5edo: Difference between revisions
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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 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). | |||
5edo is | 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]]. | |||
Despite its lack of accuracy in the 5-limit, 5edo is the second [[zeta integral edo]], after [[2edo]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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| 1 | | 1 | ||
| 240 | | 240 | ||
| | | Second-inter-third | ||
| | | | ||
| [[144/125]] (-4.969)<br>[[125/108]] (-13.076) | | [[144/125]] (-4.969)<br>[[125/108]] (-13.076) | ||
| Line 65: | Line 73: | ||
| 2 | | 2 | ||
| 480 | | 480 | ||
| | | Fourth | ||
| [[4/3]] (-18.045) | | [[4/3]] (-18.045) | ||
| | | | ||
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| 3 | | 3 | ||
| 720 | | 720 | ||
| | | Fifth | ||
| [[3/2]] (+18.045) | | [[3/2]] (+18.045) | ||
| | | | ||
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| 4 | | 4 | ||
| 960 | | 960 | ||
| | | Sixth-inter-seventh | ||
| | | | ||
| [[216/125]] (+13.076)<br>[[125/72]] (+4.969) | | [[216/125]] (+13.076)<br>[[125/72]] (+4.969) | ||
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== Approximation to JI == | == Approximation to JI == | ||
=== Selected 7-limit intervals === | === Selected 7-limit intervals === | ||
[[File: | [[File:5ed2-001.svg]] | ||
== Observations == | == Observations == | ||
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* [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid] | * [https://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid http://commons.wikimedia.org/wiki/File:5-tet_scale_on_C.mid] | ||
== Instruments == | |||
* [[Lumatone mapping for 5edo]] | |||
== Music == | == Music == | ||
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<references group="note" /> | <references group="note" /> | ||
[[Category:3-limit record edos|#]] <!-- 1-digit number --> | |||
[[Category:5-tone scales]] | [[Category:5-tone scales]] | ||