24ed5: Difference between revisions

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'''[[Ed5|Division of the 5th harmonic]] into 24 equal parts''' (24ed5) is related to the [[Miracle|miracle temperament]]. The step size about 116.0964 cents. It is similar to every third step of [[31edo]], but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.

'''[[Ed5|Division of the 5th harmonic]] into 24 equal parts''' (24ed5) is related to the [[Miracle|miracle temperament]]. The step size is about 116.0964 cents. It is similar to every third step of [[31edo]], but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.

== Theory ==

From a no-twos-or-threes point of view, 24ed5 offers a particularly good tuning of the very low-[[badness]] 5.7.11 [[subgroup temperament]] named as [[juggernaut]], tempering out 125/121. This has a CTE generator of exactly [[7/5]] (in 24ed5 approximated as 5 steps) and a period of 1\[[2ed5]] or the square root of five (which is equated to [[11/5]]). 24ed5 shares 31edo's mappings for 5 and 7, but not 11.

== Interval table ==

{| class="wikitable"

Line 135:

Line 137:

| | just major third plus two octaves

|}

== Harmonics ==

{{Harmonics in equal

| steps = 24

| num = 5

| denom = 1

}}

{{Harmonics in equal

| steps = 24

| num = 5

| denom = 1

| start = 12

| collapsed = 1

}}

==24ed5 as a generator==

24ed5 ~~can also be thought~~ of ~~as a~~ [[~~generator~~]] ~~of the 2.3.5.7.11.23 [[Subgroup temperaments|subgroup~~ temperament~~]] which tempers out 225/224~~, ~~243/242, 385/384, and 529/528, which is~~ a [[~~cluster temperament~~]] ~~with 10 clusters~~ of ~~notes in an octave~~. ~~The small chroma interval between adjacent notes in each cluster is very versatile~~, ~~representing 45~~/~~44 ~ 46/45 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by [[31edo]]~~, ~~[[82edo]], [[113edo]], and [[144edo]]~~.

One step of 24ed5 is just flat of 3\31, an acceptable generator for [[miracle]] temperament, so it can be used as a [[retraction]] of miracle. However, miracle maps 5/1 to 23 secors and 3 dieses, not 24 secors.

~~[[Category:Ed5]]~~

~~[[Category:Edonoi]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''[[Ed5|Division of the 5th harmonic]] into 24 equal parts''' (24ed5) is related to the [[Miracle|miracle temperament]]. The step size about 116.0964 cents. It is similar to every third step of [[31edo]], but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.
+'''[[Ed5|Division of the 5th harmonic]] into 24 equal parts''' (24ed5) is related to the [[Miracle|miracle temperament]]. The step size is about 116.0964 cents. It is similar to every third step of [[31edo]], but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.
+== Theory ==
+From a no-twos-or-threes point of view, 24ed5 offers a particularly good tuning of the very low-[[badness]] 5.7.11 [[subgroup temperament]] named as [[juggernaut]], tempering out 125/121. This has a CTE generator of exactly [[7/5]] (in 24ed5 approximated as 5 steps) and a period of 1\[[2ed5]] or the square root of five (which is equated to [[11/5]]). 24ed5 shares 31edo's mappings for 5 and 7, but not 11.
 == Interval table ==
 {| class="wikitable"
@@ Line 135: / Line 137: @@
 | | just major third plus two octaves
 |}
+== Harmonics ==
+{{Harmonics in equal
+| steps = 24
+| num = 5
+| denom = 1
+}}
+{{Harmonics in equal
+| steps = 24
+| num = 5
+| denom = 1
+| start = 12
+| collapsed = 1
+}}
 ==24ed5 as a generator==
-ed5 can also be thought of as a [[generator]] of the 2.3.5.7.11.23 [[Subgroup temperaments|subgroup temperament]] which tempers out 225/224, 243/242, 385/384, and 529/528, which is a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 46/45 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by [[31edo]], [[82edo]], [[113edo]], and [[144edo]].
+One step of 24ed5 is just flat of 3\31, an acceptable generator for [[miracle]] temperament, so it can be used as a [[retraction]] of miracle. However, miracle maps 5/1 to 23 secors and 3 dieses, not 24 secors.
-[[Category:Ed5]]
-[[Category:Edonoi]]