45edo: Difference between revisions

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=== Odd harmonics ===

~~=== Octave stretch ===~~

45edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[equal tuning|13ed11/9]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.

~~The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.~~

~~There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.~~

== Intervals ==

Line 589:

Line 582:

|}

== Octave stretch and compression ==

45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.

The tuning [[equal tuning|183ed17]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than 116ed6.

What follows is a comparison of compressed- and stretched-octave 45edo tunings.

; [[zpi|209zpi]]

* Step size: 26.550{{c}}, octave size: 1194.8{{c}}

Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.

{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}

{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}

; 45edo

* Step size: 26.667{{c}}, octave size: 1200.0{{c}}

Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.

{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}

{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}

; [[WE|45et, 13-limit WE tuning]]

* Step size: 26.695{{c}}, octave size: 1201.3{{c}}

Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}

{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}

; [[ed12|161ed12]]

* Step size: Octave size: 1202.4{{c}}

Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.

{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}

{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}

; [[ed6|116ed6]]

* Step size: Octave size: 1203.3{{c}}

Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.

{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}

{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}

; [[WE|45et, 7-limit WE tuning]]

* Step size: 26.745{{c}}, octave size: 1203.5{{c}}

Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.

{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}

{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}

; [[zpi|207zpi]]

* Step size: 26.762{{c}}, octave size: 1204.3{{c}}

Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.

{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}

{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}

; [[71edt]]

* Step size: 26.788{{c}}, octave size: 1205.5{{c}}

Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.

{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}

{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}

; [[equal tuning|183ed17]]

* Octave size: 1206.1{{c}}

Stretching the octave of 45edo by around 6{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 183ed17 does this.

{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}

{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}

== Instruments ==

@@ Line 12: / Line 12: @@
 === Odd harmonics ===
 {{Harmonics in equal|45}}
-=== Octave stretch ===
-edo's approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by [[equal tuning|13ed11/9]], a [[Octave stretch|stretched-octave]] version of 45edo. The trade-off is a slightly worse 2/1.
-The tuning [[126ed7]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.
-There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.
 == Intervals ==
@@ Line 589: / Line 582: @@
 |}
 <references group="note" />
+== Octave stretch and compression ==
+edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by an [[Octave stretch|stretched-octave]] version of 45edo, such as [[ed6|116ed6]]. The trade-off is a slightly worse 2/1.
+The tuning [[equal tuning|183ed17]] may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1, 13/1 ''and'' 17/1, but at the cost of a noticeably worse 2/1 than 116ed6.
+What follows is a comparison of compressed- and stretched-octave 45edo tunings.
+; [[zpi|209zpi]]
+* Step size: 26.550{{c}}, octave size: 1194.8{{c}}
+Compressing the octave of 45edo by around 5{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 11.1{{c}}. The tuning 209zpi does this.
+{{Harmonics in cet|26.550|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 209zpi}}
+{{Harmonics in cet|26.550|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 209zpi (continued)}}
+; 45edo
+* Step size: 26.667{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 45edo approximates all harmonics up to 16 within 13.0{{c}}.
+{{Harmonics in equal|45|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45edo}}
+{{Harmonics in equal|45|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45edo (continued)}}
+; [[WE|45et, 13-limit WE tuning]]
+* Step size: 26.695{{c}}, octave size: 1201.3{{c}}
+Stretching the octave of 45edo by around 1{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|26.695|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning}}
+{{Harmonics in cet|26.695|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 13-limit WE tuning (continued)}}
+; [[ed12|161ed12]]
+* Step size: Octave size: 1202.4{{c}}
+Stretching the octave of 45edo by around 2.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 12.2{{c}}. The tuning 161ed12 does this.
+{{Harmonics in equal|161|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 161ed12}}
+{{Harmonics in equal|161|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 161ed12 (continued)}}
+; [[ed6|116ed6]]
+* Step size: Octave size: 1203.3{{c}}
+Stretching the octave of 45edo by around 3{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 116ed6 does this. So does [[ed7|126ed7]] whose octave is identical within 0.1{{c}}.
+{{Harmonics in equal|116|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 116ed6}}
+{{Harmonics in equal|116|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 116ed6 (continued)}}
+; [[WE|45et, 7-limit WE tuning]]
+* Step size: 26.745{{c}}, octave size: 1203.5{{c}}
+Stretching the octave of 45edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.6{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
+{{Harmonics in cet|26.745|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning}}
+{{Harmonics in cet|26.745|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 45et, 7-limit WE tuning (continued)}}
+; [[zpi|207zpi]]
+* Step size: 26.762{{c}}, octave size: 1204.3{{c}}
+Stretching the octave of 45edo by around 4{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 207zpi does this.
+{{Harmonics in cet|26.762|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 207zpi}}
+{{Harmonics in cet|26.762|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 207zpi (continued)}}
+; [[71edt]]
+* Step size: 26.788{{c}}, octave size: 1205.5{{c}}
+Stretching the octave of 45edo by around 5.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but a worse prime 2. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 71edt does this. So do the tunings [[ed5|104ed5]] and [[equal tuning|155ed11]] whose octave is identical within 0.3{{c}}.
+{{Harmonics in equal|71|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 71edt}}
+{{Harmonics in equal|71|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 71edt (continued)}}
+; [[equal tuning|183ed17]]
+* Octave size: 1206.1{{c}}
+Stretching the octave of 45edo by around 6{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 183ed17 does this.
+{{Harmonics in equal|183|17|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 183ed17}}
+{{Harmonics in equal|183|17|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 183ed17 (continued)}}
 == Instruments ==