20edf
| ← 19edf | 20edf | 21edf → |
20 equal divisions of the perfect fifth (abbreviated 20edf or 20ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 20 equal parts of about 35.1 ¢ each. Each step represents a frequency ratio of (3/2)1/20, or the 20th root of 3/2.
Theory
20edf corresponds to 34.1902edo. It is closely related to Carlos Gamma, where Carlos Gamma, and similarly the gammic temperament, can be seen as 20edf with an independent dimension for 2 (although strictly speaking, the "canonical" optimized Carlos Gamma tuning is not exactly 20edf, with its fifth stretched by the microscopic amount of 0.016 ¢). It very accurately represents the intervals 5/4, with 11 steps, and 17/16, with 3 steps.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -6.7 | -6.7 | -13.4 | -13.6 | -13.4 | +0.6 | +15.1 | -13.4 | +14.8 | -9.8 | +15.1 |
| Relative (%) | -19.0 | -19.0 | -38.0 | -38.7 | -38.0 | +1.6 | +42.9 | -38.0 | +42.3 | -27.9 | +42.9 | |
| Steps (reduced) |
34 (14) |
54 (14) |
68 (8) |
79 (19) |
88 (8) |
96 (16) |
103 (3) |
108 (8) |
114 (14) |
118 (18) |
123 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.9 | -6.1 | +14.8 | +8.4 | +8.7 | +15.1 | -8.3 | +8.2 | -6.1 | -16.5 | +11.9 | +8.4 |
| Relative (%) | +48.1 | -17.4 | +42.3 | +23.9 | +24.9 | +42.9 | -23.8 | +23.2 | -17.4 | -46.9 | +33.8 | +23.9 | |
| Steps (reduced) |
127 (7) |
130 (10) |
134 (14) |
137 (17) |
140 (0) |
143 (3) |
145 (5) |
148 (8) |
150 (10) |
152 (12) |
155 (15) |
157 (17) | |
Intervals
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
| Degrees | 3/2.5/4.17/16 interpretation | Cents |
|---|---|---|
| 1 | 51/50 | 35.1 |
| 2 | 25/24 | 70.2 |
| 3 | 17/16 | 105.29 |
| 4 | 625/576, 867/800 | 140.39 |
| 5 | 320/289, 425/384 | 175.49 |
| 6 | 96/85 | 210.59 |
| 7 | 144/125 | 245.68 |
| 8 | 20/17 | 280.78 |
| 9 | 6/5 | 315.88 |
| 10 | 153/125, 125/102 | 350.98 |
| 11 | 5/4 | 386.075 |
| 12 | 51/40 | 421.17 |
| 13 | 125/96 | 456.27 |
| 14 | 85/64 | 491.37 |
| 15 | 576/425, 867/640 | 526.47 |
| 16 | 400/289, 864/625 | 561.56 |
| 17 | 24/17 | 596.66 |
| 18 | 36/25 | 631.76 |
| 19 | 25/17 | 666.86 |
| 20 | 3/2 | 701.955 |
| 21 | 153/100 | 737.05 |
| 22 | 25/16 | 772.15 |
| 23 | 51/32 | 807.25 |
| 24 | 625/384 | 842.35 |
| 25 | 425/256, 480/289 | 877.44 |
| 26 | 144/85 | 912.54 |
| 27 | 216/125 | 947.64 |
| 28 | 30/17 | 982.74 |
| 29 | 9/5 | 1017.835 |
| 30 | 125/68 | 1052.93 |
| 31 | 15/8 | 1088.03 |
| 32 | 153/80 | 1123.13 |
| 33 | 125/64 | 1158.23 |
| 34 | 255/128 | 1193.32 |
| 35 | 864/425 | 1228.42 |
| 36 | 600/289 | 1263.52 |
| 37 | 36/17 | 1298.62 |
| 38 | 54/25 | 1333.715 |
| 39 | 75/34 | 1368.81 |
| 40 | 9/4 | 1403.91 |
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