117edo: Difference between revisions

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~~'''117edo''' is the [[~~EDO|~~equal division of the octave]] into~~ 117 ~~parts of 10.2564102564 [[cent]]s each.~~

It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: <117 185 272 328 405| (patent val), <117 186 272 329 405| (117bd), <117 185 271 328 404| (117ce), and <117 185 272 329 405| (117d).

== Theory ==

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).

=== Commas ===

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, supporting the 7-limit [[~~Mohajira~~]] temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the [[Didymus rank three family|terpsichore ~~rank-3 temperament~~]]; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.

Using the patent val, it tempers out the syntonic comma ([[81/80]]) and {{monzo| 69 -1 -29 }} in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting 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]] temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.

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; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.

Using the 117bd val, it tempers out the 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; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-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; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.

Using the 117ce val, it tempers out the magic comma ([[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; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-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; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[~~Grosstone~~]] temperament.

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; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[grosstone]] temperament.

=== Harmonics ===

~~[[Category:Equal divisions of the octave|###]] ~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''117edo''' is the [[EDO|equal division of the octave]] into 117 parts of 10.2564102564 [[cent]]s each.
+{{EDO intro|117}}
-It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 11-limit: &lt;117 185 272 328 405| (patent val), &lt;117 186 272 329 405| (117bd), &lt;117 185 271 328 404| (117ce), and &lt;117 185 272 329 405| (117d).
 == Theory ==
+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).
 === Commas ===
-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, supporting the 7-limit [[Mohajira]] temperament; 540/539, 1344/1331, 1617/1600, and 3168/3125 in the 11-limit, supporting the [[Didymus rank three family|terpsichore rank-3 temperament]]; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.
+Using the patent val, it tempers out the syntonic comma ([[81/80]]) and {{monzo| 69 -1 -29 }} in the 5-limit; 6144/6125, 31104/30625, and 403368/390625 in the 7-limit, supporting 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]] temperament; 144/143, 196/195, 364/363, 729/715, and 3146/3125 in the 13-limit.
-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; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-limit.
+Using the 117bd val, it tempers out the 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; 169/168, 364/363, 640/637, 832/825, and 3200/3159 in the 13-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; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-limit.
+Using the 117ce val, it tempers out the magic comma ([[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; 105/104, 275/273, 1287/1280, and 2025/2002 in the 13-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; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[Grosstone]] temperament.
+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; 144/143, 640/637, 648/637, 1001/1000, and 43940/43923 in the 13-limit, supporting the 13-limit [[grosstone]] temperament.
 === Harmonics ===
 {{Harmonics in equal|117}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->