5040edo

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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

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 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