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→Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 54edo'' (2025)
 
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Ex, Fbbbbbb{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|54}}
{{ED intro}}


== Theory ==
== Theory ==
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 just sounds like a [[circulating temperament]] of [[7edo]].
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]].


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.
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]].  


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. In other words, this is the last EDO tempering out [[50/49]]. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned.
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.


Using the patent val, 54edo tempers out [[2048/2025]] in the 5-limit.
=== Odd harmonics ===
{{Harmonics in equal|54}}


The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage.
=== Octave stretch ===
===Odd harmonics===
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.
{{harmonics in equal|54}}
==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"
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.
|+Table of intervals
 
!Degree
[[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]].
!Cents
 
!Approximate Ratios
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.
![[Ups and downs notation]] using flat fifth
 
=== 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}}
|-
|-
|0
| 15
|0.000
| 333.333
|[[1/1]]
| {{UDnote|fifth=31|step=15}}
|C
| {{UDnote|step=15}}
|-
|-
|1
| 16
|22.222
| 355.556
|[[81/80]], [[64/63]]
| {{UDnote|fifth=31|step=16}}
|C#, Dbbbbbbb
| {{UDnote|step=16}}
|-
|-
|2
| 17
|44.444
| 377.778
|[[128/125]], [[36/35]]
| {{UDnote|fifth=31|step=17}}
|Cx, Dbbbbbb
| {{UDnote|step=17}}
|-
|-
|3
| 18
|66.666
| 400.000
|[[28/27]], [[25/24]]
| {{UDnote|fifth=31|step=18}}
|Cx#, Dbbbbb
| {{UDnote|step=18}}
|-
|4
|88.888
|[[19/18]], [[20/19]]
|Cxx, Dbbbb
|-
|-
|5
| 19
|111.111
| 422.222
|[[16/15]]
| {{UDnote|fifth=31|step=19}}
|Cxx#, Dbbb
| {{UDnote|step=19}}
|-
|-
|6
| 20
|133.333
| 444.444
|[[13/12]]
| {{UDnote|fifth=31|step=20}}
|Cxxx, Dbb
| {{UDnote|step=20}}
|-
|-
|7
| 21
|155.555
| 466.667
|[[12/11]], [[11/10]]
| {{UDnote|fifth=31|step=21}}
|Cxxx#, Db
| {{UDnote|step=21}}
|-
|-
|8
| 22
|177.777
| 488.889
|[[10/9]]
| {{UDnote|fifth=31|step=22}}
|D
| {{UDnote|step=22}}
|-
|-
|9
| 23
|200.000
| 511.111
|[[9/8]]
| {{UDnote|fifth=31|step=23}}
|D#, Ebbbbbbb
| {{UDnote|step=23}}
|-
|-
|10
| 24
|222.222
| 533.333
|[[8/7]], [[17/15]]
| {{UDnote|fifth=31|step=24}}
|Dx, Ebbbbbb
| {{UDnote|step=24}}
|-
|-
|11
| 25
|244.444
| 555.556
|[[15/13]], [[23/20]]
| {{UDnote|fifth=31|step=25}}
|Dx#, Ebbbbb
| {{UDnote|step=25}}
|-
|-
|12
| 26
|266.666
| 577.778
|[[7/6]]
| {{UDnote|fifth=31|step=26}}
|Dxx, Ebbbb
| {{UDnote|step=26}}
|-
|-
|13
| 27
|288.888
| 600.000
|[[13/11]], [[20/17]]
| {{UDnote|fifth=31|step=27}}
|Dxx#, Ebbb
| {{UDnote|step=27}}
|-
|-
|14
| 28
|311.111
| 622.222
|[[6/5]], [[19/16]]
| {{UDnote|fifth=31|step=28}}
|Dxxx, Ebb
| {{UDnote|step=28}}
|-
|-
|15
| 29
|333.333
| 644.444
|[[17/14]]
| {{UDnote|fifth=31|step=29}}
|Dxxx#, Eb
| {{UDnote|step=29}}
|-
|-
|16
| 30
|355.555
| 666.667
|[[11/9]], [[27/22]], [[16/13]]
| {{UDnote|fifth=31|step=30}}
|E
| {{UDnote|step=30}}
|-
|-
|17
| 31
|377.777
| 688.889
|[[5/4]]
| {{UDnote|fifth=31|step=31}}
|E#, Fbbbbbb
| {{UDnote|step=31}}
|-
|-
|18
| 32
|400.000
| 711.111
|[[5/4]], [[29/23]], [[14/11]]
| {{UDnote|fifth=31|step=32}}
|Ex, Fbbbbb
| {{UDnote|step=32}}
|-
|-
|19
| 33
|422.222
| 733.333
|[[14/11], [[9/7]]
| {{UDnote|fifth=31|step=33}}
|Ex#, Fbbbb
| {{UDnote|step=33}}
|-
|-
|20
| 34
|444.444
| 755.556
|[[9/7]], [[35/27]], [[13/10]]
| {{UDnote|fifth=31|step=34}}
|Exx, Fbbb
| {{UDnote|step=34}}
|-
|-
|21
| 35
|466.666
| 777.778
|[[21/16]]
| {{UDnote|fifth=31|step=35}}
|Exx#, Fbb
| {{UDnote|step=35}}
|-
|-
|22
| 36
|488.888
| 800.000
|[[4/3]]
| {{UDnote|fifth=31|step=36}}
|Exxx, Fb
| {{UDnote|step=36}}
|-
|-
|23
| 37
|511.111
| 822.222
|[[4/3]]
| {{UDnote|fifth=31|step=37}}
|F
| {{UDnote|step=37}}
|-
|-
|23
| 38
|533.333
| 844.444
|[[4/3]]
| {{UDnote|fifth=31|step=38}}
|F#, Gbbbbbbb
| {{UDnote|step=38}}
|-
|-
|24
| 39
|555.555
| 866.667
|[[4/3]]
| {{UDnote|fifth=31|step=39}}
|Fx, Gbbbbbb
| {{UDnote|step=39}}
|-
|-
|25
| 40
|577.777
| 888.889
|[[4/3]]
| {{UDnote|fifth=31|step=40}}
|Fxx, Gbbbbb
| {{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}}
|}
|}


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
== 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"