24edf: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 24 equal parts''' (24EDF) is related to [[41edo|41 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 0.8269 cents compressed and the step size is about 29.2481 cents. It is [[consistent]] to the 16-[[integer-limit]].
{{ED intro}}


Lookalikes: [[41edo]], [[65edt]], [[95ed5]]
== Theory ==
24edf is related to [[41edo]], but with the 3/2 rather than the [[2/1]] being just. The octave is about 0.8269 cents compressed. Like 41edo, 24edf is [[consistent]] to the [[integer limit|16-integer-limit]].


== Harmonics ==
=== Harmonics ===
{{Harmonics in equal|24|3|2|intervals=prime}}
{{Harmonics in equal|24|3|2|intervals=integer}}
{{Harmonics in equal|24|3|2|start=12|collapsed=1|intervals=prime}}
{{Harmonics in equal|24|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 65edt (continued)}}
 
=== Subsets and supersets ===
24edt is the 6th [[highly composite equal division|highly composite edt]]. Its nontrivial subsets are {{EDs|equave=t| 2, 3, 4, 6, 8, and 12 }}.


== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2"
|+ Intervals of 24edf
|+ Intervals of 24edf
|-
|-
! |
! #
! |Cents Value
! Cents
! |Approximate Ratios in the [[11-limit]]
! Approximate ratios
|-
|-
| colspan="2" style="text-align:center;" |0
| 0
| |[[1/1]]
| 0.0
| [[1/1]]
|-
|-
| style="text-align:center;" |1
| 1
| style="text-align:center;" |29.2481
| 29.2
| |[[81/80]]
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
|-
|-
| style="text-align:center;" |2
| 2
| style="text-align:center;" |58.49625
| 58.5
| |[[25/24]], [[28/27]], [[33/32]]
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
|-
|-
| style="text-align:center;" |3
| 3
| style="text-align:center;" |87.7444
| 87.7
| |[[21/20]], [[22/21]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
|-
|-
| style="text-align:center;" |4
| 4
| style="text-align:center;" |116.9925
| 117.0
| |[[16/15]], [[15/14]]
| [[14/13]], [[15/14]], [[16/15]]
|-
|-
| style="text-align:center;" |5
| 5
| style="text-align:center;" |146.2406
| 146.2
| |[[12/11]]
| [[12/11]], [[13/12]]
|-
|-
| style="text-align:center;" |6
| 6
| style="text-align:center;" |175.48875
| 175.5
| |[[10/9]], [[11/10]]
| [[10/9]], [[11/10]], [[21/19]]
|-
|-
| style="text-align:center;" |7
| 7
| style="text-align:center;" |204.7369
| 204.7
| |[[9/8]]
| [[9/8]]
|-
|-
| style="text-align:center;" |8
| 8
| style="text-align:center;" |233.985
| 234.0
| |[[8/7]]
| [[8/7]], [[15/13]]
|-
|-
| style="text-align:center;" |9
| 9
| style="text-align:center;" |263.2331
| 263.2
| |[[7/6]], [[32/25]]
| [[7/6]], [[22/19]]
|-
|-
| style="text-align:center;" |10
| 10
| style="text-align:center;" |292.48125
| 292.5
| |[[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
|-
|-
| style="text-align:center;" |11
| 11
| style="text-align:center;" |321.7293
| 321.7
| |[[6/5]]
| [[6/5]]
|-
|-
| style="text-align:center;" |12
| 12
| style="text-align:center;" |350.9775
| 351.0
| |[[11/9]],[[27/22]]
| [[11/9]], [[16/13]]
|-
|-
| style="text-align:center;" |13
| 13
| style="text-align:center;" |380.2256
| 380.2
| |[[5/4]]
| [[5/4]], [[26/21]]
|-
|-
| style="text-align:center;" |14
| 14
| style="text-align:center;" |409.47375
| 409.5
| |[[14/11]], [[81/64]]
| [[14/11]], [[19/15]], [[24/19]]
|-
|-
| style="text-align:center;" |15
| 15
| style="text-align:center;" |438.7219
| 438.7
| |[[9/7]]
| [[9/7]], [[32/25]]
|-
|-
| style="text-align:center;" |16
| 16
| style="text-align:center;" |467.97
| 468.0
| |[[21/16]]
| [[21/16]], [[13/10]]
|-
|-
| style="text-align:center;" |17
| 17
| style="text-align:center;" |497.2181
| 497.2
