5edo: Difference between revisions

5edo is the basic example of an [[equipentatonic]] scale, containing a sharp but usable [[Perfect fifth (interval region)|perfect fifth]], and can be seen as a simplified form of the familiar [[pentic]] scale. Tertian harmony is possible in 5edo, but barely: the only chords available are suspended chords, which [[Extraclassical tonality|may also be seen as]] inframinor (very flat minor) and ultramajor (very sharp major) chords, due to how sharp the fifth is. As a result, many triads will share the same three notes, so rootedness is much more important to explicitly establish.

In terms of just intonation, 5edo represents the perfect fifth 3/2 and harmonic seventh 7/4 rather accurately for how wide the steps are, with 3 being about 20 cents sharp, and 7 being about 10 cents flat. In 5edo, the perfect fifth is 3 steps, meaning it can be divided into 3 equal parts, each representing the supermajor second 8/7. This is [[slendric]] temperament. Two of these parts make the perfect fourth [[4/3]], which is [[semaphore]] temperament, and finally the harmonic seventh may be found by going up two perfect fourths, which is [[superpyth]] or "archy" temperament. This all means that 5edo contains a very simplified form of the [[2.3.7 subgroup]], and many scales in 2.3.7 take a pentatonic form.

With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third 9/7 is the same interval as the perfect fourth, which is a rather inaccurate equivalence (specifically, [[Trienstonic clan|trienstonic]] temperament). However, this can still be used as a third, as referenced in the top paragraph.  

If we extend our scope to the full 7-limit (including 5, and thus conventional major and minor thirds), then the most salient fact is that the best approximation of the major third 5/4 is extremely inaccurate, almost a full semitone sharper than just. This results in 5edo supporting several [[Exotemperament|exotemperaments]] when intervals of 5 are introduced. For example, the best 5/4 of 480 cents is in fact the same interval as 4/3, meaning that the semitone that usually separates them, [[16/15]], is [[tempered out]] (which is the very inaccurate [[father]] temperament).

Exploring more complex intervals, we find that the minor tone [[10/9]] and the minor third [[6/5]] are best mapped to the same step of 240 cents, meaning that the semitone separating them, [[27/25]], is tempered out as well - this is [[bug]] temperament, which is a little more perverse even than father.  

Because these intervals are so large, this sort of analysis is less significant with 5edo than it becomes with larger and more accurate divisions, but it still plays a role. For example, if we attempt to analyze 5edo as supporting standard [[Diatonic functional harmony|diatonic harmony]], I–IV–V–I is the same as I–III–V–I and involves triads with common intervals because of fourth-thirds equivalence.

If 5edo is taken as only a tuning of the 3-limit, we find that the circle of fifths returns to the unison after only 5 steps, rather than 12. This is called [[blackwood]] temperament, and in 5edo, this is a "good" tuning of a circle of fifths - more formally, since the comma being tempered out, the semitone 256/243, is smaller than half a step (120 cents), 5edo demonstrates [[Telicity|3-to-2 telicity]] (and is the third EDO to do so after [[1edo]] and [[2edo]]).  

5edo is the smallest edo representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo 5 to 1, 3, 5, 7 and 9 - specifically, 3 is mapped to 3 steps (720 cents), 5 is very inaccurately mapped to 2 steps (480 cents), 7 is mapped to 4 steps (960 cents), and 9 is mapped to 1 step (240 cents). However, while 2edo represents the [[3-odd-limit]] consistently, 3edo the [[5-odd-limit]], 4edo the [[7-odd-limit]] and 5edo the 9-odd-limit, to represent the [[11-odd-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]].  

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains [[5ed4]]. Multiples such as [[10edo]], [[15edo]], … up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

| [[224/195]] (-0.030)

| [[33/25]] (-0.686)

| [[50/33]] (+0.686)

[[91/60]] (+1.085)

| [[195/112]] (+0.030)

| Dicot comma, classic chroma

* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]