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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
{{Infobox ET}}
| Prime factorization = 23
{{Wikipedia|23 equal temperament}}
| Subgroup = 2.9.15.21.33.13.17
{{ED intro}}
| Step size = 52.174
| Fifth type = [[mavila]] 13\23 678.26¢
| Common uses = mavila, extreme xenharmony in e.g. metal
| Important MOS = mavila 2L5s 4334333 (13\23, 1\1)<br/>mavila 7L2s 133313333 (13\23, 1\1)<br/>sephiroth 3L4s 2525252 (7\23, 1\1)
}}


== 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]].


== Theory ==
=== 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#.


<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]].
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 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.
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.


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


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#.
=== Odd harmonics ===
{{Harmonics in equal|23}}


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.
=== 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.  


== Selected just intervals ==
=== 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.


{| class="wikitable center-all"
== 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]].
{{Sagittal chart|}}


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.
====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]].


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 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]]
 
== 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.
 
{| class="wikitable center-all"
|+Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| π
| 0.813
|-
| π/ϕ
| 0.879
|-
| ϕ
| 1.692
|}


=== Armodue Notation  ===
== 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}}


[[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]]
== 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.
 
== Scales ==
 
Important [[mos]]ses include:


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:
* 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 392: 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 416: 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 452: 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 1 3 3 3 1 3 3
| 3 3 3 1 3 3 3 3 1
|
|[[7L 2s|7L 2s (mavila superdiatonic)]]
|-
|-
| 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 586: 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]])
 
5 2 6 6 4 - Mechanical (approximated from [[31afdo]])


[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]]
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 623: Line 662:


<youtube>K4iO7k152og</youtube>
<youtube>K4iO7k152og</youtube>
 
=== Lumatone ===
See: [[Lumatone mapping for 23edo]]
 
== Music ==
== Music ==
* [https://cryptic-ruse.bandcamp.com/album/unfertile UNFERTILE] ([[Cryptic Ruse]] album)
{{Main|23edo/Music}}
* [https://www.youtube.com/watch?v=lYb4iK4Lt5Q Promethean Elegies], by [http://benfuhrman.com/ Ben Fuhrman]
{{Catrel|23edo tracks}}
* [https://soundcloud.com/overtoneshock/curiosity-finds-a-frown-23-edo Curiosity Finds a Frown, by Stephen Weigel]
* [http://soonlabel.com/xenharmonic/archives/2460 Chromatic canon, by Claudi Meneghin]
* [http://home.vicnet.net.au/%7Eepoetry/family.mp3 The Family Supper] by [[Warren Burt]]
* [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]]
== Further reading ==
[[Category:23-tone]]
[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]]
[[Category:23edo| ]]
* [[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:5/3]]
* [[343edo#Scales|Lucite23]] - [[Gordon Wery]]'s [[well temperament]] of 23edo in [[343edo]]
[[Category:Equal divisions of the octave]]
 
[[Category:23-tone scales]]
[[Category:Guitar]]
[[Category:Guitar]]
[[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]]
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