36edo: Difference between revisions

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36edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).

36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo#~~octave stretch or compression~~|octave stretch or compression]].

36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo#Octave_stretch_or_compression|octave stretch or compression]].

{{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}

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; [[21edf]]

* Step size: 33.426{{c}}, octave size: 1203.351{{c}}

Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this.

{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}

Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this.

; [[57edt]]

* Step size: 33.368{{c}}, octave size: 1201.235{{c}}

If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.

{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}

If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.

; 36edo

* Step size: 33.333{{c}}, octave size: 1200.000{{c}}

Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.

{{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}

{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}

~~Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.~~

; [[TE|36et, 13-limit TE tuning]]

* Step size: 33.304{{c}}, octave size: 1198.929{{c}}

Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.

{{Harmonics in cet|33.303596|columns=11|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et}}

{{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et (continued)}}

Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.

{| class="wikitable sortable center-all mw-collapsible mw-collapsed"

@@ Line 13: / Line 13: @@
 edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).
-edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo#octave stretch or compression|octave stretch or compression]].
+edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo#Octave_stretch_or_compression|octave stretch or compression]].
 {{Harmonics in equal|36|intervals=odd|prec=2|columns=14}}
 {{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}
@@ Line 1,020: / Line 1,020: @@
 ; [[21edf]]
 * Step size: 33.426{{c}}, octave size: 1203.351{{c}}
+Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this.
 {{Harmonics in equal|21|3|2|columns=11|collapsed=true}}
 {{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
-Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this.
 ; [[57edt]]
 * Step size: 33.368{{c}}, octave size: 1201.235{{c}}
+If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.
 {{Harmonics in equal|57|3|1|columns=11|collapsed=true}}
 {{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
-If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.
 ; 36edo
 * Step size: 33.333{{c}}, octave size: 1200.000{{c}}
+Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.
 {{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}
 {{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}
-Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.
 ; [[TE|36et, 13-limit TE tuning]]
 * Step size: 33.304{{c}}, octave size: 1198.929{{c}}
+Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.
 {{Harmonics in cet|33.303596|columns=11|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et}}
 {{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et (continued)}}
-Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.
 {| class="wikitable sortable center-all mw-collapsible mw-collapsed"