117edo: Difference between revisions

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~~117~~ is ~~a meantone edo~~ with a ~~step size~~ of 10.~~2564102564 cents~~. [[~~Category~~:~~Stub~~|~~#stub~~]]

117edo is in[[consistent]] to the [[5-odd-limit]] and higher odd limits, with four mappings possible for the [[11-limit]]: {{val| 117 185 272 328 405 }} ([[patent val]]), {{val| 117 186 '''272''' 329 '''405''' }} (117bd), {{val| 117 185 '''271''' 328 '''404''' }} (117ce), and {{val| 117 185 272 '''329''' 405 }} (117d).

Using the patent val, it [[tempering out|tempers out]] 81/80 ([[syntonic comma]]) and {{monzo| 69 -1 -29 }} in the 5-limit; [[6144/6125]], 31104/30625, and 403368/390625 in the 7-limit, [[support]]ing the 7-limit [[mohajira]] temperament; [[540/539]], 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the rank-3 [[Didymus rank three family #Terpsichore|terpsichore]] temperament; [[144/143]], [[196/195]], [[364/363]], 729/715, and 3146/3125 in the 13-limit.

Using the 117d val, it tempers out [[126/125]], [[225/224]], and {{monzo| 29 3 0 -12 }} in the 7-limit; [[99/98]], [[176/175]], [[441/440]], and 12582912/12400927 in the 11-limit; 144/143, [[640/637]], 648/637, [[1001/1000]], and [[Kuragesma|43940/43923]] in the 13-limit, supporting the 13-limit [[grosstone]] temperament.

Using the 117ce val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| -31 24 -3 }} in the 5-limit; [[2401/2400]], 3645/3584, and [[4375/4374]] in the 7-limit; [[243/242]], 441/440, and 1815/1792 in the 11-limit; [[105/104]], [[275/273]], [[1287/1280]], and 2025/2002 in the 13-limit.

Using the 117bd val, it tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 34 -17 -3 }} in the 5-limit; [[245/243]], [[3136/3125]], and 51200/50421 in the 7-limit; 176/175, 1232/1215, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], 640/637, [[832/825]], and 3200/3159 in the 13-limit.

== Odd harmonics ==

== Octave stretch ==

117edo’s approximations of 3/1, 5/1, 7/1 and 17/1 are all noticeably improved by [[Gallery of arithmetic pitch sequences#APS of hekts|APS7hekt]], a [[Octave shrinking|compressed-octave]] version of 117edo. The trade-off is an unnoticeably worse 2/1 and 11/1, but noticeably worse 13/1.

There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 696zpi, 697zpi, 698zpi, 699zpi, 700zpi, 701zpi and 702zpi.

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 ==

Since 117 factors into {{factorization|117}}, 117edo has subset edos {{EDOs| 3, 9, 13, and 39 }}. [[234edo]], which doubles it, provides a correction for the approximation to harmonic 3.

== Intervals ==

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-is a meantone edo with a step size of 10.2564102564 cents. [[Category:Stub|#stub]]
+{{Infobox ET}}
+{{ED intro}}
+edo is in[[consistent]] to the [[5-odd-limit]] and higher odd limits, with four mappings possible for the [[11-limit]]: {{val| 117 185 272 328 405 }} ([[patent val]]), {{val| 117 186 '''272''' 329 '''405''' }} (117bd), {{val| 117 185 '''271''' 328 '''404''' }} (117ce), and {{val| 117 185 272 '''329''' 405 }} (117d).
+Using the patent val, it [[tempering out|tempers out]] 81/80 ([[syntonic comma]]) and {{monzo| 69 -1 -29 }} in the 5-limit; [[6144/6125]], 31104/30625, and 403368/390625 in the 7-limit, [[support]]ing the 7-limit [[mohajira]] temperament; [[540/539]], 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the rank-3 [[Didymus rank three family #Terpsichore|terpsichore]] temperament; [[144/143]], [[196/195]], [[364/363]], 729/715, and 3146/3125 in the 13-limit.
+Using the 117d val, it tempers out [[126/125]], [[225/224]], and {{monzo| 29 3 0 -12 }} in the 7-limit; [[99/98]], [[176/175]], [[441/440]], and 12582912/12400927 in the 11-limit; 144/143, [[640/637]], 648/637, [[1001/1000]], and [[Kuragesma|43940/43923]] in the 13-limit, supporting the 13-limit [[grosstone]] temperament.
+Using the 117ce val, it tempers out 3125/3072 ([[magic comma]]) and {{monzo| -31 24 -3 }} in the 5-limit; [[2401/2400]], 3645/3584, and [[4375/4374]] in the 7-limit; [[243/242]], 441/440, and 1815/1792 in the 11-limit; [[105/104]], [[275/273]], [[1287/1280]], and 2025/2002 in the 13-limit.
+Using the 117bd val, it tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 34 -17 -3 }} in the 5-limit; [[245/243]], [[3136/3125]], and 51200/50421 in the 7-limit; 176/175, 1232/1215, [[1375/1372]], and 2560/2541 in the 11-limit; [[169/168]], [[364/363]], 640/637, [[832/825]], and 3200/3159 in the 13-limit.
+== Odd harmonics ==
+{{Harmonics in equal|117}}
+== Octave stretch ==
+edo’s approximations of 3/1, 5/1, 7/1 and 17/1 are all noticeably improved by [[Gallery of arithmetic pitch sequences#APS of hekts|APS7hekt]], a [[Octave shrinking|compressed-octave]] version of 117edo. The trade-off is an unnoticeably worse 2/1 and 11/1, but noticeably worse 13/1.
+There are also several nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 696zpi, 697zpi, 698zpi, 699zpi, 700zpi, 701zpi and 702zpi.
+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 ==
+Since 117 factors into {{factorization|117}}, 117edo has subset edos {{EDOs| 3, 9, 13, and 39 }}. [[234edo]], which doubles it, provides a correction for the approximation to harmonic 3.
+== Intervals ==
+{{Interval table}}