31920edo: Difference between revisions
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{{Infobox ET|Consistency=41|Distinct consistency=41}} | |||
{{ED intro}} | |||
31920edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}. | |||
It is also the smallest multiple of [[12edo]] to be [[purely consistent]] in the 31-odd-limit (i.e. all odd harmonics up to and including 31 are approximated with no greater than 25% relative error). | |||
The simplest of the commas under the 43-limit it tempers out are 47916/47915, 52480/52479, 58311/58310, 60516/60515, 67600/67599, 68783/68782, 72501/72500, 75141/75140, 76875/76874, 81549/81548, 81796/81795, 82944/82943, 88320/88319, 93093/93092, 93500/93499, 96876/96875 and 98736/98735. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|31920|prec=5|intervals=prime|columns=9}} | |||
{{Harmonics in equal|31920|prec=5|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31920edo (continued)}} | |||
=== Subsets and supersets === | |||
31920 is a very composite number, with many divisors: 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, and 15960. These facts make it a good candidate for an [[interval size measure]], and one step of it may be called an [[imp]], so that the cent is 26.6 imps, and a [[12edo]] semitone is 2660 imps. A single step of [[15edo]] is 2128 imps, of [[19edo]] 1680 imps, of [[84edo]] 380 imps, of [[140edo]] 228 imps, of [[152edo]] 210 imps, of [[190edo]] 168 imps, and of 665edo 48 imps. | |||
Latest revision as of 14:05, 30 July 2025
| ← 31919edo | 31920edo | 31921edo → |
31920 equal divisions of the octave (abbreviated 31920edo or 31920ed2), also called 31920-tone equal temperament (31920tet) or 31920 equal temperament (31920et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 31920 equal parts of about 0.0376 ¢ each. Each step represents a frequency ratio of 21/31920, or the 31920th root of 2.
31920edo is distinctly consistent through the 41-odd-limit, with a smaller 41-limit relative error than any smaller distinctly consistent division. Its 3rd harmonic derives from 665edo. It is also enfactored in the 5-limit, with the same tuning as 15960edo, which is an atomic tuning, tempering out Kirnberger's atom, [161 -84 -12⟩.
It is also the smallest multiple of 12edo to be purely consistent in the 31-odd-limit (i.e. all odd harmonics up to and including 31 are approximated with no greater than 25% relative error).
The simplest of the commas under the 43-limit it tempers out are 47916/47915, 52480/52479, 58311/58310, 60516/60515, 67600/67599, 68783/68782, 72501/72500, 75141/75140, 76875/76874, 81549/81548, 81796/81795, 82944/82943, 88320/88319, 93093/93092, 93500/93499, 96876/96875 and 98736/98735.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | -0.00011 | +0.00208 | +0.00868 | -0.00215 | -0.00135 | +0.00700 | +0.00578 | -0.00367 |
| Relative (%) | +0.0 | -0.3 | +5.5 | +23.1 | -5.7 | -3.6 | +18.6 | +15.4 | -9.8 | |
| Steps (reduced) |
31920 (0) |
50592 (18672) |
74116 (10276) |
89611 (25771) |
110425 (14665) |
118118 (22358) |
130472 (2792) |
135594 (7914) |
144392 (16712) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00927 | +0.00202 | +0.00934 | -0.00226 | -0.01395 | -0.01790 | -0.00830 | +0.00127 | -0.00511 |
| Relative (%) | +24.7 | +5.4 | +24.9 | -6.0 | -37.1 | -47.6 | -22.1 | +3.4 | -13.6 | |
| Steps (reduced) |
155067 (27387) |
158138 (30458) |
166286 (6686) |
171013 (11413) |
173206 (13606) |
177302 (17702) |
182835 (23235) |
187774 (28174) |
189309 (29709) | |
Subsets and supersets
31920 is a very composite number, with many divisors: 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, and 15960. These facts make it a good candidate for an interval size measure, and one step of it may be called an imp, so that the cent is 26.6 imps, and a 12edo semitone is 2660 imps. A single step of 15edo is 2128 imps, of 19edo 1680 imps, of 84edo 380 imps, of 140edo 228 imps, of 152edo 210 imps, of 190edo 168 imps, and of 665edo 48 imps.