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{{interwiki
{{interwiki
| de =  
| de = 23edo
| en = 23edo
| en = 23edo
| es =  
| es = 23 EDO
| ja = 23平均律
| ja = 23平均律
}}
}}
{{Infobox ET}}
{{Wikipedia|23 equal temperament}}
{{ED intro}}


== Theory ==
== 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. In this sense, it can be thought of as every other step of [[46edo]]. 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]].


<b>23-TET</b>, or <b>23-EDO</b>, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].
=== Mapping ===
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#.


23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
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.


23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics 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, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].
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.


As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic 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 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.
See ''[[Harmony of 23edo]]'' for more details.  


However, one can also map 3/2 to 14 degrees of 23-EDO 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|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO 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 23-EDO 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.
=== Odd harmonics ===
{{Harmonics in equal|23}}


== Selected just intervals ==
=== Subsets and supersets ===
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.


{| class="wikitable center-all"
=== Miscellany ===
23edo was proposed by ethnomusicologist {{w|Erich von Hornbostel}} as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
 
== Intervals ==
{| class="wikitable center-1 right-2 left-10"
|-
! [[Degree]]
! [[Cent]]s
! Approximate Ratios*
! Comments
|-
| 0
| 0.0
| [[1/1]]
|
|-
| 1
| 52.2
| [[33/32]], [[34/33]]
|
|-
| 2
| 104.3
| [[17/16]], [[16/15]], [[18/17]]
| Less than 1 cent off [[17/16]]
|-
| 3
| 156.5
| [[11/10]], [[12/11]], [[35/32]]
|
|-
| 4
| 208.7
| [[9/8]], [[44/39]]
|
|-
| 5
| 260.9
| [[7/6]], [[15/13]], [[29/25]]
|
|-
| 6
| 313.0
| [[6/5]]
| Much better 6/5 than 12-edo
|-
| 7
| 365.2
| [[16/13]], [[21/17]], [[26/21]]
|
|-
| 8
| 417.4
| [[14/11]], [[33/26]]
| Practically just 14/11
|-
| 9
| 469.6
| [[21/16]], [[17/13]]
|
|-
| 10
| 521.7
| [[23/17]], [[27/20]], [[88/65]]
|
|-
| 11
| 573.9
| [[7/5]], [[32/23]], [[46/33]]
|
|-
| 12
| 626.1
| [[10/7]], [[23/16]], [[33/23]]
|
|-
| 13
| 678.3
| [[34/23]], [[40/27]], [[65/44]]
| Great Hornbostel generator
|-
| 14
| 730.4
| [[32/21]], [[26/17]]
|
|-
| 15
| 782.6
| [[11/7]], [[52/33]]
| Practically just [[11/7]]
|-
| 16
| 834.8
| [[13/8]], [[34/21]], [[21/13]]
|
|-
| 17
| 887.0
| [[5/3]]
| Much better [[5/3]] than 12-edo
|-
| 18
| 939.1
| [[12/7]], [[26/15]], [[50/29]]
|
|-
|-
|+ Approximation of [[primary interval]]s in 23 EDO
| 19
| 991.3
| [[16/9]], [[39/22]]
|
|-
|-
! colspan="2" | Prime number
| 20
! 3
| 1043.5
! 5
| [[11/6]], [[20/11]], [[64/35]]
! 7
|
! 11
! 13
! 17
! 19
! 23
|-
|-
! rowspan="2" | Error
| 21
! absolute ([[cent|¢]])
| 1095.7
| -23.69
| [[15/8]], [[17/9]], [[32/17]]
| -21.10
| Less than 1 cent off 32/17
| +22.48
| +22.60
| -5.75
| -0.61
| +15.53
| -2.19
|-
|-
! [[Relative error|relative]] (%)
| 22
| -45.4
| 1147.8
| -40.4
| [[33/17]], [[64/33]]
| +43.1
|
| +43.3
| -11.0
| -1.2
| +29.8
| -4.2
|-
|-
! colspan="2" | Degree ([[octave reduction|reduced]])
| 23
| 36 (13)
| 1200.0
| 53 (7)
| [[2/1]]
| 65 (19)
|
| 80 (11)
| 85 (16)
| 94 (2)
| 98 (6)
| 104 (12)
|}
|}
*Based on treating 23edo as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.


== Notation ==
== Notation ==
===Conventional notation ===
{{Mavila}}
===Sagittal notation===
====Best fifth notation====
This notation uses the same sagittal sequence as EDOs [[28edo#Sagittal notation|28]] and [[33edo#Sagittal notation|33]].


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 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
{{Sagittal chart|}}


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".
====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]].


{{Sagittal chart||23b}}
=== Armodue notation  ===
Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names.
Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names.


