42edo: Difference between revisions

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

42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.

~~The following table compares three stretched tunings of 42edo:~~

~~{| class="wikitable mw-collapsible mw-collapsed"~~

|-

~~! Tuning~~

~~! [[42ed257/128]]~~

~~! [[11ed6/5]]~~

~~! [[zpi|189zpi]]~~

~~! [[42edo]]~~

|-

~~! Steps / octave~~

~~| ~41.77~~

~~| ~41.81~~

~~| ~41.83~~

~~| 42.00~~

|-

~~! Approximation of harmonics~~

~~| great: 5 good: 2, 3, 7 okay: bad: 11, 13~~

~~| great: 5 good: 2, 3 okay: 7, 11, 13 bad:~~

~~| great: 5 good: 2, 13 okay: 3, 11 bad: 7~~

~~| great: 2, 7 good: okay: 11 bad: 3, 5, 13~~

|-

| colspan="5" | <div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div>

|}

~~The following table compares three compressed tunings of 42edo:~~

~~{| class="wikitable mw-collapsible mw-collapsed"~~

|-

~~! Tuning~~

~~! [[42edo]]~~

~~! [[Equal-step tuning|34ed7/4]]~~

~~! [[191zpi]]~~

~~! [[AS|AS123/121]]~~

|-

~~! Steps / octave~~

~~| 42.00~~

~~| ~42.10~~

~~| ~42.19~~

~~| ~42.24~~

|-

~~! Approximation of harmonics~~

~~| great: 2, 7 good: okay: 11 bad: 3, 5, 13~~

~~| great: 2 good: 5, 7, 13 okay: 3, 11 bad:~~

~~| great: 5, 11, 13 good: 2, 3 okay: bad: 7~~

~~| great: 3, 5, 11 good: 2 okay: 13 bad: 7~~

|-

| colspan="5" | <div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div>

|}

=== Subsets and supersets ===

Line 425:

Line 372:

| 11.42

|}

== Octave stretch or compression ==

42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.

What follows is a comparison of stretched- and compressed-octave 42edo tunings.

; [[ed6|108ed6]]

* Octave size: 1206.3{{c}}

Stretching the octave of 42edo by around 6{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 108ed6 does this. So does the tuning [[ed5|97ed5]] whose octave differs by only 0.1{{c}}.

{{Harmonics in equal|108|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 108ed6}}

{{Harmonics in equal|108|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 108ed6 (continued)}}

; [[zpi|189zpi]]

* Step size: 28.689{{c}}, octave size: 1204.9{{c}}

Stretching the octave of 42edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. The tuning 189zpi does this.

{{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}

{{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}

; [[ed12|150ed12]]

* Octave size: 1204.5{{c}}

Stretcing the octave of 42edo by around 4.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.6{{c}}. The tuning 150ed12 does this.

{{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}

{{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}

; [[equal tuning|145ed11]]

* Octave size: 1202.5{{c}}

Stretching the octave of 42edo by around 2.5{{c}} results in improved primes 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 145ed11 does this.

{{Harmonics in equal|145|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}

{{Harmonics in equal|145|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}

; 42edo

* Step size: 28.571{{c}}, octave size: 1200.0{{c}}

Pure-octaves 42edo approximates all harmonics up to 16 within 13.7{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.

{{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}

{{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}

; [[ed7|118ed7]]

* Step size: Octave size: 1199.1{{c}}

Compressing the octave of 42edo by around 1{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 13.2{{c}}. The tuning 118ed7 does this.

{{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}

{{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}

; [[WE|42et, 13-limit WE tuning]]

* Step size: 28.534{{c}}, octave size: 1198.4{{c}}

Compressing the octave of 42edo by around 1.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

Of the tunings discussed in this section, 13-limit WE and TE are the only ones to approximate all harmonics up to 10 within 10 cents, making them a good all-round choice.

{{Harmonics in cet|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}

{{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}

; [[ed12|151ed12]]

* Step size: Octave size: 1196.6{{c}}

Compressing the octave of 42edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 151ed12 does this. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.

{{Harmonics in equal|151|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}

{{Harmonics in equal|151|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}

; [[ed6|109ed6]]

* Octave size: 1195.2{{c}}

Compressing the octave of 42edo by around 5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 109ed6 does this.

{{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}

{{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}

; [[zpi|191zpi]]

* Step size: 28.444{{c}}, octave size: 1194.6{{c}}

Compressing the octave of 42edo by around 5.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.4{{c}}. The tuning 191zpi does this.

{{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}

{{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}

; [[67edt]]

* Step size: 28.387{{c}}, octave size: 1192.3{{c}}

Compressing the octave of 42edo by around 7.5{{c}} results in improved primes 3, 5 and 11, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 67edt does this.

