189edo: Difference between revisions

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~~The~~ equal temperament [[tempering out|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 [[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.

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 ===

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=== Subsets and supersets ===

Since 189 factors into {{factorization|189}}, 189edo contains {{EDOs| 3, 7, 9, 21, 27, and 63 }} as its subsets.

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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 {{EDO intro}}
-The equal temperament [[tempering out|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 [[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.
+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 ===
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 === Subsets and supersets ===
-Since 189 factors into {{factorization|189}}, 189edo contains {{EDOs| 3, 7, 9, 21, 27, and 63 }} as its subsets.
+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.