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~~. 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~~.

'''[[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.

==Intervals==

== Theory ==

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 unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.

29edo for comparison:

=== 67ed5 as a generator ===

67ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a [[cluster temperament]] with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by [[29edo]], [[202edo]], and [[231edo]].

== Intervals ==

{| class="wikitable mw-collapsible"

|+ Intervals of 67ed5

|-

! | degree

Line 351:

Line 368:

|}

~~== Harmonics ==~~

[[Category:29edo]]

~~{{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==~~

67ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a cluster temperament with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by [[29edo]], [[202edo]], and [[231edo]].

[[Category:~~Ed5]]~~

~~[[Category:Edonoi~~]]

~~{{todo|expand}}~~

@@ 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. 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.
+'''[[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.
-==Intervals==
+== Theory ==
+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 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}}
+edo for comparison:
+{{Harmonics in equal|29|columns=11}}
+=== 67ed5 as a generator ===
+ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a [[cluster temperament]] with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by [[29edo]], [[202edo]], and [[231edo]].
+== Intervals ==
 {| class="wikitable mw-collapsible"
+|+ Intervals of 67ed5
 |-
 ! | degree
@@ Line 351: / Line 368: @@
 |}
-== Harmonics ==
+[[Category:29edo]]
-{{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==
-ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.19 [[Subgroup temperaments|subgroup temperament]] which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a cluster temperament with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by [[29edo]], [[202edo]], and [[231edo]].
-[[Category:Ed5]]
-[[Category:Edonoi]]
-{{todo|expand}}