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.

5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.

Theory

Approximation of prime harmonics in 5040edo

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 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