169edo: Difference between revisions

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=~~<span style~~=~~"color: #76653a; font-family: 'Times New Roman',Times,serif; font-size: 113%;">169 equal divisions per octave</span>~~=

~~'''169edo''' is the [[EDO|equal division of the octave]] into 169 parts of 7.1006 [[cent]]s each.~~ It is inconsistent to the 5-limit and higher limit, with two mappings possible for the 5-limit: <169 268 392| (patent val) and <169 268 393| (169c). Using the patent val, it tempers out the [[Sycamore family|sycamore comma]], 48828125/47775744 and the rodan comma, 131072000/129140163 in the 5-limit; 245/243, 1029/1024, and 9765625/9633792 in the 7-limit; 385/384, 441/440, 896/891, and 312500/307461 in the 11-limit; 676/675, 975/968, and 1625/1617 in the 13-limit. Using the 169cdf val, it tempers out the valentine comma, 1990656/1953125 and the [[Vulture family|vulture comma]], 10485760000/10460353203 in the 5-limit; 1728/1715, 5120/5103, and 235298/234375 in the 7-limit; 176/175, 540/539, 8019/8000, and 43923/43904 in the 11-limit; 351/350, 352/351, 640/637, and 676/675 in the 13-limit, supporting the [[Hemifamity temperaments|buzzard temperament]]. Using the 169d val, it tempers out 225/224, 51200/50421, and 1071875/1062882 in the 7-limit; 2200/2187, 2420/2401, 2560/2541, and 6000/5929 in the 11-limit; 169/168, 364/363, 640/637, and 676/675 in the 13-limit.

==Theory==

It is inconsistent to the 5-limit and higher limit, with two mappings possible for the 5-limit: <169 268 392| (patent val) and <169 268 393| (169c). Using the patent val, it tempers out the [[Sycamore family|sycamore comma]], 48828125/47775744 and the rodan comma, 131072000/129140163 in the 5-limit; 245/243, 1029/1024, and 9765625/9633792 in the 7-limit; 385/384, 441/440, 896/891, and 312500/307461 in the 11-limit; 676/675, 975/968, and 1625/1617 in the 13-limit. Using the 169cdf val, it tempers out the valentine comma, 1990656/1953125 and the [[Vulture family|vulture comma]], 10485760000/10460353203 in the 5-limit; 1728/1715, 5120/5103, and 235298/234375 in the 7-limit; 176/175, 540/539, 8019/8000, and 43923/43904 in the 11-limit; 351/350, 352/351, 640/637, and 676/675 in the 13-limit, supporting the [[Hemifamity temperaments|buzzard temperament]]. Using the 169d val, it tempers out 225/224, 51200/50421, and 1071875/1062882 in the 7-limit; 2200/2187, 2420/2401, 2560/2541, and 6000/5929 in the 11-limit; 169/168, 364/363, 640/637, and 676/675 in the 13-limit.

=== Harmonics ===

[[Category:Equal divisions of the octave]]

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-=<span style="color: #76653a; font-family: 'Times New Roman',Times,serif; font-size: 113%;">169 equal divisions per octave</span>=
+{{EDO intro|169}}
-'''169edo''' is the [[EDO|equal division of the octave]] into 169 parts of 7.1006 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with two mappings possible for the 5-limit: &lt;169 268 392| (patent val) and &lt;169 268 393| (169c). Using the patent val, it tempers out the [[Sycamore family|sycamore comma]], 48828125/47775744 and the rodan comma, 131072000/129140163 in the 5-limit; 245/243, 1029/1024, and 9765625/9633792 in the 7-limit; 385/384, 441/440, 896/891, and 312500/307461 in the 11-limit; 676/675, 975/968, and 1625/1617 in the 13-limit. Using the 169cdf val, it tempers out the valentine comma, 1990656/1953125 and the [[Vulture family|vulture comma]], 10485760000/10460353203 in the 5-limit; 1728/1715, 5120/5103, and 235298/234375 in the 7-limit; 176/175, 540/539, 8019/8000, and 43923/43904 in the 11-limit; 351/350, 352/351, 640/637, and 676/675 in the 13-limit, supporting the [[Hemifamity temperaments|buzzard temperament]]. Using the 169d val, it tempers out 225/224, 51200/50421, and 1071875/1062882 in the 7-limit; 2200/2187, 2420/2401, 2560/2541, and 6000/5929 in the 11-limit; 169/168, 364/363, 640/637, and 676/675 in the 13-limit.
+==Theory==
+It is inconsistent to the 5-limit and higher limit, with two mappings possible for the 5-limit: &lt;169 268 392| (patent val) and &lt;169 268 393| (169c). Using the patent val, it tempers out the [[Sycamore family|sycamore comma]], 48828125/47775744 and the rodan comma, 131072000/129140163 in the 5-limit; 245/243, 1029/1024, and 9765625/9633792 in the 7-limit; 385/384, 441/440, 896/891, and 312500/307461 in the 11-limit; 676/675, 975/968, and 1625/1617 in the 13-limit. Using the 169cdf val, it tempers out the valentine comma, 1990656/1953125 and the [[Vulture family|vulture comma]], 10485760000/10460353203 in the 5-limit; 1728/1715, 5120/5103, and 235298/234375 in the 7-limit; 176/175, 540/539, 8019/8000, and 43923/43904 in the 11-limit; 351/350, 352/351, 640/637, and 676/675 in the 13-limit, supporting the [[Hemifamity temperaments|buzzard temperament]]. Using the 169d val, it tempers out 225/224, 51200/50421, and 1071875/1062882 in the 7-limit; 2200/2187, 2420/2401, 2560/2541, and 6000/5929 in the 11-limit; 169/168, 364/363, 640/637, and 676/675 in the 13-limit.
+=== Harmonics ===
+{{Harmonics in equal|169}}
 [[Category:Equal divisions of the octave]]