{| class="wikitable center-all right-1 right-3 left-10"
{| class="wikitable center-all right-2"
|-
|-
! colspan="2" | [[Degree]] <ref>The dots indicate which frets on a 23-edo guitar would have dots.</ref>
! #
! [[Cent]]s
! [[Cent]]s
! 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
! colspan="2" | Major wider <br> than minor
! colspan="2" | Major narrower <br> than minor
! colspan="2" | Major narrower <br> than minor
! Armodue <br> Notation
! Armodue <br> Notation
! Notes
|-
|-
| 0 ||
| 0
| 0.000
| 0.0
| 1/1
| P1 || D
| P1 || D
| P1 || D
| P1 || D
| 1
| 1
|
|-
|-
| 1 ||
| 1
| 52.174
| 52.2
| 33/32, 34/33
| A1 || D#
| A1 || D#
| d1 || Db
| d1 || Db
| 2b
| 2b
|
|-
|-
| 2 ||
| 2
| 104.348
| 104.3
| 17/16, 16/15, 18/17
| d2 || Eb
| d2 || Eb
| A2 || E#
| A2 || E#
| 1#
| 1#
| Less than 1 cent off [[17/16]]
|-
|-
| 3 ||
| 3
| 156.522
| 156.5
| 11/10, 12/11, 35/32
| m2 || E
| m2 || E
| M2 || E
| M2 || E
| 2
| 2
|
|-
|-
| 4 || &bull;
| 4
| 208.696
| 208.7
| 9/8, 44/39
| M2 || E#
| M2 || E#
| m2 || Eb
| m2 || Eb
| 3b
| 3b
|
|-
|-
| 5 ||
| 5
| 260.870
| 260.9
| 7/6, 15/13, 29/25
| A2, d3 || Ex, Fbb
| A2, d3 || Ex, Fbb
| d2, A3 || Ebb, Fx
| d2, A3 || Ebb, Fx
| 2#
| 2#
|
|-
|-
| 6 ||
| 6
| 313.043
| 313.0
| 6/5
| m3 || Fb
| m3 || Fb
| M3 || F#
| M3 || F#
| 3
| 3
| Much better [[6/5]] than 12-edo
|-
|-
| 7 || &bull;
| 7
| 365.217
| 365.2
| 16/13, 21/17, 26/21
| M3 || F
| M3 || F
| m3 || F
| m3 || F
| 4b
| 4b
|
|-
|-
| 8 ||
| 8
| 417.391
| 417.4
| 14/11, 33/26
| A3 || F#
| A3 || F#
| d3 || Fb
| d3 || Fb
| 3#
| 3#
| Practically just [[14/11]]
|-
|-
| 9 ||
| 9
| 469.565
| 469.6
| 21/16, 17/13
| d4 || Gb
| d4 || Gb
| A4 || G#
| A4 || G#
| 4
| 4
|
|-
|-
| 10 || &bull;
| 10
| 521.739
| 521.7
| 23/17, 88/65, 256/189
| P4 || G
| P4 || G
| P4 || G
| P4 || G
| 5
| 5
|
|-
|-
| 11 ||
| 11
| 573.913
| 573.9
| 7/5, 32/23, 46/33
| A4 || G#
| A4 || G#
| d4 || Gb
| d4 || Gb
| 6b
| 6b
|
|-
|-
| 12 ||
| 12
| 626.087
| 626.1
| 10/7, 23/16, 33/23
| d5 || Ab
| d5 || Ab
| A5 || A#
| A5 || A#
| 5#
| 5#
|
|-
|-
| 13 || &bull;
| 13
| 678.261
| 678.3
| 34/23, 65/44, 189/128
| P5 || A
| P5 || A
| P5 || A
| P5 || A
| 6
| 6
| Great Hornbostel generator
|-
|-
| 14 ||
| 14
| 730.435
| 730.4
| 32/21, 26/17
| A5 || A#
| A5 || A#
| d5 || Ab
| d5 || Ab
| 7b
| 7b
|
|-
|-
| 15 ||
| 15
| 782.609
| 782.6
| 11/7, 52/33
| d6 || Bb
| d6 || Bb
| A6 || B#
| A6 || B#
| 6#
| 6#
| Practically just [[11/7]]
|-
|-
| 16 || &bull;
| 16
| 834.783
| 834.8
| 13/8, 34/21, 21/13
| m6 || B
| m6 || B
| M6 || B
| M6 || B
| 7
| 7
|
|-
|-
| 17 ||
| 17
| 886.957
| 887.0
| 5/3
| M6 || B#
| M6 || B#
| m6 || Bb
| m6 || Bb
| 8b
| 8b
| Much better [[5/3]] than 12-edo
|-
|-
| 18 ||
| 18
| 939.130
| 939.1
| 12/7, 26/15, 50/29
| A6, d7 || Bx, Cbb
| A6, d7 || Bx, Cbb
| d6, A7 || Bbb, Cx
| d6, A7 || Bbb, Cx
| 7#
| 7#
|
|-
|-
| 19 || &bull;
| 19
| 991.304
| 991.3
| 16/9, 39/22
| m7 || Cb
| m7 || Cb
| M7 || C#
| M7 || C#
| 8
| 8
|
|-
|-
| 20 ||
| 20
| 1043.478
| 1043.5
| 11/6, 20/11, 64/35
| M7 || C
| M7 || C
| m7 || C
| m7 || C
| 9b
| 9b
|
|-
|-
| 21 ||
| 21
| 1095.652
| 1095.7
| 15/8, 17/9, 32/17
| A7 || C#
| A7 || C#
| d7 || Cb
| d7 || Cb
| 8#
| 8#
| Less than 1 cent off [[32/17]]
|-
|-
| 22 ||
| 22
| 1147.826
| 1147.8
| 33/17, 64/33
| d8 || Db
| d8 || Db
| A8 || D#
| A8 || D#
| 9
| 9
|
|-
|-
| 23 || &bull;&bull;
| 23
| 1200.000
| 1200.0
| 2/1
| P8 || D
| P8 || D
| P8 || D
| P8 || D
| 1
| 1
|
|}
|}


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


=== Armodue Notation  ===
== Approximation to irrational intervals ==
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.


