5040edo: Difference between revisions
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|+Approximation of prime harmoniics in 5040edo | |||
! colspan="2" |Harmonic (prime ''p'') | |||
!2 | |||
!3 | |||
!5 | |||
!7 | |||
!11 | |||
!13 | |||
!17 | |||
!19 | |||
!23 | |||
!29 | |||
|- | |||
! rowspan="2" |Error | |||
!absolute (c) | |||
| +0.000 | |||
| -0.050 | |||
| +0.115 | |||
| -0.016 | |||
| +0.111 | |||
| -0.051 | |||
| +0.045 | |||
| +0.106 | |||
| +0.059 | |||
| -0.053 | |||
|- | |||
!relative (%) | |||
| +0 | |||
| -21 | |||
| +48 | |||
| -7 | |||
| +46 | |||
| -22 | |||
| +19 | |||
| +45 | |||
| +25 | |||
| -22 | |||
|- | |||
! colspan="2" |Steps | |||
(reduced) | |||
|5040 | |||
(0) | |||
|7988 | |||
(2948) | |||
|11703 | |||
(1623) | |||
|14149 | |||
(4069) | |||
|17436 | |||
(2316) | |||
|18650 | |||
(3530) | |||
|20601 | |||
(441) | |||
|21410 | |||
(1250) | |||
|22799 | |||
(2639) | |||
|24484 | |||
(4324) | |||
|- | |||
! colspan="2" |Contorsion order | |||
for 2.''p'' subgroup | |||
|5040 | |||
|4 | |||
|3 | |||
|1 | |||
|12 | |||
|10 | |||
|63 | |||
|10 | |||
|7 | |||
|4 | |||
|} | |} | ||
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. | 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. | ||
Revision as of 09:26, 31 January 2022
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
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.
5040 is a sum of 43 consecutive primes, running from 23 to 229 inclusive.
Theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.050 | +0.115 | -0.016 | +0.111 | -0.051 | +0.045 | +0.106 | +0.059 | -0.053 |
| Relative (%) | +0.0 | -21.1 | +48.2 | -6.9 | +46.5 | -21.6 | +18.7 | +44.5 | +24.8 | -22.4 | |
| Steps (reduced) |
5040 (0) |
7988 (2948) |
11703 (1623) |
14149 (4069) |
17436 (2316) |
18650 (3530) |
20601 (441) |
21410 (1250) |
22799 (2639) |
24484 (4324) | |
| Prime p | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 |
|---|---|---|---|---|---|---|---|---|---|
| Contorsion order | 5040 | 4 | 3 | 1 | 12 | 10 | 63 | 10 | 7 |
| Harmonic (prime p) | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (c) | +0.000 | -0.050 | +0.115 | -0.016 | +0.111 | -0.051 | +0.045 | +0.106 | +0.059 | -0.053 |
| relative (%) | +0 | -21 | +48 | -7 | +46 | -22 | +19 | +45 | +25 | -22 | |
| Steps
(reduced) |
5040
(0) |
7988
(2948) |
11703
(1623) |
14149
(4069) |
17436
(2316) |
18650
(3530) |
20601
(441) |
21410
(1250) |
22799
(2639) |
24484
(4324) | |
| Contorsion order
for 2.p subgroup |
5040 | 4 | 3 | 1 | 12 | 10 | 63 | 10 | 7 | 4 | |
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.
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, ⟨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.
It tempers out 9801/9800 in the 11-limit.
Scales
- Consecutive[43]
References
- Wikipedia Contributors. 5040 (number)
- https://mathworld.wolfram.com/PlatosNumbers.html