12edo: Difference between revisions
No edit summary |
ET parameter name, cleanup |
||
| Line 6: | Line 6: | ||
}} | }} | ||
{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2<sup>2</sup> | | Prime factorization = 2<sup>2</sup> × 3 | ||
| Step size = 100.000¢ | | Step size = 100.000¢ | ||
| Fifth | | Fifth = 7\12 = 700¢ | ||
| Major 2nd = 2\12 = 200¢ | | Major 2nd = 2\12 = 200¢ | ||
| Minor 2nd = 1\12 = 100¢ | | Minor 2nd = 1\12 = 100¢ | ||
| Line 16: | Line 16: | ||
== Theory == | == Theory == | ||
{| class="wikitable | {| class="wikitable center-all" | ||
! | ! | ||
! prime 2 | ! prime 2 | ||
| Line 24: | Line 24: | ||
! prime 11 | ! prime 11 | ||
! prime 13 | ! prime 13 | ||
!prime 17 | ! prime 17 | ||
!prime 19 | ! prime 19 | ||
|- | |- | ||
! Error (¢) | ! Error (¢) | ||
| 0 | | 0 | ||
| | | -2.0 | ||
| | | +13.7 | ||
| | | +31.2 | ||
| | | +48.7 | ||
| | | -40.5 | ||
| | | -5.0 | ||
| | | +2.5 | ||
|- | |- | ||
!Error (%) | ! Error (%) | ||
|0 | | 0 | ||
| -1.96 | | -1.96 | ||
| +13.7 | | +13.7 | ||
| Line 44: | Line 44: | ||
| +48.7 | | +48.7 | ||
| -40.5 | | -40.5 | ||
| | | -5.0 | ||
| | | +2.5 | ||
|- | |- | ||
![[Patent val|nearest edomapping]] | ! [[Patent val|nearest edomapping]] | ||
|12 | | 12 | ||
|7 | | 7 | ||
|4 | | 4 | ||
|10 | | 10 | ||
|6 | | 6 | ||
|8 | | 8 | ||
|1 | | 1 | ||
|3 | | 3 | ||
|- | |- | ||
![[Fifthspan]] | ! [[Fifthspan]] | ||
| 0 | | 0 | ||
| | | +1 | ||
| | | +4 | ||
| | | -2 | ||
| | | +6 | ||
| | | -4 | ||
| -5 | | -5 | ||
| -3 | | -3 | ||
|} | |} | ||
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. | 12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. | ||
| Line 75: | Line 76: | ||
== Intervals == | == Intervals == | ||
{| class="wikitable right-1 right-2" | {| class="wikitable right-1 right-2" | ||
! | ! Steps | ||
! Cents | ! Cents | ||
! colspan="3" | Interval | ! colspan="3" | Interval | ||