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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. 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}}
 
=== Octave stretch ===
Some approximations can be improved by octave stretching. See ''[[23edo and octave stretching]]'' for more details.
 
=== 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 {{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.


== Selected just intervals ==
== Selected just intervals ==
{{Q-odd-limit intervals|23}}
== Notation ==
===Conventional notation ===
{{Mavila}}


{| class="wikitable center-all"
===Sagittal notation===
|-
====Best fifth notation====
|+ 23-EDO Approximation of Primary Intervals
This notation uses the same sagittal sequence as EDOs [[28edo#Sagittal notation|28]] and [[33edo#Sagittal notation|33]].
|-
! colspan="2" | Prime number
! 3
! 5
! 7
! 11
! 13
! 17
! 19
! 23
|-
! rowspan="2" | Error
! absolute ([[cent|¢]])
| -23.69
| -21.10
| +22.48
| +22.60
| -5.75
| -0.61
| +15.53
| -2.19
|-
! [[Relative error|relative]] (%)
| -45.4
| -40.4
| +43.1
| +43.3
| -11.0
| -1.2
| +29.8
| -4.2
|-
! colspan="2" | Degree ([[octave reduction|reduced]])
| 36 (13)
| 53 (7)
| 65 (19)
| 80 (11)
| 85 (16)
| 94 (2)
| 98 (6)
| 104 (12)
|}


== Notation ==
<imagemap>
File:23-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
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>


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".
<imagemap>
File:23b_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
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>


=== 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-1 right-3 left-10"
|-
|-
! colspan="2" | [[Degree]] <ref>The dots indicate which frets on a 23-edo guitar would have dots.</ref>
! [[Degree]]
! [[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>
! 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>
Line 84: Line 78:
! Notes
! Notes
|-
|-
| 0 ||
| 0
| 0.000
| 0.000
| 1/1
| 1/1
Line 92: Line 86:
|
|
|-
|-
| 1 ||
| 1
| 52.174
| 52.174
| 33/32, 34/33
| 33/32, 34/33
Line 100: Line 94:
|
|
|-
|-
| 2 ||
| 2
| 104.348
| 104.348
| 17/16, 16/15, 18/17
| 17/16, 16/15, 18/17
Line 108: Line 102:
| Less than 1 cent off [[17/16]]
| Less than 1 cent off [[17/16]]
|-
|-
| 3 ||
| 3
| 156.522
| 156.522
| 11/10, 12/11, 35/32
| 11/10, 12/11, 35/32
Line 116: Line 110:
|
|
|-
|-
| 4 || &bull;
| 4
| 208.696
| 208.696
| 9/8, 44/39
| 9/8, 44/39
Line 124: Line 118:
|
|
|-
|-
| 5 ||
| 5
| 260.870
| 260.870
| 7/6, 15/13, 29/25
| 7/6, 15/13, 29/25
Line 132: Line 126:
|
|
|-
|-
| 6 ||
| 6
| 313.043
| 313.043
| 6/5
| 6/5
Line 140: Line 134:
| Much better [[6/5]] than 12-edo
| Much better [[6/5]] than 12-edo
|-
|-
| 7 || &bull;
| 7
| 365.217
| 365.217
| 16/13, 21/17, 26/21
| 16/13, 21/17, 26/21
Line 148: Line 142:
|
|
|-
|-
| 8 ||
| 8
| 417.391
| 417.391
| 14/11, 33/26
| 14/11, 33/26
Line 156: Line 150:
| Practically just [[14/11]]
| Practically just [[14/11]]
|-
|-
| 9 ||
| 9
| 469.565
| 469.565
| 21/16, 17/13
| 21/16, 17/13
Line 164: Line 158:
|
|
|-
|-
| 10 || &bull;
| 10
| 521.739
| 521.739
| 23/17, 88/65, 256/189
| 23/17, 88/65, 256/189
Line 172: Line 166:
|
|
|-
|-
| 11 ||
| 11
| 573.913
| 573.913
| 7/5, 32/23, 46/33
| 7/5, 32/23, 46/33
Line 180: Line 174:
|
|
|-
|-
| 12 ||
| 12
| 626.087
| 626.087
| 10/7, 23/16, 33/23
| 10/7, 23/16, 33/23
Line 188: Line 182:
|
|
|-
|-
| 13 || &bull;
| 13
| 678.261
| 678.261
| 34/23, 65/44, 189/128
| 34/23, 65/44, 189/128
Line 196: Line 190:
| Great Hornbostel generator
| Great Hornbostel generator
|-
|-
| 14 ||
| 14
| 730.435
| 730.435
| 32/21, 26/17
| 32/21, 26/17
Line 204: Line 198:
|
|
|-
|-
| 15 ||
| 15
| 782.609
| 782.609
| 11/7, 52/33
| 11/7, 52/33
Line 212: Line 206:
| Practically just [[11/7]]
| Practically just [[11/7]]
|-
|-
| 16 || &bull;
| 16
| 834.783
| 834.783
| 13/8, 34/21, 21/13
| 13/8, 34/21, 21/13
Line 220: Line 214:
|
|
|-
|-
| 17 ||
| 17
| 886.957
| 886.957
| 5/3
| 5/3
Line 228: Line 222:
| Much better [[5/3]] than 12-edo
| Much better [[5/3]] than 12-edo
|-
|-
| 18 ||
| 18
| 939.130
| 939.130
| 12/7, 26/15, 50/29
| 12/7, 26/15, 50/29
Line 236: Line 230:
|
|
|-
|-
| 19 || &bull;
| 19
| 991.304
| 991.304
| 16/9, 39/22
| 16/9, 39/22
Line 244: Line 238:
|
|
|-
|-
| 20 ||
| 20
| 1043.478
| 1043.478
| 11/6, 20/11, 64/35
| 11/6, 20/11, 64/35
Line 252: Line 246:
|
|
|-
|-
| 21 ||
| 21
| 1095.652
| 1095.652
| 15/8, 17/9, 32/17
| 15/8, 17/9, 32/17
Line 260: Line 254:
| Less than 1 cent off [[32/17]]
| Less than 1 cent off [[32/17]]
|-
|-
| 22 ||
| 22
| 1147.826
| 1147.826
| 33/17, 64/33
| 33/17, 64/33
Line 268: Line 262:
|
|
|-
|-
| 23 || &bull;&bull;
| 23
| 1200.000
| 1200.000
| 2/1
| 2/1
Line 279: Line 273:
<references/>
<references/>


=== Armodue Notation  ===


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


== Commas ==
== Approximation to irrational intervals ==
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).
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 left-2 right-3"
{| class="wikitable center-all"
|+Direct approximation
|-
|-
! Interval
! Error (abs, [[Cent|¢]])
|-
| π
| 0.813
|-
| π/ϕ
| 0.879
|-
| ϕ
| 1.692
|}
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=23}}
=== 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"
|-
! [[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 ==
== Scales ==


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:
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 397:
|-
|-
| 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 415:
|-
|-
| 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 439:
|-
|-
| 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 523:
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]])


[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]]
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]])


== Instruments ==
== Instruments ==
Line 614: Line 589:


<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]]
* [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])
* [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)


[[Category:11/7]]
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[[Category:23-tone]]
[[Category:23edo| ]]
[[Category:5/3]]
[[Category:Edo]]
[[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]]
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