54edo: Difference between revisions
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'''54-EDO''' is an equal temperament that divides the octave into 54 equal parts, each 22.2222 [[cent]]s in size | '''54-EDO''' is an equal temperament that divides the octave into 54 equal parts, each 22.2222 [[cent]]s in size. | ||
The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage. | The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage. | ||
== Theory == | == Theory == | ||
54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. | |||
It's a rare temperament which adds 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 is the highest [[EDO]] in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]]. | |||
{| class="wikitable" | |||
|+Table of intervals | |||
!Degree | |||
!Name | |||
!Cents | |||
!Approximate Ratios | |||
|- | |||
|0 | |||
|Natural Unison | |||
|0.000 | |||
| | |||
|- | |||
|1 | |||
|Ninth-tone | |||
|22.222 | |||
| | |||
|- | |||
|2 | |||
|Extreme bayati quarter-tone | |||
|44.444 | |||
| | |||
|- | |||
|3 | |||
|Third-tone | |||
|66.666 | |||
| | |||
|- | |||
|4 | |||
| | |||
|88.888 | |||
| | |||
|- | |||
|5 | |||
| | |||
|111.111 | |||
| | |||
|- | |||
|6 | |||
|Extreme bayati neutral second | |||
|133.333 | |||
| | |||
|- | |||
|7 | |||
| | |||
|155.555 | |||
| | |||
|- | |||
|8 | |||
|Minor whole tone | |||
|177.777 | |||
|[[10/9]] | |||
|- | |||
|9 | |||
|Symmetric whole tone | |||
|200.000 | |||
|[[9/8]] | |||
|- | |||
|10 | |||
|Extreme bayati whole tone | |||
|222.222 | |||
| | |||
|- | |||
|12 | |||
|Septimal submajor third | |||
|266.666 | |||
|[[7/6]] | |||
|- | |||
|17 | |||
|Classical major third | |||
|377.777 | |||
|[[5/4]] | |||
|- | |||
|18 | |||
|Symmetric major third | |||
|400.000 | |||
|[[29/23]], [[19/16]] | |||
|- | |||
|25 | |||
|Undecimal superfourth | |||
|555.555 | |||
|[[11/8]] | |||
|- | |||
|26 | |||
|Septimal minor tritone | |||
|577.777 | |||
|[[7/5]] | |||
|- | |||
|27 | |||
|Symmetric tritone | |||
|600.000 | |||
| | |||
|- | |||
|28 | |||
|Septimal major tritone | |||
|633.333 | |||
|[[10/7]] | |||
|- | |||
|36 | |||
|Symmetric augmented fifth | |||
|800.000 | |||
| | |||
|- | |||
|44 | |||
|Harmonic seventh | |||
|977.777 | |||
|[[7/4]] | |||
|- | |||
|54 | |||
|Octave | |||
|1200.000 | |||
|Exact 2/1 | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||