5edo: Difference between revisions

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

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.

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.

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.

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.

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.

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.

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).

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).

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.

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