169edo: Difference between revisions

(10 intermediate revisions by 6 users not shown)

Line 1:

~~=<span style="color: #b6a140; 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 5.38117 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with two mappings possible for the 5-limit: <169 268 392| (optimal patent val) and <169 268 393| (169c). Using the optimal 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. Using the 169c 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 using the alternative 169cd val.

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

169edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with two mappings possible for the 5-limit: {{val| 169 268 392 }} ([[patent val]]) and {{val| 169 268 393 }} (169c).

Using the patent val, it tempers out the [[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 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.

Using the 169cdf val, it tempers out the [[valentine comma]], 1990656/1953125 and the [[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, [[support]]ing the [[buzzard]] temperament.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 169 factors into {{factorization|169}}, 169edo contains [[13edo]] as its only nontrivial subset. [[338edo]], which doubles it, provides good correction for the approximations of harmonics [[5/1|5]] and [[7/1|7]].

@@ Line 1: / Line 1: @@
-=<span style="color: #b6a140; font-family: 'Times New Roman',Times,serif; font-size: 113%;">169 equal divisions per octave</span>=
+{{Infobox ET}}
-'''169edo''' is the [[EDO|equal division of the octave]] into 169 parts of 5.38117 [[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| (optimal patent val) and &lt;169 268 393| (169c). Using the optimal 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. Using the 169c 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 using the alternative 169cd val.
+{{ED intro}}
-[[Category:Edo]]
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with two mappings possible for the 5-limit: {{val| 169 268 392 }} ([[patent val]]) and {{val| 169 268 393 }} (169c).
+Using the patent val, it tempers out the [[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 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.
+Using the 169cdf val, it tempers out the [[valentine comma]], 1990656/1953125 and the [[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, [[support]]ing the [[buzzard]] temperament.
+=== Prime harmonics ===
+{{Harmonics in equal|169}}
+=== Subsets and supersets ===
+Since 169 factors into {{factorization|169}}, 169edo contains [[13edo]] as its only nontrivial subset. [[338edo]], which doubles it, provides good correction for the approximations of harmonics [[5/1|5]] and [[7/1|7]].