Xenharmonic Wiki

23edo: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,264 edits
Overthink (talk | contribs)
autopatrolled
2,790 edits
 
(4 intermediate revisions by 4 users not shown)
Line 10: Line 10:


== 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 561: 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 ==
Retrieved from "https://en.xen.wiki/w/23edo"