[[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]]
{| class="wikitable center-all"
|+Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| π
| 0.813
|-
| π/ϕ
| 0.879
|-
| ϕ
| 1.692
|}
 
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|23}}
{{Q-odd-limit intervals|22.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 23de val mapping}}
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=23}}


== Commas ==
=== Commas ===
23 EDO tempers out the following [[commas]]. (Note: This assumes the val &lt; 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).
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="wikitable center-all left-2 right-3"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]
! [[Ratio]]
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color notation/Temperament Names|Color Name]]
! [[Color name]]
! Name 1
! Name(s)
! Name 2
! Name 3
|-
|-
| 135/128
| 5
| {{Monzo| -7 3 1 }}  
| [[135/128]]
| {{monzo| -7 3 1 }}  
| 92.18
| 92.18
| Layobi
| Layobi
| Major Chroma
| Mavila comma, major chroma
| Major Limma
| Pelogic Comma
|-
|-
| 15625/15552
| 5
| {{Monzo| -6 -5 6 }}  
| [[15625/15552]]
| {{monzo| -6 -5 6 }}  
| 8.11
| 8.11
| Tribiyo
| Tribiyo
| Kleisma
| Kleisma, semicomma majeur
| Semicomma Majeur
|
|-
|-
| 36/35
| 7
| {{Monzo| 2 2 -1 -1 }}  
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}  
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal Quarter Tone
| Mint comma, septimal quartertone
|
|
|-
|-
| 525/512
| 7
| {{Monzo| -9 1 2 1 }}  
| [[525/512]]
| {{monzo| -9 1 2 1 }}  
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma
| Avicennma, Avicenna's enharmonic diesis
| Avicenna's Enharmonic Diesis
|
|-
|-
| 4000/3969
| 7
| {{Monzo| 5 -4 3 -2 }}  
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}  
| 13.47
| 13.47
| Rurutriyo
| Rurutriyo
| Octagar
| Octagar comma
|
|
|-
|-
| 6144/6125
| 7
| {{Monzo| 11 1 -3 -2 }}  
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}  
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|
|
|-
|-
| 100/99
| 11
| {{Monzo| 2 -2 2 0 -1 }}  
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}  
| 17.40
| 17.40
| Luyoyo
| Luyoyo
| Ptolemisma
| Ptolemisma
|
|
|-
|-
| 441/440
| 11
| {{Monzo| -3 2 -1 2 -1 }}  
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}  
| 3.93
| 3.93
| Luzozogu
| Luzozogu
| Werckisma
| Werckisma
|
|
|}
|}


== MOS Scales ==
== Octave stretch or compression ==
{{main|23edo and octave stretching}}
 
23edo is not often taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.
 
However, when using a slightly [[stretched tuning|stretched octave]] of around 1206 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.
 
Stretched-23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments.


The chart below shows some of the [[MOSScales|Moment of Symmetry (MOS)]] modes of [[Mavila]] available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note MOSs:
== Scales ==
 
Important [[mos]]ses include:
 
* Mavila 2L5s 4334333 (13\23, 1\1)
* Mavila 7L2s 133313333 (13\23, 1\1)
* Sephiroth 3L4s 2525252 (7\23, 1\1)
* [[Semiquartal]] 5L4s 332323232 (5\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:


[[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]]
[[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]]


