View source for 54edo

{{Infobox ET}}
{{ED intro}}

== Theory ==
54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This scale is so [[soft]], with {{nowrap|L/s {{=}} 8/7}}, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].

The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. 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{{c}} apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. 

The 54cd val makes for an excellent tuning of 7-limit [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.

=== Odd harmonics ===
{{Harmonics in equal|54}}

=== Octave stretch ===
54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[Gallery of arithmetic pitch sequences#APS of mérides|APS4/5méride]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1.

If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s  3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1. 

[[ed255/128#54ed255/128|54ed255/128]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].

There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings.

=== Subsets and supersets ===
Since 54 factors into {{factorization|54}}, 54edo has subset edos {{EDOs| 2, 3, 6, 9, 18, and 27 }}.

== Intervals ==
Using the sharp fifth as a [[generator]], 54edo requires up to quadruple ups and downs to notate. But using the flat fifth as a generator, it requires up to septuple sharps and flats. 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"
|+ style="font-size: 105%;" | Table of intervals in 54edo
|-
! rowspan="2" | Degree
! rowspan="2" | Cents
! colspan="2" | [[Ups and downs notation]]
|-
! Flat fifth (31\54)
! Sharp fifth (16\27)
|-
| 0
| 0.000
| {{UDnote|fifth=31|step=0}}
| {{UDnote|step=0}}
|-
| 1
| 22.222
| {{UDnote|fifth=31|step=1}}
| {{UDnote|step=1}}
|-
| 2
| 44.444
| {{UDnote|fifth=31|step=2}}
| {{UDnote|step=2}}
|-
| 3
| 66.667
| {{UDnote|fifth=31|step=3}}
| {{UDnote|step=3}}
|-
| 4
| 88.889
| {{UDnote|fifth=31|step=4}}
| {{UDnote|step=4}}
|-
| 5
| 111.111
| {{UDnote|fifth=31|step=5}}
| {{UDnote|step=5}}
|-
| 6
| 133.333
| {{UDnote|fifth=31|step=6}}
| {{UDnote|step=6}}
|-
| 7
| 155.556
| {{UDnote|fifth=31|step=7}}
| {{UDnote|step=7}}
|-
| 8
| 177.778
| {{UDnote|fifth=31|step=8}}
| {{UDnote|step=8}}
|-
| 9
| 200.000
| {{UDnote|fifth=31|step=9}}
| {{UDnote|step=9}}
|-
| 10
| 222.222
| {{UDnote|fifth=31|step=10}}
| {{UDnote|step=10}}
|-
| 11
| 244.444
| {{UDnote|fifth=31|step=11}}
| {{UDnote|step=11}}
|-
| 12
| 266.667
| {{UDnote|fifth=31|step=12}}
| {{UDnote|step=12}}
|-
| 13
| 288.889
| {{UDnote|fifth=31|step=13}}
| {{UDnote|step=13}}
|-
| 14
| 311.111
| {{UDnote|fifth=31|step=14}}
| {{UDnote|step=14}}
|-
| 15
| 333.333
| {{UDnote|fifth=31|step=15}}
| {{UDnote|step=15}}
|-
| 16
| 355.556
| {{UDnote|fifth=31|step=16}}
| {{UDnote|step=16}}
|-
| 17
| 377.778
| {{UDnote|fifth=31|step=17}}
| {{UDnote|step=17}}
|-
| 18
| 400.000
| {{UDnote|fifth=31|step=18}}
| {{UDnote|step=18}}
|-
| 19
| 422.222
| {{UDnote|fifth=31|step=19}}
| {{UDnote|step=19}}
|-
| 20
| 444.444
| {{UDnote|fifth=31|step=20}}
| {{UDnote|step=20}}
|-
| 21
| 466.667
| {{UDnote|fifth=31|step=21}}
| {{UDnote|step=21}}
|-
| 22
| 488.889
| {{UDnote|fifth=31|step=22}}
| {{UDnote|step=22}}
|-
| 23
| 511.111
| {{UDnote|fifth=31|step=23}}
| {{UDnote|step=23}}
|-
| 24
| 533.333
| {{UDnote|fifth=31|step=24}}
| {{UDnote|step=24}}
|-
| 25
