54edo

54-EDO is an equal temperament that divides the octave into 54 equal parts, each 22.2222 cents in size.

The immediate close presence of 53edo obscures 54edo and puts this temperament out of popular usage.

Theory

54edo is suitable for usage with dual-fifth tuning systems, or alternately, no-fifth tuning systems.

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 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 hexe temperament, while the bdf val does higher limit muggles about as well as it can be tuned.

Using the patent val, 54edo tempers out 2048/2025 in the 5-limit.

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	19/18, 20/19
5		111.111	16/15
6	Extreme bayati neutral second	133.333	13/12
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	8/7, 17/15
11		244.444	15/13, 23/20
12	Septimal submajor third	266.666	7/6
13	Gothic minor third	288.888	13/11, 20/17
14	Classical minor third	311.111	6/5, 19/16
15		333.333	17/14
16		355.555	11/9, 16/13
17	Classical major third	377.777	5/4
18	Symmetric major third	400.000	29/23
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