# 5040edo

← 5039edo | 5040edo | 5041edo → |

^{4}× 3^{2}× 5 × 7**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. 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.

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 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

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
---|---|---|---|---|---|---|---|---|---|---|---|

Error | absolute (¢) | +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) |

### 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.

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

- Wikipedia Contributors. 5040 (number)
- https://mathworld.wolfram.com/PlatosNumbers.html