170edo: Difference between revisions

(7 intermediate revisions by 6 users not shown)

Line 1:

'''170edo''' is the [[EDO|equal division of the octave]] into 170 parts of 7.0588 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: <170 269 395 477| (optimal patent val), <170 270 395 477| (170b), <170 270 395 478| (170bd), and <170 269 394 477| (170c). Using the patent val, it tempers out the valentine comma, 1990656/1953125 and 3486784401/3355443200 in the 5-limit; 126/125, 1029/1024, and 215233605/210827008 in the 7-limit; 540/539, 1944/1925, 2835/2816, and 43923/43904 in the 11-limit. Using the 170b val, it tempers out the [[Diaschismic family|diaschisma]], 2048/2025 and the [[kleisma]], 15625/15552 in the 5-limit; 33614/32805, 50421/50000, and 84035/82944 in the 7-limit; 385/384, 1232/1215, 1331/1323, and 6250/6237 in the 11-limit. Using the 170bd val, it tempers out 16875/16807, 51200/50421, and 420175/419904 in the 7-limit; 176/175, 896/891, and 6875/6804 in the 11-limit. Using the alternative 170bde val, it tempers out 540/539, 1375/1372, 4375/4356, and 8192/8085 in the 11-limit. Using the 170c val, it tempers out the double large green comma, 43046721/41943040 and the [[Sycamore family|sycamore comma]], 48828125/47775744 in the 5-limit; 1029/1024, 4375/4374, and 78125/76832 in the 7-limit; 385/384, 441/440, and 8019/8000 in the 11-limit.

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

170edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with four mappings possible for the 7-limit: {{val| 170 269 395 477 }} ([[patent val]]), {{val| 170 '''270''' 395 477 }} (170b), {{val| 170 '''270''' 395 '''478''' }} (170bd), and {{val| 170 269 '''394''' 477 }} (170c).

Using the patent val, it [[tempering out|tempers out]] the [[valentine comma]], 1990656/1953125 and 3486784401/3355443200 in the 5-limit; [[126/125]], [[1029/1024]], and 215233605/210827008 in the 7-limit, [[support]]ing the 7-limit [[valentine]] temperament; [[540/539]], 1944/1925, 2835/2816, and 43923/43904 in the 11-limit; [[847/845]], [[1188/1183]], [[1287/1280]], and [[1575/1573]] in the 13-limit.

Using the 170c val, it tempers out the [[python comma]], 43046721/41943040 and the [[sycamore comma]], 48828125/47775744 in the 5-limit; 1029/1024, [[4375/4374]], and 78125/76832 in the 7-limit; [[385/384]], [[441/440]], and [[8019/8000]] in the 11-limit, supporting the 11-limit [[unidec]] temperament; 975/968, 1188/1183, 1625/1617, and 3159/3136 in the 13-limit.

The 170b val is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[34edo]], tempering out the [[diaschisma]], 2048/2025 and the [[15625/15552|kleisma]], 15625/15552. It tempers out 33614/32805, [[50421/50000]], and 84035/82944 in the 7-limit; 385/384, 1232/1215, 1331/1323, and [[6250/6237]] in the 11-limit; [[196/195]], [[364/363]], 572/567, and 3146/3125 in the 13-limit.

Using the 170bd val, it tempers out [[16875/16807]], 51200/50421, and [[420175/419904]] in the 7-limit; [[176/175]], 896/891, and 6875/6804 in the 11-limit; [[169/168]], [[640/637]], [[1001/1000]], 2704/2673, and 4235/4212 in the 13-limit. Using the alternative 170bdef val, it tempers out 540/539, 1375/1372, 4375/4356, and 8192/8085 in the 11-limit; [[325/324]], 364/363, [[512/507]], [[625/624]], and 1625/1617 in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 170 factors into {{factorization|170}}, 170edo has subset edos {{EDOs| 2, 5, 10, 17, 34, and 85 }}.

