24ed5: Difference between revisions
Created page with "'''Division of the 5th harmonic into 24 equal parts''' (24ed5) is related to the miracle temperament. The step size about 116.0964 cents. It is similar to every th..." Tags: Mobile edit Mobile web edit |
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'''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. | {{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. | |||
== 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" | ||
|- | |- | ||
| Line 133: | Line 137: | ||
| | 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]]. | ||