54edo: Difference between revisions

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=== 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.

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.

[[~~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]].

[[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.

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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 ==

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

== Music ==

; [[Bryan Deister]]

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

@@ Line 13: / Line 13: @@
 === 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 [[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.
+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.
-[[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]].
+[[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.
@@ Line 362: / Line 362: @@
 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 369: / Line 375: @@
 [[Category:Todo:add rank 2 temperaments table]]
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/Bi5-YQUQHek ''microtonal improvisation in 54edo''] (2025)