2513edo: Difference between revisions
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{{ | 2513edo is a very strong 5-limit system, with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any edo until we reach the cosmically excellent [[4296edo]]. 2513 = {{factorization|2513}}, and it shares its [[harmonic]] [[3/1|3]] with [[359edo]]. A basis for its 5-limit commas consists of senior, {{monzo| -17 62 -35 }}, and fortune, {{monzo| -107 47 14 }}; it also [[tempering out|tempers out]] pirate, {{monzo| -90 -15 49 }}. It is uniquely [[consistent]] through to the [[11-odd-limit]], and tempers out 420175/419904 in the 7-limit and 151263/151250 and 234375/234256 in the 11-limit. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|2513|prec=4}} | |||
Latest revision as of 07:02, 20 February 2025
| ← 2512edo | 2513edo | 2514edo → |
2513 equal divisions of the octave (abbreviated 2513edo or 2513ed2), also called 2513-tone equal temperament (2513tet) or 2513 equal temperament (2513et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2513 equal parts of about 0.478 ¢ each. Each step represents a frequency ratio of 21/2513, or the 2513th root of 2.
2513edo is a very strong 5-limit system, with a lower 5-limit relative error than any edo until we reach the cosmically excellent 4296edo. 2513 = 7 × 359, and it shares its harmonic 3 with 359edo. A basis for its 5-limit commas consists of senior, [-17 62 -35⟩, and fortune, [-107 47 14⟩; it also tempers out pirate, [-90 -15 49⟩. It is uniquely consistent through to the 11-odd-limit, and tempers out 420175/419904 in the 7-limit and 151263/151250 and 234375/234256 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0051 | -0.0025 | +0.0559 | +0.2141 | -0.0979 | +0.0983 | -0.0200 | +0.1379 | -0.0507 | +0.0500 |
| Relative (%) | +0.0 | -1.1 | -0.5 | +11.7 | +44.8 | -20.5 | +20.6 | -4.2 | +28.9 | -10.6 | +10.5 | |
| Steps (reduced) |
2513 (0) |
3983 (1470) |
5835 (809) |
7055 (2029) |
8694 (1155) |
9299 (1760) |
10272 (220) |
10675 (623) |
11368 (1316) |
12208 (2156) |
12450 (2398) | |