User:MisterShafXen/65edt
Jump to navigation
Jump to search
Prime factorization
5 × 13
Step size
29.2608 ¢
Octave
41\65edt (1199.69 ¢)
(convergent)
Consistency limit
16
Distinct consistency limit
10
| ← 64edt | 65edt | 66edt → |
(convergent)
65 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 65edt or 65ed3), is a nonoctave tuning system that divides the interval of 3/1 into 65 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 31/65, or the 65th root of 3.
Theory
65edt, being a multiple of 13, contains the Bohlen–Pierce scale as a subset. Owing to the fact that Bohlen-Pierce is effectively every 5th step of 41edo, 65edt has a near-pure octave at 41 steps of 65edt, making this a formidable no-5 11-limit system which maintains less than 15% relative error (and less than 5¢ absolute error) on all subgroup elements.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.3 | +0.0 | -0.6 | -6.5 | -0.3 | -3.8 | -0.9 | +0.0 | -6.8 | +3.7 | -0.6 |
| Relative (%) | -1.0 | +0.0 | -2.1 | -22.3 | -1.0 | -13.1 | -3.1 | +0.0 | -23.4 | +12.7 | -2.1 | |
| Steps (reduced) |
41 (41) |
65 (0) |
82 (17) |
95 (30) |
106 (41) |
115 (50) |
123 (58) |
130 (0) |
136 (6) |
142 (12) |
147 (17) | |
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 29.3 | 20 | |
| 2 | 58.5 | 40 | 29/28, 30/29, 31/30, 32/31 |
| 3 | 87.8 | 60 | 20/19, 21/20 |
| 4 | 117 | 80 | 15/14, 31/29 |
| 5 | 146.3 | 100 | |
| 6 | 175.6 | 120 | 21/19, 31/28 |
| 7 | 204.8 | 140 | 9/8 |
| 8 | 234.1 | 160 | 8/7 |
| 9 | 263.3 | 180 | 7/6 |
| 10 | 292.6 | 200 | 13/11, 32/27 |
| 11 | 321.9 | 220 | 35/29 |
| 12 | 351.1 | 240 | 11/9, 27/22 |
| 13 | 380.4 | 260 | |
| 14 | 409.7 | 280 | 19/15, 33/26 |
| 15 | 438.9 | 300 | 9/7 |
| 16 | 468.2 | 320 | 17/13, 21/16 |
| 17 | 497.4 | 340 | 4/3 |
| 18 | 526.7 | 360 | 19/14, 23/17 |
| 19 | 556 | 380 | 29/21 |
| 20 | 585.2 | 400 | 7/5 |
| 21 | 614.5 | 420 | 10/7 |
| 22 | 643.7 | 440 | 29/20 |
| 23 | 673 | 460 | 28/19, 31/21, 34/23 |
| 24 | 702.3 | 480 | 3/2 |
| 25 | 731.5 | 500 | 26/17, 29/19, 32/21 |
| 26 | 760.8 | 520 | 31/20 |
| 27 | 790 | 540 | 30/19 |
| 28 | 819.3 | 560 | |
| 29 | 848.6 | 580 | 18/11, 31/19 |
| 30 | 877.8 | 600 | |
| 31 | 907.1 | 620 | 22/13, 27/16 |
| 32 | 936.3 | 640 | 12/7 |
| 33 | 965.6 | 660 | 7/4 |
| 34 | 994.9 | 680 | 16/9 |
| 35 | 1024.1 | 700 | |
| 36 | 1053.4 | 720 | 11/6 |
| 37 | 1082.7 | 740 | 28/15 |
| 38 | 1111.9 | 760 | 19/10 |
| 39 | 1141.2 | 780 | 29/15, 31/16 |
| 40 | 1170.4 | 800 | |
| 41 | 1199.7 | 820 | 2/1 |
| 42 | 1229 | 840 | |
| 43 | 1258.2 | 860 | 29/14, 31/15 |
| 44 | 1287.5 | 880 | 21/10 |
| 45 | 1316.7 | 900 | 15/7 |
| 46 | 1346 | 920 | |
| 47 | 1375.3 | 940 | 31/14 |
| 48 | 1404.5 | 960 | 9/4 |
| 49 | 1433.8 | 980 | 16/7 |
| 50 | 1463 | 1000 | 7/3 |
| 51 | 1492.3 | 1020 | 26/11 |
| 52 | 1521.6 | 1040 | |
| 53 | 1550.8 | 1060 | 22/9, 27/11 |
| 54 | 1580.1 | 1080 | |
| 55 | 1609.3 | 1100 | 33/13 |
| 56 | 1638.6 | 1120 | 18/7 |
| 57 | 1667.9 | 1140 | 21/8, 34/13 |
| 58 | 1697.1 | 1160 | 8/3 |
| 59 | 1726.4 | 1180 | 19/7 |
| 60 | 1755.7 | 1200 | |
| 61 | 1784.9 | 1220 | 14/5 |
| 62 | 1814.2 | 1240 | 20/7 |
| 63 | 1843.4 | 1260 | 29/10 |
| 64 | 1872.7 | 1280 | |
| 65 | 1902 | 1300 | 3/1 |