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 preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 is not E. Chord names are different because C - E - G is not P1 - M3 - P5.

The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly".

===Sagittal 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. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 is not E. Chord names are different because C - E - G is not P1 - M3 - P5.
+{{Mavila}}
-The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly".
 ===Sagittal notation===
@@ Line 278: / 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 296: / 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 491: / 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]]
 |}