2547047edo: Difference between revisions
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{{Mathematical interest}} | |||
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Despite being less practical than many smaller [[ | Despite being less practical than many smaller [[edo]]s, it is a strong higher-limit system, especially in the [[35-odd-limit]] and [[odd prime sum limit|36-OPSL]]. An interesting quirk, though, is that all prime harmonics up to 41 are tuned sharp except for 19 which is only slightly flat. Also, the only two prime factors of the number of notes per octave appear to be unusually close together. | ||
{{Harmonics in equal|steps=2547047|columns= | {{Harmonics in equal|steps=2547047|columns=9}} | ||
{{Harmonics in equal|steps=2547047|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2547047edo (continued)}} | |||
Latest revision as of 22:07, 10 August 2025
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 2547046edo | 2547047edo | 2547048edo → |
2547047 equal divisions of the octave (abbreviated 2547047edo or 2547047ed2), also called 2547047-tone equal temperament (2547047tet) or 2547047 equal temperament (2547047et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2547047 equal parts of about 0.000471 ¢ each. Each step represents a frequency ratio of 21/2547047, or the 2547047th root of 2.
Despite being less practical than many smaller edos, it is a strong higher-limit system, especially in the 35-odd-limit and 36-OPSL. An interesting quirk, though, is that all prime harmonics up to 41 are tuned sharp except for 19 which is only slightly flat. Also, the only two prime factors of the number of notes per octave appear to be unusually close together.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000000 | +0.000008 | +0.000006 | +0.000032 | +0.000035 | +0.000055 | +0.000015 | -0.000014 | +0.000043 |
| Relative (%) | +0.0 | +1.7 | +1.2 | +6.8 | +7.4 | +11.7 | +3.3 | -2.9 | +9.1 | |
| Steps (reduced) |
2547047 (0) |
4036974 (1489927) |
5914060 (819966) |
7150465 (2056371) |
8811335 (1170194) |
9425194 (1784053) |
10410960 (222772) |
10819671 (631483) |
11521725 (1333537) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000038 | +0.000071 | +0.000204 | +0.000113 | -0.000172 | +0.000061 | +0.000184 | -0.000156 | +0.000121 |
| Relative (%) | +8.0 | +15.0 | +43.3 | +23.9 | -36.5 | +12.9 | +39.0 | -33.1 | +25.7 | |
| Steps (reduced) |
12373506 (2185318) |
12618571 (2430383) |
13268723 (533488) |
13645937 (910702) |
13820951 (1085716) |
14147799 (1412564) |
14589283 (1854048) |
14983368 (2248133) |
15105867 (2370632) | |