10edo: Difference between revisions

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Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament, which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

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 Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.
-By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.
+By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.
 edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].