 0 5 6 4 9|, <0 3 -5 -6 -1 -10|] | ||
EDOs: 15f, 17c, 32c, 49c | EDOs: {{EDOs|15f, 17c, 32c, 49c}} | ||
Badness: 0.0327 | Badness: 0.0327 | ||
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[[Category:Clan]] | [[Category:Clan]] | ||
[[Category:Temperament]] | [[Category:Temperament]] | ||
[[Category:Rank 2]] | |||