5040edo

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← 5039edo5040edo5041edo →
Prime factorization 24 × 32 × 5 × 7
Step size 0.238095¢
Fifth 2948\5040 (701.905¢) (→737\1260)
Semitones (A1:m2) 476:380 (113.3¢ : 90.48¢)
Consistency limit 3
Distinct consistency limit 3
Special properties

5040 equal divisions of the octave (5040edo), or 5040-tone equal temperament (5040tet), 5040 equal temperament (5040et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 5040 equal parts of about 0.238 ¢ each.

Theory

5040 is a factorial (7! = 1·2·3·4·5·6·7), superabundant, and a highly composite number. The abundancy index of this number is about 2.84, or exactly 298/105.

5040's divisors besides 1 and itself are 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520 - which is a total of 58. This sequence does include some notable temperaments. For example, 12edo, which is the dominant tuning system in the world today, 15edo, notable for its usage by Easley Blackwood Jr., 72edo, notable for use in Gregorian chants and known for use by Joe Maneri. Other notable systems which are divisors of 5040 are all the EDOs from 1 to 10 inclusive, and also 140edo.

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. To provide some reference of how notable 5040 is, due to its profound divisibility the number has found some applied historical use. Ancient Greek philosopher Plato suggested that 5040 is the ideal number of people in a city, citing its high divisibility properties, and also the fact that the number that is two below 5040, 5038, is divisible by 11.

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

Prime harmonics

Approximation of prime harmonics in 5040edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.000 -0.050 +0.115 -0.016 +0.111 -0.051 +0.045 +0.106 +0.059 -0.053 -0.036
relative (%) +0 -21 +48 -7 +46 -22 +19 +45 +25 -22 -15
Steps
(reduced)
5040
(0)
7988
(2948)
11703
(1623)
14149
(4069)
17436
(2316)
18650
(3530)
20601
(441)
21410
(1250)
22799
(2639)
24484
(4324)
24969
(4809)

Regular temperament theory

From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. It does offer unique harmonies, although they are not simple. Since having errors less than 25% guarantees "naive consistency" to distance 1, this means that the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.47.61.67.

There's an interesting property that arises in a stricter subgroup, 2.7.13.17.29.31.41.47.61.67. In this subgroup, it makes a rank two temperament with 1111edo (1111 & 5040), which brings two notable number classes together - a repunit and a highly composite number.

Using the patent val, 5040edo tempers out 9801/9800 in the 11-limit.

Contorsion table

This table shows from what EDOs 5040edo inherits its prime harmonics.

Contorsion table for 5040 equal divisions of the octave
Harmonic Contorsion order Meaning that it comes from
3 4 1260edo
5 3 1680edo
7 1 5040edo (maps to a coprime step)
11 12 420edo
13 10 504edo
17 63 80edo
19 10 504edo
23 7 720edo
29 4 1260edo
31 21 240edo
37 48 105edo
41 2 2520edo
43 12 420edo
47 5 1008edo
53 3 1680edo
59 3 1680edo
61 1 5040edo (maps to a coprime step)
67 9 560edo

Regular temperament properties

Tempered commas

2.3.7 subgroup: [81  41 -52, [-110 119 -28

2.3.7.13 subgroup: [8 15 -10 -1, [-41 34 2 -5, [-52 6 -2 13

2.3.7.13.17 subgroup: 5832/5831, 31213/31212, [-44 28 5 -5 1, [-50  9 -6 12  2.

2.3.7.13.17.23 subgroup: 5832/5831, 213003/212992, 279888/279841, [-25 -3 -9 14 3 -2, [-27 -6 -5 15 1 -2

2.3.7.13.17.23.29 subgroup: 4914/4913, 5832/5831, 107406/107387, 116928/116909, 3234573/3233984, 2871098559212689/2870492063072256

Scales

  • Consecutive[43] - 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

References