5edo: Difference between revisions

'''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]].

{| class="wikitable center-all"

| 5

|}

If 5EDO is regarded as a temperament, which is to say as 5-TET, then the most salient fact is that 16/15 is tempered out. This means in 5-TET the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[Trienstonic clan|father temperament]].

Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]], after 2EDO. It also is the smallest equal division representing the [[9-odd-limit|9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo|4EDO]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while 2EDO represents the [[3-limit]] consistently, 3EDO the [[5-limit]], 4EDO the [[7-limit]] and 5EDO the 9-limit, to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo|22EDO]]. Nevertheless, because the comma tempered out for this EDO's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second EDO to demonstrate 3-to-2 [[telicity]].

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.

[[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.

* By its cardinality, 5EDO is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s.

* Due to the interest around the "fifth" interval size, there are many [[nonoctave]] "stretch sisters" to 5EDO: square root of 4/3, cube root of 3/2, 8th root of 3, etc.

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.

In contrast to other EDOs, all of the notes can be used at once in order to get a functioning scale. (As in Blackwood in [[10edo|10EDO]]).

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.

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

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

* [[Paul Rubenstein]]: various, with electric guitars in 10- and 15EDO

[[Category:Prime EDO]]