=== 23 tone MOS scales ===
=== 23-tone mos scales ===


{| class="wikitable"
{| class="wikitable"
Line 383: Line 468:
|-
|-
| 7 7 7 2
| 7 7 7 2
|
|-
| 7 2 7 7
|
|
|-
|-
| 6 6 6 5
| 6 6 6 5
|
|-
| 6 5 6 6
|
|
|-
|-
| 5 4 5 5 4
| 5 4 5 5 4
| [[3L 2s|3L 2s (father)]]
| [[3L 2s|3L 2s (oneiro-pentatonic)]]
|-
|-
| 5 4 5 4 5
| 5 4 5 4 5
Line 407: Line 486:
|-
|-
| 5 5 5 5 3
| 5 5 5 5 3
| [[4L 1s|4L 1s (bug)]]
| [[4L 1s|4L 1s (bug pentatonic)]]
|-
| 5 3 5 5 5
|
|-
|-
| 4 4 4 4 4 3
| 4 4 4 4 4 3
| [[5L 1s|5L 1s (Grumpy hexatonic)]]
| [[5L 1s|5L 1s (machinoid)]]
|-
| 4 3 4 4 4 4
|
|-
|-
| 5 1 5 1 5 1 5
| 5 1 5 1 5 1 5
| [[4L 3s|4L 3s (mish)]]
| [[4L 3s|4L 3s (smitonic)]]
|-
|-
| 3 3 3 5 3 3 3
| 3 3 3 5 3 3 3
| [[1L 6s|1L 6s (Happy heptatonic)]]
| [[1L 6s|1L 6s (antiarcheotonic)]]
|-
|-
| 4 3 3 3 3 3 4
| 4 3 3 3 3 3 4
| [[2L 5s|2L 5s (mavila, anti-diatonic)]]
|  
|-
| 3 4 3 3 4 3 3
|
|-
|-
| 3 3 4 3 3 3 4
| 3 3 4 3 3 3 4
|
| [[2L 5s|2L 5s (mavila, anti-diatonic)]]
|-
| 3 3 3 4 3 3 4
|
|-
|-
| 3 3 3 4 3 4 3
| 4 3 3 3 3 4 3
|
|
|-
|-
Line 443: Line 510:
|-
|-
| 4 1 4 4 1 4 4 1
| 4 1 4 4 1 4 4 1
| [[5L 3s|5L 3s (unfair father)]]
| [[5L 3s|5L 3s (oneirotonic)]]
|-
|-
| 3 3 3 3 3 3 3 2
| 3 3 3 3 3 3 3 2
| [[7L 1s|7L 1s (Grumpy octatonic)]]
| [[7L 1s|7L 1s (porcupoid)]]
|-
|-
| 3 2 3 3 3 3 3 3
| 3 3 3 1 3 3 3 3 1
|
|[[7L 2s|7L 2s (mavila superdiatonic)]]
|-
| '''3 3 3 1 3 3 3 3 1'''
| [[7L 2s|7L 2s (mavila superdiatonic)]]
|-
| 3 3 1 3 3 3 1 3 3
|
|-
|-
| 3 2 3 2 3 2 3 2 3
| 3 2 3 2 3 2 3 2 3
| [[5L 4s|5L 4s (unfair bug)]]
| [[5L 4s|5L 4s (bug semiquartal)]]
|-
| 2 2 2 3 2 2 3 2 2 3
| Mode Keter
|-
| 2 2 3 2 2 3 2 2 3 2
| Chesed
|-
| 2 3 2 2 3 2 2 3 2 2
| Netzach
|-
|-
| 3 2 2 3 2 2 3 2 2 2
| 3 2 2 3 2 2 3 2 2 2
| Malkuth
| [[3L 7s|3L 7s (sephiroid)]]
|-
|-
| 2 2 3 2 2 3 2 2 2 3
| 4 1 1 4 1 1 4 1 1 4 1
| Binah
| [[4L 7s|4L 7s (kleistonic)]]
|-
|-
| 2 3 2 2 3 2 2 2 3 2
| 3 1 3 1 3 1 3 1 3 1 3
| Tiferet
| Palestine 11
|-
| 3 2 2 3 2 2 2 3 2 2
| Yesod
|-
| 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
|-
|-
| '''3 1 3 1 3 1 3 1 3 1 3'''
| 3 1 1 3 1 3 1 1 3 1 3 1 1
| Palestine 11
| [[5L 8s|5L 8s (ateamtonic)]]
|-
|-
| 2 2 2 2 1 2 2 2 1 2 2 2 1
| 2 2 2 2 1 2 2 2 1 2 2 2 1
| Mode Tishrei
| [[10L 3s|10L 3s (luachoid)]]
|-
| 2 2 2 1 2 2 2 1 2 2 2 1 2
| Cheshvan
|-
| 2 2 1 2 2 2 1 2 2 2 1 2 2
| Kislev
|-
| 2 1 2 2 2 1 2 2 2 1 2 2 2
| Tevet
|-
| 1 2 2 2 1 2 2 2 1 2 2 2 2
| Shvat
|-
| 2 2 2 1 2 2 2 1 2 2 2 2 1
| Adar minor
|-
| 2 2 1 2 2 2 1 2 2 2 2 1 2
| Adar major
|-
| 2 1 2 2 2 1 2 2 2 2 1 2 2
| Nisan
|-
| 1 2 2 2 1 2 2 2 2 1 2 2 2
| Iyar
|-
| 2 2 2 1 2 2 2 2 1 2 2 2 1
| Sivan
|-
| 2 2 1 2 2 2 2 1 2 2 2 1 2
| Tammuz
|-
| 2 1 2 2 2 2 1 2 2 2 1 2 2
| Av
|-
| 1 2 2 2 2 1 2 2 2 1 2 2 1
| Elul
|-
|-
| 2 2 1 2 2 1 2 2 1 2 2 1 2 1
| 2 2 1 2 2 1 2 2 1 2 2 1 2 1
|
| [[9L 5s]] (Brittle [[Titanium]])
|-
|-
| '''2 1 2 2 1 2 2 1 2 2 1 2 2 1'''
| 2 1 2 2 1 2 2 1 2 2 1 2 2 1
| Palestine 14
| Palestine 14
|-
|-
| 1 1 1 4 1 1 1 1 4 1 1 1 1 4
| 1 1 1 4 1 1 1 1 4 1 1 1 1 4
|
| [[3L 11s]]
|-
| 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1
| [[4L 11s|4L 11s (mynoid)]]
|-
|-
| 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
| 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
|
| [[8L 7s]]
|-
| 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 1
| [[7L 9s|7L 9s (mavila chromatic)]]
|-
|-
| '''2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1'''
| 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1
| Palestine 17
| Palestine 17
|-
| 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1
| [[5L 13s]]
|-
|-
| 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
| 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1
|
| [[4L 15s]]
|}
|}
While [[35edo]] is the largest edo without a nondegenerate [[5L 2s]] scale, it has both degenerate cases (the equalised 7edo and the collapsed 5edo).
23edo is the largest edo without any form of 5L 2s, including the degenerate cases.


=== Kosmorsky's Sephiroth modes ===
=== Kosmorsky's Sephiroth modes ===


I would argue 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.
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:
Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows:
Line 577: Line 594:
3 2 2 2 3 2 2 3 2 2 - Hod
3 2 2 2 3 2 2 3 2 2 - Hod


== Books ==
=== Miscellaneous ===
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]])
 
5 4 2 2 4 2 4 - Lost phantom (approximated from [[Mavila]] in [[30edo]])
 
6 4 2 1 5 1 4 - [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]])


[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]]
5 2 6 6 4 - Mechanical (approximated from [[31afdo]])
 
5 4 4 2 8 - Mushroom (approximated from [[30afdo]])
 
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]])
 
''More listed in: [[User:BudjarnLambeth/Quasipelog theory#Scales]]''


== Instruments ==
== Instruments ==
Line 614: Line 662:


<youtube>K4iO7k152og</youtube>
<youtube>K4iO7k152og</youtube>
 
=== Lumatone ===
See: [[Lumatone mapping for 23edo]]
 
