5edo: Difference between revisions
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'''5 equal divisions of the octave''' (or '''5edo''') is the [[tuning system]] derived by dividing the [[octave]] into 5 equal steps of 240 [[cent]]s each, or the fifth root of two. 5edo is the third [[prime edo]], after [[ | '''5 equal divisions of the octave''' (or '''5edo''') is the [[tuning system]] derived by dividing the [[octave]] into 5 equal steps of 240 [[cent]]s each, or the fifth root of two. 5edo is the third [[prime edo]], after [[2edo]] and [[3edo]]. Most importantly, 5edo is the smallest [[edo]] containing xenharmonic intervals — 1edo, 2edo, 3edo, and 4edo are all subsets of [[12edo]]. | ||
== Theory == | == Theory == | ||
{{Harmonics in equal|5|intervals=odd}} | {{Harmonics in equal|5|intervals=odd}} | ||
If 5edo is regarded as a temperament, which is to say as | If 5edo is regarded as a temperament, which is to say as 5tet, then the most salient fact is that 16/15 is tempered out. This means in 5-TET the major third and the fourth, and the minor sixth and the fifth, are not distinguished; this is 5-limit [[father]] temperament. | ||
Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[bug]] temperament, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence. | Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[bug]] temperament, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as I-III-V-I and involves triads with common intervals because of fourth-thirds equivalence. | ||
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Despite its lack of accuracy, 5edo is the second [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]], after [[2edo]]. It also is the smallest equal division representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second EDO to demonstrate 3-to-2 [[telicity]] — that is, when not counting the comparatively trivial [[1edo]]. | Despite its lack of accuracy, 5edo is the second [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]], after [[2edo]]. It also is the smallest equal division representing the [[9-odd-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The_Seven_Limit_Symmetrical_Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5edo. However, while 2edo represents the [[3-limit]] consistently, 3edo the [[5-limit]], 4edo the [[7-limit]] and 5edo the 9-limit, to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. Nevertheless, because the comma tempered out for this edo's circle of fifths is [[256/243]], and since this interval is smaller than half a step, 5edo is the second EDO to demonstrate 3-to-2 [[telicity]] — that is, when not counting the comparatively trivial [[1edo]]. | ||
In addition, considering 5edo as a no-5s temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next | In addition, considering 5edo as a no-5s temperament improves its standing significantly. It is especially prominent as a simple 2.3.7 temperament with high relative accuracy (the next edo doing it better being [[17edo|17]]), and is the optimal patent val for the no-5s [[Trienstonic clan|trienstonic]] (or [[Color notation/Temperament Names|Zo]]) temperament. | ||
== Intervals == | == Intervals == | ||
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== Observations == | == Observations == | ||
=== Related scales === | === Related scales === | ||
* By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s. | * By its cardinality, 5edo is related to other [[pentatonic]] scales, and it is especially close in sound to many Indonesian [[slendro]]s. | ||
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** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5 = 7.985099. | ** Make a chain of five "bigger fifths" (50/33), which makes three octaves 3.227¢ flat. (50/33)^5 = 7.985099. | ||
=== Cycles, | === Cycles, divisions === | ||
5 is a prime number so 5edo contains no sub-edos. Only simple cycles: | 5 is a prime number so 5edo contains no sub-edos. Only simple cycles: | ||
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<references/> | <references/> | ||
== Ear | == Ear training == | ||
5edo ear-training exercises by Alex Ness available here: | 5edo ear-training exercises by Alex Ness available here: | ||
* https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web | * https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web | ||
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[[Category:5-tone scales]] | [[Category:5-tone scales]] | ||
[[Category:7-limit]] | [[Category:7-limit]] | ||
[[Category:9-limit]] | [[Category:9-odd-limit]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Macrotonal]] | [[Category:Macrotonal]] | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||