5040edo: Difference between revisions
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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. | '''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 == | == Number history == | ||
5040 is a factorial (7! = | 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. | 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. | ||
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== Theory == | == Theory == | ||
{{ | {{Harmonics in equal|5040|columns=10}} | ||
{| class="wikitable" | {| class="wikitable" style="text-align:center;" | ||
!Prime ''p'' | |+ Contorsion order for 2.''p'' subgroup | ||
|2 | ! Prime ''p'' | ||
|3 | | 2 | ||
|5 | | 3 | ||
|7 | | 5 | ||
|11 | | 7 | ||
|13 | | 11 | ||
|17 | | 13 | ||
|19 | | 17 | ||
|23 | | 19 | ||
| 23 | |||
|- | |- | ||
!Contorsion | ! Contorsion order | ||
order | | 5040 | ||
| 4 | |||
| 3 | |||
|5040 | | 1 | ||
|4 | | 12 | ||
|3 | | 10 | ||
|1 | | 63 | ||
|12 | | 10 | ||
|10 | | 7 | ||
|63 | |||
|10 | |||
|7 | |||
|} | |} | ||
5040 is both a superabundant and a highly composite number, meaning | 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. | The best subgroup in the patent val for 5040edo is 2.7.13.17.29.31.41.47.61.67. | ||
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== Scales == | == Scales == | ||
* Consecutive[43] | * Consecutive[43] | ||
== References == | == References == | ||
* Wikipedia Contributors. [[Wikipedia:5040 (number)|5040 (number)]] | * Wikipedia Contributors. [[Wikipedia:5040 (number)|5040 (number)]] | ||
* https://mathworld.wolfram.com/PlatosNumbers.html | * https://mathworld.wolfram.com/PlatosNumbers.html | ||
[[Category:Equal divisions of the octave]] | |||
[[Category:Highly melodic]] | [[Category:Highly melodic]] | ||