54edo: Difference between revisions

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=== 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 ===

Line 362:

Line 353:

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.

== Octave stretch or compression ==

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 [[ed6|139ed6]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.

If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[ed5|126ed5]] and [[ed7|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.

What follows is a comparison of stretched- and compressed-octave 54edo tunings.

; [[ed6|139ed6]]

* Octave size: 1205.08{{c}}

Stretching the octave of 54edo by around 5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.

{{Harmonics in equal|139|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed6}}

{{Harmonics in equal|139|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed6 (continued)}}

; [[ed7|151ed7]]

* Octave size: 1204.75{{c}}

Stretching the octave of 54edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12{{c}}. The tuning 151ed7 does this.

{{Harmonics in equal|151|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed7}}

{{Harmonics in equal|151|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed7 (continued)}}

; [[ed12|193ed12]]

* Octave size: 1203.66{{c}}

Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.

{{Harmonics in equal|193|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 193ed12}}

{{Harmonics in equal|193|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 193ed12 (continued)}}

; [[zpi|263zpi]]

* Step size: 22.243{{c}}, octave size: 1201.12{{c}}

Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.

{{Harmonics in cet|22.243|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 263zpi}}

{{Harmonics in cet|22.243|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 263zpi (continued)}}

; 54edo

* Step size: 22.222{{c}}, octave size: 1200.00{{c}}

Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.

{{Harmonics in equal|54|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54edo}}

{{Harmonics in equal|54|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54edo (continued)}}

; [[WE|54et, 13-limit WE tuning]]

* Step size: 22.198{{c}}, octave size: 1198.69{{c}}

Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.

{{Harmonics in cet|22.198|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning}}

{{Harmonics in cet|22.198|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}}

; [[zpi|264zpi]]

* Step size: 22.175{{c}}, octave size: 1197.45{{c}}

Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.

{{Harmonics in cet|22.175|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 264zpi}}

{{Harmonics in cet|22.175|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 264zpi (continued)}}

; [[ed7|152ed7]]

* Octave size: 1196.82{{c}}

Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.

{{Harmonics in equal|152|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed7}}

{{Harmonics in equal|152|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed7 (continued)}}

; [[ed6|140ed6]]

* Octave size: 1196.47{{c}}

Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.

{{Harmonics in equal|140|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 140ed6}}

{{Harmonics in equal|140|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 140ed6 (continued)}}

; [[ed5|126ed5]]

* Octave size: 1194.13{{c}}

Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.

{{Harmonics in equal|126|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}

{{Harmonics in equal|126|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}}

== Scales ==

@@ Line 11: / Line 11: @@
 === 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 ===
@@ Line 362: / Line 353: @@
 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.
+== Octave stretch or compression ==
+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 [[ed6|139ed6]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.
+If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[ed5|126ed5]] and [[ed7|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.
+What follows is a comparison of stretched- and compressed-octave 54edo tunings.
+; [[ed6|139ed6]]
+* Octave size: 1205.08{{c}}
+Stretching the octave of 54edo by around 5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.
+{{Harmonics in equal|139|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 139ed6}}
+{{Harmonics in equal|139|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed6 (continued)}}
+; [[ed7|151ed7]]
+* Octave size: 1204.75{{c}}
+Stretching the octave of 54edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12{{c}}. The tuning 151ed7 does this.
+{{Harmonics in equal|151|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed7}}
+{{Harmonics in equal|151|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed7 (continued)}}
+; [[ed12|193ed12]]
+* Octave size: 1203.66{{c}}
+Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.
+{{Harmonics in equal|193|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 193ed12}}
+{{Harmonics in equal|193|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 193ed12 (continued)}}
+; [[zpi|263zpi]]
+* Step size: 22.243{{c}}, octave size: 1201.12{{c}}
+Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.
+{{Harmonics in cet|22.243|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 263zpi}}
+{{Harmonics in cet|22.243|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 263zpi (continued)}}
+; 54edo
+* Step size: 22.222{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.
+{{Harmonics in equal|54|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54edo}}
+{{Harmonics in equal|54|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54edo (continued)}}
+; [[WE|54et, 13-limit WE tuning]]
+* Step size: 22.198{{c}}, octave size: 1198.69{{c}}
+Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.
+{{Harmonics in cet|22.198|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning}}
+{{Harmonics in cet|22.198|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}}
+; [[zpi|264zpi]]
+* Step size: 22.175{{c}}, octave size: 1197.45{{c}}
+Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.
+{{Harmonics in cet|22.175|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 264zpi}}
+{{Harmonics in cet|22.175|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 264zpi (continued)}}
+; [[ed7|152ed7]]
+* Octave size: 1196.82{{c}}
+Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.
+{{Harmonics in equal|152|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 152ed7}}
+{{Harmonics in equal|152|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 152ed7 (continued)}}
+; [[ed6|140ed6]]
+* Octave size: 1196.47{{c}}
+Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.
+{{Harmonics in equal|140|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 140ed6}}
+{{Harmonics in equal|140|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 140ed6 (continued)}}
+; [[ed5|126ed5]]
+* Octave size: 1194.13{{c}}
+Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
+{{Harmonics in equal|126|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 126ed5}}
+{{Harmonics in equal|126|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}}
 == Scales ==