User:BudjarnLambeth/Sandbox2: Difference between revisions

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 [[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]], [[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]].

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}}

 

 

* 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}}.

 

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)}}

 

 

 

 

{{Harmonics in equal|40|5|3|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 126ed5 (continued)}}