34ed7
| ← 33ed7 | 34ed7 | 35ed7 → |
34 equal divisions of the 7th harmonic (abbreviated 34ed7) is a nonoctave tuning system that divides the interval of 7/1 into 34 equal parts of about 99.1 ¢ each. Each step represents a frequency ratio of 71/34, or the 34th root of 7.
Theory
34ed7 is related to 12edo, but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents. It is consistent to the 11-integer-limit, but not to the 12-integer-limit. In comparison, 12edo is only consistent up to the 10-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.0 | -19.4 | -22.0 | -12.0 | -30.4 | +0.0 | -33.0 | -38.8 | -23.0 | +10.2 | -41.4 |
| Relative (%) | -11.1 | -19.6 | -22.2 | -12.1 | -30.7 | +0.0 | -33.3 | -39.1 | -23.2 | +10.3 | -41.8 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (31) |
34 (0) |
36 (2) |
38 (4) |
40 (6) |
42 (8) |
43 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.2 | -11.0 | -31.4 | -44.0 | +49.2 | +49.3 | -44.3 | -34.0 | -19.4 | -0.8 | +21.3 | +46.7 |
| Relative (%) | +18.4 | -11.1 | -31.7 | -44.4 | +49.7 | +49.8 | -44.7 | -34.3 | -19.6 | -0.8 | +21.5 | +47.1 | |
| Steps (reduced) |
45 (11) |
46 (12) |
47 (13) |
48 (14) |
50 (16) |
51 (17) |
51 (17) |
52 (18) |
53 (19) |
54 (20) |
55 (21) |
56 (22) | |
Subsets and supersets
Since 34 factors into 2 × 17, 34ed7 contains 2ed7 and 17ed7 as subsets.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 99.1 | 21/20 |
| 2 | 198.2 | 9/8 |
| 3 | 297.2 | 6/5 |
| 4 | 396.3 | 5/4 |
| 5 | 495.4 | 4/3 |
| 6 | 594.5 | 7/5 |
| 7 | 693.6 | 3/2 |
| 8 | 792.7 | 8/5 |
| 9 | 891.7 | 5/3 |
| 10 | 990.8 | 7/4 |
| 11 | 1089.9 | 15/8 |
| 12 | 1189.0 | 2/1 |
| 13 | 1288.1 | 21/10 |
| 14 | 1387.2 | 9/4 |
| 15 | 1486.2 | 7/3 |
| 16 | 1585.3 | 5/2 |
| 17 | 1684.4 | 8/3 |
| 18 | 1783.5 | 14/5 |
| 19 | 1882.6 | 3/1 |
| 20 | 1981.7 | 22/7 |
| 21 | 2080.7 | 10/3 |
| 22 | 2179.8 | 7/2 |
| 23 | 2278.9 | 15/4 |
| 24 | 2378.0 | 4/1 |
| 25 | 2477.1 | 21/5 |
| 26 | 2576.2 | 9/2 |
| 27 | 2675.2 | 14/3 |
| 28 | 2774.3 | 5/1 |
| 29 | 2873.4 | 16/3 |
| 30 | 2972.5 | 28/5 |
| 31 | 3071.6 | 6/1 |
| 32 | 3170.7 | 25/4 |
| 33 | 3269.7 | 20/3 |
| 34 | 3368.8 | 7/1 |
Regular temperaments
34ed7 can also be thought of as a generator of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a cluster temperament with 12 clusters of notes in an octave (quintupole temperament). This temperament is supported by 12edo, 109edo, and 121edo among others.