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23edo: Difference between revisions

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Split interval table; rearrange JI intervals section; remove extra precision; link intervals; replace overly complex 189/128 and 256/189 with 40/27 and 27/20 which are still in the subgroup
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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 51: Line 178:
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]]
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| 1.692
| 1.692
|}
|}
== Approximation to JI ==
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|23}}


== Regular temperament properties ==
== Regular temperament properties ==
Retrieved from "https://en.xen.wiki/w/23edo"