23edo: Difference between revisions

| en = 23edo

{{Wikipedia|23 equal temperament}}

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]].  

As with [[9edo]], [[16edo]], and [[25edo]], one way to treat 23edo is as a tuning of the [[mavila]] temperament, tempering out the "comma" of [[135/128]] and equating three acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping "[[3/2]]" to 13 degrees of 23, and results in a 7-note [[2L 5s|antidiatonic]] scale of 3–3–4–3–3–3–4 (in steps of 23edo), which extends to a 9-note [[7L 2s|superdiatonic]] scale (3–3–3–1–3–3–3–3–1). One can notate 23edo using the [[Armodue]] system, but just like notating 17edo with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23edo, the "Armodue 6th" is sharper than it is in 16edo, just like the diatonic 5th in 17edo is sharper than in 12edo. In other words, 2b is lower in pitch than 1#, just like how in 17edo Eb is lower than D#.

However, one can also map 3/2 to 14 degrees of 23edo without significantly increasing the error, taking us to a [[7-limit]] temperament where two broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo]] and [[18edo]] and produces [[mos scale]]s of 5 and 8 notes: 5–5–4–5–4 ([[3L 2s|antipentic]]) and 4–1–4–1–4–4–1–4 (the "quartertone" version of the [[Easley Blackwood Jr.|Blackwood]]/[[Paul Rapoport|Rapoport]]/[[Erv Wilson|Wilson]] 13edo "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23edo a sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0–4–9 degrees of 23edo as an approximation to 16:18:21, and 0–5–9 as 1/(16:18:21); both of these triads are abundant in the 8-note mos scale.

23edo has good approximations for [[5/3]], [[11/7]], 13 and 17, among many others, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of no-19's [[23-limit]] [[46edo]], the larger no-19's 23-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.33.13.17.23 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does no-19's 23-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. If one dares to take advantage of this harmony by using 23edo as a period, you get [[icositritonic]], a [[23rd-octave temperaments|23rd-octave temperament]], so that the harmony of 23edo is adequately explained by what harmonies you can achieve using only periods and zero generators.

See ''[[Harmony of 23edo]]'' for more details.  

=== Odd harmonics ===

{{Harmonics in equal|23}}

=== Octave stretch ===

Some approximations can be improved by octave stretching. See ''[[23edo and octave stretching]]'' for more details.

23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]], so it does not contain any nontrivial subset edos, though it contains [[23ed4]]. 46edo, which doubles it, considerably improves most of its approximations of lower harmonics.  

=== Miscellany ===

== Selected just intervals ==

{{Q-odd-limit intervals|23}}

== Notation ==

===Conventional notation ===

===Sagittal notation===

<imagemap>

File:23-EDO_Sagittal.svg

rect 367 0 527 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 367 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]

default [[File:23-EDO_Sagittal.svg]]

</imagemap>

====Second-best fifth notation====

This notation uses the same sagittal sequence as EDOs [[30edo#Sagittal notation|30]], [[37edo#Sagittal notation|37]], and [[44edo#Sagittal notation|44]].

File:23b_Sagittal.svg

rect 375 0 535 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 375 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

default [[File:23b_Sagittal.svg]]

</imagemap>

{| class="wikitable center-all right-1 right-3 left-10"

! Approximate <br> Ratios <ref>Based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.</ref>

! colspan="2" | Major wider <br> than minor

! Armodue <br> Notation

| 1/1

| 33/32, 34/33

|-

| 104.348

| 17/16, 16/15, 18/17

| Less than 1 cent off [[17/16]]

|-

| 11/10, 12/11, 35/32

|-

| 9/8, 44/39

|-

| 5

| 7/6, 15/13, 29/25

|-

| 313.043

| 6/5

| Much better [[6/5]] than 12-edo

|-

| 16/13, 21/17, 26/21

|-

| 14/11, 33/26

| Practically just [[14/11]]

|-

| 469.565

| 21/16, 17/13

|-

| 23/17, 88/65, 256/189

|-

| 573.913

| 7/5, 32/23, 46/33

|-

| 10/7, 23/16, 33/23

| 5#

|-

| 13

| 678.261

| 34/23, 65/44, 189/128

|-

| 32/21, 26/17

|-

| 11/7, 52/33

| Practically just [[11/7]]

|-

| 13/8, 34/21, 21/13

|-

| 5/3

| Much better [[5/3]] than 12-edo

|-

| 12/7, 26/15, 50/29

|-

| 16/9, 39/22

|-

| 1043.478

| 11/6, 20/11, 64/35

|-

| 15/8, 17/9, 32/17

| Less than 1 cent off [[32/17]]

|-

| 1147.826

| 33/17, 64/33

|-

| 2/1

<references/>

 

 

[[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]]

 

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.

 

{| class="wikitable center-all"

|-

! Error (abs, [[Cent|¢]])

|-

|-

|-

 

== Regular temperament properties ==

=== Uniform maps ===

 

=== Commas ===

23et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 23 36 53 65 80 85 }}. Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27).

