13edo: Difference between revisions
m State the primality in infobox |
ET parameter name, cleanup |
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| Prime factorization = 13 (prime) | | Prime factorization = 13 (prime) | ||
| Step size = 92.308¢ | | Step size = 92.308¢ | ||
| Fifth | | Fifth = 8\13 = 738.46¢ | ||
| Major 2nd = 3\13 = 277¢ | | Major 2nd = 3\13 = 277¢ | ||
| Minor 2nd = -1\13 = -92¢ | | Minor 2nd = -1\13 = -92¢ | ||
| Line 17: | Line 17: | ||
== Theory == | == Theory == | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
! colspan="2" | Prime interval | |||
! colspan="2" |Prime | ! 2 | ||
!2 | ! 3 | ||
!3 | ! 5 | ||
!5 | ! 7 | ||
!7 | ! 11 | ||
!11 | ! 13 | ||
!13 | ! 17 | ||
!17 | ! 19 | ||
!19 | ! 23 | ||
!23 | |||
|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" | Error | ||
!absolute ([[Cent|¢]]) | ! absolute ([[Cent|¢]]) | ||
|0 | | 0 | ||
|36. | | +36.5 | ||
| -17.1 | | -17.1 | ||
| -45.7 | | -45.7 | ||
|2.5 | | +2.5 | ||
| -9.8 | | -9.8 | ||
| -12.6 | | -12.6 | ||
| -20.6 | | -20.6 | ||
|17.9 | | +17.9 | ||
|- | |- | ||
![[Relative error|relative]] (%) | ! [[Relative error|relative]] (%) | ||
|0 | | 0 | ||
|40 | | +40 | ||
| -19 | | -19 | ||
| -50 | | -50 | ||
|3 | | +3 | ||
| -11 | | -11 | ||
| -14 | | -14 | ||
| -22 | | -22 | ||
|19 | | +19 | ||
|- | |- | ||
! colspan="2" |[[nearest edomapping]] | ! colspan="2" | [[nearest edomapping]] | ||
|13 | | 13 | ||
|8 | | 8 | ||
|4 | | 4 | ||
|10 | | 10 | ||
|6 | | 6 | ||
|9 | | 9 | ||
|1 | | 1 | ||
|3 | | 3 | ||
|7 | | 7 | ||
|- | |- | ||
! colspan="2" |[[fifthspan]] | ! colspan="2" | [[fifthspan]] | ||
|0 | | 0 | ||
| +1 | | +1 | ||
| +7 | | +7 | ||
| Line 75: | Line 74: | ||
| -4 | | -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. | ||