58edo: Difference between revisions

Line 658:

58et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).

== Octave stretch or compression ==

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].

What follows is a comparison of stretched- and compressed-octave 58edo tunings.

; [[zpi|288zpi]]

* Step size: 20.736{{c}}, octave size: 1202.69{{c}}

Stretching the octave of 58edo by around 2.5{{c}} results in improved primes 11, 13, 19 and 23, but worse primes 2, 3, 5, 7 and 17. This approximates all harmonics up to 16 within 9.98{{c}}. The tuning 288zpi does this.

{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}

{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}

; 58edo

* Step size: 20.690{{c}}, octave size: 1200.00{{c}}

Pure-octaves 58edo approximates all harmonics up to 16 within 8.28{{c}}.

{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}

{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}

; [[150ed6]]

* Step size: 20.680{{c}}, octave size: 1199.42{{c}}

Compressing the octave of 58edo by around half a cent results in improved primes 3, 5, 7, 11 and 13 but a worse prime 2. This approximates all harmonics up to 16 within 6.02{{c}}. The tuning 150ed6 does this.

{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}

{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}

; [[92edt]]

* Step size: 20.673{{c}}, octave size: 1199.06{{c}}

Compressing the octave of 58edo by around 1{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 4.60{{c}}. The tuning 92edt does this.

{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}

{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}

; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]

* Step size: 20.666{{c}}, octave size: 1198.63{{c}}

Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 5.49{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.

{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}

{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}

; [[WE|58et, 13-limit WE tuning]]

* Step size: 20.663{{c}}, octave size: 1198.45{{c}}

Compressing the octave of 58edo by just over 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 6.18{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}

{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}

== Scales ==

@@ Line 658: / Line 658: @@
 et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).
+== Octave stretch or compression ==
+edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].
+What follows is a comparison of stretched- and compressed-octave 58edo tunings.
+; [[zpi|288zpi]]
+* Step size: 20.736{{c}}, octave size: 1202.69{{c}}
+Stretching the octave of 58edo by around 2.5{{c}} results in improved primes 11, 13, 19 and 23, but worse primes 2, 3, 5, 7 and 17. This approximates all harmonics up to 16 within 9.98{{c}}. The tuning 288zpi does this.
+{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}
+{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}
+; 58edo
+* Step size: 20.690{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 58edo approximates all harmonics up to 16 within 8.28{{c}}.
+{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}
+{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}
+; [[150ed6]]
+* Step size: 20.680{{c}}, octave size: 1199.42{{c}}
+Compressing the octave of 58edo by around half a cent results in improved primes 3, 5, 7, 11 and 13 but a worse prime 2. This approximates all harmonics up to 16 within 6.02{{c}}. The tuning 150ed6 does this.
+{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}
+{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}
+; [[92edt]]
+* Step size: 20.673{{c}}, octave size: 1199.06{{c}}
+Compressing the octave of 58edo by around 1{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 4.60{{c}}. The tuning 92edt does this.
+{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}
+{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}
+; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]
+* Step size: 20.666{{c}}, octave size: 1198.63{{c}}
+Compressing the octave of 58edo by just under 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 5.49{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.
+{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}
+{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}
+; [[WE|58et, 13-limit WE tuning]]
+* Step size: 20.663{{c}}, octave size: 1198.45{{c}}
+Compressing the octave of 58edo by just over 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 6.18{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}
+{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}
 == Scales ==