| |[[4/3]]
| [[4/3]]
|-
|-
| style="text-align:center;" |18
| 18
| style="text-align:center;" |526.46625
| 526.5
| |[[15/11]], [[27/20]]
| [[15/11]], [[19/14]], [[27/20]]
|-
|-
| style="text-align:center;" |19
| 19
| style="text-align:center;" |556.7144
| 556.7
| |[[11/8]]
| [[11/8]], [[18/13]], [[26/19]]
|-
|-
| style="text-align:center;" |20
| 20
| style="text-align:center;" |584.9625
| 585.0
| |[[7/5]]
| [[7/5]], [[45/32]]
|-
|-
| style="text-align:center;" |21
| 21
| style="text-align:center;" |614.2106
| 614.2
| |[[10/7]]
| [[10/7]], [[64/45]]
|-
|-
| style="text-align:center;" |22
| 22
| style="text-align:center;" |643.45875
| 643.5
| |[[16/11]]
| [[13/9]], [[16/11]], [[19/13]]
|-
|-
| style="text-align:center;" |23
| 23
| style="text-align:center;" |671.7069
| 671.7
| |[[22/15]], [[40/27]]
| [[22/15]], [[28/19]], [[40/27]]
|-
|-
| style="text-align:center;" |24
| 24
| style="text-align:center;" |701.955
| 702.0
| |[[3/2]]
| [[3/2]]
|-
|-
| style="text-align:center;" |25
| 25
| style="text-align:center;" |731.2031
| 731.2
| |[[32/21]]
| [[20/13]], [[32/21]]
|-
|-
| style="text-align:center;" |26
| 26
| style="text-align:center;" |760.45125
| 760.5
| |[[14/9]], [[25/16]]
| [[14/9]], [[25/16]]
|-
|-
| style="text-align:center;" |27
| 27
| style="text-align:center;" |789.6994
| 789.7
| |[[11/7]], [[128/81]]
| [[11/7]], [[19/12]], [[30/19]]
|-
|-
| style="text-align:center;" |28
| 28
| style="text-align:center;" |818.9475
| 818.9
| |[[8/5]]
| [[8/5]]
|-
|-
| style="text-align:center;" |29
| 29
| style="text-align:center;" |848.1956
| 848.2
| |[[18/11]], [[44/27]]
| [[13/8]], [[18/11]]
|-
|-
| style="text-align:center;" |30
| 30
| style="text-align:center;" |877.44375
| 877.4
| |[[5/3]]
| [[5/3]]
|-
|-
| style="text-align:center;" |31
| 31
| style="text-align:center;" |906.6919
| 906.7
| |[[27/16]]
| [[22/13]], [[27/16]], [[32/19]]
|-
|-
| style="text-align:center;" |32
| 32
| style="text-align:center;" |935.94
| 935.9
| |[[12/7]]
| [[12/7]], [[19/11]]
|-
|-
| style="text-align:center;" |33
| 33
| style="text-align:center;" |965.1881
| 965.2
| |[[7/4]]
| [[7/4]], [[26/15]]
|-
|-
| style="text-align:center;" |34
| 34
| style="text-align:center;" |994.43625
| 994.4
| |[[16/9]]
| [[16/9]]
|-
|-
| style="text-align:center;" |35
| 35
| style="text-align:center;" |1023.6844
| 1023.7
| |[[9/5]], [[20/11]]
| [[9/5]]
|-
|-
| style="text-align:center;" |36
| 36
| style="text-align:center;" |1052.9325
| 1052.9
| |[[11/6]]
| [[11/6]]
|-
|-
| style="text-align:center;" |37
| 37
| style="text-align:center;" |1082.1806
| 1082.2
| |[[15/8]]
| [[13/7]], [[15/8]]
|-
|-
| style="text-align:center;" |38
| 38
| style="text-align:center;" |1111.42875
| 1111.4
| |[[40/21]], [[21/11]]
| [[19/10]], [[21/11]]
|-
|-
| style="text-align:center;" |39
| 39
| style="text-align:center;" |1140.6769
| 1140.7
| |[[48/25]], [[27/14]], [[64/33]]
| [[27/14]], [[35/18]]
|-
|-
| style="text-align:center;" |40
| 40
| style="text-align:center;" |1169.925
| 1169.9
| |[[160/81]]
| [[49/25]], [[56/28]], [[63/32]]
|-
|-
| style="text-align:center;" |41
| 41
| style="text-align:center;" |1199.1731
| 1199.2
| |2/1
| 2/1
|-
|-
|42
| 42
|1228.42125
| 1228.4
|81/40
| [[45/22]], [[49/24]], [[55/27]], [[81/40]]
|-
|-
|43
| 43
|1257.6694
| 1257.7
|25/12, 56/27, 33/16
| [[25/12]], [[33/16]]
|-
|-
|44
| 44
|1286.9175
| 1286.9
|21/10, 44/21
| [[19/9]], [[21/10]]
|-
|-
|45
| 45
|1316.1656
| 1316.2
|32/15, 15/7
| [[15/7]]
|-
|-
|46
| 46
|1345.41375
| 1345.4
|24/11
| [[13/6]]
|-
|-
|47
| 47
|1374.6619
| 1374.7
|20/9, 11/5
| [[11/5]]
|-
|-
|48
| 48
|1403.91
| 1403.9
|9/4
| [[9/4]]
|}
|}


