32ed12
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| ← 31ed12 | 32ed12 | 33ed12 → |
32 equal divisions of the 12th harmonic (abbreviated 32ed12) is a nonoctave tuning system that divides the interval of 12/1 into 32 equal parts of about 134 ¢ each. Each step represents a frequency ratio of 121/32, or the 32nd root of 12.
Theory
32ed12 is similar to 9edo, but has the 12th harmonic tuned just instead of the octave, which stretches the octave by about 9.9 ¢. It also approximates Pelog tunings in Indonesian gamelan music very well, since Pelog is well-approximated by 9edo with stretched octaves.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.9 | -19.8 | +19.8 | +36.8 | -9.9 | -7.9 | +29.8 | -39.7 | +46.8 | +16.2 | +0.0 |
| Relative (%) | +7.4 | -14.8 | +14.8 | +27.4 | -7.4 | -5.9 | +22.1 | -29.5 | +34.8 | +12.1 | +0.0 | |
| Steps (reduced) |
9 (9) |
14 (14) |
18 (18) |
21 (21) |
23 (23) |
25 (25) |
27 (27) |
28 (28) |
30 (30) |
31 (31) |
32 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.1 | +2.0 | +17.0 | +39.7 | -65.3 | -29.8 | +11.1 | +56.7 | -27.8 | +26.1 | -50.8 | +9.9 |
| Relative (%) | -3.1 | +1.5 | +12.6 | +29.5 | -48.5 | -22.1 | +8.2 | +42.2 | -20.7 | +19.4 | -37.8 | +7.4 | |
| Steps (reduced) |
33 (1) |
34 (2) |
35 (3) |
36 (4) |
36 (4) |
37 (5) |
38 (6) |
39 (7) |
39 (7) |
40 (8) |
40 (8) |
41 (9) | |
Subsets and supersets
Since 32 factors into primes as 25, 32ed6 contains subset ed6's 2, 4, 8, and 16.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 134.4 | 13/12, 14/13 |
| 2 | 268.9 | 7/6, 27/23 |
| 3 | 403.3 | 19/15, 24/19 |
| 4 | 537.7 | 15/11, 19/14, 26/19 |
| 5 | 672.2 | 22/15, 28/19 |
| 6 | 806.6 | 8/5, 27/17 |
| 7 | 941.1 | 12/7, 19/11 |
| 8 | 1075.5 | 13/7, 28/15 |
| 9 | 1209.9 | |
| 10 | 1344.4 | 13/6, 24/11 |
| 11 | 1478.8 | |
| 12 | 1613.2 | 28/11 |
| 13 | 1747.7 | 11/4 |
| 14 | 1882.1 | |
| 15 | 2016.5 | 16/5 |
| 16 | 2151 | |
| 17 | 2285.4 | 15/4 |
| 18 | 2419.8 | |
| 19 | 2554.3 | |
| 20 | 2688.7 | 19/4 |
| 21 | 2823.2 | |
| 22 | 2957.6 | 11/2 |
| 23 | 3092 | |
| 24 | 3226.5 | |
| 25 | 3360.9 | 7/1 |
| 26 | 3495.3 | 15/2 |
| 27 | 3629.8 | |
| 28 | 3764.2 | |
| 29 | 3898.6 | 19/2 |
| 30 | 4033.1 | |
| 31 | 4167.5 | |
| 32 | 4302 | 12/1 |