35edf
| ← 34edf | 35edf | 36edf → |
35 equal divisions of the perfect fifth (abbreviated 35edf or 35ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 35 equal parts of about 20.1 ¢ each. Each step represents a frequency ratio of (3/2)1/35, or the 35th root of 3/2.
Theory
35edf corresponds to 59.8329…edo and is practically identical to every sixth step of 359edo. It is related to 60edo, but with the perfect fifth rather than the octave being just. The octave is stretched by about 3.35 cents.
The patent val has a generally sharp tendency for prime harmonics up to 17, with the exception for 13. Unlike 60edo, which is consistent to the 10-integer-limit, 35edf is only consistent up to the 7-integer-limit, with discrepancy for the 8th harmonic (three octaves).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.35 | +3.35 | +6.70 | +1.45 | +6.70 | +0.56 | -10.00 | +6.70 | +4.80 | +0.24 | -10.00 |
| Relative (%) | +16.7 | +16.7 | +33.4 | +7.2 | +33.4 | +2.8 | -49.9 | +33.4 | +23.9 | +1.2 | -49.9 | |
| Steps (reduced) |
60 (25) |
95 (25) |
120 (15) |
139 (34) |
155 (15) |
168 (28) |
179 (4) |
190 (15) |
199 (24) |
207 (32) |
214 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8.18 | +3.91 | +4.80 | -6.65 | +8.73 | -10.00 | -3.33 | +8.15 | +3.91 | +3.60 | +6.86 | -6.65 |
| Relative (%) | -40.8 | +19.5 | +23.9 | -33.2 | +43.5 | -49.9 | -16.6 | +40.7 | +19.5 | +17.9 | +34.2 | -33.2 | |
| Steps (reduced) |
221 (11) |
228 (18) |
234 (24) |
239 (29) |
245 (0) |
249 (4) |
254 (9) |
259 (14) |
263 (18) |
267 (22) |
271 (26) |
274 (29) | |
Subsets and supersets
Since 35 factors into primes as 5 × 7, 35edf has subset edfs 5edf and 7edf.
Intervals
| # | Cents | Approximate ratios in the 2.3.5.13 subgroup |
Additional ratios of 7 and 11 (assuming flat values for primes) |
|---|---|---|---|
| 0 | 0.0 | ||
| 1 | 20.1 | 81/80 | |
| 2 | 40.1 | ||
| 3 | 60.2 | 28/27, 27/26 | |
| 4 | 80.2 | 21/20 | |
| 5 | 100.3 | ||
| 6 | 120.3 | 16/15 | |
| 7 | 140.4 | ||
| 8 | 160.4 | 12/11, 11/10 | |
| 9 | 180.5 | 10/9 | |
| 10 | 200.6 | 9/8 | |
| 11 | 220.6 | ||
| 12 | 240.7 | 15/13 | 8/7 |
| 13 | 260.8 | 7/6 | |
| 14 | 280.8 | ||
| 15 | 300.8 | ||
| 16 | 320.9 | 6/5 | |
| 17 | 340.9 | 11/9 | |
| 18 | 361.0 | 16/13 | |
| 19 | 381.1 | 5/4 | |
| 20 | 401.1 | ||
| 21 | 421.2 | 14/11 | |
| 22 | 441.2 | 9/7 | |
| 23 | 461.3 | 13/10 | |
| 24 | 481.3 | ||
| 25 | 501.4 | 4/3 | |
| 26 | 521.5 | ||
| 27 | 541.5 | 11/8, 15/11 | |
| 28 | 561.6 | 18/13 | |
| 29 | 581.6 | 7/5 | |
| 30 | 601.7 | ||
| 31 | 621.7 | 10/7 | |
| 32 | 641.8 | 13/9 | |
| 33 | 661.8 | 16/11, 22/15 | |
| 34 | 681.9 | ||
| 35 | 702.0 | 3/2 | |
| 36 | 722.0 | ||
| 37 | 742.1 | 20/13 | |
| 38 | 762.1 | 14/9 | |
| 39 | 782.2 | 11/7 | |
| 40 | 802.2 | ||
| 41 | 822.3 | 8/5 | |
| 42 | 842.3 | 13/8 | |
| 43 | 862.4 | 18/11 | |
| 44 | 882.5 | 5/3 | |
| 45 | 902.5 | ||
| 46 | 922.6 | ||
| 47 | 942.6 | 12/7 | |
| 48 | 962.7 | 26/15 | 7/4 |
| 49 | 982.7 | ||
| 50 | 1002.8 | 16/9 | |
| 51 | 1022.8 | 9/5 | |
| 52 | 1042.9 | 11/6, 20/11 | |
| 53 | 1063.0 | ||
| 54 | 1083.0 | 15/8 | |
| 55 | 1103.1 | ||
| 56 | 1123.1 | ||
| 57 | 1143.2 | ||
| 58 | 1163.2 | ||
| 59 | 1183.3 | ||
| 60 | 1203.4 | ||
| 61 | 1223.4 | 81/40 | |
| 62 | 1243.5 | ||
| 63 | 1263.5 | 56/27, 27/13 | |
| 64 | 1283.6 | 21/10 | |
| 65 | 1303.6 | ||
| 66 | 1323.7 | 32/15 | |
| 67 | 1343.7 | ||
| 68 | 1363.8 | 24/11, 11/5 | |
| 69 | 1383.9 | 20/9 | |
| 70 | 1403.9 | 9/4 |