== See also ==
* [[41edo]] – relative edo
* [[65edt]] – relative edt
* [[95ed5]] – relative ed5
* [[106ed6]] – relative ed6
* [[147ed12]] – relative ed12
* [[361ed448]] – close to the zeta-optimized tuning for 41edo


{{stub}}
[[Category:41edo]]
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 12:50, 19 June 2025

← 23edf 24edf 25edf →
Prime factorization 23 × 3
Step size 29.2481 ¢ 
Octave 41\24edf (1199.17 ¢)
(convergent)
Twelfth 65\24edf (1901.13 ¢)
(convergent)
Consistency limit 16
Distinct consistency limit 10
Special properties

24 equal divisions of the perfect fifth (abbreviated 24edf or 24ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 24 equal parts of about 29.2 ¢ each. Each step represents a frequency ratio of (3/2)1/24, or the 24th root of 3/2.

Theory

24edf is related to 41edo, but with the 3/2 rather than the 2/1 being just. The octave is about 0.8269 cents compressed. Like 41edo, 24edf is consistent to the 16-integer-limit.

Harmonics

Approximation of harmonics in 24edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.8 -0.8 -1.7 -7.7 -1.7 -5.3 -2.5 -1.7 -8.6 +1.9 -2.5
Relative (%) -2.8 -2.8 -5.7 -26.5 -5.7 -18.1 -8.5 -5.7 -29.3 +6.6 -8.5
Steps
(reduced) 		41
(17) 		65
(17) 		82
(10) 		95
(23) 		106
(10) 		115
(19) 		123
(3) 		130
(10) 		136
(16) 		142
(22) 		147
(3)
Approximation of harmonics in 24edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 -2.5 -8.3 -9.4 -6.1 +1.1 +11.9 -3.3
Relative (%) +17.7 -20.9 -29.3 -11.3 +29.8 -8.5 -28.5 -32.1 -20.9 +3.7 +40.6 -11.3
Steps
(reduced) 		152
(8) 		156
(12) 		160
(16) 		164
(20) 		168
(0) 		171
(3) 		174
(6) 		177
(9) 		180
(12) 		183
(15) 		186
(18) 		188
(20)

Subsets and supersets

24edt is the 6th highly composite edt. Its nontrivial subsets are 2, 3, 4, 6, 8, and 12.

Intervals

Intervals of 24edf
# Cents Approximate ratios
0 0.0 1/1
1 29.2 49/48, 50/49, 64/63, 81/80
2 58.5 25/24, 28/27, 33/32, 36/35
3 87.7 19/18, 20/19, 21/20, 22/21
4 117.0 14/13, 15/14, 16/15
5 146.2 12/11, 13/12
6 175.5 10/9, 11/10, 21/19
7 204.7 9/8
8 234.0 8/7, 15/13
9 263.2 7/6, 22/19
10 292.5 13/11, 19/16, 32/27
11 321.7 6/5
12 351.0 11/9, 16/13
13 380.2 5/4, 26/21
14 409.5 14/11, 19/15, 24/19
15 438.7 9/7, 32/25
16 468.0 21/16, 13/10
17 497.2 4/3
18 526.5 15/11, 19/14, 27/20
19 556.7 11/8, 18/13, 26/19
20 585.0 7/5, 45/32
21 614.2 10/7, 64/45
22 643.5 13/9, 16/11, 19/13
23 671.7 22/15, 28/19, 40/27
24 702.0 3/2
25 731.2 20/13, 32/21
26 760.5 14/9, 25/16
27 789.7 11/7, 19/12, 30/19
28 818.9 8/5
29 848.2 13/8, 18/11
30 877.4 5/3
31 906.7 22/13, 27/16, 32/19
32 935.9 12/7, 19/11
33 965.2 7/4, 26/15
34 994.4 16/9
35 1023.7 9/5
36 1052.9 11/6
37 1082.2 13/7, 15/8
38 1111.4 19/10, 21/11
39 1140.7 27/14, 35/18
40 1169.9 49/25, 56/28, 63/32
41 1199.2 2/1
42 1228.4 45/22, 49/24, 55/27, 81/40
43 1257.7 25/12, 33/16
44 1286.9 19/9, 21/10
45 1316.2 15/7
46 1345.4 13/6
47 1374.7 11/5
48 1403.9 9/4

See also

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