{{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}

{{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}

== Scales ==

@@ Line 13: / Line 13: @@
 {{Harmonics in equal|42}}
-=== Octave stretch ===
-edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.
-The following table compares three stretched tunings of 42edo:
-{| class="wikitable mw-collapsible mw-collapsed"
-|-
-! Tuning
-! [[42ed257/128]]
-! [[11ed6/5]]
-! [[zpi|189zpi]]
-! [[42edo]]
-|-
-! Steps / octave
-| ~41.77
-| ~41.81
-| ~41.83
-| 42.00
-|-
-! Approximation<br>of harmonics
-| great: 5<br>good: 2, 3, 7 <br>okay: <br>bad: 11, 13
-| great: 5<br>good: 2, 3 <br>okay: 7, 11, 13 <br>bad:
-| great: 5<br>good: 2, 13 <br>okay: 3, 11 <br>bad: 7
-| great: 2, 7<br>good: <br>okay: 11 <br>bad: 3, 5, 13
-|-
-| colspan="5" | <span style="font-size: 0.75em;"><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></span>
-|}
-The following table compares three compressed tunings of 42edo:
-{| class="wikitable mw-collapsible mw-collapsed"
-|-
-! Tuning
-! [[42edo]]
-! [[Equal-step tuning|34ed7/4]]
-! [[191zpi]]
-! [[AS|AS123/121]]
-|-
-! Steps / octave
-| 42.00
-| ~42.10
-| ~42.19
-| ~42.24
-|-
-! Approximation <br>of harmonics
-| great: 2, 7 <br>good: <br>okay: 11 <br>bad: 3, 5, 13
-| great: 2 <br>good: 5, 7, 13 <br>okay: 3, 11 <br>bad:
-| great: 5, 11, 13 <br>good: 2, 3 <br>okay: <br>bad: 7
-| great: 3, 5, 11 <br>good: 2 <br>okay: 13 <br>bad: 7
-|-
-| colspan="5" | <span style="font-size: 0.75em;"><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></span>
-|}
 === Subsets and supersets ===
@@ Line 425: / Line 372: @@
 | 11.42
 |}
+== Octave stretch or compression ==
+edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.
+What follows is a comparison of stretched- and compressed-octave 42edo tunings.
+; [[ed6|108ed6]]
+* Octave size: 1206.3{{c}}
+Stretching the octave of 42edo by around 6{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 108ed6 does this. So does the tuning [[ed5|97ed5]] whose octave differs by only 0.1{{c}}.
+{{Harmonics in equal|108|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 108ed6}}
+{{Harmonics in equal|108|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 108ed6 (continued)}}
+; [[zpi|189zpi]]
+* Step size: 28.689{{c}}, octave size: 1204.9{{c}}
+Stretching the octave of 42edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. The tuning 189zpi does this.
+{{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}
+{{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}
+; [[ed12|150ed12]]
+* Octave size: 1204.5{{c}}
+Stretcing the octave of 42edo by around 4.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.6{{c}}. The tuning 150ed12 does this.
+{{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}
+{{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}
+; [[equal tuning|145ed11]]
+* Octave size: 1202.5{{c}}
+Stretching the octave of 42edo by around 2.5{{c}} results in improved primes 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 145ed11 does this.
+{{Harmonics in equal|145|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}
+{{Harmonics in equal|145|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}
+; 42edo
+* Step size: 28.571{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 42edo approximates all harmonics up to 16 within 13.7{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.
+{{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}
+{{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}
+; [[ed7|118ed7]]
+* Step size: Octave size: 1199.1{{c}}
+Compressing the octave of 42edo by around 1{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 13.2{{c}}. The tuning 118ed7 does this.
+{{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}
+{{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}
+; [[WE|42et, 13-limit WE tuning]]
+* Step size: 28.534{{c}}, octave size: 1198.4{{c}}
+Compressing the octave of 42edo by around 1.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+Of the tunings discussed in this section, 13-limit WE and TE are the only ones to approximate all harmonics up to 10 within 10 cents, making them a good all-round choice.
+{{Harmonics in cet|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}
+{{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}
+; [[ed12|151ed12]]
+* Step size: Octave size: 1196.6{{c}}
+Compressing the octave of 42edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 151ed12 does this. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.
+{{Harmonics in equal|151|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}
+{{Harmonics in equal|151|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}
+; [[ed6|109ed6]]
+* Octave size: 1195.2{{c}}
+Compressing the octave of 42edo by around 5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 109ed6 does this.
+{{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}
+{{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}
+; [[zpi|191zpi]]
+* Step size: 28.444{{c}}, octave size: 1194.6{{c}}
+Compressing the octave of 42edo by around 5.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.4{{c}}. The tuning 191zpi does this.
+{{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}
+{{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}
+; [[67edt]]
+* Step size: 28.387{{c}}, octave size: 1192.3{{c}}
+Compressing the octave of 42edo by around 7.5{{c}} results in improved primes 3, 5 and 11, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 67edt does this.
+{{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}
+{{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}
 == Scales ==