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<b>54-EDO</b> is an equal temperament that divides the octave into 54 equal parts, each 22.2222 [[cent]]s in size. It's a rare temperament which adds approximations of the 11th and 15th harmonics, as well as an alternate (flat) mapping for the 5th, to [[27edo|27edo]] which it doubles. It is the highest edo in which the best mappings of the major 3rd (17\54) and harmonic 7th (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]].
{{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 [[38ed5/3]], 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.
 
[[40ed5/3]] 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)
Retrieved from "https://en.xen.wiki/w/54edo"