| 555.556
| {{UDnote|fifth=31|step=25}}
| {{UDnote|step=25}}
|-
| 26
| 577.778
| {{UDnote|fifth=31|step=26}}
| {{UDnote|step=26}}
|-
| 27
| 600.000
| {{UDnote|fifth=31|step=27}}
| {{UDnote|step=27}}
|-
| 28
| 622.222
| {{UDnote|fifth=31|step=28}}
| {{UDnote|step=28}}
|-
| 29
| 644.444
| {{UDnote|fifth=31|step=29}}
| {{UDnote|step=29}}
|-
| 30
| 666.667
| {{UDnote|fifth=31|step=30}}
| {{UDnote|step=30}}
|-
| 31
| 688.889
| {{UDnote|fifth=31|step=31}}
| {{UDnote|step=31}}
|-
| 32
| 711.111
| {{UDnote|fifth=31|step=32}}
| {{UDnote|step=32}}
|-
| 33
| 733.333
| {{UDnote|fifth=31|step=33}}
| {{UDnote|step=33}}
|-
| 34
| 755.556
| {{UDnote|fifth=31|step=34}}
| {{UDnote|step=34}}
|-
| 35
| 777.778
| {{UDnote|fifth=31|step=35}}
| {{UDnote|step=35}}
|-
| 36
| 800.000
| {{UDnote|fifth=31|step=36}}
| {{UDnote|step=36}}
|-
| 37
| 822.222
| {{UDnote|fifth=31|step=37}}
| {{UDnote|step=37}}
|-
| 38
| 844.444
| {{UDnote|fifth=31|step=38}}
| {{UDnote|step=38}}
|-
| 39
| 866.667
| {{UDnote|fifth=31|step=39}}
| {{UDnote|step=39}}
|-
| 40
| 888.889
| {{UDnote|fifth=31|step=40}}
| {{UDnote|step=40}}
|-
| 41
| 911.111
| {{UDnote|fifth=31|step=41}}
| {{UDnote|step=41}}
|-
| 42
| 933.333
| {{UDnote|fifth=31|step=42}}
| {{UDnote|step=42}}
|-
| 43
| 955.556
| {{UDnote|fifth=31|step=43}}
| {{UDnote|step=43}}
|-
| 44
| 977.778
| {{UDnote|fifth=31|step=44}}
| {{UDnote|step=44}}
|-
| 45
| 1000.000
| {{UDnote|fifth=31|step=45}}
| {{UDnote|step=45}}
|-
| 46
| 1022.222
| {{UDnote|fifth=31|step=46}}
| {{UDnote|step=46}}
|-
| 47
| 1044.444
| {{UDnote|fifth=31|step=47}}
| {{UDnote|step=47}}
|-
| 48
| 1066.667
| {{UDnote|fifth=31|step=48}}
| {{UDnote|step=48}}
|-
| 49
| 1088.889
| {{UDnote|fifth=31|step=49}}
| {{UDnote|step=49}}
|-
| 50
| 1111.111
| {{UDnote|fifth=31|step=50}}
| {{UDnote|step=50}}
|-
| 51
| 1133.333
| {{UDnote|fifth=31|step=51}}
| {{UDnote|step=51}}
|-
| 52
| 1155.556
| {{UDnote|fifth=31|step=52}}
| {{UDnote|step=52}}
|-
| 53
| 1177.778
| {{UDnote|fifth=31|step=53}}
| {{UDnote|step=53}}
|-
| 54
| 1200.000
| {{UDnote|fifth=31|step=54}}
| {{UDnote|step=54}}
|}

== Notation ==
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:
{{Sharpness-sharp8}}
Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals can be used to fill in the gap.

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[61edo#Sagittal notation|61-EDO]], and is a superset of the notation for [[27edo#Sagittal notation|27-EDO]].

==== Evo flavor ====
<imagemap>
File:54-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:54-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:54-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 650 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:54-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:54-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 642 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
rect 360 80 480 106 [[27/26]]
default [[File:54-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

== Scales ==
* Approximations of [[gamelan]] scales:
** 5-tone pelog: 5 7 19 4 19
** 7-tone pelog: 5 7 11 8 4 13 6
** 5-tone slendro: 11 11 10 11 11

== Instruments ==
; Lumatone

See [[Lumatone mapping for 54edo]]

[[Category:Todo:add rank 2 temperaments table]]

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Bi5-YQUQHek ''microtonal improvisation in 54edo''] (2025)