20edf: Difference between revisions
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Carlos Gamma as we know today is non-octave, even tho it was originally intended to include the octaves |
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== Theory == | == Theory == | ||
20edf corresponds to 34.1902edo. It is closely related to [[Carlos Gamma]] | 20edf corresponds to 34.1902edo. It is closely related to [[Carlos Gamma]] and the [[gammic]] temperament, which adds an independent dimension for the [[2/1|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{{c}}). It very accurately represents the intervals [[5/4]], with 11 steps, and [[17/16]], with 3 steps. | ||
=== Harmonics === | === Harmonics === | ||
| Line 31: | Line 31: | ||
|- | |- | ||
|4 | |4 | ||
| | |625/576, 867/800 | ||
|140.39 | |140.39 | ||
|- | |- | ||
|5 | |5 | ||
| | |320/289, 425/384 | ||
|175.49 | |175.49 | ||
|- | |- | ||
| Line 75: | Line 75: | ||
|- | |- | ||
|15 | |15 | ||
| | |576/425, 867/640 | ||
|526.47 | |526.47 | ||
|- | |- | ||
|16 | |16 | ||
| | |400/289, 864/625 | ||
|561.56 | |561.56 | ||
|- | |- | ||
| Line 111: | Line 111: | ||
|- | |- | ||
|24 | |24 | ||
| | |625/384 | ||
|842.35 | |842.35 | ||
|- | |- | ||
|25 | |25 | ||
| | |425/256, 480/289 | ||
|877.44 | |877.44 | ||
|- | |- | ||
| Line 155: | Line 155: | ||
|- | |- | ||
|35 | |35 | ||
| | |864/425 | ||
|1228.42 | |1228.42 | ||
|- | |- | ||
|36 | |36 | ||
| | |600/289 | ||
|1263.52 | |1263.52 | ||
|- | |- | ||
Latest revision as of 11:40, 6 August 2025
| ← 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.
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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