189edo: Difference between revisions

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'''189edo''' is the [[EDO|equal division of the octave]] into 189 parts of 6.3492 cents each. It tempers out 15625/15552 (kleisma) and 9007199254740992/8578797170610375 in the 5-limit; 4000/3969, 6144/6125, and 537824/531441 in the 7-limit, supporting the [[Kleismic family|hemikleismic temperament]]. Using the patent val, it tempers out 896/891, 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; 169/168, 352/351, 364/363, and 1001/1000 in the 13-limit.

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

189edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]]s [[3/1|3]] and [[7/1|7]] are about halfway between its steps. It has good approximations to [[5/1|5]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[21/1|21]], making it suitable for a 2.9.5.21.11.19 [[subgroup]] interpretation.

Using the full 13-limit [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 53 -29 -3 }} in the 5-limit; [[4000/3969]], [[6144/6125]], and 537824/531441 in the 7-limit, supporting the [[hemikleismic]] temperament. It tempers out [[896/891]], 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; [[169/168]], [[352/351]], [[364/363]], and [[1001/1000]] in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 189 factors into {{factorization|189}}, 189edo contains {{EDOs| 3, 7, 9, 21, 27, and 63 }} as its subsets. [[378edo]], which doubles it, provides a good correction for the approximation of 3 and 7.

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-'''189edo''' is the [[EDO|equal division of the octave]] into 189 parts of 6.3492 cents each. It tempers out 15625/15552 (kleisma) and 9007199254740992/8578797170610375 in the 5-limit; 4000/3969, 6144/6125, and 537824/531441 in the 7-limit, supporting the [[Kleismic family|hemikleismic temperament]]. Using the patent val, it tempers out 896/891, 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; 169/168, 352/351, 364/363, and 1001/1000 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave]]
+edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]]s [[3/1|3]] and [[7/1|7]] are about halfway between its steps. It has good approximations to [[5/1|5]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[21/1|21]], making it suitable for a 2.9.5.21.11.19 [[subgroup]] interpretation.
+Using the full 13-limit [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 53 -29 -3 }} in the 5-limit; [[4000/3969]], [[6144/6125]], and 537824/531441 in the 7-limit, supporting the [[hemikleismic]] temperament. It tempers out [[896/891]], 1331/1323, 1375/1372, and 16896/16807 in the 11-limit; [[169/168]], [[352/351]], [[364/363]], and [[1001/1000]] in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|189}}
+=== Subsets and supersets ===
+Since 189 factors into {{factorization|189}}, 189edo contains {{EDOs| 3, 7, 9, 21, 27, and 63 }} as its subsets. [[378edo]], which doubles it, provides a good correction for the approximation of 3 and 7.