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 and the gammic temperament, which adds an independent dimension for the octave (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.
20edf is closely approximated by every 5 steps of 171edo, while its full 23-limit mapping is maybe better interpreted by every 3 steps of 103edo 103h val.
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 | 103h val interpretation | Cents |
|---|---|---|---|
| 1 | 51/50 | 35.1 | |
| 2 | 25/24 | 28/27, 26/25, 25/24, 24/23 | 70.2 |
| 3 | 17/16 | 17/16 | 105.29 |
| 4 | 625/576, 867/800 | 13/12, 25/23 | 140.39 |
| 5 | 320/289, 425/384 | 21/19 | 175.49 |
| 6 | 96/85 | 26/23 | 210.59 |
| 7 | 144/125 | 23/20, 15/13 | 245.68 |
| 8 | 20/17 | 20/17 | 280.78 |
| 9 | 6/5 | 6/5 | 315.88 |
| 10 | 153/125, 125/102 | 11/9, 27/22 | 350.98 |
| 11 | 5/4 | 5/4 | 386.075 |
| 12 | 51/40 | 14/11 | 421.17 |
| 13 | 125/96 | 13/10, 30/23 | 456.27 |
| 14 | 85/64 | 491.37 | |
| 15 | 576/425, 867/640 | 23/17, 19/14 | 526.47 |
| 16 | 400/289, 864/625 | 18/13 | 561.56 |
| 17 | 24/17 | 38/27, 24/17 | 596.66 |
| 18 | 36/25 | 23/16, 36/25 | 631.76 |
| 19 | 25/17 | 22/15, 25/17 | 666.86 |
| 20 | 3/2 | 3/2 | 701.955 |
| 21 | 153/100 | 32/21, 26/17 | 737.05 |
| 22 | 25/16 | 14/9, 25/16, 36/23 | 772.15 |
| 23 | 51/32 | 807.25 | |
| 24 | 625/384 | 34/21, 13/8 | 842.35 |
| 25 | 425/256, 480/289 | 877.44 | |
| 26 | 144/85 | 32/19, 22/13 | 912.54 |
| 27 | 216/125 | 19/11 | 947.64 |
| 28 | 30/17 | 44/25, 30/17 | 982.74 |
| 29 | 9/5 | 34/19, 9/5 | 1017.835 |
| 30 | 125/68 | 11/6 | 1052.93 |
| 31 | 15/8 | 28/15, 15/8 | 1088.03 |
| 32 | 153/80 | 40/21, 21/11, 44/23 | 1123.13 |
| 33 | 125/64 | 1158.23 | |
| 34 | 255/128 | 1193.32 | |
| 35 | 864/425 | 1228.42 | |
| 36 | 600/289 | 27/13 | 1263.52 |
| 37 | 36/17 | 40/19, 19/9, 36/17 | 1298.62 |
| 38 | 54/25 | 28/13, 54/25 | 1333.715 |
| 39 | 75/34 | 46/21, 11/5 | 1368.81 |
| 40 | 9/4 | 56/25, 9/4 | 1403.91 |
See also
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