20edo: Difference between revisions
m Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot |
added the template, moved the primes-error table up to the top |
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| ja = 20平均律 | | ja = 20平均律 | ||
}} | }} | ||
{{Infobox ET | |||
| Prime factorization = 2<sup>2</sup> * 5 | |||
| Step size = 60¢ | |||
| Fifth type = 12\20 = 720¢ = [[5-edo]] | |||
| Major 2nd = 4\20 = 240¢ | |||
| Minor 2nd = 0\20 = 0¢ | |||
| Augmented 1sn = 4\20 = 240¢ | |||
}} | |||
== Theory == | == Theory == | ||
20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 [[cent]]s each. It contains smaller [[EDO|edo]]s [[2edo|2]], [[4edo|4]], [[5edo|5]], and [[10edo|10]] and is part of the 5n Family of equal divisions of the octave. 20 edo fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 & 15 (from [[10edo]]), 19 & 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15. As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0 7 9 steps) and it's utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the Black/White keys on a 20 tone keyboard. Treating the generator as 11/20 creates the same scale, but the primary triads are now 13:16:19 (0 6 11 steps) and it's inversion instead. | {| class="wikitable center-all" | ||
! colspan="2" | | |||
! prime 2 | |||
! prime 3 | |||
! prime 5 | |||
! prime 7 | |||
! prime 11 | |||
! prime 13 | |||
! prime 17 | |||
! prime 19 | |||
|- | |||
! rowspan="2" |Error | |||
! absolute (¢) | |||
| 0.00 | |||
| +18.04 | |||
| -26.3 | |||
| -8.8 | |||
| -11.3 | |||
| -0.5 | |||
| +15.0 | |||
| +2.5 | |||
|- | |||
![[Relative error|relative]] (%) | |||
| 0 | |||
| +30 | |||
| -44 | |||
| -15 | |||
| -19 | |||
| -0.8 | |||
| +25 | |||
| +4.1 | |||
|- | |||
! colspan="2" |[[nearest edomapping]] | |||
|20 | |||
|12 | |||
|6 | |||
|16 | |||
|9 | |||
|14 | |||
|2 | |||
|5 | |||
|} | |||
20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 [[cent]]s each. It contains smaller [[EDO|edo]]s [[2edo|2]], [[4edo|4]], [[5edo|5]], and [[10edo|10]] and is part of the 5n Family of equal divisions of the octave. 20 edo fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 & 15 (from [[10edo]]), 19 & 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15. As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0 7 9 steps) and it's utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the Black/White keys on a 20 tone keyboard. Treating the generator as 11/20 creates the same scale, but the primary triads are now 13:16:19 (0 6 11 steps) and it's inversion instead. | |||
Alternately, 20edo can be used as a [[Blackwood]] temperament, combining minor and major thirds to generate a highly symmetrical decatonic scale where every note is root to a major or minor triad and 7-limit tetrad that are heavily tempered, but in a useful way, as you can easily modulate to anywhere in the small cycle of 5ths, and build extended chords that use every note in the scale without clashing. Either of these works better than trying to force 20 into a diatonic framework. | Alternately, 20edo can be used as a [[Blackwood]] temperament, combining minor and major thirds to generate a highly symmetrical decatonic scale where every note is root to a major or minor triad and 7-limit tetrad that are heavily tempered, but in a useful way, as you can easily modulate to anywhere in the small cycle of 5ths, and build extended chords that use every note in the scale without clashing. Either of these works better than trying to force 20 into a diatonic framework. | ||
== Intervals == | == Intervals == | ||
| Line 238: | Line 285: | ||
=== Selected just intervals === | === Selected just intervals === | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+Direct mapping (even if inconsistent) | |+Direct mapping (even if inconsistent) | ||