31edf: Difference between revisions
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== Theory == | == Theory == | ||
31edf is almost identical to [[53edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1|7]], the [[5/1|5]], [[11/1|11]], [[13/1|13]], and [[19/1|19]] remain flat but significantly less so than in 53edo. | 31edf is almost identical to [[53edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1]] being [[just]]. The octave is [[stretched and compressed tuning|stretched]] by about 0.117 [[cents]]. Like 53edo, 31edf is [[consistent]] to the [[integer limit|10-integer-limit]]. While the [[3-limit]] part is tuned sharp plus a sharper [[7/1|7]], the [[5/1|5]], [[11/1|11]], [[13/1|13]], and [[19/1|19]] remain flat but significantly less so than in 53edo. | ||
The [[The Riemann zeta function and tuning|local zeta peak]] around 53 is located at 52.996829, which has the octave stretched by 0.0718{{c}}; the octave of 31edf comes extremely close (differing by only {{sfrac|1|22}}{{c}}), thus minimizing relative error as much as possible. | |||
=== Harmonics === | === Harmonics === | ||
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* [[84edt]] – relative edt | * [[84edt]] – relative edt | ||
* [[137ed6]] – relative ed6 | * [[137ed6]] – relative ed6 | ||
[[Category:53edo]] | |||
[[Category:Zeta-optimized tunings]] | |||
Latest revision as of 13:50, 18 June 2025
| ← 30edf | 31edf | 32edf → |
(convergent)
(convergent)
31 equal divisions of the perfect fifth (abbreviated 31edf or 31ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 31 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of (3/2)1/31, or the 31st root of 3/2.
Theory
31edf is almost identical to 53edo, but with the perfect fifth rather than the 2/1 being just. The octave is stretched by about 0.117 cents. Like 53edo, 31edf is consistent to the 10-integer-limit. While the 3-limit part is tuned sharp plus a sharper 7, the 5, 11, 13, and 19 remain flat but significantly less so than in 53edo.
The local zeta peak around 53 is located at 52.996829, which has the octave stretched by 0.0718 ¢; the octave of 31edf comes extremely close (differing by only 1/22 ¢), thus minimizing relative error as much as possible.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.12 | +0.12 | +0.23 | -1.14 | +0.23 | +5.09 | +0.35 | +0.23 | -1.02 | -7.52 | +0.35 |
| Relative (%) | +0.5 | +0.5 | +1.0 | -5.0 | +1.0 | +22.5 | +1.5 | +1.0 | -4.5 | -33.2 | +1.5 | |
| Steps (reduced) |
53 (22) |
84 (22) |
106 (13) |
123 (30) |
137 (13) |
149 (25) |
159 (4) |
168 (13) |
176 (21) |
183 (28) |
190 (4) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.36 | +5.20 | -1.02 | +0.47 | +8.73 | +0.35 | -2.68 | -0.90 | +5.20 | -7.40 | +6.22 | +0.47 |
| Relative (%) | -10.4 | +23.0 | -4.5 | +2.1 | +38.6 | +1.5 | -11.8 | -4.0 | +23.0 | -32.7 | +27.5 | +2.1 | |
| Steps (reduced) |
196 (10) |
202 (16) |
207 (21) |
212 (26) |
217 (0) |
221 (4) |
225 (8) |
229 (12) |
233 (16) |
236 (19) |
240 (23) |
243 (26) | |
Subsets and supersets
31edf is the 11th prime edf, following 29edf and coming before 37edf. It does not contain any nontrivial subset edfs.