18edf
| ← 17edf | 18edf | 19edf → |
18 equal divisions of the perfect fifth (abbreviated 18edf or 18ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 18 equal parts of about 39 ¢ each. Each step represents a frequency ratio of (3/2)1/18, or the 18th root of 3/2.
Theory
18edf is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; with 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.9 | +8.9 | +17.8 | -17.5 | +17.8 | -15.0 | -12.2 | +17.8 | -8.6 | -17.6 | -12.2 |
| Relative (%) | +22.9 | +22.9 | +45.8 | -44.9 | +45.8 | -38.6 | -31.4 | +45.8 | -22.0 | -45.1 | -31.4 | |
| Steps (reduced) |
31 (13) |
49 (13) |
62 (8) |
71 (17) |
80 (8) |
86 (14) |
92 (2) |
98 (8) |
102 (12) |
106 (16) |
110 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | -6.1 | -8.6 | -3.3 | +8.7 | -12.2 | +11.2 | +0.4 | -6.1 | -8.7 | -7.6 | -3.3 |
| Relative (%) | +13.3 | -15.7 | -22.0 | -8.5 | +22.4 | -31.4 | +28.6 | +0.9 | -15.7 | -22.2 | -19.5 | -8.5 | |
| Steps (reduced) |
114 (6) |
117 (9) |
120 (12) |
123 (15) |
126 (0) |
128 (2) |
131 (5) |
133 (7) |
135 (9) |
137 (11) |
139 (13) |
141 (15) | |
Subsets and supersets
Since 18 factors into primes as 2 × 32, 18edf has subset edfs 2, 3, 6, and 9.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 39.0 | 33/32, 36/35, 49/48, 50/49, 64/63 |
| 2 | 78.0 | 21/20, 22/21, 25/24, 28/27 |
| 3 | 117.0 | 15/14, 16/15 |
| 4 | 156.0 | 11/10, 12/11 |
| 5 | 195.0 | 9/8, 10/9 |
| 6 | 234.0 | 8/7 |
| 7 | 273.0 | 7/6 |
| 8 | 312.0 | 6/5 |
| 9 | 351.0 | 11/9, 16/13 |
| 10 | 390.0 | 5/4 |
| 11 | 429.0 | 9/7, 14/11 |
| 12 | 468.0 | 13/10, 21/16 |
| 13 | 507.0 | 4/3 |
| 14 | 546.0 | 11/8, 15/11 |
| 15 | 585.0 | 7/5 |
| 16 | 624.0 | 10/7 |
| 17 | 663.0 | 16/11, 22/15 |
| 18 | 702.0 | 3/2 |
| 19 | 741.0 | 20/13, 32/21 |
| 20 | 780.0 | 11/7, 14/9 |
| 21 | 818.9 | 8/5 |
| 22 | 857.9 | 18/11 |
| 23 | 896.9 | 5/3 |
| 24 | 935.9 | 12/7 |
| 25 | 974.9 | 7/4 |
| 26 | 1013.9 | 9/5 |
| 27 | 1052.9 | 11/6 |
| 28 | 1091.9 | 15/8 |
| 29 | 1130.9 | 27/14 |
| 30 | 1169.9 | 35/18, 49/25, 63/32 |
| 31 | 1208.9 | 2/1 |
| 32 | 1247.9 | 33/16, 45/22, 49/24, 55/27 |
| 33 | 1286.9 | 21/10, 25/12 |
| 34 | 1325.9 | 15/7 |
| 35 | 1364.9 | 11/5 |
| 36 | 1403.9 | 9/4 |
Related regular temperaments
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
7-limit 31 & 369
Commas: 2401/2400, 8589934592/8544921875
POTE generator: ~5/4 = 386.997
Mapping: [⟨1 19 2 7], ⟨0 -54 1 -13]]
11-limit 31 & 369
Commas: 2401/2400, 5632/5625, 46656/46585
POTE generator: ~5/4 = 386.999
Mapping: [⟨1 19 2 7 37], ⟨0 -54 1 -13 -104]]
EDOs: 31, 369, 400, 431, 462
13-limit 31 & 369
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
POTE generator: ~5/4 = 387.003
Mapping: [⟨1 19 2 7 37 -35], ⟨0 -54 1 -13 -104 120]]
EDOs: 31, 369, 400, 431, 462
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Todo: cleanup , expand say what the temperaments are like and why one would want to use them, and for what |