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== Theory ==
== Theory ==
{{Odd harmonics in edo|edo=23}}
{{primes in equal|23}}


<b>23-TET</b>, or <b>23-EDO</b>, is a musical system which divides the [[octave]] into 23 equal parts of approximately 52.2 cents. 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]].
<b>23-TET</b>, or <b>23-EDO</b>, is a musical system which divides the [[octave]] into 23 equal parts of approximately 52.2 cents. 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]].
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{| 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>
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! Notes
! Notes
|-
|-
| 0 ||
| 0
| 0.000
| 0.000
| 1/1
| 1/1
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|
|
|-
|-
| 1 ||
| 1
| 52.174
| 52.174
| 33/32, 34/33
| 33/32, 34/33
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|
|
|-
|-
| 2 ||
| 2
| 104.348
| 104.348
| 17/16, 16/15, 18/17
| 17/16, 16/15, 18/17
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| 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
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|
|
|-
|-
| 4 || &bull;
| 4
| 208.696
| 208.696
| 9/8, 44/39
| 9/8, 44/39
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|
|
|-
|-
| 5 ||
| 5
| 260.870
| 260.870
| 7/6, 15/13, 29/25
| 7/6, 15/13, 29/25
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|
|
|-
|-
| 6 ||
| 6
| 313.043
| 313.043
| 6/5
| 6/5
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| 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
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|
|
|-
|-
| 8 ||
| 8
| 417.391
| 417.391
| 14/11, 33/26
| 14/11, 33/26
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| Practically just [[14/11]]
| Practically just [[14/11]]
|-
|-
| 9 ||
| 9
| 469.565
| 469.565
| 21/16, 17/13
| 21/16, 17/13
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|
|
|-
|-
| 10 || &bull;
| 10
| 521.739
| 521.739
| 23/17, 88/65, 256/189
| 23/17, 88/65, 256/189
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|
|
|-
|-
| 11 ||
| 11
| 573.913
| 573.913
| 7/5, 32/23, 46/33
| 7/5, 32/23, 46/33
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|
|
|-
|-
| 12 ||
| 12
| 626.087
| 626.087
| 10/7, 23/16, 33/23
| 10/7, 23/16, 33/23
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|
|
|-
|-
| 13 || &bull;
| 13
| 678.261
| 678.261
| 34/23, 65/44, 189/128
| 34/23, 65/44, 189/128
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| Great Hornbostel generator
| Great Hornbostel generator
|-
|-
| 14 ||
| 14
| 730.435
| 730.435
| 32/21, 26/17
| 32/21, 26/17
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|
|
|-
|-
| 15 ||
| 15
| 782.609
| 782.609
| 11/7, 52/33
| 11/7, 52/33
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| 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
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|
|
|-
|-
| 17 ||
| 17
| 886.957
| 886.957
| 5/3
| 5/3
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| 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
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|
|
|-
|-
| 19 || &bull;
| 19
| 991.304
| 991.304
| 16/9, 39/22
| 16/9, 39/22
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|
|
|-
|-
| 20 ||
| 20
| 1043.478
| 1043.478
| 11/6, 20/11, 64/35
| 11/6, 20/11, 64/35
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|
|
|-
|-
| 21 ||
| 21
| 1095.652
| 1095.652
| 15/8, 17/9, 32/17
| 15/8, 17/9, 32/17
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| 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
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|
|
|-
|-
| 23 || &bull;&bull;
| 23
| 1200.000
| 1200.000
| 2/1
| 2/1
Retrieved from "https://en.xen.wiki/w/23edo"