== Music ==
== Music ==
{{Main|23edo/Music}}
{{Catrel|23edo tracks}}


* [https://www.youtube.com/watch?v=lYb4iK4Lt5Q Promethean Elegies], by [http://benfuhrman.com/ Ben Fuhrman]
== Further reading ==
* [https://soundcloud.com/overtoneshock/curiosity-finds-a-frown-23-edo Curiosity Finds a Frown, by Stephen Weigel]
[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]]
* [http://soonlabel.com/xenharmonic/archives/2460 Chromatic canon, by Claudi Meneghin]
* [[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.
* [http://home.vicnet.net.au/%7Eepoetry/family.mp3 The Family Supper] by [[Warren Burt]]
* [[343edo#Scales|Lucite23]] - [[Gordon Wery]]'s [[well temperament]] of 23edo in [[343edo]]
* [http://www.youtube.com/watch?v=Hqst8MaRiYM Icositriphonic Heptatonic MOS] by [[Igliashon Jones]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20His%20Wandering%20Kinship%20with%20Ashes.mp3 His Wandering Kinship with Ashes] by Igliashon Jones
* [http://www.nonoctave.com/tunes/CosmicChamber.mp3 Cosmic Chamber] by [[X. J. Scott]]
* [http://www.nonoctave.com/tunes/Daisies.mp3 Daisies on the Beach] by X. J. Scott
* [http://www.akjmusic.com/audio/boogie_pie.mp3 Boogie Pie] by [[Aaron Krister Johnson]]
* [http://clones.soonlabel.com/public/micro/23edo/daily20110619_23edo_23_chilled.mp3 23 Chilled] by [[Chris Vaisvil]]
* [http://www.seraph.it/dep/det/DesertWinds.mp3 Desert Winds] by [[Carlo Serafini]]  ([http://www.seraph.it/blog_files/926007c7483e4abc5a48d582c0667947-105.html blog entry])
* [http://www.seraph.it/dep/det/23Laments.mp3 23 Laments] by Carlo Serafini ([http://www.seraph.it/blog_files/b2bf6f252efd467ee36ecc332a4872ac-106.html blog entry])
* [http://www.seraph.it/dep/det/Doomsday23.mp3 Doomsday 23] by Carlo Serafini ([http://www.seraph.it/blog_files/add481fdf4ae8c3afe56a0d2cb6dd672-164.html blog entry])
* [https://soundcloud.com/muesk/doomsday-clock Doomsday Clock (23 edo, remastered) &#124; SoundCloud] by [[müesk]]
* [http://www.seraph.it/dep/int/Adagio23ForStrings.mp3 Barber’s Adagio For Strings in 23ED2] by Carlo Serafini ([http://www.seraph.it/blog_files/9e630d3f8ba93ab8264a3862dac950ce-192.html blog entry])
* [http://www.seraph.it/dep/det/Nubian%20Dance.mp3 Nubian Dance] by Carlo Serafini ([http://www.seraph.it/blog_files/694f0a26d29cd2a215f37754dd8428c3-237.html blog entry])
* [http://www.seraph.it/dep/det/AroundTheBonfire.mp3 Around the bonfire] by Carlo Serafini ([http://www.seraph.it/blog_files/9c6c54c593bc4720c8bd775fd5e244f4-261.html blog entry])
* ''Allegro Moderato'' by Easley Blackwood
* [http://andrewheathwaite.bandcamp.com/track/pentaswing Pentaswing] [[Technical Notes for Newbeams#Track notes:-Pentaswing|Notes]] by [[Andrew Heathwaite]]
* [http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 Mindaugas Rex Lithuaniae] by [http://chrisvaisvil.com/?p=2267 Chris Vaisvil] (in Superpelog-9 tuning)
* [http://micro.soonlabel.com/23edo/Tutim_Dennsuul/T%fatim%20Dennsuul%20-%20Indigorange.mp3 Indigorange] by [[Tutim Dennsuul]]
* [http://micro.soonlabel.com/23edo/Tutim_Dennsuul/T%fatim%20Dennsuul%20-%20Wignud.mp3 Wignud] by Tutim Dennsuul
* [http://micro.soonlabel.com/23edo/Tutim_Dennsuul/Tutim%20Dennsuul%20-%20%20Harid.mp3 Harid] by Tutim Dennsuul
* [https://soundcloud.com/nanovibrationalrelations/a-rest-in-the-desert-23edo A Rest In The Desert] [http://micro.soonlabel.com/gene_ward_smith/Others/Mcandrew/A%20Rest%20In%20The%20Desert%20(23edo).mp3 play] by [[Gary Mcandrew]]
* [http://spectropolrecords.bandcamp.com/track/jacky-ligon-numenoctagon Numenoctagon] by [[Jacky Ligon]] (on spectropolrecords)
* [https://soundcloud.com/ism-studio/sets/ligon-sevish-dubshot-23 Ligon / Sevish / Dubshot ~ 23] album by Jacky Ligon, Sevish &amp; Tony Dubshot
* [https://soundcloud.com/shunya-kiyokawa/23edo-klavier8 23EDO Klavier8] by [[Shunya Kiyokawa]]
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments The Mavila Experiments &#45; 23&#45;EDO Version &#124; SoundCloud] by [[Mike Battaglia]] (6 remapped classic pieces)
* [https://soundcloud.com/nanoharmony/sun-ice-23edo-guitar Sun ice (23edo guitar) &#124; SoundCloud] by [[NanoHarmony]]
* [https://www.youtube.com/watch?v=1fopTqKjBZY Grand Theft Octave (23 EDO) &#45; YouTube] by [[E8 Heterotic]]


