20edf: Difference between revisions
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!Degrees | !Degrees | ||
!3/2.5/4.17/16 interpretation | |||
!Cents | !Cents | ||
|- | |- | ||
|1 | |1 | ||
|51/50 | |||
|35.1 | |35.1 | ||
|- | |- | ||
|2 | |2 | ||
|25/24 | |||
|70.2 | |70.2 | ||
|- | |- | ||
|3 | |3 | ||
|17/16 | |||
|105.29 | |105.29 | ||
|- | |- | ||
|4 | |4 | ||
| | |||
|140.39 | |140.39 | ||
|- | |- | ||
|5 | |5 | ||
| | |||
|175.49 | |175.49 | ||
|- | |- | ||
|6 | |6 | ||
|96/85 | |||
|210.59 | |210.59 | ||
|- | |- | ||
|7 | |7 | ||
|144/125 | |||
|245.68 | |245.68 | ||
|- | |- | ||
|8 | |8 | ||
|20/17 | |||
|280.78 | |280.78 | ||
|- | |- | ||
|9 | |9 | ||
|6/5 | |||
|315.88 | |315.88 | ||
|- | |- | ||
|10 | |10 | ||
|153/125, 125/102 | |||
|350.98 | |350.98 | ||
|- | |- | ||
|11 | |11 | ||
|5/4 | |||
|386.075 | |386.075 | ||
|- | |- | ||
|12 | |12 | ||
|51/40 | |||
|421.17 | |421.17 | ||
|- | |- | ||
|13 | |13 | ||
|125/96 | |||
|456.27 | |456.27 | ||
|- | |- | ||
|14 | |14 | ||
|85/64 | |||
|491.37 | |491.37 | ||
|- | |- | ||
|15 | |15 | ||
| | |||
|526.47 | |526.47 | ||
|- | |- | ||
|16 | |16 | ||
| | |||
|561.56 | |561.56 | ||
|- | |- | ||
|17 | |17 | ||
|24/17 | |||
|596.66 | |596.66 | ||
|- | |- | ||
|18 | |18 | ||
|36/25 | |||
|631.76 | |631.76 | ||
|- | |- | ||
|19 | |19 | ||
|25/17 | |||
|666.86 | |666.86 | ||
|- | |- | ||
|20 | |'''20''' | ||
|701.955 | |'''3/2''' | ||
|'''701.955''' | |||
|- | |- | ||
|21 | |21 | ||
|153/100 | |||
|737.05 | |737.05 | ||
|- | |- | ||
|22 | |22 | ||
|25/16 | |||
|772.15 | |772.15 | ||
|- | |- | ||
|23 | |23 | ||
|51/32 | |||
|807.25 | |807.25 | ||
|- | |- | ||
|24 | |24 | ||
| | |||
|842.35 | |842.35 | ||
|- | |- | ||
|25 | |25 | ||
| | |||
|877.44 | |877.44 | ||
|- | |- | ||
|26 | |26 | ||
|144/85 | |||
|912.54 | |912.54 | ||
|- | |- | ||
|27 | |27 | ||
|216/125 | |||
|947.64 | |947.64 | ||
|- | |- | ||
|28 | |28 | ||
|30/17 | |||
|982.74 | |982.74 | ||
|- | |- | ||
|29 | |29 | ||
|9/5 | |||
|1017.835 | |1017.835 | ||
|- | |- | ||
|30 | |30 | ||
|125/68 | |||
|1052.93 | |1052.93 | ||
|- | |- | ||
|31 | |31 | ||
|15/8 | |||
|1088.03 | |1088.03 | ||
|- | |- | ||
|32 | |32 | ||
|153/80 | |||
|1123.13 | |1123.13 | ||
|- | |- | ||
|33 | |33 | ||
|125/64 | |||
|1158.23 | |1158.23 | ||
|- | |- | ||
|34 | |34 | ||
|255/128 | |||
|1193.32 | |1193.32 | ||
|- | |- | ||
|35 | |35 | ||
| | |||
|1228.42 | |1228.42 | ||
|- | |- | ||
|36 | |36 | ||
| | |||
|1263.52 | |1263.52 | ||
|- | |- | ||
|37 | |37 | ||
|36/17 | |||
|1298.62 | |1298.62 | ||
|- | |- | ||
|38 | |38 | ||
|54/25 | |||
|1333.715 | |1333.715 | ||
|- | |- | ||
|39 | |39 | ||
|75/34 | |||
|1368.81 | |1368.81 | ||
|- | |- | ||
|40 | |'''40''' | ||
|1403.91 | |'''9/4''' | ||
|'''1403.91''' | |||
|} | |} | ||
{{stub}} | {{stub}} | ||
Revision as of 22:41, 5 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, 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 | 140.39 | |
| 5 | 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 | 526.47 | |
| 16 | 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 | 842.35 | |
| 25 | 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 | 1228.42 | |
| 36 | 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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