60edf: Difference between revisions
Create the page for 60edf in a rush because music now exists for it |
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=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|60|3|2|intervals=integer|columns=11}} | 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]]. | ||
{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of | |||
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-n|dual]]-2, dual-3, dual-13 [[31-limit]] tuning. Extending it to the [[37-limit]] could also be an option. | |||
{{Harmonics in equal|60|3|2|intervals=prime|columns=13|title=Approximation of primes in 60edf (continued)}} | |||
{{Harmonics in equal|60|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of integers in 60edf (continued)}} | |||
{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of integers in 60edf (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Revision as of 22:43, 17 October 2025
| ← 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
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Todo: complete section |
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
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Todo: complete section |
Music
- 60ed(3/2) improv (2025)
See also
- 103edo – relative edo
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Todo: complete section |
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