5edo: Difference between revisions

{{Infobox ET

| Prime factorization = 5 (prime)

| Fifth = 3\5 = 720¢

| Major 2nd = 1\5 = 240¢

| Semitones = 1 : 0

| Consistency = 9

'''5 equal divisions of the octave''' (or '''5edo''') is the [[tuning system]] derived by dividing the [[octave]] into 5 equal steps of 240 [[cent]]s each, or the fifth root of two. 5edo is the third [[prime edo]], after [[2edo|2edo]] and [[3edo|3edo]]. Most importantly, 5edo is the smallest [[edo]] containing xenharmonic intervals — 1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo|12edo]].

{{Harmonics in equal|5|intervals=odd}}

In addition, considering 5edo as a no-5s temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next EDO doing it better being [[17edo|17]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament.

{| class="wikitable center-all"

|-

| '''1/1'''

|-

| +8.826¢ from septimal second [[8/7]] <br>-4.969¢ from diminished third [[144/125]] <br>-13.076¢ from augmented second [[125/108]] <br>-26.871¢ from septimal minor third [[7/6]]

|-

| 2

| +9.219¢ from narrow fourth [[21/16]] <br>-0.686¢ from smaller fourth [[33/25]] <br>-18.045¢ from just fourth [[4/3]]

|-

| 3

| +18.045¢ from just fifth [[3/2]] <br>+0.686¢ from bigger fifth [[50/33]] <br>-9.219¢ from wide fifth [[32/21]]

|-

| 960

| +26.871¢ from septimal major sixth [[12/7]] <br>+13.076¢ from diminished seventh [[216/125]] <br>+4.969¢ from augmented sixth [[125/72]] <br>-8.826¢ from septimal seventh [[7/4]]

|-

|}

[[:File:5ed2-001.svg|5ed2-001.svg]]

[[Kite Giedraitis]] has proposed a pentatonic notation that retains the appearance of heptatonic names, to avoid the confusion caused by one's lifelong association of "fourth" with 4/3, not 3/2. The interval names are unisoid, subthird, fourthoid, fifthoid, subseventh and octoid, or 1d s3 4d 5d s7 8d. When notating larger edos such as 8 or 13, there are major or minor sub3rds and sub7ths. Note that 15/8 is an octoid.

5 is a prime number so 5edo contains no sub-edos. Only simple cycles:

5edo does not have any strong consonance nor dissonance. The 240 cent interval can serve as either a major second or minor third, and the 960 cent interval as either a major sixth or minor seventh. The fourth is about 18 cents flat of a just fourth, making it rather "dirty" but recognizable. The fifth is likewise about 18 cents sharp of a just fifth, dissonant but still easily recognizable.

Important chords:

* 0+2+3+4

Smallest edo that can be used for melodies in a "standard" way. The relatively large step of 240 cents can be used as major second for the melody construction. The scale has whole-tone as well as pentatonic character.

Either way, it is hard to wander very far from where you start. However, it has the scale-like feature that there are (barely) enough notes to create melody, in the form of an equal version of pentatonic.

5edo [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 5 8 12 14 17 19 }}.  

{| class="commatable wikitable center-1 center-2 right-4 center-5"

|-

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

|-

| [[256/243]]

|-

| [[27/25]]

| Large limma

|-

| 5

|-

| 5

|-

| [[Vulture]]

|-

| {{monzo| 2 2 -1 -1 }}

| 48.770

|-

| 7

| {{monzo| -4 -1 0 2 }}

|-

| {{monzo| 6 -2 0 -1 }}

| Septimal comma, Archytas' comma, Leipziger Komma

|-

|-

| 7

| {{monzo| 6 3 -1 -3 }}

| 13.074

|-

| 7

| [[1029/1024]]

| 8.433

|-

| {{monzo| -4 9 -2 -2 }}

|-

| [[5120/5103]]

| {{monzo| 10 -6 1 -1 }}

| 5.758

|-

| {{monzo| -26 -1 1 9 }}

|-

|-

| [[11/10]]

| 165.004

| Large undecimal neutral 2nd

|-

| 11

| [[99/98]]

|-

| 11

|-

| [[385/384]]

| 4.503

|-

| 11

| [[441/440]]

|-

| 11

| [[3025/3024]]

| Lehmerisma

|-

|-

|-

| 13

| [[676/675]]

|}

5edo ear-training exercises by Alex Ness available here:

For any musician, there is no substitute for the experience of a particular xenharmonic sound. The user going by the name Hyacinth on Wikipedia and Wikimedia Commons has many xenharmonic MIDI's and has graciously copylefted them! This is his 5-TET scale MIDI:

{| class="wikitable sortable"

!Title

!Year

!Genre

|

|-

|NullPointerException Music

|2020

|

|-

|Vince Kaichan

|2021

|}

There is also much 5edo-like world music, just search for "gyil" or "amadinda" or "slendro".