User:BudjarnLambeth/Sandbox2: Difference between revisions

Line 8:

; 32edo

* Step size: 37.500{{c}}, octave size: 1200.00{{c}}

* Step size: 37.500{{c}}, octave size: 1200.0{{c}}

Pure-octaves 32edo approximates all harmonics up to 16 within NNN{{c}}.

{{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}

Line 14:

; [[WE|32et, 13-limit WE tuning]]

* Step size: 37.481{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.481{{c}}, octave size: 1199.4{{c}}

Compressing the octave of 32edo 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-linut [[TE]] tuning both do this.

Compressing the octave of 32edo by around half a cent 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-linut [[TE]] tuning both do this.

{{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}

{{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; [[WE|32et, 11-limit WE tuning]]

* Step size: 37.453{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.453{{c}}, octave size: 1198.5{{c}}

Compressing the octave of 32edo 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.

Compressing the octave of 32edo by around 1.5{{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.

{{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}}

{{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}}

; [[90ed7]]

* Step size: ~~NNN~~{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.431{{c}}, octave size: 1197.8{{c}}

Compressing the octave of 32edo by around ~~NNN~~{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.

Compressing the octave of 32edo by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.

{{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}

{{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}

Line 34:

* Step size: 37.418{{c}}, octave size: 1197.375{{c}}

Compressing the octave of 32edo 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 133zpi does this.

{{Harmonics in cet|~~100~~|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}

{{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}

{{Harmonics in cet|~~100~~|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}

{{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}

Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.

Line 41:

; [[51edt]]

* Step size: ~~NNN~~{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.293{{c}}, octave size: 1193.4{{c}}

Compressing the octave of 32edo 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 51edt does this.

Compressing the octave of 32edo by around 6.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 51edt does this.

{{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}}

{{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}}

; [[zpi|134zpi]]

* Step size: 37.176{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.176{{c}}, octave size: 1189.6{{c}}

Compressing the octave of 134zpi 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 134zpi does this.

Compressing the octave of 134zpi by around 10.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 134zpi does this.

{{Harmonics in cet|37.176|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 134zpi}}

{{Harmonics in cet|37.176|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 134zpi (continued)}}

; [[75ed5]]

* Step size: ~~NNN~~{{c}}, octave size: ~~NNN~~{{c}}

* Step size: 37.151{{c}}, octave size: 1188.8{{c}}

Compressing the octave of 32edo 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 75ed5 does this.

Compressing the octave of 32edo by around 11{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 75ed5 does this.

{{Harmonics in equal|75|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 75ed5}}

{{Harmonics in equal|75|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 75ed5 (continued)}}

; [[WE|ETNAME, SUBGROUP WE tuning]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing 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 SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.

{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}

{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; [[EDONOI]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing 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}}. The tuning EDONOI does this.

{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}

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

; [[zpi|ZPINAME]]

* Step size: NNN{{c}}, octave size: NNN{{c}}

_ing 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}}. The tuning ZPINAME does this.

{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}

{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}

= Title2 =

@@ Line 8: / Line 8: @@
 ; 32edo
-* Step size: 37.500{{c}}, octave size: 1200.00{{c}}
+* Step size: 37.500{{c}}, octave size: 1200.0{{c}}
 Pure-octaves 32edo approximates all harmonics up to 16 within NNN{{c}}.
 {{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}
@@ Line 14: / Line 14: @@
 ; [[WE|32et, 13-limit WE tuning]]
-* Step size: 37.481{{c}}, octave size: NNN{{c}}
+* Step size: 37.481{{c}}, octave size: 1199.4{{c}}
-Compressing the octave of 32edo 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-linut [[TE]] tuning both do this.
+Compressing the octave of 32edo by around half a cent 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-linut [[TE]] tuning both do this.
 {{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}
 {{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
 ; [[WE|32et, 11-limit WE tuning]]
-* Step size: 37.453{{c}}, octave size: NNN{{c}}
+* Step size: 37.453{{c}}, octave size: 1198.5{{c}}
-Compressing the octave of 32edo 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.
+Compressing the octave of 32edo by around 1.5{{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.
 {{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}}
 {{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}}
 ; [[90ed7]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: 37.431{{c}}, octave size: 1197.8{{c}}
-Compressing the octave of 32edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.
+Compressing the octave of 32edo by around 2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.
 {{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}
 {{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}
@@ Line 34: / Line 34: @@
 * Step size: 37.418{{c}}, octave size: 1197.375{{c}}
 Compressing the octave of 32edo 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 133zpi does this.
-{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}
+{{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}
-{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}
+{{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}
 Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.
@@ Line 41: / Line 41: @@
 ; [[51edt]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: 37.293{{c}}, octave size: 1193.4{{c}}
-Compressing the octave of 32edo 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 51edt does this.
+Compressing the octave of 32edo by around 6.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 51edt does this.
 {{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}}
 {{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}}
 ; [[zpi|134zpi]]
-* Step size: 37.176{{c}}, octave size: NNN{{c}}
+* Step size: 37.176{{c}}, octave size: 1189.6{{c}}
-Compressing the octave of 134zpi 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 134zpi does this.
+Compressing the octave of 134zpi by around 10.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 134zpi does this.
 {{Harmonics in cet|37.176|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 134zpi}}
 {{Harmonics in cet|37.176|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 134zpi (continued)}}
 ; [[75ed5]]
-* Step size: NNN{{c}}, octave size: NNN{{c}}
+* Step size: 37.151{{c}}, octave size: 1188.8{{c}}
-Compressing the octave of 32edo 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 75ed5 does this.
+Compressing the octave of 32edo by around 11{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 75ed5 does this.
 {{Harmonics in equal|75|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 75ed5}}
 {{Harmonics in equal|75|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 75ed5 (continued)}}
+; [[WE|ETNAME, SUBGROUP WE tuning]]
+* Step size: NNN{{c}}, octave size: NNN{{c}}
+_ing 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 SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
+{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
+{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}
+; [[EDONOI]]
+* Step size: NNN{{c}}, octave size: NNN{{c}}
+_ing 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}}. The tuning EDONOI does this.
+{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
+{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
+; [[zpi|ZPINAME]]
+* Step size: NNN{{c}}, octave size: NNN{{c}}
+_ing 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}}. The tuning ZPINAME does this.
+{{Harmonics in cet|100|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
+{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
 = Title2 =