5edo: Difference between revisions

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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]].

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]].

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 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]]).
-edo 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]].
+edo 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]].