117edo: Difference between revisions

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'''117edo''' is the [[EDO|equal division of the octave]] into 117 parts of 10.2564102564 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: <117 185 272 328 405| (optimal patent val), <117 186 272 329 405| (117bd), <117 185 271 328 404| (117ce), and <117 185 272 329 405| (117d). Using the patent val, it tempers out the [[syntonic comma]], 81/80 and |69 -1 -29> in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, providing the [[optimal patent val]] for 7-limit [[Meantone family|mohajira temperament]]; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, providing the optimal patent val for the [[Didymus rank three family|terpsichore rank-3 temperament]]. Using the 117bd val, it tempers out the [[Kleismic family|kleisma]], 15625/15552 and 17179869184/16142520375 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. Using the 117ce val, it tempers out the [[Magic family|small diesis]], 3125/3072 and 282429536481/268435456000 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. Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit, supporting the 11-limit [[Meantone family|meantone temperament]].

[[~~Category~~:~~Edo~~]]

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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-'''117edo''' is the [[EDO|equal division of the octave]] into 117 parts of 10.2564102564 cents each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: &lt;117 185 272 328 405| (optimal patent val), &lt;117 186 272 329 405| (117bd), &lt;117 185 271 328 404| (117ce), and &lt;117 185 272 329 405| (117d). Using the patent val, it tempers out the [[syntonic comma]], 81/80 and |69 -1 -29&gt; in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, providing the [[optimal patent val]] for 7-limit [[Meantone family|mohajira temperament]]; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, providing the optimal patent val for the [[Didymus rank three family|terpsichore rank-3 temperament]]. Using the 117bd val, it tempers out the [[Kleismic family|kleisma]], 15625/15552 and 17179869184/16142520375 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. Using the 117ce val, it tempers out the [[Magic family|small diesis]], 3125/3072 and 282429536481/268435456000 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. Using the 117d val, it tempers out 126/125, 225/224, and 14495514624/13841287201 in the 7-limit; 99/98, 176/175, 441/440, and 12582912/12400927 in the 11-limit, supporting the 11-limit [[Meantone family|meantone temperament]].
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+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}}