75ed4
Jump to navigation
Jump to search
| ← 73ed4 | 75ed4 | 77ed4 → |
75 equal divisions of the 4th harmonic (abbreviated 75ed4) is a nonoctave tuning system that divides the interval of 4/1 into 75 equal parts of exactly 32 ¢ each. Each step represents a frequency ratio of 41/75, or the 75th root of 4. As an ED4 system, it is equivalent to taking every other tone of 75edo.
Theory
This tuning tempers out 99/98 in the 11-limit; 27/26 and 91/88 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 69/68 and 95/92 in the 23-limit; 75/74 in the 37-limit; and 41/40 in the 41-limit.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 32 | |
| 2 | 64 | |
| 3 | 96 | 18/17, 19/18, 37/35 |
| 4 | 128 | 14/13 |
| 5 | 160 | 34/31 |
| 6 | 192 | 19/17 |
| 7 | 224 | 33/29 |
| 8 | 256 | 29/25, 36/31 |
| 9 | 288 | 13/11 |
| 10 | 320 | |
| 11 | 352 | 38/31 |
| 12 | 384 | |
| 13 | 416 | 14/11 |
| 14 | 448 | |
| 15 | 480 | 33/25 |
| 16 | 512 | |
| 17 | 544 | |
| 18 | 576 | |
| 19 | 608 | |
| 20 | 640 | |
| 21 | 672 | |
| 22 | 704 | |
| 23 | 736 | |
| 24 | 768 | |
| 25 | 800 | |
| 26 | 832 | |
| 27 | 864 | |
| 28 | 896 | |
| 29 | 928 | 29/17 |
| 30 | 960 | |
| 31 | 992 | |
| 32 | 1024 | |
| 33 | 1056 | 35/19 |
| 34 | 1088 | |
| 35 | 1120 | |
| 36 | 1152 | 35/18, 37/19 |
| 37 | 1184 | |
| 38 | 1216 | |
| 39 | 1248 | 35/17, 37/18 |
| 40 | 1280 | 23/11 |
| 41 | 1312 | |
| 42 | 1344 | 37/17 |
| 43 | 1376 | 31/14 |
| 44 | 1408 | |
| 45 | 1440 | 23/10 |
| 46 | 1472 | |
| 47 | 1504 | 31/13 |
| 48 | 1536 | 17/7 |
| 49 | 1568 | |
| 50 | 1600 | |
| 51 | 1632 | 18/7 |
| 52 | 1664 | 34/13 |
| 53 | 1696 | |
| 54 | 1728 | 19/7 |
| 55 | 1760 | 36/13 |
| 56 | 1792 | 31/11 |
| 57 | 1824 | |
| 58 | 1856 | 38/13 |
| 59 | 1888 | |
| 60 | 1920 | |
| 61 | 1952 | 34/11 |
| 62 | 1984 | |
| 63 | 2016 | |
| 64 | 2048 | |
| 65 | 2080 | |
| 66 | 2112 | |
| 67 | 2144 | 38/11 |
| 68 | 2176 | |
| 69 | 2208 | |
| 70 | 2240 | |
| 71 | 2272 | |
| 72 | 2304 | |
| 73 | 2336 | |
| 74 | 2368 | |
| 75 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.89 | +0.27 | +0.00 | +4.80 | -8.62 | -8.83 | +8.89 | +0.53 | -4.09 | +8.68 | +0.27 |
| Relative (%) | +50.0 | +1.5 | +0.0 | +27.0 | -48.5 | -49.6 | +50.0 | +3.0 | -23.0 | +48.8 | +1.5 | |
| Steps (reduced) |
68 (68) |
107 (107) |
135 (0) |
157 (22) |
174 (39) |
189 (54) |
203 (68) |
214 (79) |
224 (89) |
234 (99) |
242 (107) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.92 | +0.06 | +5.06 | +0.00 | +1.71 | -8.35 | +4.71 | +4.80 | -8.56 | -0.21 | -6.05 |
| Relative (%) | +22.0 | +0.4 | +28.5 | +0.0 | +9.6 | -47.0 | +26.5 | +27.0 | -48.1 | -1.2 | -34.0 | |
| Steps (reduced) |
250 (115) |
257 (122) |
264 (129) |
270 (0) |
276 (6) |
281 (11) |
287 (17) |
292 (22) |
296 (26) |
301 (31) |
305 (35) | |
Music
- 4/1 Equave Blues (2024)