12edo: Difference between revisions
m Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot |
moved the prime-errors table up to the top |
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| Line 9: | Line 9: | ||
| Subgroup = 2.3.5.7.17.19 | | Subgroup = 2.3.5.7.17.19 | ||
| Step size = 100.000 | | Step size = 100.000 | ||
| Fifth type = Meantone 7\12 700¢ | | Fifth type = Meantone 7\12 700¢ | ||
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)<br/>pentatonic (meantone) 2L3s 22323 (generator = 7\12)<br/>diminished 4L4s 12121212 (generator = 1\12, period = 3\12) | | Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)<br/>pentatonic (meantone) 2L3s 22323 (generator = 7\12)<br/>diminished 4L4s 12121212 (generator = 1\12, period = 3\12) | ||
}} | }} | ||
| Line 15: | Line 15: | ||
== Theory == | == Theory == | ||
{| class="wikitable" style="text-align:center;" | |||
! | |||
! prime 2 | |||
! prime 3 | |||
! prime 5 | |||
! prime 7 | |||
! prime 11 | |||
! prime 13 | |||
!prime 17 | |||
!prime 19 | |||
|- | |||
! Error (¢) | |||
| 0 | |||
| -1.96 | |||
| +13.7 | |||
| +31.2 | |||
| +48.7 | |||
| -40.5 | |||
| +5.0 | |||
| -2.5 | |||
|- | |||
!Error (%) | |||
|0 | |||
| -1.96 | |||
| +13.7 | |||
| +31.2 | |||
| +48.7 | |||
| -40.5 | |||
| +5.0 | |||
| -2.5 | |||
|- | |||
![[Patent val|nearest edomapping]] | |||
|12 | |||
|7 | |||
|4 | |||
|10 | |||
|6 | |||
|8 | |||
|1 | |||
|3 | |||
|- | |||
![[Fifthspan]] | |||
| 0 | |||
| +1 | |||
| +4 | |||
| -2 | |||
| +6 | |||
| -4 | |||
| -5 | |||
| -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 122: | Line 173: | ||
== Just approximation == | == Just approximation == | ||
=== Selected just intervals by | === Selected just intervals by erro === | ||
==== 15-odd-limit interval mappings ==== | ==== 15-odd-limit interval mappings ==== | ||