 

{| class="commatable wikitable center-all left-3 right-4 left-6"

| [[135/128]]

| 92.18

| [[15625/15552]]

| {{monzo| -6 -5 6 }}

| 8.11

| [[36/35]]

| 48.77

| 43.41

| [[4000/3969]]

| {{monzo| 5 -4 3 -2 }}

|-

| [[6144/6125]]

| {{monzo| 11 1 -3 -2 }}

| 5.36

|-

| {{monzo| 2 -2 2 0 -1 }}

|-

 

== Scales ==

 

 

* Mavila 7L2s 133313333 (13\23, 1\1)

 

The chart below shows some of the mos modes of [[mavila]] available in 23edo, mainly Pentatonic (5-note), antidiatonic (7-note), 9- and 16-note mosses. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note mosses:

 

 

=== 23-tone mos scales ===

 

{| class="wikitable"

| 10 10 3

| 9 9 5

| 8 8 7

| 7 7 7 2

| 6 6 6 5

| 5 4 5 5 4

| 5 4 5 4 5

| 7 1 7 7 1

| 7 1 7 1 7

| 5 5 5 5 3

| 4 4 4 4 4 3

| 5 1 5 1 5 1 5

| 3 3 3 5 3 3 3

| 4 3 3 3 3 3 4

| 3 3 4 3 3 3 4

| 4 3 3 3 3 4 3

| 4 1 4 4 1 4 4 1

| 3 3 3 3 3 3 3 2

| 3 3 3 1 3 3 3 3 1

| 3 2 3 2 3 2 3 2 3

| 3 2 2 3 2 2 3 2 2 2

| 3 1 3 1 3 1 3 1 3 1 3

| 3 1 1 3 1 3 1 1 3 1 3 1 1

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

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

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

| 1 1 1 4 1 1 1 1 4 1 1 1 1 4

| 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1

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

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

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

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

 

While [[35edo]] is the largest edo without a nondegenerate [[5L 2s]] scale, it has both degenerate cases (the equalised 7edo and the collapsed 5edo).

 

 

=== Kosmorsky's Sephiroth modes ===

 

Kosmorsky has argued that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale ([[3L 7s|3L 7s fair mosh]]); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence.

 

Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows:

 

 

 

 

 

 

 

 

2 2 3 2 2 2 3 2 2 3 - Chokmah

 

2 3 2 2 2 3 2 2 3 2 - Gevurah

 

3 2 2 2 3 2 2 3 2 2 - Hod

 

5 5 1 2 5 5 - [[Antipental blues]] (approximated from [[Dwarf17marv]])

 

7 2 4 6 4 - Arcade (approximated from [[32afdo]])

 

6 4 1 2 2 6 2 - [[Blackened skies]] (approximated from [[Compton]] in [[72edo]])

 

5 5 3 7 3 - Geode (approximated from [[6afdo]])

 

 

 

 

 

6 4 3 7 3 - Nightdrive (approximated from [[Mavila]] in [[30edo]])

 

6 4 1 2 6 4 - Pelagic (approximated from [[Mavila]] in [[30edo]])

 

2 3 8 2 8 - Approximation of [[Pelog]] lima

 

4 3 6 6 4 - Springwater (approximated from [[8afdo]])

 

2 5 2 4 6 4 - Starship (approximated from [[68ifdo]])

 

2 4 6 1 10 - Tightrope (this is the original/default tuning)

 

6 7 4 2 4 - Underpass (approximated from [[10afdo]])

 

2 5 6 6 4 - Volcanic (approximated from [[16afdo]])

 

 

[[File:Icositriphonic_Guitar.PNG|alt=Icositriphonic_Guitar.PNG|601x305px|Icositriphonic_Guitar.PNG]]

 

 

[[File:Bajo_23-EDD.jpg|alt=Bajo 23-EDD.jpg|800x250px|Bajo 23-EDD.jpg]]

 

''23-EDD Bass by Osmiorisbendi.''

 

[[File:Baritarra_23-EDD.jpg|alt=Baritarra 23-EDD.jpg|800x258px|Baritarra 23-EDD.jpg]]

 

''23-EDD Baritar by Osmiorisbendi.''

 

[[File:Icositritar_1.png|alt=Icositritar 1.png|640x271px|Icositritar 1.png]]

 

''23-EDD 5-string Acoustic Guitar by Tútim Dennsuul Wafiil (RIP).''

 

[[File:Teclado_Icositrifónico.PNG|alt=Teclado Icositrifónico.PNG|607x323px|Teclado Icositrifónico.PNG]]

 

''Illustrative 23-EDD Keyboard''

 

 

[[File:playing.jpg|alt=playing.jpg|playing.jpg]]

 

 

[[File:complette.jpg|alt=complette.jpg|complette.jpg]]

 

This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion.

 

<youtube>K4iO7k152og</youtube>

 

=== Lumatone ===

 

{{Main|23edo/Music}}

 

[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]]

* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitriphonic Scales for Guitar: A Repository of Theory, Reference Materials, and Scale Charts for Xentonal Families]''. 2010.

 

[[Category:23-tone scales]]

[[Category:Mavila]]

[[Category:Modes]]

[[Category:Twentuning]]