146edo: Difference between revisions

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The '''146 edo''' divides the octave into 146 equal parts of 8.219178 [[cent]]s each. It has an accurate major third, only 0.012344 cents compressed from just [[5/4]] interval. 146edo is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. However, it also provides the optimal patent val for the 11-limit [[Semicomma family|newspeak temperament]]. It tempers out the [[semicomma]], 2109375/2097152 and 129140163/125000000 in the 5-limit; 225/224, 1728/1715, and 100442349/97656250 in the 7-limit; 441/440, 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; 1001/1000, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.

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

146edo has an accurate [[harmonic]] [[5/1|5]], compressed by only 0.012344{{c}} from just. 146 is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. Combined with fairly accurate approximations of [[7/1|7]], [[9/1|9]], [[11/1|11]], [[17/1|17]], and [[19/1|19]], it commends itself as a 2.9.5.7.11.13.17.19 [[subgroup]] system.

However, it also provides the [[optimal patent val]] for the 11-limit [[newspeak]] temperament. Using the [[patent val]], it [[tempering out|tempers out]] the 2109375/2097152 ([[semicomma]]), and {{monzo| -6 17 -9 }} in the 5-limit; [[225/224]], [[1728/1715]], and 100442349/97656250 in the 7-limit; [[441/440]], 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; [[1001/1000]], [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 146 factors into {{factorization|146}}, 146edo contains [[2edo]] and [[73edo]] as its subsets.

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-The '''146 edo''' divides the octave into 146 equal parts of 8.219178 [[cent]]s each. It has an accurate major third, only 0.012344 cents compressed from just [[5/4]] interval. 146edo is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. However, it also provides the optimal patent val for the 11-limit [[Semicomma family|newspeak temperament]]. It tempers out the [[semicomma]], 2109375/2097152 and 129140163/125000000 in the 5-limit; 225/224, 1728/1715, and 100442349/97656250 in the 7-limit; 441/440, 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; 1001/1000, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave]]
+edo has an accurate [[harmonic]] [[5/1|5]], compressed by only 0.012344{{c}} from just. 146 is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. Combined with fairly accurate approximations of [[7/1|7]], [[9/1|9]], [[11/1|11]], [[17/1|17]], and [[19/1|19]], it commends itself as a 2.9.5.7.11.13.17.19 [[subgroup]] system.
+However, it also provides the [[optimal patent val]] for the 11-limit [[newspeak]] temperament. Using the [[patent val]], it [[tempering out|tempers out]] the 2109375/2097152 ([[semicomma]]), and {{monzo| -6 17 -9 }} in the 5-limit; [[225/224]], [[1728/1715]], and 100442349/97656250 in the 7-limit; [[441/440]], 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; [[1001/1000]], [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|146}}
+=== Subsets and supersets ===
+Since 146 factors into {{factorization|146}}, 146edo contains [[2edo]] and [[73edo]] as its subsets.