34edf
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| ← 33edf | 34edf | 35edf → |
34 equal divisions of the perfect fifth (abbreviated 34edf or 34ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 34 equal parts of about 20.6 ¢ each. Each step represents a frequency ratio of (3/2)1/34, or the 34th root of 3/2.
Theory
34edf corresponds to 58.1234…edo. It is related to 58edo, but with the 3/2 rather than the 2/1 being just. The octave is compressed by about 2.5474 cents.
The patent val has a generally flat tendency for harmonics up to 16 (four octaves), with the exception for 5. Unlike 58edo, it is only consistent up to the 15-integer-limit, with discrepancy for the 16th harmonic.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.55 | -2.55 | -5.09 | +0.86 | -5.09 | -3.57 | -7.64 | -5.09 | -1.69 | -1.53 | -7.64 |
| Relative (%) | -12.3 | -12.3 | -24.7 | +4.2 | -24.7 | -17.3 | -37.0 | -24.7 | -8.2 | -7.4 | -37.0 | |
| Steps (reduced) |
58 (24) |
92 (24) |
116 (14) |
135 (33) |
150 (14) |
163 (27) |
174 (4) |
184 (14) |
193 (23) |
201 (31) |
208 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.69 | -6.12 | -1.69 | -10.19 | +8.73 | -7.64 | +1.98 | -4.23 | -6.12 | -4.07 | +1.55 | -10.19 |
| Relative (%) | -8.2 | -29.6 | -8.2 | -49.4 | +42.3 | -37.0 | +9.6 | -20.5 | -29.6 | -19.7 | +7.5 | -49.4 | |
| Steps (reduced) |
215 (11) |
221 (17) |
227 (23) |
232 (28) |
238 (0) |
242 (4) |
247 (9) |
251 (13) |
255 (17) |
259 (21) |
263 (25) |
266 (28) | |
Subsets and supersets
Since 34 factors into primes as 2 × 17, 34edf contains 2edf and 17edf as subset edfs.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 20.6 | 56/55, 64/63, 81/80, 91/90, 105/104 |
| 2 | 41.3 | 36/35, 40/39, 45/44, 49/48, 50/49, 55/54 |
| 3 | 61.9 | 26/25, 27/26, 28/27, 33/32 |
| 4 | 82.6 | 21/20, 22/21, 25/24 |
| 5 | 103.2 | 16/15, 17/16, 18/17 |
| 6 | 123.9 | 14/13, 15/14 |
| 7· | 144.5 | 12/11, 13/12 |
| 8 | 165.2 | 11/10 |
| 9 | 185.8 | 10/9 |
| 10 | 206.5 | 9/8 |
| 11 | 227.1 | 8/7 |
| 12· | 248.7 | 15/13 |
| 13 | 268.4 | 7/6 |
| 14 | 289.0 | 13/11, 20/17 |
| 15 | 309.7 | 6/5 |
| 16 | 330.3 | 17/14, 40/33 |
| 17· | 351.0 | 11/9, 16/13 |
| 18 | 371.6 | 21/17, 26/21 |
| 19 | 392.3 | 5/4 |
| 20 | 412.9 | 14/11 |
| 21 | 433.6 | 9/7 |
| 22· | 455.2 | 13/10, 17/13, 22/17 |
| 23 | 474.9 | 21/16 |
| 24 | 495.5 | 4/3 |
| 25 | 516.1 | 27/20 |
| 26 | 536.8 | 15/11 |
| 27 | 557.4 | 11/8, 18/13 |
| 28 | 578.1 | 7/5 |
| 29 | 598.7 | 17/12, 24/17 |
| 30 | 619.4 | 10/7 |
| 31 | 640.0 | 13/9, 16/11 |
| 32 | 660.7 | 22/15 |
| 33 | 681.3 | 40/27 |
| 34 | 702.0 | 3/2 |
| 35 | 722.6 | 32/21 |
| 36 | 743.2 | 17/11, 20/13, 26/17 |
| 37 | 763.9 | 14/9 |
| 38 | 784.5 | 11/7 |
| 39 | 805.2 | 8/5 |
| 40 | 825.8 | 21/13, 34/21 |
| 41 | 846.5 | 13/8, 18/11 |
| 42 | 867.1 | 28/17, 33/20 |
| 43 | 887.8 | 5/3 |
| 44 | 908.4 | 17/10, 22/13 |
| 45 | 929.1 | 12/7 |
| 46 | 949.7 | 26/15 |
| 47 | 970.3 | 7/4 |
| 48 | 991.0 | 16/9 |
| 49 | 1011.7 | 9/5 |
| 50 | 1032.3 | 20/11 |
| 51 | 1052.9 | 11/6 |
| 52 | 1073.6 | 13/7 |
| 53 | 1094.2 | 15/8, 17/9 |
| 54 | 1114.9 | 21/11 |
| 55 | 1135.5 | 25/13, 27/14 |
| 56 | 1156.2 | 35/18, 39/20, 49/25 |
| 57 | 1176.8 | 55/28, 63/32 |
| 58 | 1197.5 | 2/1 |
| 59 | 1218.1 | 81/40, 91/45, 105/52 |
| 60 | 1238.7 | 45/22, 49/24, 55/27 |
| 61 | 1259.4 | 27/13, 33/16 |
| 62 | 1280.0 | 21/10, 25/12 |
| 63 | 1300.7 | 17/8 |
| 64 | 1321.3 | 15/7 |
| 65 | 1342.0 | 13/6 |
| 66 | 1362.6 | 11/5 |
| 67 | 1383.4 | 20/9 |
| 68 | 1403.9 | 9/4 |