16edf
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| ← 15edf | 16edf | 17edf → |
16 equal divisions of the perfect fifth (abbreviated 16edf or 16ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 16 equal parts of about 43.9 ¢ each. Each step represents a frequency ratio of (3/2)1/16, or the 16th root of 3/2.
Theory
16edf corresponds to 27.3522…edo. It is similar to every third step of 82edo but not quite similar to 27edo; the octave is compressed by 15.45 ¢, a small but significant deviation. It contains good approximations of the 7th and 13th harmonics.
It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.5 | -15.5 | +13.0 | +21.5 | +13.0 | +9.3 | -2.5 | +13.0 | +6.1 | +16.5 | -2.5 |
| Relative (%) | -35.2 | -35.2 | +29.6 | +49.0 | +29.6 | +21.3 | -5.7 | +29.6 | +13.8 | +37.7 | -5.7 | |
| Steps (reduced) |
27 (11) |
43 (11) |
55 (7) |
64 (0) |
71 (7) |
77 (13) |
82 (2) |
87 (7) |
91 (11) |
95 (15) |
98 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.4 | -6.1 | +6.1 | -17.9 | +8.7 | -2.5 | -8.3 | -9.4 | -6.1 | +1.1 | +11.9 | -17.9 |
| Relative (%) | -21.5 | -13.9 | +13.8 | -40.9 | +19.9 | -5.7 | -19.0 | -21.4 | -13.9 | +2.5 | +27.1 | -40.9 | |
| Steps (reduced) |
101 (5) |
104 (8) |
107 (11) |
109 (13) |
112 (0) |
114 (2) |
116 (4) |
118 (6) |
120 (8) |
122 (10) |
124 (12) |
125 (13) | |
Subsets and supersets
Since 16 factors into primes as 24, 16edf contains subset edfs 2, 4, and 8.
Intervals
| # | Cents | Approximate ratios | Halftone[6] notation (using ups and downs) |
Comments |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | C | |
| 1 | 43.9 | 40/39, 39/38 | ^C | |
| 2 | 87.7 | 20/19 | Db | |
| 3 | 131.6 | 55/51, (27/25) | vD | |
| 4 | 175.5 | (21/19) | D | |
| 5 | 219.4 | vE | ||
| 6 | 263.2 | (7/6) | E | |
| 7 | 307.1 | Fb | ||
| 8 | 351.0 | 60/49, 49/40 | vF | |
| 9 | 394.8 | (44/35) | F | |
| 10 | 438.7 | (9/7) | Ab | |
| 11 | 482.6 | vA | ||
| 12 | 526.5 | (19/14) | A | |
| 13 | 570.3 | (25/18), 153/110, 112/81 | B | |
| 14 | 614.2 | (10/7) | Cb | |
| 15 | 658.1 | 19/13 | vC | |
| 16 | 702.0 | 3/2 | C | Just perfect fifth |
| 17 | 745.8 | 20/13 | ||
| 18 | 789.7 | 30/19 | ||
| 19 | 833.6 | 55/34 | ||
| 20 | 877.4 | |||
| 21 | 921.3 | |||
| 22 | 965.2 | 7/4 | ||
| 23 | 1009.0 | |||
| 24 | 1052.9 | 90/49, (11/6) | ||
| 25 | 1096.8 | (66/35) | ||
| 26 | 1140.7 | |||
| 27 | 1184.5 | |||
| 28 | 1228.4 | 128/63 | ||
| 29 | 1272.3 | 25/12 | ||
| 30 | 1316.2 | 15/7 | ||
| 31 | 1360.0 | 57/26 | ||
| 32 | 1403.9 | 9/4 | Pythagorean major ninth |
Music
- Neptune (2021)
- schizophrenic lullaby fugue (2011)