21edt
| ← 20edt | 21edt | 22edt → |
21 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 21edt or 21ed3), is a nonoctave tuning system that divides the interval of 3/1 into 21 equal parts of about 90.6 ¢ each. Each step represents a frequency ratio of 31/21, or the 21st root of 3.
Intervals
| Degrees | Cents | Hekts | Approximate Ratio |
|---|---|---|---|
| 0 | 1/1 | ||
| 1 | 90.569 | 61.905 | 21/20, 135/128 |
| 2 | 181.139 | 123.81 | 10/9 |
| 3 | 271.708 | 185.714 | 7/6 |
| 4 | 362.277 | 247.619 | 16/13 |
| 5 | 452.846 | 309.524 | 13/10 |
| 6 | 543.416 | 371.429 | 15/11, 11/8 |
| 7 | 633.985 | 433.333 | 13/9 |
| 8 | 724.554 | 495.238 | 35/23 |
| 9 | 815.124 | 557.143 | 8/5 |
| 10 | 905.693 | 619.048 | 27/16 |
| 11 | 996.262 | 680.952 | 16/9 |
| 12 | 1086.831 | 742.857 | 15/8 |
| 13 | 1177.401 | 804.762 | 69/35 |
| 14 | 1267.97 | 866.667 | 27/13 |
| 15 | 1358.539 | 928.571 | 11/5 (11/10 plus an octave), 24/11 (12/11 plus an octave) |
| 16 | 1449.109 | 990.476 | 30/13 (15/13 plus an octave) |
| 17 | 1539.678 | 1052.381 | 39/16 |
| 18 | 1630.247 | 1114.286 | 18/7 (9/7 plus an octave) |
| 19 | 1720.816 | 1076.19 | 27/10 |
| 20 | 1811.386 | 1238.095 | 20/7, 128/45 |
| 21 | 1901.955 | 1300 | 3/1 |
21edt contains 6 intervals from 7edt and 2 intervals from 3edt, meaning that it introduces 12 new intervals not available in lower edt's. These new intervals allow for construction of strange chords like 9:10:13:16:22:27:30...
21edt contains a 7L7s MOS similar to Whitewood, which I call Ivory. It has a period of 1/7 of the tritave and the generator is one step. The major scale is LsLsLsLsLsLsLs, and the minor scale is sLsLsLsLsLsLsL.
21edt also contains a 4L5s MOS similar to BP, with a 4:1 ratio of large to small; quite exaggerated from the optimal 2:1. Although the 7/3 is a little off, the 4L+5s BP scale is pretty. However, one of the star scales in 21edt is the 3L+6s (ssLssLssL and modes thereof) which is very harmonically rich, the cornerstone of which is the approximate 9:13:19 chord (which is just the 3edt essentially tempered chord).
Harmonics
Not the best approximations but all within 20 cents: it has 5th (+20¢), 7th (−16¢), 10th (+2¢), 11th (+15¢), 13th (−3¢), 17th (−14¢), 23rd (+6¢), and 37th (−2¢) harmonics. For a lower division of the tritave that's quite a constellation! The chord is a little out of tune but it works, you can really sink into it.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -22.6 | +0.0 | -45.2 | +21.3 | -22.6 | -17.8 | +22.8 | +0.0 | -1.3 | +14.9 | -45.2 |
| Relative (%) | -25.0 | +0.0 | -49.9 | +23.6 | -25.0 | -19.6 | +25.1 | +0.0 | -1.4 | +16.4 | -49.9 | |
| Steps (reduced) |
13 (13) |
21 (0) |
26 (5) |
31 (10) |
34 (13) |
37 (16) |
40 (19) |
42 (0) |
44 (2) |
46 (4) |
47 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | -40.4 | +21.3 | +0.2 | -14.2 | -22.6 | -25.6 | -23.9 | -17.8 | -7.7 | +5.9 |
| Relative (%) | -2.9 | -44.6 | +23.6 | +0.2 | -15.7 | -25.0 | -28.3 | -26.3 | -19.6 | -8.5 | +6.5 | |
| Steps (reduced) |
49 (7) |
50 (8) |
52 (10) |
53 (11) |
54 (12) |
55 (13) |
56 (14) |
57 (15) |
58 (16) |
59 (17) |
60 (18) | |