5040edo: Difference between revisions

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== Theory ==

5040 is a factorial (7! = 1·2·3·4·5·6·7), superabundant, and a highly composite number. ~~Its divisors besides 1 and itself are {{EDOs|~~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~~.

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

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's divisors besides 1 and itself are {{EDOs|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 ~~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.~~

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.

~~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.

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 == Theory ==
-is a factorial (7! = 1·2·3·4·5·6·7), superabundant, and a highly composite number. Its divisors besides 1 and itself are {{EDOs|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.
+is a factorial (7! = 1·2·3·4·5·6·7), [[Highly composite equal division|superabundant, and a highly composite]] number. The [[abundancy index]] of this number is about 2.84, or exactly 298/105.
-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.
+'s divisors besides 1 and itself are {{EDOs|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]].
-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.
+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.
-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.
 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.