34ed7: Difference between revisions
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Revision as of 23:35, 10 April 2025
| This page is nominated for deletion. Reason: 12edo octave stretching is addressed and should be addressed in "octave stretch" section of the 12edo page. Neighboring ed7s do not have articles One of the operators will take care of it shortly. |
| ← 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.