5040edo
5040 equal divisions of the octave divides the octave into steps of 238 millicents each, or exactly 5/21 of a cent.
Number history
5040 is a factorial (7! = 1 2 3 4 5 6 7), superabundant, and a highly composite number. 5040 is the 19th superabundant and highly composite EDO, and it marks the end of the sequence where superabundant and highly composite numbers are the same - 7560 is the first highly composite that isn't superabundant.
Ancient Greek philosopher Plato suggested that 5040 is the ideal number of people in a city, owing to it's large divisibility and a bunch of other traits.
5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.
Theory
| Prime number | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.050 | +0.115 | -0.016 | +0.111 | -0.051 | +0.045 | +0.106 | +0.059 | -0.053 | -0.036 | +0.085 | -0.015 | -0.089 | -0.030 | +0.067 | +0.114 | +0.020 | -0.021 | +0.065 |
| relative (%) | +0 | -21 | +48 | -7 | +46 | -22 | +19 | +45 | +25 | -22 | -15 | +36 | -6 | -37 | -13 | +28 | +48 | +8 | -9 | +27 | |
| Steps (reduced) | 5040 (0) | 7988 (2948) | 11703 (1623) | 14149 (4069) | 17436 (2316) | 18650 (3530) | 20601 (441) | 21410 (1250) | 22799 (2639) | 24484 (4324) | 24969 (4809) | 26256 (1056) | 27002 (1802) | 27348 (2148) | 27995 (2795) | 28869 (3669) | 29649 (4449) | 29891 (4691) | 30573 (333) | 30995 (755) | |
| Prime p | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 |
|---|---|---|---|---|---|---|---|---|---|
| Contorsion
order for 2.p subgroup |
5040 | 4 | 3 | 1 | 12 | 10 | 63 | 10 | 7 |
5040 is both a superabundant and a highly composite number, meaning it's amount of symmetrical chords and subscales increases to a record, and the amount of notes which make up those scales, if stretched end-to-end, also is largest relative to the number's size.
The best subgroup in the patent val for 5040edo is 2.7.13.17.29.31.41.47.61.67.
5040 is contorted order-4 in the 3-limit and contorted order-2 in the 5-limit in the 5040c val. In the 5040cdd val, ⟨5040 7988 11072 14148], it is contorted order 2 in the 7-limit and tempers out 2401/2400 and 4375/4374. Under such a val, the 5th harmonic comes from 315edo, and the 7th ultimately derives from 140edo.
It tempers out 9801/9800 in the 11-limit.
Scales
- Consecutive[43]
References
- Wikipedia Contributors. 5040 (number)
- https://mathworld.wolfram.com/PlatosNumbers.html