@@ Line 1: / Line 1: @@
-'''170edo''' is the [[EDO|equal division of the octave]] into 170 parts of 7.0588 [[cent]]s each. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: &lt;170 269 395 477| (optimal patent val), &lt;170 270 395 477| (170b), &lt;170 270 395 478| (170bd), and &lt;170 269 394 477| (170c). Using the patent val, it tempers out the valentine comma, 1990656/1953125 and 3486784401/3355443200 in the 5-limit; 126/125, 1029/1024, and 215233605/210827008 in the 7-limit; 540/539, 1944/1925, 2835/2816, and 43923/43904 in the 11-limit. Using the 170b val, it tempers out the [[Diaschismic family|diaschisma]], 2048/2025 and the [[kleisma]], 15625/15552 in the 5-limit; 33614/32805, 50421/50000, and 84035/82944 in the 7-limit; 385/384, 1232/1215, 1331/1323, and 6250/6237 in the 11-limit. Using the 170bd val, it tempers out 16875/16807, 51200/50421, and 420175/419904 in the 7-limit; 176/175, 896/891, and 6875/6804 in the 11-limit. Using the alternative 170bde val, it tempers out 540/539, 1375/1372, 4375/4356, and 8192/8085 in the 11-limit. Using the 170c val, it tempers out the double large green comma, 43046721/41943040 and the [[Sycamore family|sycamore comma]], 48828125/47775744 in the 5-limit; 1029/1024, 4375/4374, and 78125/76832 in the 7-limit; 385/384, 441/440, and 8019/8000 in the 11-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with four mappings possible for the 7-limit: {{val| 170 269 395 477 }} ([[patent val]]), {{val| 170 '''270''' 395 477 }} (170b), {{val| 170 '''270''' 395 '''478''' }} (170bd), and {{val| 170 269 '''394''' 477 }} (170c).
+Using the patent val, it [[tempering out|tempers out]] the [[valentine comma]], 1990656/1953125 and 3486784401/3355443200 in the 5-limit; [[126/125]], [[1029/1024]], and 215233605/210827008 in the 7-limit, [[support]]ing the 7-limit [[valentine]] temperament; [[540/539]], 1944/1925, 2835/2816, and 43923/43904 in the 11-limit; [[847/845]], [[1188/1183]], [[1287/1280]], and [[1575/1573]] in the 13-limit.
+Using the 170c val, it tempers out the [[python comma]], 43046721/41943040 and the [[sycamore comma]], 48828125/47775744 in the 5-limit; 1029/1024, [[4375/4374]], and 78125/76832 in the 7-limit; [[385/384]], [[441/440]], and [[8019/8000]] in the 11-limit, supporting the 11-limit [[unidec]] temperament; 975/968, 1188/1183, 1625/1617, and 3159/3136 in the 13-limit.
+The 170b val is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[34edo]], tempering out the [[diaschisma]], 2048/2025 and the [[15625/15552|kleisma]], 15625/15552. It tempers out 33614/32805, [[50421/50000]], and 84035/82944 in the 7-limit; 385/384, 1232/1215, 1331/1323, and [[6250/6237]] in the 11-limit; [[196/195]], [[364/363]], 572/567, and 3146/3125 in the 13-limit.
+Using the 170bd val, it tempers out [[16875/16807]], 51200/50421, and [[420175/419904]] in the 7-limit; [[176/175]], 896/891, and 6875/6804 in the 11-limit; [[169/168]], [[640/637]], [[1001/1000]], 2704/2673, and 4235/4212 in the 13-limit. Using the alternative 170bdef val, it tempers out 540/539, 1375/1372, 4375/4356, and 8192/8085 in the 11-limit; [[325/324]], 364/363, [[512/507]], [[625/624]], and 1625/1617 in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|170}}
+=== Subsets and supersets ===
+Since 170 factors into {{factorization|170}}, 170edo has subset edos {{EDOs| 2, 5, 10, 17, 34, and 85 }}.