5edo: Difference between revisions

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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|9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo|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|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]].

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.

=== Differences between distributionally-even scales and smaller EDOs ===

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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|9-limit]] [[consistent|consistently]], giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo|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|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]].
+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.
 === Differences between distributionally-even scales and smaller EDOs ===