67ed5: Difference between revisions

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'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo~~|29 edo~~]], 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 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.

=== 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.

29edo for comparison:

==Intervals==

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| | just major third plus two octaves

|}

~~== Harmonics ==~~

~~{{Harmonics in equal~~

~~| steps = 67~~

~~| num = 5~~

~~| denom = 1~~

}}

~~{{Harmonics in equal~~

~~| steps = 67~~

~~| num = 5~~

~~| denom = 1~~

~~| start = 12~~

~~| collapsed = 1~~

}}

==67ed5 as a generator==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''[[Ed5|Division of the 5th harmonic]] into 67 equal parts''' (67ed5) is related to [[29edo|29 edo]], 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 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.
+=== 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.
+{{Harmonics in equal|67|5|1|intervals=prime|columns=11}}
+edo for comparison:
+{{Harmonics in equal|29|columns=11}}
 ==Intervals==
@@ Line 354: / Line 364: @@
 | | just major third plus two octaves
 |}
-== Harmonics ==
-{{Harmonics in equal
-| steps = 67
-| num = 5
-| denom = 1
-}}
-{{Harmonics in equal
-| steps = 67
-| num = 5
-| denom = 1
-| start = 12
-| collapsed = 1
-}}
 ==67ed5 as a generator==