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== 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]].  
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]].  


=== Mapping ===
=== Mapping ===
Line 30: Line 30:
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.
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 ==
== Intervals ==
{{Q-odd-limit intervals|23}}
{| 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]]
|
|-
| 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 ==
== Notation ==
Line 41: Line 168:
This notation uses the same sagittal sequence as EDOs [[28edo#Sagittal notation|28]] and [[33edo#Sagittal notation|33]].
This notation uses the same sagittal sequence as EDOs [[28edo#Sagittal notation|28]] and [[33edo#Sagittal notation|33]].


<imagemap>
{{Sagittal chart|}}
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>


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


<imagemap>
{{Sagittal chart||23b}}
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  ===
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"
|-
|-
! [[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>
! 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
| 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
| 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
| 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
| 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
| 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
| 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
| 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]]
[[File:Ciclo_Icositrifonía.png|alt=Ciclo Icositrifonía.png|491x490px|link=Harmony_of_23edo]]
Line 291: Line 351:
| 1.692
| 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 ==
== Regular temperament properties ==
Line 368: Line 433:
{{main|23edo and octave stretching}}
{{main|23edo and octave stretching}}


23edo is not typically 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.
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.
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.
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.
What follows is a comparison of compressed- and stretched-octave 23edo tunings.
; [[zpi|86zpi]]
* Step size: 51.653{{c}}, octave size: 1188.0{{c}}
* Approximates all harmonics <9 within 24.0{{c}}
Compressing the octave of 23edo by around 12{{c}} results in improved primes 5, 11 and 13, but worse primes 2, 3 and 7. The tuning 86zpi does this.
{{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 86zpi}}
{{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 86zpi (continued)}}
; [[60ed6]]
* Step size: 51.700{{c}}, octave size: 1189.1{{c}}
* Approximates all harmonics <9 within 21.8{{c}}
Compressing the octave of 23edo by around 11{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2. The tuning 60ed6 does this. So does the tuning [[equal tuning|105ed23]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60ed6}}
{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}
; [[zpi|85zpi]]
* Step size: 52.114{{c}}, octave size: 1198.6{{c}}
* Approximates all harmonics <9 within 24.9{{c}}
Compressing the octave of 23edo by around 1.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. The tuning 85zpi does this. So does the tuning [[ed9|73ed9]] whose octave is identical within 0.02{{c}}.
{{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 85zpi}}
{{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 85zpi (continued)}}
; 23edo
* Step size: 52.174{{c}}, octave size: 1200.0{{c}}
* Approximates all harmonics <9 within 23.7{{c}}
Pure-octaves 23edo.
{{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23edo}}
{{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23edo (continued)}}
; [[WE|23et, 13-limit WE tuning]]
* Step size: 52.237{{c}}, octave size: 1201.5{{c}}
* Approximates all harmonics <9 within 25.7{{c}}
Stretching the octave of 23edo by around 1.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning|85ed13]] whose octave is identical within 0.1{{c}}.
{{Harmonics in cet|52.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning}}
{{Harmonics in cet|52.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 13-limit WE tuning (continued)}}
; [[WE|23et, 2.3.5.13 WE tuning]]
* Step size: 52.447{{c}}, octave size: 1206.3{{c}}
* Approximates all harmonics <9 within 18.8{{c}}
Stretching the octave of 23edo by around 6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. Its 2.3.5.13 WE tuning and 2.3.5.13 [[TE]] tuning both do this. So does the tuning [[ed10|76ed10]] whose octave is identical within 0.01{{c}}.
{{Harmonics in cet|52.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning}}
{{Harmonics in cet|52.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 23et, 2.3.5.13 WE tuning (continued)}}
; [[59ed6]]
* Step size: 52.575{{c}}, octave size: 1209.2{{c}}
* Approximates all harmonics <9 within 24.9{{c}}
Stretching the octave of 23edo by around 9{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 59ed6 does this. So does the tuning [[ed5|53ed5]] whose octave is identical within 0.01{{c}}.
{{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 59ed6}}
{{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 59ed6 (continued)}}
; [[zpi|84zpi]]
* Step size: 52.615{{c}}, octave size: 1210.1{{c}}
* Approximates all harmonics <9 within 22.2{{c}}
Stretching the octave of 23edo by around 10{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. The tuning 84zpi does this.
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84zpi}}
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84zpi (continued)}}
; [[36edt]]
* Step size: 52.832{{c}}, octave size: 1215.1{{c}}
* Approximates all harmonics <9 within 22.6{{c}}
Stretching the octave of 23edo by around 15{{c}} results in improved primes 3, 5, 7 and 13, but a worse prime 2. The tuning 36edt does this.
{{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36edt}}
{{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edt (continued)}}
; [[equal tuning|84ed13]]
* Step size: 52.863{{c}}, octave size: 1215.9{{c}}
* Approximates all harmonics <9 within 21.1{{c}}
Stretching the octave of 23edo by around 16{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. The tuning 84ed13 does this.
{{Harmonics in equal|84|13|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 84ed13}}
{{Harmonics in equal|84|13|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 84ed13 (continued)}}


== Scales ==
== Scales ==
Line 633: Line 626:


2 5 6 6 4 - Volcanic (approximated from [[16afdo]])
2 5 6 6 4 - Volcanic (approximated from [[16afdo]])
''More listed in: [[User:BudjarnLambeth/Quasipelog theory#Scales]]''


== Instruments ==
== Instruments ==
Line 678: Line 673:
[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]]
[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG|thumb|''Icosikaitriphonic Scales for Guitar'' cover art.]]
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitriphonic Scales for Guitar: A Repository of Theory, Reference Materials, and Scale Charts for Xentonal Families]''. 2010.
* [[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.
* [[343edo#Scales|Lucite23]] - [[Gordon Wery]]'s [[well temperament]] of 23edo in [[343edo]]


[[Category:23-tone scales]]
[[Category:23-tone scales]]
Retrieved from "https://en.xen.wiki/w/23edo"