107edo: Difference between revisions

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== Theory ==

107edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, ~~with four~~ [[~~mapping~~]]s ~~possible for~~ the 7-limit: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).

107edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 [[subgroup]] can be used. For the full 7-limit, it has four possible [[mapping]]s: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).

Using the patent val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| 28 -22 3 }} in the 5-limit; [[1029/1024]], [[2240/2187]], and [[3125/3087]] in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit.

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=== Odd harmonics ===

=== Octave stretch ===

107edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by [[AS|1ed175/174]], a [[Octave stretch|stretched-octave]] version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.

There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve some harmonics at the expense of others: 622zpi, 623zpi, 624zpi, 625zpi, 626zpi, 627zpi, 628zpi and 629zpi.

The details of each of those ZPI tunings are visible in [[User:Contribution]]’s gallery of [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)|Zeta Peak Indexes (1 - 10 000)]]. Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.

=== Subsets and supersets ===

107edo is the 28th [[prime edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.

107edo is the 28th [[prime edo]], following [[103edo]] and before [[109edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|107}}
+{{ED intro}}
 == Theory ==
-edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with four [[mapping]]s possible for the 7-limit: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 [[subgroup]] can be used. For the full 7-limit, it has four possible [[mapping]]s: {{val| 107 170 248 300 }} ([[patent val]]), {{val| 107 '''169''' 248 300 }} (107b), {{val| 107 170 '''249''' 300 }} (107c), and {{val| 107 170 '''249''' '''301''' }} (107cd).
 Using the patent val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| 28 -22 3 }} in the 5-limit; [[1029/1024]], [[2240/2187]], and [[3125/3087]] in the 7-limit; [[100/99]], 1232/1215, and 1331/1323 in the 11-limit.
@@ Line 15: / Line 15: @@
 === Odd harmonics ===
 {{Harmonics in equal|107}}
+=== Octave stretch ===
+edo’s approximations of 3/1, 5/1, 7/1, 13/1, 17/1 and 19/1 are all improved by [[AS|1ed175/174]], a [[Octave stretch|stretched-octave]] version of 107edo. The trade-off is a slightly worse 2/1 and 11/1.
+There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve some harmonics at the expense of others: 622zpi, 623zpi, 624zpi, 625zpi, 626zpi, 627zpi, 628zpi and 629zpi.
+The details of each of those ZPI tunings are visible in [[User:Contribution]]’s gallery of [[User:Contribution/Gallery of Zeta Peak Indexes (1 - 10 000)|Zeta Peak Indexes (1 - 10 000)]]. Warning: due to its length, that page may slow down your device while it is open. The effect will go away after you close the page.
 === Subsets and supersets ===
-edo is the 28th [[prime edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.
+edo is the 28th [[prime edo]], following [[103edo]] and before [[109edo]]. [[214edo]], which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.
 == Intervals ==