80edo: Difference between revisions
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| Line 35: | Line 35: | ||
| 0 | | 0 | ||
| 0 | | 0 | ||
| 1/1 | | [[1/1]] | ||
|- | |- | ||
| 1 | | 1 | ||
| 15 | | 15 | ||
| 64/63 | | [[64/63]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 30 | | 30 | ||
| 81/80 | | [[81/80]], [[50/49]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 45 | | 45 | ||
| 34/33 | | [[36/35]], [[49/48]], [[34/33]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 60 | | 60 | ||
| | | [[28/27]], [[33/32]], [[26/25]], [[35/34]] | ||
|- | |- | ||
| 5 | | 5 | ||
| 75 | | 75 | ||
| 22/21, | | [[25/24]], [[22/21]], [[27/26]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 90 | | 90 | ||
| 19/18, 20/19 | | [[21/20]], [[19/18]], [[20/19]] | ||
|- | |- | ||
| 7 | | 7 | ||
| 105 | | 105 | ||
| 16/15, 17/16, 18/17 | | [[16/15]], [[17/16]], [[18/17]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 120 | | 120 | ||
| 14/13 | | [[15/14]], [[14/13]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 135 | | 135 | ||
| 13/12 | | [[13/12]] | ||
|- | |- | ||
| 10 | | 10 | ||
| 150 | | 150 | ||
| 12/11 | | [[12/11]] | ||
|- | |- | ||
| 11 | | 11 | ||
| 165 | | 165 | ||
| 11/10 | | [[11/10]] | ||
|- | |- | ||
| 12 | | 12 | ||
| 180 | | 180 | ||
| 10/9, 21/19 | | [[10/9]], [[21/19]] | ||
|- | |- | ||
| 13 | | 13 | ||
| 195 | | 195 | ||
| 19/17 | | [[19/17]] | ||
|- | |- | ||
| 14 | | 14 | ||
| 210 | | 210 | ||
| 9/8, 17/15 | | [[9/8]], [[17/15]] | ||
|- | |- | ||
| 15 | | 15 | ||
| 225 | | 225 | ||
| 8/7 | | [[8/7]] | ||
|- | |- | ||
| 16 | | 16 | ||
| Line 103: | Line 103: | ||
| 17 | | 17 | ||
| 255 | | 255 | ||
| 15/13, 22/19 | | [[81/70]], [[15/13]], [[22/19]] | ||
|- | |- | ||
| 18 | | 18 | ||
| 270 | | 270 | ||
| 7/6 | | [[7/6]] | ||
|- | |- | ||
| 19 | | 19 | ||
| 285 | | 285 | ||
| 13/11, 20/17 | | [[13/11]], [[20/17]] | ||
|- | |- | ||
| 20 | | 20 | ||
| 300 | | 300 | ||
| 19/16 | | [[25/21]], [[19/16]] | ||
|- | |- | ||
| 21 | | 21 | ||
| 315 | | 315 | ||
| 6/5 | | [[6/5]] | ||
|- | |- | ||
| 22 | | 22 | ||
| 330 | | 330 | ||
| 17/14 | | [[17/14]] | ||
|- | |- | ||
| 23 | | 23 | ||
| 345 | | 345 | ||
| 11/9 | | [[11/9]] | ||
|- | |- | ||
| 24 | | 24 | ||
| 360 | | 360 | ||
| 16/13 | | [[16/13]] | ||
|- | |- | ||
| 25 | | 25 | ||
| 375 | | 375 | ||
| | | [[21/17]] | ||
|- | |- | ||
| 26 | | 26 | ||
| 390 | | 390 | ||
| 5/4 | | [[5/4]] | ||
|- | |- | ||
| 27 | | 27 | ||
| 405 | | 405 | ||
| 19/15 | | [[24/19]], [[19/15]] | ||
|- | |- | ||
| 28 | | 28 | ||
| 420 | | 420 | ||
| 14/11 | | [[14/11]] | ||
|- | |- | ||
| 29 | | 29 | ||
| 435 | | 435 | ||
| 9/7 | | [[9/7]] | ||
|- | |- | ||
| 30 | | 30 | ||
| 450 | | 450 | ||
| 13/10, 22/17 | | [[13/10]], [[22/17]] | ||
|- | |- | ||
| 31 | | 31 | ||
| 465 | | 465 | ||
| 17/13 | | [[17/13]] | ||
|- | |- | ||
| 32 | | 32 | ||
| 480 | | 480 | ||
| 21/16, 25/19 | | [[21/16]], [[25/19]] | ||
