5040edo
5040 equal divisions of the octave (5040edo) 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.
Theory
| Harmonic (prime p) | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (c) | +0.000 | -0.050 | +0.115 | -0.016 | +0.111 | -0.051 | +0.045 | +0.106 | +0.059 | -0.053 |
| relative (%) | +0 | -21 | +48 | -7 | +46 | -22 | +19 | +45 | +25 | -22 | |
| Steps
(reduced) |
5040
(0) |
7988
(2948) |
11703
(1623) |
14149
(4069) |
17436
(2316) |
18650
(3530) |
20601
(441) |
21410
(1250) |
22799
(2639) |
24484
(4324) | |
| Contorsion order
for 2.p subgroup |
5040 | 4 | 3 | 1 | 12 | 10 | 63 | 10 | 7 | 4 | |
5040 is both a superabundant and a highly composite number, meaning its 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 abundance index of 5040 is about 2.84, or exactly 298/105.
The best subgroup in the patent val for 5040edo is 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which is from a numerical standpoint quite amusing - a repunit and a highly composite number.
It tempers out 9801/9800 in the 11-limit.
5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.
Scales
- Consecutive[43]
References
- Wikipedia Contributors. 5040 (number)
- https://mathworld.wolfram.com/PlatosNumbers.html