5040edo

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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

Approximation of prime intervals in 5040 EDO
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