24ed5: Difference between revisions
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{{Infobox ET}} | {{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 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]]). | |||
== Interval table == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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| | just major third plus two octaves | | | 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 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]]. | 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]]. | ||