20edo: Difference between revisions

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{{Infobox ET

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{| class="wikitable center-all"

! colspan="2" | <!-- empty cell -->

! prime 5

! prime 7

! prime 11

! prime 13

! prime 19

! absolute (¢)

| 0.0

| +18.0

| -11.3

| -0.5

| +2.5

|-

! [[Relative error|relative]] (%)

| -44

| -15

| -0.8

! colspan="2" | [[nearest edomapping]]

| 20

20-tone equal temperament, or 20edo, divides the octave into exactly 20 equal steps of 60 [[cent]]s each. It contains smaller {{EDOs|legend=1| 2, 4, 5, and 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 &amp; 15 (from [[10edo]]), 19 &amp; 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.  

! Approximate Ratios

! colspan="3" | [[Ups and Downs Notation]]

! [[Armodue]]/Balzano Notation

! Heptatonic Major 2nd Generated

! Nearest Harmonic

| [[15/14]], 14/13

| double-up unison, mid 2nd

| [[16/13]], 5/4

| [[14/11]], [[51/40]]

| [[15/11]], 11/8

| [[22/15]], 16/11

| [[13/8]], 8/5

| [[28/15]], 15/8, 13/7

=== Selected just intervals ===

 

== Chord Names ==

For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. Because many intervals have several names, many chords do too.

== Modes ==

20 tone equal modes:

{| class="wikitable"

| 3 1 3 1 3 1 3 1 3 1

| Balzano Nine-tone ([[2L_7s|fair mavila]], [[score9]]) <ref>[https://www.jstor.org/stable/833490?seq=1 Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach]</ref>

|}

== Commas ==

20 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 20 32 46 56 69 74 }}.)

! [[Harmonic limit|Prime<br>Limit]]

| [[Limma]], Pythagorean Minor 2nd

| Negri Comma, Double Augmentation Diesis

| [[Ampersand]], Ampersand's Comma

| Avicennma, Avicenna's Enharmonic Diesis

| Slendro Diesis

| Tritonic Diesis, Jubilisma

| Septimal Comma, Archytas' Comma, Leipziger Komma

| Septimal Kleisma, Marvel Comma

| Mirkwai

| [[Linus]]

| Superleap

| Parizeksma

<references/>

Like other members of the 5EDO family, 20-EDO lends itself well to guitar (and other fretted string instruments), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-4-4-2-0-0 (low to high), an "A" chord would be 0-0-4-4-2-0, and a "D" chord would be 2-0-0-4-4-2.