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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|5040}}
{{ED intro}}


== Theory ==
== Theory ==
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 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'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 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.
5040 is also a sum of 43 consecutive primes, running from 23 to 229 inclusive.
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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.
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 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.
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.
Using the patent val, 5040edo tempers out [[9801/9800]] in the 11-limit.
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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
!Meaning that it comes from
|-
|-
|3
! Harmonic
|4
! Contorsion order
|[[1260edo]]
! Meaning that it comes from
|-
|-
|5
| 3
|3
| 4
|[[1680edo]]
| [[1260edo]]
|-
|-
|7
| 5
|1
| 3
|5040edo (maps to a coprime step)
| [[1680edo]]
|-
|-
|11
| 7
|12
| 1
|[[420edo]]
| 5040edo (maps to a coprime step)
|-
|-
|13
| 11
|10
| 12
|[[504edo]]
| [[420edo]]
|-
|-
|17
| 13
|63
| 10
|[[80edo]]
| [[504edo]]
|-
|-
|19
| 17
|10
| 63
|504edo
| [[80edo]]
|-
|-
|23
| 19
|7
| 10
|[[720edo]]
| 504edo
|-
|-
|29
| 23
|4
| 7
|1260edo
| [[720edo]]
|-
|-
|31
| 29
|21
| 4
|[[240edo]]
| 1260edo
|-
|-
|37
| 31
|48
| 21
|[[105edo]]
| [[240edo]]
|-
|-
|41
| 37
|2
| 48
|[[2520edo]]
| [[105edo]]
|-
|-
|43
| 41
|12
| 2
|[[420edo]]
| [[2520edo]]
|-
|-
|47
| 43
|5
| 12
|[[1008edo]]
| [[420edo]]
|-
|-
|53
| 47
|3
| 5
|[[1680edo]]
| [[1008edo]]
|-
|-
|59
| 53
|3
| 3
|[[1680edo]]
| [[1680edo]]
|-
|-
|61
| 59
|1
| 3
|5040edo (maps to a coprime step)
| [[1680edo]]
|-
|-
|67
| 61
|9
| 1
|[[560edo]]
| 5040edo (maps to a coprime step)
|-
| 67
| 9
| [[560edo]]
|}
|}


== Regular temperament properties ==
== Regular temperament properties ==
=== Tempered commas ===
=== Tempered commas ===
2.3.7 subgroup: {{Monzo|81  41 -52}}, {{Monzo|-110 119 -28}}
2.3.7 subgroup: {{Monzo|81  41 -52}}, {{Monzo|-110 119 -28}}
Retrieved from "https://en.xen.wiki/w/5040edo"