67ed5: Difference between revisions

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'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 6.0164 cents stretched and the step size is about 41.5868 cents.

'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo]], but with the [[5/1]] rather than the [[2/1]] being [[just]]. The octave is about 6.0164 cents [[Stretched octave|stretched]] and the step size is about 41.5868 cents.

== Theory ==

~~The patent val~~ has a generally sharp tendency for ~~harmonics~~ up to 28. Unlike 29edo, it is only consistent up to the 8-[[integer-limit]], with discrepancy for the 9th harmonic. ~~This tuning~~ tempers out 49/48 in the 7-limit; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.

67ed5 has a generally sharp tendency for [[harmonic]]s up to 28. Unlike 29edo, it is only [[consistent]] up to the 8-[[integer-limit]], with discrepancy for the 9th harmonic. As an equal temperament, it [[tempering out|tempers out]] 49/48 in the [[7-limit]]; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.

=== Prime harmonics ===

Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.

The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that ~~horrible~~ 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.

The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.

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~~[[Category:Ed5]]~~

~~[[Category:Edonoi]]~~

[[Category:29edo]]

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 {{Infobox ET}}
-'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 6.0164 cents stretched and the step size is about 41.5868 cents.
+'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo]], but with the [[5/1]] rather than the [[2/1]] being [[just]]. The octave is about 6.0164 cents [[Stretched octave|stretched]] and the step size is about 41.5868 cents.
 == Theory ==
-The patent val has a generally sharp tendency for harmonics up to 28. Unlike 29edo, it is only consistent up to the 8-[[integer-limit]], with discrepancy for the 9th harmonic. This tuning tempers out 49/48 in the 7-limit; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.
+ed5 has a generally sharp tendency for [[harmonic]]s up to 28. Unlike 29edo, it is only [[consistent]] up to the 8-[[integer-limit]], with discrepancy for the 9th harmonic. As an equal temperament, it [[tempering out|tempers out]] 49/48 in the [[7-limit]]; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.
 === Prime harmonics ===
 Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.
-The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that horrible 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.
+The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.
 {{Harmonics in equal|67|5|1|intervals=prime|columns=11}}
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 |}
-[[Category:Ed5]]
-[[Category:Edonoi]]
 [[Category:29edo]]