|- | |- | ||
| 33 | | 33 | ||
| 495 | | 495 | ||
| 4/3 | | [[4/3]] | ||
|- | |- | ||
| 34 | | 34 | ||
| Line 175: | Line 175: | ||
| 35 | | 35 | ||
| 525 | | 525 | ||
| 19/14 | | [[19/14]] | ||
|- | |- | ||
| 36 | | 36 | ||
| 540 | | 540 | ||
| 26/19 | | [[26/19]] | ||
|- | |- | ||
| 37 | | 37 | ||
| 555 | | 555 | ||
| 11/8 | | [[11/8]] | ||
|- | |- | ||
| 38 | | 38 | ||
| 570 | | 570 | ||
| 18/13 | | [[18/13]] | ||
|- | |- | ||
| 39 | | 39 | ||
| 585 | | 585 | ||
| 7/5 | | [[7/5]] | ||
|- | |- | ||
| 40 | | 40 | ||
| 600 | | 600 | ||
| 17/12, 24/17 | | [[17/12]], [[24/17]] | ||
|- | |||
| … | |||
| … | |||
| … | |||
|} | |} | ||
<nowiki>*</nowiki> based on treating 80edo as a [[19-limit]] temperament; other approaches are possible. | <nowiki>*</nowiki> based on treating 80edo as a [[19-limit]] temperament; other approaches are possible. | ||
[[Category:Edo]] | |||
[[Category:19-limit]] | [[Category:19-limit]] | ||
Revision as of 06:48, 5 October 2020
The 80 equal temperament, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 cents. 80et is the first equal temperament that represents the 19-limit tonality diamond consistently (it barely manages to do so).
80et tempers out 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.
80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:
31&80 <<7 6 15 27 -24 -23 -20 ... ||
72&80 <<24 30 40 24 32 24 0 ... ||
34&80 <<2 -4 -50 22 16 2 -40 ... ||
46&80 <<2 -4 30 22 16 2 40 ... ||
29&80 <<3 34 45 33 24 -37 20 ... ||
12&80 <<4 -8 -20 -36 32 4 0 ... ||
22&80 <<6 -10 12 -14 -32 6 -40 ... ||
58&80 <<6 -10 12 -14 -32 6 40 ... ||
41&80 <<7 26 25 -3 -24 -33 20 ... ||
In each case, the numbers joined by an ampersand represent 19-limit patent vals (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.
Intervals
| Degree | Cents | Approximate Ratios* |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 15 | 64/63 |
| 2 | 30 | 81/80, 50/49 |
| 3 | 45 | 36/35, 49/48, 34/33 |
| 4 | 60 | 28/27, 33/32, 26/25, 35/34 |
| 5 | 75 | 25/24, 22/21, 27/26 |
| 6 | 90 | 21/20, 19/18, 20/19 |
| 7 | 105 | 16/15, 17/16, 18/17 |
| 8 | 120 | 15/14, 14/13 |
| 9 | 135 | 13/12 |
| 10 | 150 | 12/11 |
| 11 | 165 | 11/10 |
| 12 | 180 | 10/9, 21/19 |
| 13 | 195 | 19/17 |
| 14 | 210 | 9/8, 17/15 |
| 15 | 225 | 8/7 |
| 16 | 240 | |
| 17 | 255 | 81/70, 15/13, 22/19 |
| 18 | 270 | 7/6 |
| 19 | 285 | 13/11, 20/17 |
| 20 | 300 | 25/21, 19/16 |
| 21 | 315 | 6/5 |
| 22 | 330 | 17/14 |
| 23 | 345 | 11/9 |
| 24 | 360 | 16/13 |
| 25 | 375 | 21/17 |
| 26 | 390 | 5/4 |
| 27 | 405 | 24/19, 19/15 |
| 28 | 420 | 14/11 |
| 29 | 435 | 9/7 |
| 30 | 450 | 13/10, 22/17 |
| 31 | 465 | 17/13 |
| 32 | 480 | 21/16, 25/19 |
| 33 | 495 | 4/3 |
| 34 | 510 | |
| 35 | 525 | 19/14 |
| 36 | 540 | 26/19 |
| 37 | 555 | 11/8 |
| 38 | 570 | 18/13 |
| 39 | 585 | 7/5 |
| 40 | 600 | 17/12, 24/17 |
| … | … | … |
* based on treating 80edo as a 19-limit temperament; other approaches are possible.