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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.
For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].
Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.
What follows is a comparison of stretched- and compressed-octave 41edo tunings.
What follows is a comparison of stretched- and compressed-octave 41edo tunings.


; [[184zpi]] / [[WE|41et, 11-limit WE tuning]]  
; [[147ed12]] / [[106ed6]] / [[65edt]]
* Step size: 29.277{{c}}, octave size: NNN{{c}}
* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
Stretching the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, which is identical to WE within 1/1000 of a cent.
* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}
* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning (continued)}}
Compressing the octave of 41edo by around 0.2{{c}} results in just slightly improved primes 3, 11 and 13, but just slightly worse primes , 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this.
{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}


; 41edo
; 41edo
* Step size: 29.268{{c}}, octave size: 1200.0{{c}}  
* Step size: 29.268{{c}}, octave size: 1200.00{{c}}  
Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}.
Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}


; [[WE|41et, 13-limit WE tuning]]  
; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]  
* Step size: 29.267{{c}}, octave size: NNN{{c}}
* Step size: 29.277{{c}}, octave size: 1200.35{{c}}
Compressing the octave of 41edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Stretching the octave of 41edo by around 0.5{{c}} results in just slightly improved primes 5 and 7, but just slightly worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.
{{Harmonics in cet|29.267|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}
{{Harmonics in cet|29.267|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}
 
; [[147ed12]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 41edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 147ed12 does this.
{{Harmonics in equal|147|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 147ed12}}
{{Harmonics in equal|147|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 147ed12 (continued)}}
 
; [[106ed6]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 41edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 106ed6 does this.
{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}


; [[65edt]]  
; [[WE|41et, 7-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 29.288{{c}}, octave size: 1200.81{{c}}
Compressing the octave of 41edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 65edt does this.
Stretching the octave of 41edo by just under 1{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
{{Harmonics in equal|65|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65edt}}
{{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}
{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65edt (continued)}}
{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}
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