60edf
| ← 59edf | 60edf | 61edf → |
60 equal divisions of the perfect fifth (abbreviated 60edf or 60ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 60 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of (3/2)1/60, or the 60th root of 3/2.
Theory
60edf can be thought of as a very octave stretched version of 103edo, or a very compressed version of 102edo, but it actually inherits few properties from either.
It makes available dual versions of primes 2 and 3 from both systems. Yet its mappings of primes 5, 7, 11, 13 and up are actually all different from either of those edos. For example mapping prime 5 to the 238th step (not 237 as in 102edo, nor 239 as in 103edo).
60edf is very similar to 205ed4.
Harmonics
60edf's approximations of primes are strange. Because of its small step size, it's difficult not to hear primes 2, 3, or even 13, even though they have a lot of relative error.
60edf is much more accurate on higher primes than on smaller primes. It approximates all primes from 17 through 31 with less than 29% relative error, but has over 43% rel. err. on 2, 3 and 13.
So perhaps a reasonable - if clunky - way to interpret 60edf, is as a dual-2, dual-3, dual-13 31-limit tuning. Extending it to the 37-limit could also be an option.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.02 | +5.02 | -1.89 | +0.56 | +1.92 | +5.19 | -2.97 | +3.36 | +0.18 | -3.35 | -1.82 | -3.94 | +5.53 |
| Relative (%) | +42.9 | +42.9 | -16.2 | +4.8 | +16.4 | +44.3 | -25.4 | +28.7 | +1.5 | -28.6 | -15.5 | -33.7 | +47.2 | |
| Steps (reduced) |
103 (43) |
163 (43) |
238 (58) |
288 (48) |
355 (55) |
380 (20) |
419 (59) |
436 (16) |
464 (44) |
498 (18) |
508 (28) |
534 (54) |
550 (10) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.02 | +5.02 | -1.65 | -1.89 | -1.65 | +0.56 | +3.37 | -1.65 | +3.13 | +1.92 | +3.37 |
| Relative (%) | +42.9 | +42.9 | -14.1 | -16.2 | -14.1 | +4.8 | +28.8 | -14.1 | +26.8 | +16.4 | +28.8 | |
| Steps (reduced) |
103 (43) |
163 (43) |
205 (25) |
238 (58) |
265 (25) |
288 (48) |
308 (8) |
325 (25) |
341 (41) |
355 (55) |
368 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.19 | +5.58 | +3.13 | -3.31 | -2.97 | +3.37 | +3.36 | -3.55 | +5.58 | -4.76 | +0.18 | -3.31 |
| Relative (%) | +44.3 | +47.7 | +26.8 | -28.3 | -25.4 | +28.8 | +28.7 | -30.3 | +47.7 | -40.7 | +1.5 | -28.3 | |
| Steps (reduced) |
380 (20) |
391 (31) |
401 (41) |
410 (50) |
419 (59) |
428 (8) |
436 (16) |
443 (23) |
451 (31) |
457 (37) |
464 (44) |
470 (50) | |
Subsets and supersets
As a highly composite edf, 60edf includes as subsets: 2edf, 3edf, 5edf, 6edf, 9edf, 10edf, 12edf, 15edf, 20edf and 30edf.
This makes it potentially a good polymicrotonal system for using multiple edfs (or their stretched/compressed relative edos/ed4s) simultaneously.
The relative edos/ed4s of its subsets are: 7ed4, 5edo, 9edo/17ed4, 21ed4, 31ed4, 17edo, 21edo/41ed4, 26edo, 34edo and 51edo/103ed4.
The simplest supersets of 60edf are 120edf and 180edf.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 11.7 | |
| 2 | 23.4 | |
| 3 | 35.1 | |
| 4 | 46.8 | 35/34 |
| 5 | 58.5 | 28/27 |
| 6 | 70.2 | |
| 7 | 81.9 | 21/20, 22/21 |
| 8 | 93.6 | |
| 9 | 105.3 | |
| 10 | 117 | 15/14, 31/29 |
| 11 | 128.7 | 14/13 |
| 12 | 140.4 | 25/23 |
| 13 | 152.1 | |
| 14 | 163.8 | 11/10, 34/31 |
| 15 | 175.5 | 21/19 |
| 16 | 187.2 | |
| 17 | 198.9 | |
| 18 | 210.6 | 35/31 |
| 19 | 222.3 | |
| 20 | 234 | |
| 21 | 245.7 | 15/13 |
| 22 | 257.4 | 22/19, 29/25 |
| 23 | 269.1 | |
| 24 | 280.8 | 33/28 |
| 25 | 292.5 | 13/11 |
| 26 | 304.2 | |
| 27 | 315.9 | |
| 28 | 327.6 | 23/19, 35/29 |
| 29 | 339.3 | |
| 30 | 351 | |
| 31 | 362.7 | |
| 32 | 374.4 | 31/25 |
| 33 | 386.1 | |
| 34 | 397.8 | 29/23 |
| 35 | 409.5 | 19/15, 33/26 |
| 36 | 421.2 | 14/11 |
| 37 | 432.9 | |
| 38 | 444.6 | |
| 39 | 456.3 | 13/10 |
| 40 | 468 | |
| 41 | 479.7 | |
| 42 | 491.4 | |
| 43 | 503.1 | |
| 44 | 514.8 | 31/23 |
| 45 | 526.5 | 19/14, 23/17 |
| 46 | 538.2 | 15/11 |
| 47 | 549.9 | |
| 48 | 561.6 | |
| 49 | 573.3 | |
| 50 | 585 | 7/5 |
| 51 | 596.7 | |
| 52 | 608.4 | |
| 53 | 620.1 | 10/7 |
| 54 | 631.8 | |
| 55 | 643.5 | |
| 56 | 655.2 | 19/13 |
| 57 | 666.9 | 22/15, 25/17 |
| 58 | 678.6 | 34/23 |
| 59 | 690.3 | |
| 60 | 702 | 3/2 |
Instruments
A Lumatone mapping for 60edf is now available.
Music
- 60ed(3/2) improv (2025)