24edf: Difference between revisions
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=== Subsets and supersets === | === Subsets and supersets === | ||
24edf is the 6th [[highly composite equal division|highly composite edf]]. Its nontrivial subsets are {{EDs|equave=t| 2, 3, 4, 6, 8, and 12 }}. | |||
== Intervals == | == Intervals == | ||
Latest revision as of 23:37, 17 February 2026
| ← 23edf | 24edf | 25edf → |
(convergent)
(convergent)
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
| 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) | |
| 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
24edf is the 6th highly composite edf. Its nontrivial subsets are 2, 3, 4, 6, 8, and 12.
Intervals
| # | 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 |