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{{Infobox ET}}
{{Infobox ET}}
'''5040 equal divisions of the octave''' ('''5040edo''') divides the octave into steps of 238 millicents each, or exactly 5/21 of a [[cent]].
{{ED intro}}


== Theory ==
== 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 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 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'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]].


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


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|5040}}
{{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 ===
=== 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.
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.


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


=== Contorsion table ===
=== Contorsion table ===
This table shows from what EDOs 5040edo inherits its prime harmonics.  
This table shows from what EDOs 5040edo inherits its prime harmonics.  
{| class="wikitable"
{| class="wikitable"
|+Contorsion table for 5040 equal divisions of the octave
|+ style="font-size: 105%;" | Contorsion table for 5040 equal divisions of the octave
!Harmonic
|-
!Contorsion order
! Harmonic
!Meaning that it comes from
! 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
|-
|-
|3
| 31
|4
| 21
|[[1260edo]]
| [[240edo]]
|-
|-
|5
| 37
|3
| 48
|[[1680edo]]
| [[105edo]]
|-
|-
|7
| 41
|1
| 2
|5040edo (maps to a coprime step)
| [[2520edo]]
|-
|-
|11
| 43
|12
| 12
|[[420edo]]
| [[420edo]]
|-
|-
|13
| 47
|10
| 5
|[[504edo]]
| [[1008edo]]
|-
|-
|17
| 53
|63
| 3
|[[80edo]]
| [[1680edo]]
|-
|-
|19
| 59
|10
| 3
|504edo
| [[1680edo]]
|-
|-
|23
| 61
|7
| 1
|[[720edo]]
| 5040edo (maps to a coprime step)
|-
|-
|29
| 67
|4
| 9
|1260edo
| [[560edo]]
|}
|}


== Regular temperament properties ==
== Regular temperament properties ==
Using the patent val, 5040edo tempers out [[9801/9800]] in the 11-limit.
=== Tempered commas ===
2.3.7 subgroup: {{Monzo|81  41 -52}}, {{Monzo|-110 119 -28}}
 
2.3.7.13 subgroup: {{monzo|8 15 -10 -1}}, {{monzo|-41 34  2 -5}}, {{monzo|-52  6  -2 13}}
 
2.3.7.13.17 subgroup: [[5832/5831]], [[31213/31212]], {{monzo|-44 28 5 -5 1}}, {{monzo|-50  9 -6 12  2}}.
 
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}}


A comma basis for the 2.3.7.13.17 subgroup is [[5832/5831]], [[31213/31212]], {{monzo|-44 28 5 -5 1}}, {{monzo|-50  9 -6 12  2}}.
2.3.7.13.17.23.29 subgroup: 4914/4913, 5832/5831, 107406/107387, 116928/116909, 3234573/3233984, 2871098559212689/2870492063072256


== Scales ==
== Scales ==
* Consecutive[43]
* 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 ==
== References ==
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* https://mathworld.wolfram.com/PlatosNumbers.html
* https://mathworld.wolfram.com/PlatosNumbers.html


[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
{{Todo|review}}
[[Category:Highly composite]]
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