5edo: Difference between revisions

| en = 5edo

5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{Cent}}, being 18{{C}} sharp of a [[just]]ly tuned [[3/2]] ratio at 702{{C}}. As such, it is the smallest edo where elements of traditional music theory begin to make sense.

The 720{{C}} fifth generates an [[equalized]] tuning of the [[pentic]] (2L 3s) scale, where every step is the same size at 240{{C}}, or one step of 5edo. It also generates a [[collapsed]] tuning of the [[diatonic]] (5L 2s) scale, where the [[diatonic semitone]] or minor second is mapped to 0 steps, meaning that E and F as well as B and C are the same note in 5edo.

5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.

{{W|Tertian harmony}} is also possible in 5edo, but barely: the only chords available are suspended chords, which may also be seen as inframinor (very flat minor) and ultramajor (very sharp major) chords, also known as [[Extraclassical tonality|arto and tendo]] 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.

 

With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third [[9/7]] is mapped very sharply to 480{{C}}, which is the same interval as the perfect fourth. Thus [[28/27]] is tempered out, leading to the rather inaccurate [[Trienstonic clan|trienstonic]] temperament. However, this interval can still be used as a third, as referenced above.

 

If we attempt to add prime [[5/1|5]] to the mix and extend 5et to the full [[7-limit]], then the major third [[5/4]] is mapped very sharply to 2 steps (480{{C}}), almost a full semitone sharper than the just 5/4 at 386.3{{C}}. This results in 5edo supporting several [[exotemperament]]s when intervals of 5 are introduced. For example, the best 5/4 is the same interval as 4/3, meaning that the semitone that separates them in JI, [[16/15]], is tempered out, leading to 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 5edo's step is so large, such 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 major thirds and fourths are equivalent.

 

If 5edo is taken as only a tuning of the [[3-limit]], we find that the circle of fifths closes after only 5 steps, rather than 12, meaning [[256/243]] is tempered out. 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 256/243 semitone at 90.2{{C}}, is smaller than half a step at 120{{C}}, 5edo demonstrates [[Telicity|3-to-2 telicity]], and is in fact 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 [[octave-reduced]] step to harmonics 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]].

 

Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]].

 

 

5edo is the 3rd [[prime edo]], after [[2edo]] and [[3edo]] and before [[7edo]]. It does not contain any nontrivial subset edos, though it contains 5 equal divisions of the double octave [[4/1]], or [[5ed4]]. Multiples of 5edo, such as [[10edo]], [[15edo]], …, up to [[35edo]], share the same tuning of the perfect fifth as 5edo, while improving on other intervals.

 

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Intervals of 5edo

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])

| [[23/20]] (-1.960)<br>[[31/27]] (+0.829)<br>[[224/195]] (-0.030)

| [[File:0-240 second, third (5-EDO).mp3|frameless]]

| [[4/3]] (-18.045)

| [[File:0-480 fourth (5-EDO).mp3|frameless]]

| [[3/2]] (+18.045)

| [[32/21]] (-9.219)

| [[50/33]] (+0.686)<br>[[91/60]] (+1.085)

| [[File:0-720 fifth (5-EDO).mp3|frameless]]

| [[12/7]] (+26.871)<br>[[7/4]] (-8.826)

| [[40/23]] (+1.960)<br>[[54/31]] (-0.829)<br>[[195/112]] (+0.030)

| [[File:0-1200 octave.mp3|frameless]]

== Notation ==

The usual [[Musical notation|notation system]] for 5edo is the heptatonic [[chain-of-fifths notation]], which is directly derived from the standard notation used in [[12edo]]. The [[enharmonic unison]] is the minor 2nd, thus E and F are the same pitch.

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

! colspan="2" | [[Chain-of-fifths notation]]

|-

|-

| 1

|-

|-

|-

|-

* [[ups and downs notation]] is identical to circle-of-fifths notation;

* mixed [[sagittal notation]] is identical to circle-of-fifths notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===

This notation uses the same sagittal sequence as EDOs [[12edo#Sagittal notation|12]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], and is a subset of the notations for EDOs [[10edo#Sagittal notation|10]], [[15edo#Sagittal notation|15]], [[20edo#Sagittal notation|20]], [[25edo#Sagittal notation|25]], [[30edo#Sagittal notation|30]], and [[35edo#Second-best fifth notation|35b]].

Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.

=== Alternative notations ===

 

 

 

 

 

 

 

 

=== Harmony ===

5edo does not have any strong consonance nor dissonance. It could be considered [[omniconsonant scale|omniconsonant]]. 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]]).

* 0+1+3

* 0+2+3

* 0+1+3+4

* 0+2+3+4

 

 

 

 

 

 

 

 

 

 

 

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

== See also ==

* [[Alpha, beta, and gamma family of equal divisions]]

<references group="note" />

[[Category:3-limit record edos|#]] <!-- 1-digit number -->

[[Category:5-tone scales]]