9ed9/8: Difference between revisions
Jump to navigation
Jump to search
m →15-odd-limit mappings: minor adjustments on table to improve readability of wikitext (add space around meta chars) |
m →15-odd-limit mappings: linebreak in table title looks not that great (revert my idea) |
||
| Line 235: | Line 235: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
|+ Direct mapping | |+ Direct mapping (even if inconsistent) | ||
|- | |- | ||
! Interval(s) | ! Interval(s) | ||
Revision as of 11:48, 4 December 2021
9ED9/8 is the equal division of the Pythagorean whole tone into nine parts of 22.6567 cents each, corresponding to 52.9645 edo. This tuning is used in Ottoman classical music theory, in which ninth root of the 9/8 whole tone is treated as the minimum interval.
Intervals
| degree | cents value | ratio |
|---|---|---|
| 0 | 0.0000 | 1/1 |
| 1 | 22.6567 | (9/8)1/9 |
| 2 | 45.3133 | (9/8)2/9 |
| 3 | 67.9700 | (9/8)1/3 |
| 4 | 90.6267 | (9/8)4/9 |
| 5 | 113.2833 | (9/8)5/9 |
| 6 | 135.9400 | (9/8)2/3 |
| 7 | 158.5967 | (9/8)7/9 |
| 8 | 181.2533 | (9/8)8/9 |
| 9 | 203.9100 | 9/8 |
| 10 | 226.5667 | (9/8)10/9 |
| 11 | 249.2233 | (9/8)11/9 |
| 12 | 271.8800 | (9/8)4/3 |
| 13 | 294.5367 | (9/8)13/9 |
| 14 | 317.1933 | (9/8)14/9 |
| 15 | 339.8500 | (9/8)5/3 |
| 16 | 362.5067 | (9/8)16/9 |
| 17 | 385.1633 | (9/8)17/9 |
| 18 | 407.8200 | (9/8)2 = 81/64 |
| 19 | 430.4767 | (9/8)19/9 |
| 20 | 453.1333 | (9/8)20/9 |
| 21 | 475.7900 | (9/8)7/3 |
| 22 | 498.4467 | (9/8)22/9 |
| 23 | 521.1033 | (9/8)23/9 |
| 24 | 543.7600 | (9/8)8/3 |
| 25 | 566.4167 | (9/8)25/9 |
| 26 | 589.0733 | (9/8)26/9 |
| 27 | 611.7300 | (9/8)3 = 729/512 |
| 28 | 634.3867 | (9/8)28/9 |
| 29 | 657.0433 | (9/8)29/9 |
| 30 | 679.7000 | (9/8)10/3 |
| 31 | 702.3567 | (9/8)31/9 |
| 32 | 725.0133 | (9/8)32/9 |
| 33 | 747.6700 | (9/8)11/3 |
| 34 | 770.3267 | (9/8)34/9 |
| 35 | 792.9833 | (9/8)35/9 |
| 36 | 815.6400 | (9/8)4 = 6561/4096 |
| 37 | 838.2967 | (9/8)37/9 |
| 38 | 860.9533 | (9/8)38/9 |
| 39 | 883.6100 | (9/8)13/3 |
| 40 | 906.2667 | (9/8)40/9 |
| 41 | 928.9233 | (9/8)41/9 |
| 42 | 951.5800 | (9/8)14/3 |
| 43 | 974.2367 | (9/8)43/9 |
| 44 | 996.8933 | (9/8)44/9 |
| 45 | 1019.5500 | (9/8)5 = 59049/32768 |
| 46 | 1042.2067 | (9/8)46/9 |
| 47 | 1064.8633 | (9/8)47/9 |
| 48 | 1087.5200 | (9/8)16/3 |
| 49 | 1110.1767 | (9/8)49/9 |
| 50 | 1132.8333 | (9/8)50/9 |
| 51 | 1155.4900 | (9/8)17/3 |
| 52 | 1178.1467 | (9/8)52/9 |
| 53 | 1200.8033 | (9/8)53/9 |
| 54 | 1223.4600 | (9/8)6 = 531441/262144 |
Just approximation
15-odd-limit mappings
The following table shows how 15-odd-limit intervals are represented in 9ed9/8 (ordered by absolute error).
| Interval(s) | Error (abs, ¢) |
|---|---|
| 9/8 | 0.000 |
| 3/2, 4/3 | 0.402 |
| 26/15 | 0.679 |
| 15/8, 5/3 | 0.749 |
| 16/9 | 0.803 |
| 13/10 | 1.081 |
| 5/4, 10/9 | 1.150 |
| 15/13 | 1.482 |
| 6/5, 16/15 | 1.552 |
| 20/13 | 1.884 |
| 9/5, 8/5 | 1.954 |
| 13/8, 13/9 | 2.231 |
| 13/12 | 2.633 |
| 16/13, 18/13 | 3.034 |
| 24/13 | 3.436 |
| 12/7 | 4.206 |
| 22/13 | 4.524 |
| 9/7, 8/7 | 4.607 |
| 7/6 | 5.009 |
| 13/11 | 5.327 |
| 7/4, 14/9 | 5.411 |
| 10/7 | 5.758 |
| 22/15 | 6.006 |
| 15/14 | 6.159 |
| 11/10 | 6.408 |
| 7/5 | 6.561 |
| 15/11 | 6.809 |
| 13/7 | 6.838 |
| 28/15 | 6.963 |
| 11/6 | 7.156 |
| 20/11 | 7.211 |
| 11/9, 11/8 | 7.558 |
| 14/13 | 7.642 |
| 12/11 | 7.960 |
| 18/11, 16/11 | 8.361 |
| 14/11 | 9.688 |
| 11/7 | 10.491 |