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'''5040 equal divisions of the octave''' divides the octave into steps of 0.238 cents each.
{{Infobox ET}}
{{ED intro}}


== Number history ==
== Theory ==
5040 is a factorial (7! = 1 2 3 4 5 6 7), superabundant, and a highly composite number.
5040 is a factorial ({{nowrap|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'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 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 also a sum of 43 consecutive primes, running from 23 to 229 inclusive.
 
=== Prime harmonics ===
{{Harmonics in equal|5040}}
{{Harmonics in equal|84|columns=15|start=12|collapsed=1|title=Approximation of prime harmonics in 5040edo (continued)}}
 
=== Regular temperament theory ===
From the regular temperament theory perspective, 5040edo is not as impressive, as for example zeta EDOs. Using the 25% error cutoff, the best subgroup for 5040edo is 2.3.7.13.17.23.29.31.41.43.53.59.61.73.89.


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.
There's an interesting property that arises in the 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.


5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.  
Using the patent val, 5040edo tempers out [[9801/9800]] in the 11-limit.


== Theory ==
=== Contorsion table ===
{{Primes in edo|5040|columns=20}}
This table shows from what EDOs 5040edo inherits its prime harmonics.
{| class="wikitable"
{| class="wikitable"
!Prime ''p''
|+ style="font-size: 105%;" | Contorsion table for 5040 equal divisions of the octave
|2
|-
|3
! Harmonic
|5
! Contorsion order
|7
! Meaning that it comes from
|11
|-
|13
| 3
|17
| 4
|19
| [[1260edo]]
|23
|-
| 5
| 3
| [[1680edo]]
|-
| 7
| 1
| 5040edo (maps to a coprime step)
|-
| 11
| 12
| [[420edo]]
|-
| 13
| 10
| [[504edo]]
|-
| 17
| 63
| [[80edo]]
|-
| 19
| 10
| 504edo
|-
|-
!Contorsion
| 23
order for 2.''p'' 
| 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: {{Monzo|81  41 -52}}, {{Monzo|-110 119 -28}}


subgroup
2.3.7.13 subgroup: {{monzo|8 15 -10 -1}}, {{monzo|-41 34  2 -5}}, {{monzo|-52  6  -2 13}}
|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.  
2.3.7.13.17 subgroup: [[5832/5831]], [[31213/31212]], {{monzo|-44 28 5 -5 1}}, {{monzo|-50  9 -6 12  2}}.


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, {{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]].
2.3.7.13.17.23 subgroup: 5832/5831, 213003/212992, 279888/279841, {{Monzo|-25 -3 -9 14 3 -2}}, {{Monzo|-27 -6 -5 15 1 -2}}


It tempers out [[9801/9800]] in the 11-limit.
2.3.7.13.17.23.29 subgroup: 4914/4913, 5832/5831, 107406/107387, 116928/116909, 3234573/3233984, 2871098559212689/2870492063072256


== Scales ==
== 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
* 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
{{Todo|review}}
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