[[Category:11/7]]
[[Category:23-tone scales]]
[[Category:23-tone]]
[[Category:23edo| ]]
[[Category:5/3]]
[[Category:Equal divisions of the octave]]
[[Category:Guitar]]
[[Category:Guitar]]
[[Category:Intervals]]
[[Category:Keyboard]]
[[Category:Listen]]
[[Category:Mavila]]
[[Category:Mavila]]
[[Category:Modes]]
[[Category:Modes]]
[[Category:Prime EDO]]
[[Category:Subgroup]]
[[Category:Theory]]
[[Category:Twentuning]]
[[Category:Twentuning]]
[[Category:Todo:improve synopsis]]

Latest revision as of 23:49, 7 May 2026

← 22edo 23edo 24edo →
Prime factorization 23 (prime)
Step size 52.1739 ¢ 
Fifth 13\23 (678.261 ¢)
Semitones (A1:m2) -1:4 (-52.17 ¢ : 208.7 ¢)
Dual sharp fifth 14\23 (730.435 ¢)
Dual flat fifth 13\23 (678.261 ¢)
Dual major 2nd 4\23 (208.696 ¢)
(semiconvergent)
Consistency limit 5
Distinct consistency limit 5
English Wikipedia has an article on:

23 equal divisions of the octave (abbreviated 23edo or 23ed2), also called 23-tone equal temperament (23tet) or 23 equal temperament (23et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 23 equal parts of about 52.2 ¢ each. Each step represents a frequency ratio of 21/23, or the 23rd root of 2.

Theory

23edo is significant in that it is the last edo that has no diatonic perfect fifths and not even 5edo or 7edo fifths. It is also the last edo that fails to approximate the 3rd, 5th, 7th, and 11th harmonics 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. In this sense, it can be thought of as every other step of 46edo. The lowest harmonics well-approximated by 23edo are 9, 13, 15, 17, 21, 23, 31, 33 and 35.

Mapping

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 antidiatonic scale of 3–3–4–3–3–3–4 (in steps of 23edo), which extends to a 9-note 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 scales of 5 and 8 notes: 5–5–4–5–4 (antipentic) and 4–1–4–1–4–4–1–4 (the "quartertone" version of the Blackwood/Rapoport/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 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 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

Approximation of odd harmonics in 23edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -23.7 -21.1 +22.5 +4.8 +22.6 -5.7 +7.4 -0.6 +15.5 -1.2 -2.2
Relative (%) -45.4 -40.4 +43.1 +9.2 +43.3 -11.0 +14.2 -1.2 +29.8 -2.3 -4.2
Steps
(reduced) 		36
(13) 		53
(7) 		65
(19) 		73
(4) 		80
(11) 		85
(16) 		90
(21) 		94
(2) 		98
(6) 		101
(9) 		104
(12)

Subsets and supersets

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

23edo was proposed by ethnomusicologist Erich von Hornbostel as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

Intervals

Degree Cents Approximate Ratios* Comments
0 0.0 1/1
1 52.2 33/32, 34/33
2 104.3 17/16, 16/15, 18/17 Less than 1 cent off 17/16
3 156.5 11/10, 12/11, 35/32
4 208.7 9/8, 44/39
5 260.9 7/6, 15/13, 29/25
6 313.0 6/5 Much better 6/5 than 12-edo
7 365.2 16/13, 21/17, 26/21
8 417.4 14/11, 33/26 Practically just 14/11
9 469.6 21/16, 17/13
10 521.7 23/17, 27/20, 88/65
11 573.9 7/5, 32/23, 46/33
12 626.1 10/7, 23/16, 33/23
13 678.3 34/23, 40/27, 65/44 Great Hornbostel generator
14 730.4 32/21, 26/17
15 782.6 11/7, 52/33 Practically just 11/7
16 834.8 13/8, 34/21, 21/13
17 887.0 5/3 Much better 5/3 than 12-edo
18 939.1 12/7, 26/15, 50/29
19 991.3 16/9, 39/22
20 1043.5 11/6, 20/11, 64/35
21 1095.7 15/8, 17/9, 32/17 Less than 1 cent off 32/17
22 1147.8 33/17, 64/33
23 1200.0 2/1
  • Based on treating 23edo as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.

Notation

Conventional notation

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

The first, melodic notation, defines sharp/flat, major/minor, and aug/dim in terms of the 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. M2 + M2 is not M3, and D + M2 is not E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Note that the notes that form chords are different from in diatonic: for example, a major chord, P1–M3–P5, is approximately 4:5:6 as would be expected, but is notated C–E♯–G on C.

Alternatively, one can essentially pretend the 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, known as harmonic notation. The primary purpose of doing this is to allow music notated in 12edo or another diatonic system to be directly translated on the fly, or to allow support for 23edo in tools that only allow chain-of-fifths notation, and it carries over the way interval arithmetic works from diatonic notation, at the cost of notating the sizes of intervals and the shapes of chords incorrectly: that is, a major chord, P1–M3–P5, is notated C–E–G on C, but is no longer ~4:5:6 (since the third is closer to a minor third).

For the sake of clarity, the first notation is commonly called melodic notation, and the second is called harmonic notation, but this is a bit of a misnomer as both preserve different features of the notation of harmony.

Comparison of notations
Notation P1–M3–P5 ~ 4:5:6 P1–M3–P5 = C–E–G on C
Diatonic No Yes
Antidiatonic Yes No

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as EDOs 28 and 33.

