41edo: Difference between revisions

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== 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.

; [[147ed12]] / [[106ed6]] / [[65edt]]

* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}

* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}

* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}

Compressing the octave of 41edo by around 0.2{{c}} results in [[JND|unnoticeably]] improved primes 3, 11 and 13, but unnoticeably worse primes 2, 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

* Step size: 29.268{{c}}, octave size: 1200.00{{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=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}

; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]

* Step size: 29.277{{c}}, octave size: 1200.35{{c}}

Stretching the octave of 41edo by around 0.5{{c}} results in [[JND|unnoticeably]] improved primes 5 and 7, but unnoticeably 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|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}

{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}

; [[WE|41et, 7-limit WE tuning]]

* Step size: 29.288{{c}}, octave size: 1200.81{{c}}

Stretching the octave of 41edo by just under 1{{c}} results in [[JND|just-noticeably]] improved primes 5 and 7, but just-noticeably worse primes 11 and 13. This approximates all harmonics up to 16 within 11.2{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.

{{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}

{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}

== Scales and modes ==

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+== 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.
+; [[147ed12]] / [[106ed6]] / [[65edt]]
+* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
+* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
+* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
+Compressing the octave of 41edo by around 0.2{{c}} results in [[JND|unnoticeably]] improved primes 3, 11 and 13, but unnoticeably worse primes 2, 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
+* Step size: 29.268{{c}}, octave size: 1200.00{{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=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}
+; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]
+* Step size: 29.277{{c}}, octave size: 1200.35{{c}}
+Stretching the octave of 41edo by around 0.5{{c}} results in [[JND|unnoticeably]] improved primes 5 and 7, but unnoticeably 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|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}
+{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}
+; [[WE|41et, 7-limit WE tuning]]
+* Step size: 29.288{{c}}, octave size: 1200.81{{c}}
+Stretching the octave of 41edo by just under 1{{c}} results in [[JND|just-noticeably]] improved primes 5 and 7, but just-noticeably worse primes 11 and 13. This approximates all harmonics up to 16 within 11.2{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
+{{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}
+{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}
 == Scales and modes ==