13edo: Difference between revisions
mNo edit summary |
added M2, m2 and A1 to the template, made the primes-error table |
||
| Line 8: | Line 8: | ||
| Prime factorization = 13 | | Prime factorization = 13 | ||
| Subgroup = 2.5.9.11.13.21 | | Subgroup = 2.5.9.11.13.21 | ||
| Step size = 92. | | Step size = 92.308¢ | ||
| Fifth type = father 8\13 738.46¢ | | Fifth type = father 8\13 = 738.46¢ | ||
| Major 2nd = 3\13 = 277¢ | |||
| Minor 2nd = -1\13 = -92¢ | |||
| Augmented 1sn = 4\13 = 369¢ | |||
| Common uses = distorted 12edo | | Common uses = distorted 12edo | ||
| Important MOS = [[oneirotonic]] ([[A-Team]]/[[Petrtri]]) 5L3s 22122121 (5\13, 1\1)<br/>archeotonic 6L1s 2222221 (2\13, 1\1)<br/>[[sephiroth]] 3L4s 3131311 (4\13, 1\1)<br/>[[lovecraft]] 4L5s 212121211 (3\13, 1\1) | | Important MOS = [[oneirotonic]] ([[A-Team]]/[[Petrtri]]) 5L3s 22122121 (5\13, 1\1)<br/>archeotonic 6L1s 2222221 (2\13, 1\1)<br/>[[sephiroth]] 3L4s 3131311 (4\13, 1\1)<br/>[[lovecraft]] 4L5s 212121211 (3\13, 1\1) | ||
| Line 20: | Line 23: | ||
== Theory == | == Theory == | ||
{| class="wikitable" | |||
|+ | |||
! colspan="2" |Prime number ---> | |||
!2 | |||
!3 | |||
!5 | |||
!7 | |||
!11 | |||
!13 | |||
!17 | |||
!19 | |||
!23 | |||
|- | |||
! rowspan="2" |Error | |||
!absolute ([[Cent|¢]]) | |||
|0 | |||
|36.51 | |||
| -17.1 | |||
| -45.7 | |||
|2.5 | |||
| -9.8 | |||
| -12.6 | |||
| -20.6 | |||
|17.9 | |||
|- | |||
![[Relative error|relative]] (%) | |||
|0 | |||
|40 | |||
| -19 | |||
| -50 | |||
|3 | |||
| -11 | |||
| -14 | |||
| -22 | |||
|19 | |||
|- | |||
! colspan="2" |[[nearest edomapping]] | |||
|13 | |||
|8 | |||
|4 | |||
|10 | |||
|6 | |||
|9 | |||
|1 | |||
|3 | |||
|7 | |||
|- | |||
! colspan="2" |[[fifthspan]] | |||
|0 | |||
| +1 | |||
| +7 | |||
| -2 | |||
| +4 | |||
| +6 | |||
| +5 | |||
| +2 | |||
| -4 | |||
|} | |||
As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the '''2.5.9.11.13.17.19.21''' subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size. | As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the '''2.5.9.11.13.17.19.21''' subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size. | ||