23-EDO__Sagittal.svg

Second-best fifth notation

This notation uses the same sagittal sequence as EDOs 30, 37, and 44.

23b-EDO__Sagittal.svg

Armodue notation

Armodue notation is a nonatonic notation that uses the numbers 1-9 as note names.

# Cents Major wider
than minor 		Major narrower
than minor 		Armodue
Notation
0 0.0 P1 D P1 D 1
1 52.2 A1 D# d1 Db 2b
2 104.3 d2 Eb A2 E# 1#
3 156.5 m2 E M2 E 2
4 208.7 M2 E# m2 Eb 3b
5 260.9 A2, d3 Ex, Fbb d2, A3 Ebb, Fx 2#
6 313.0 m3 Fb M3 F# 3
7 365.2 M3 F m3 F 4b
8 417.4 A3 F# d3 Fb 3#
9 469.6 d4 Gb A4 G# 4
10 521.7 P4 G P4 G 5
11 573.9 A4 G# d4 Gb 6b
12 626.1 d5 Ab A5 A# 5#
13 678.3 P5 A P5 A 6
14 730.4 A5 A# d5 Ab 7b
15 782.6 d6 Bb A6 B# 6#
16 834.8 m6 B M6 B 7
17 887.0 M6 B# m6 Bb 8b
18 939.1 A6, d7 Bx, Cbb d6, A7 Bbb, Cx 7#
19 991.3 m7 Cb M7 C# 8
20 1043.5 M7 C m7 C 9b
21 1095.7 A7 C# d7 Cb 8#
22 1147.8 d8 Db A8 D# 9
23 1200.0 P8 D P8 D 1

Ciclo Icositrifonía.png

Approximation to irrational intervals

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

Direct approximation
Interval Error (abs, ¢)
π 0.813
π/ϕ 0.879
ϕ 1.692

Approximation to JI

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 23edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 23edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 0.117 0.2
5/3, 6/5 2.598 5.0
9/8, 16/9 4.786 9.2
13/8, 16/13 5.745 11.0
11/6, 12/11 5.885 11.3
7/6, 12/7 6.001 11.5
15/8, 16/15 7.383 14.2
11/10, 20/11 8.482 16.3
7/5, 10/7 8.599 16.5
13/9, 18/13 10.531 20.2
15/13, 26/15 13.129 25.2
15/14, 28/15 15.095 28.9
15/11, 22/15 15.212 29.2
13/10, 20/13 15.351 29.4
9/7, 14/9 17.693 33.9
11/9, 18/11 17.809 34.1
13/12, 24/13 17.949 34.4
5/4, 8/5 21.096 40.4
7/4, 8/7 22.478 43.1
11/8, 16/11 22.595 43.3
3/2, 4/3 23.694 45.4
13/11, 22/13 23.834 45.7
13/7, 14/13 23.950 45.9
9/5, 10/9 25.882 49.6
15-odd-limit intervals in 23edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 0.117 0.2
5/3, 6/5 2.598 5.0
13/8, 16/13 5.745 11.0
13/10, 20/13 15.351 29.4
13/12, 24/13 17.949 34.4
5/4, 8/5 21.096 40.4
7/4, 8/7 22.478 43.1
11/8, 16/11 22.595 43.3
3/2, 4/3 23.694 45.4
9/5, 10/9 26.292 50.4
13/7, 14/13 28.223 54.1
13/11, 22/13 28.340 54.3
15/13, 26/15 39.045 74.8
13/9, 18/13 41.643 79.8
7/5, 10/7 43.575 83.5
11/10, 20/11 43.691 83.7
15/8, 16/15 44.790 85.8
7/6, 12/7 46.173 88.5
11/6, 12/11 46.289 88.7
9/8, 16/9 47.388 90.8
15/14, 28/15 67.269 128.9
15/11, 22/15 67.386 129.2
9/7, 14/9 69.867 133.9
11/9, 18/11 69.983 134.1
15-odd-limit intervals by 23de val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 0.117 0.2
5/3, 6/5 2.598 5.0
13/8, 16/13 5.745 11.0
11/6, 12/11 5.885 11.3
7/6, 12/7 6.001 11.5
11/10, 20/11 8.482 16.3
7/5, 10/7 8.599 16.5
15/14, 28/15 15.095 28.9
15/11, 22/15 15.212 29.2
13/10, 20/13 15.351 29.4
9/7, 14/9 17.693 33.9
11/9, 18/11 17.809 34.1
13/12, 24/13 17.949 34.4
5/4, 8/5 21.096 40.4
3/2, 4/3 23.694 45.4
13/11, 22/13 23.834 45.7
13/7, 14/13 23.950 45.9
9/5, 10/9 26.292 50.4
11/8, 16/11 29.579 56.7
7/4, 8/7 29.695 56.9
15/13, 26/15 39.045 74.8
13/9, 18/13 41.643 79.8
15/8, 16/15 44.790 85.8
9/8, 16/9 47.388 90.8

Regular temperament properties

Uniform maps

13-limit uniform maps between 22.8 and 23.2
Min. size Max. size Wart notation Map
22.6916 22.8351 23deff ⟨23 36 53 64 79 84]
22.8351 22.9754 23de ⟨23 36 53 64 79 85]
22.9754 22.9807 23e ⟨23 36 53 65 79 85]
22.9807 23.0289 23 ⟨23 36 53 65 80 85]
23.0289 23.0412 23b ⟨23 37 53 65 80 85]
23.0412 23.1054 23bc ⟨23 37 54 65 80 85]
23.1054 23.2697 23bcf ⟨23 37 54 65 80 86]

Commas

23et tempers out the following commas. This assumes the 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).

