23edo: Difference between revisions

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== Theory ==

23edo is significant in that it is the last edo that has no [[5L 2s|diatonic]] perfect fifths and not even [[5edo]] or [[7edo]] fifths. It is also the last edo that fails to approximate the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/5]], [[11/7]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[55/32]], [[77/64]]) very well. The lowest harmonics well-approximated by 23edo are [[9/1|9]], [[13/1|13]], [[15/1|15]], [[17/1|17]], [[21/1|21]], [[23/1|23]], [[31/1|31]], [[33/1|33]] and [[35/1|35]].

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== Notation ==

===Conventional notation ===

~~23edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.~~

The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Chord names don't follow diatonic nominals because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}.

The second approach is to essentially pretend 23edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. This allows music notated in 12edo or another diatonic system to be directly translated to 23edo "on the fly", and it carries over the way interval arithmetic and chord names work from diatonic notation.

===Sagittal notation===

====Best fifth notation====

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== Approximation to irrational intervals ==

~~=== Acoustic π and ϕ ===~~

23edo has good approximations of [[acoustic phi]] on 16\23, and [[pi]] on 38\23. Not until [[72edo|72]] do we find a better edo in terms of absolute error, and not until [[749edo|749]] do we find one in terms of relative error.

23edo has ~~a very close approximation~~ of [[~~11/7#Proximity with π/2|~~acoustic ~~π/2~~]] on 15\23 and ~~a very close approximation of~~ [[~~acoustic phi~~]] on ~~the step just above (16~~\23).

{| class="wikitable center-all"

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| 1.692

|}

~~Not until [[72edo|72]] do we find a better edo in terms of absolute error, and not until [[749edo|749]] do we find one in terms of relative error.~~

== Regular temperament properties ==

=== Uniform maps ===

=== Commas ===

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|-

| 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1

| ~~Pathological~~ 5L 13s ~~(ateamtonic[18~~])

| [[5L 13s]]

|-

| 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1

| ~~Pathological~~ [[4L 15s~~|<nowiki>4L 15s (mynoid[19]]</nowiki>~~]]

| [[4L 15s]]

|}

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 {{Wikipedia|23 equal temperament}}
 {{ED intro}}
 == Theory ==
 edo is significant in that it is the last edo that has no [[5L 2s|diatonic]] perfect fifths and not even [[5edo]] or [[7edo]] fifths. It is also the last edo that fails to approximate the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/5]], [[11/7]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[55/32]], [[77/64]]) very well. The lowest harmonics well-approximated by 23edo are [[9/1|9]], [[13/1|13]], [[15/1|15]], [[17/1|17]], [[21/1|21]], [[23/1|23]], [[31/1|31]], [[33/1|33]] and [[35/1|35]].
@@ Line 37: / Line 38: @@
 == Notation ==
 ===Conventional notation ===
-edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.
+{{Mavila}}
-The first defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Chord names don't follow diatonic nominals because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}.
-The second approach is to essentially pretend 23edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. This allows music notated in 12edo or another diatonic system to be directly translated to 23edo "on the fly", and it carries over the way interval arithmetic and chord names work from diatonic notation.
 ===Sagittal notation===
 ====Best fifth notation====
@@ Line 279: / Line 277: @@
 == Approximation to irrational intervals ==
-=== Acoustic π and ϕ ===
+edo has good approximations of [[acoustic phi]] on 16\23, and [[pi]] on 38\23. Not until [[72edo|72]] do we find a better edo in terms of absolute error, and not until [[749edo|749]] do we find one in terms of relative error.
-edo has a very close approximation of [[11/7#Proximity with π/2|acoustic π/2]] on 15\23 and a very close approximation of [[acoustic phi]] on the step just above (16\23).
 {| class="wikitable center-all"
@@ Line 297: / Line 294: @@
 | 1.692
 |}
-Not until [[72edo|72]] do we find a better edo in terms of absolute error, and not until [[749edo|749]] do we find one in terms of relative error.
 == Regular temperament properties ==
 === Uniform maps ===
-{{Uniform map|13|22.5|23.5}}
+{{Uniform map|edo=23}}
 === Commas ===
@@ Line 492: / Line 487: @@
 |-
 | 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1
-| Pathological 5L 13s (ateamtonic[18])
+| [[5L 13s]]
 |-
 | 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
-| Pathological [[4L 15s|<nowiki>4L 15s (mynoid[19]]</nowiki>]]
+| [[4L 15s]]
 |}