54edo: Difference between revisions

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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.

The 54cdd 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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.

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 ===

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== Notation ==

=== Ups and downs notation ===

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

54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down).

It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:

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.

Line 357:

Line 362:

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.

If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[zpi|264zpi]] is a good choice, improving 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 7: / Line 7: @@
 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.
+The 54cdd 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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.
+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 ===
@@ Line 306: / Line 308: @@
 == Notation ==
 === Ups and downs notation ===
-Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:
+edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down).
+{{Ups and downs sharpness}}
+It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:
 {{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.
@@ Line 357: / Line 362: @@
 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.
+If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[zpi|264zpi]] is a good choice, improving 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 ==