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[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]
[[File:5edo scale.mp3|thumb|A chromatic 5edo scale on C.]]


5edo is the smallest edo that contains a usable (albeit very sharp) [[Perfect fifth (interval region)|perfect fifth]]. 5edo can be seen as a simplified form of the familiar pentatonic scale, which has the [[mos]] pattern [[2L 3s]] (known as ''pentic''), with 5edo specifically being the [[equalized]] tuning of 2L 3s. 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.
5edo is the smallest edo that contains a usable [[Perfect fifth (interval region)|perfect fifth]] at 720{{C}}, 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.


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 18 [[cent]]s sharp, and 7 being about 9 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.
5edo is the basic example of an [[equipentatonic]] scale, as in 5edo all steps are exactly the same size.


With more complex intervals, however, 5edo becomes increasingly inaccurate. For example, the supermajor third 9/7 is mapped to 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.  
{{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.


If we extend our scope to the full [[7-limit]] (including prime [[5/1|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).
In terms of just intonation, besides the perfect fifth, 5edo also contains a relatively accurate approximation the harmonic seventh [[7/4]] at 4 steps (960{{C}}), being 8.8{{C}} flat of just. 5edo can thus be used as a simplified version of the [[2.3.7 subgroup]], and defines much of its underlying structure. For example, 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 known as [[slendric]] temperament, where [[1029/1024]], the gamelisma, is tempered out. Two intervals of [[7/6]] or 8/7 make the perfect fourth [[4/3]], tempering out [[49/48]], known as [[semaphore]] temperament. Finally, the harmonic seventh may be found by going up two perfect fourths, tempering out [[64/63]], which is [[superpyth]] temperament (sometimes known as ''archy'' in the 2.3.7 subgroup).


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.  
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.  


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 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.  


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]]).  
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.


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


== Intervals ==
== Intervals ==
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