Prime
limit 		Ratio Monzo Cents Color name Name(s)
5 135/128 [-7 3 1⟩ 92.18 Layobi Mavila comma, major chroma
5 15625/15552 [-6 -5 6⟩ 8.11 Tribiyo Kleisma, semicomma majeur
7 36/35 [2 2 -1 -1⟩ 48.77 Rugu Mint comma, septimal quartertone
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 4000/3969 [5 -4 3 -2⟩ 13.47 Rurutriyo Octagar comma
7 6144/6125 [11 1 -3 -2⟩ 5.36 Sarurutrigu Porwell comma
11 100/99 [2 -2 2 0 -1⟩ 17.40 Luyoyo Ptolemisma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozogu Werckisma

Octave stretch or compression

Main article: 23edo and octave stretching

23edo is not often taken seriously as a tuning except by those interested in extreme xenharmony. Its fifths are significantly flat, and is neighbors 22edo and 24edo generally get more attention.

However, when using a slightly stretched octave of around 1206 cents, 23edo looks much better, and it approximates the perfect fifth (and various other intervals involving the 5th, 7th, 11th, and 13th harmonics) to within 18 cents or so. If we can tolerate errors around this size in 12edo, we can probably tolerate them in stretched-23 as well.

Stretched-23edo is one of the best tunings to use for exploring the antidiatonic scale since its fifth is more consonant and less "wolfish" than fifths in other pelogic family temperaments.

Scales

Important mosses include:

  • Mavila 2L5s 4334333 (13\23, 1\1)
  • Mavila 7L2s 133313333 (13\23, 1\1)
  • Sephiroth 3L4s 2525252 (7\23, 1\1)
  • Semiquartal 5L4s 332323232 (5\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:

23edoMavilaMOS.jpg

23-tone mos scales

MOS scale Name
10 10 3
9 9 5
8 8 7
7 7 7 2
6 6 6 5
5 4 5 5 4 3L 2s (oneiro-pentatonic)
5 4 5 4 5
7 1 7 7 1
7 1 7 1 7
5 5 5 5 3 4L 1s (bug pentatonic)
4 4 4 4 4 3 5L 1s (machinoid)
5 1 5 1 5 1 5 4L 3s (smitonic)
3 3 3 5 3 3 3 1L 6s (antiarcheotonic)
4 3 3 3 3 3 4
3 3 4 3 3 3 4 2L 5s (mavila, anti-diatonic)
4 3 3 3 3 4 3
2 5 2 5 2 5 2 3L 4s (mosh)
4 1 4 4 1 4 4 1 5L 3s (oneirotonic)
3 3 3 3 3 3 3 2 7L 1s (porcupoid)
3 3 3 1 3 3 3 3 1 7L 2s (mavila superdiatonic)
3 2 3 2 3 2 3 2 3 5L 4s (bug semiquartal)
3 2 2 3 2 2 3 2 2 2 3L 7s (sephiroid)
4 1 1 4 1 1 4 1 1 4 1 4L 7s (kleistonic)
3 1 3 1 3 1 3 1 3 1 3 Palestine 11
3 1 1 3 1 3 1 1 3 1 3 1 1 5L 8s (ateamtonic)
2 2 2 2 1 2 2 2 1 2 2 2 1 10L 3s (luachoid)
2 2 1 2 2 1 2 2 1 2 2 1 2 1 9L 5s (Brittle Titanium)
2 1 2 2 1 2 2 1 2 2 1 2 2 1 Palestine 14
1 1 1 4 1 1 1 1 4 1 1 1 1 4 3L 11s
3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 4L 11s (mynoid)
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 8L 7s
2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 1 7L 9s (mavila chromatic)
2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 Palestine 17
2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 5L 13s
2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 4L 15s

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

23edo is the largest edo without any form of 5L 2s, including the degenerate cases.

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 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 2 3 2 2 3 2 2 3 - Mode Keter

2 2 3 2 2 3 2 2 3 2 - Chesed

2 3 2 2 3 2 2 3 2 2 - Netzach

3 2 2 3 2 2 3 2 2 2 - Malkuth

2 2 3 2 2 3 2 2 2 3 - Binah

2 3 2 2 3 2 2 2 3 2 - Tiferet

3 2 2 3 2 2 2 3 2 2 - Yesod

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

Miscellaneous

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)

5 4 2 2 4 2 4 - Lost phantom (approximated from Mavila in 30edo)

6 4 2 1 5 1 4 - Lost spirit (approximated from Meantone in 31edo)

5 2 6 6 4 - Mechanical (approximated from 31afdo)

5 4 4 2 8 - Mushroom (approximated from 30afdo)

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)

More listed in: User:BudjarnLambeth/Quasipelog theory#Scales

Instruments

Icositriphonic_Guitar.PNG

23-EDD 8-string Guitar by Ron Sword.

Bajo 23-EDD.jpg

23-EDD Bass by Osmiorisbendi.

Baritarra 23-EDD.jpg

23-EDD Baritar by Osmiorisbendi.

Icositritar 1.png

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

Teclado Icositrifónico.PNG

Illustrative 23-EDD Keyboard

Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.

playing.jpg

Here is a still shot of the completed instrument.

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.

Load video
YouTube

Lumatone

See: Lumatone mapping for 23edo

Music

Main article: 23edo/Music
See also: Category:23edo tracks

Further reading

Libro_Icositrifónico.PNG
Icosikaitriphonic Scales for Guitar cover art.
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