54edo: Difference between revisions
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== Theory == | == Theory == | ||
54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. | 54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. 54edo has an ultrahard diatonic scale using the sharp fifth of [[27edo]] and an ultrasoft diatonic using the flat fifth. The soft diatonic scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but now just sounds like a [[circulating temperament]] of [[7edo]]. | ||
It's a rare temperament which adds better approximations of the 11th and 15th harmonics from [[27edo]], which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale. | It's a rare temperament which adds better approximations of the 11th and 15th harmonics from [[27edo]], which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale. | ||
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{{harmonics in equal|54}} | {{harmonics in equal|54}} | ||
==Intervals== | ==Intervals== | ||
Using the sharp fifth as a generator, 54edo require a large amount of ups and downs to notate, and using the flat fifth as a generator, 54edo requires a large amount of sharps and flats to notate. Because the flat fifth generates a diatonic scale with a chroma of 1 step, ups and downs are not needed in notation if the flat fifth is used. | |||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+Table of intervals | |+Table of intervals | ||
!Degree | !Degree | ||
!Cents | !Cents | ||
!Approximate Ratios | !Approximate Ratios | ||
![[Ups and downs notation]] using flat fifth | |||
|- | |- | ||
|0 | |0 | ||
|0.000 | |0.000 | ||
| | |[[1/1]] | ||
|C | |||
|- | |- | ||
|1 | |1 | ||
|22.222 | |22.222 | ||
| | |[[81/80]], [[64/63]] | ||
|C#, Dbbbbbbb | |||
|- | |- | ||
|2 | |2 | ||
|44.444 | |44.444 | ||
| | [[81/80]] | ||
|Cx, Dbbbbbb | |||
|- | |- | ||
|3 | |3 | ||
|66.666 | |66.666 | ||
| | |[[28/27]], [[25/24]] | ||
|- | |Cx#, Dbbbbb | ||
|- | |||
|4 | |4 | ||
|88.888 | |88.888 | ||
|[[19/18]], [[20/19]] | |[[19/18]], [[20/19]] | ||
|Cxx, Dbbbb | |||
|- | |- | ||
|5 | |5 | ||
|111.111 | |111.111 | ||
|[[16/15]] | |[[16/15]] | ||
|Cxx#, Dbbb | |||
|- | |- | ||
|6 | |6 | ||
|133.333 | |133.333 | ||
|[[13/12]] | |[[13/12]] | ||
|Cxxx, Dbb | |||
|- | |- | ||
|7 | |7 | ||
|155.555 | |155.555 | ||
| | |[[12/11]], [[11/10]] | ||
|Cxxx#, Db | |||
|- | |- | ||
|8 | |8 | ||
| Line 66: | Line 68: | ||
|177.777 | |177.777 | ||
|[[10/9]] | |[[10/9]] | ||
| | |||
|- | |- | ||
|9 | |9 | ||