54edo: Difference between revisions

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== Theory ==

54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a 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 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]].

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 ~~cents~~ 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 are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.

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 ===

=== 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 ~~require an incredibly large amount of~~ ups and downs to notate~~, and~~ using the flat fifth as a generator, ~~54edo~~ requires ~~an incredibly 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.

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"

|+Table of intervals

|+ style="font-size: 105%;" | Table of intervals in 54edo

! Degree

|-

! Cents

! rowspan="2" | Degree

! [[Ups and downs notation~~|Ups and Downs Notation~~]]~~<br>~~(~~Flat Fifth~~ 31\54)

! rowspan="2" | Cents

! ~~[[Ups and downs notation|Ups and Downs Notation]]<br>~~(~~Sharp Fifth~~ 16\27)

! colspan="2" | [[Ups and downs notation]]

|-

! Flat fifth (31\54)

! Sharp fifth (16\27)

|-

| 0

Line 295:

Line 312:

| {{UDnote|step=54}}

|}

== Notation ==

=== Ups and downs notation ===

Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:

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 ====

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 ====

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 ====

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 ==

Line 302:

Line 375:

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

== Music ==

; [[Bryan Deister]]

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|54}}
+{{ED intro}}
 == Theory ==
-edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a 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 L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].
+edo 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 cents 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 are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.
+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 ===
+edo’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 require an incredibly large amount of ups and downs to notate, and using the flat fifth as a generator, 54edo requires an incredibly 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.
+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"
-|+Table of intervals
+|+ style="font-size: 105%;" | Table of intervals in 54edo
-! Degree
+|-
-! Cents
+! rowspan="2" | Degree
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Flat Fifth 31\54)
+! rowspan="2" | Cents
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Sharp Fifth 16\27)
+! colspan="2" | [[Ups and downs notation]]
+|-
+! Flat fifth (31\54)
+! Sharp fifth (16\27)
 |-
 | 0
@@ Line 295: / Line 312: @@
 | {{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 ==
@@ Line 302: / Line 375: @@
 [[Category:Todo:add rank 2 temperaments table]]
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/Bi5-YQUQHek ''microtonal improvisation in 54edo''] (2025)