99edo: Difference between revisions

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<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.

What follows is a comparison of stretched- and compressed-octave 99edo tunings.

; [[zpi|567zpi]]

* Step size: 12.138{{c}}, octave size: 1201.66{{c}}

Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.

{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}

{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}

; [[WE|99et, 13-limit WE tuning]]

* Step size: 12.123{{c}}, octave size: 1200.18{{c}}

Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}

{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}

; 99edo

* Step size: 12.121{{c}}, octave size: 1200.00{{c}}

Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.

{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}

{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}

; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]

* Step size: 12.117{{c}}, octave size: 1199.58{{c}}

Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.

{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}

{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}

; [[zpi|568zpi]]

* Step size: 12.115{{c}}, octave size: 1199.39{{c}}

Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.

{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}

{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}

; [[157edt]] / [[ed5|230ed5]]

* Step size: 12.114{{c}}, octave size: 1199.32{{c}}

Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.

{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}

{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}

; [[58edf]]

* Step size: 12.103{{c}}, octave size: 1199.16{{c}}

Compressing the octave of 99edo by around 1{{c}} results in improved primes 11, 13, 17 and 19 but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.98{{c}}. The tuning 58edf does this.

{{Harmonics in equal|58|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edf}}

{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}}

== Scales ==

@@ Line 182: / Line 182: @@
 |}
 <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Octave stretch or compression ==
+edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
+What follows is a comparison of stretched- and compressed-octave 99edo tunings.
+; [[zpi|567zpi]]
+* Step size: 12.138{{c}}, octave size: 1201.66{{c}}
+Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
+{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}
+{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}
+; [[WE|99et, 13-limit WE tuning]]
+* Step size: 12.123{{c}}, octave size: 1200.18{{c}}
+Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}
+{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}
+; 99edo
+* Step size: 12.121{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
+{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}
+{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}
+; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
+* Step size: 12.117{{c}}, octave size: 1199.58{{c}}
+Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
+{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}
+{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}
+; [[zpi|568zpi]]
+* Step size: 12.115{{c}}, octave size: 1199.39{{c}}
+Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
+{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}
+{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}
+; [[157edt]] / [[ed5|230ed5]]
+* Step size: 12.114{{c}}, octave size: 1199.32{{c}}
+Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
+{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}
+{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}
+; [[58edf]]
+* Step size: 12.103{{c}}, octave size: 1199.16{{c}}
+Compressing the octave of 99edo by around 1{{c}} results in improved primes 11, 13, 17 and 19 but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.98{{c}}. The tuning 58edf does this.
+{{Harmonics in equal|58|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edf}}
+{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}}
 == Scales ==