72ed5: Difference between revisions
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== Theory == | == Theory == | ||
72ed5 is related to [[31edo]], but with the 5/1 rather than the [[2/1]] being just. The octave is slightly compressed (about 0.3372 cents). | 72ed5 is related to [[31edo]], but with the 5/1 rather than the [[2/1]] being just. The octave is slightly compressed (about 0.3372 cents). Like 31edo, 72ed5 is [[consistent]] through the [[integer limit|12-integer-limit]], but it has a flat tendency, with [[prime harmonic]]s 2, [[3/1|3]], [[7/1|7]], and [[11/1|11]] all tuned flat. It [[support]]s [[meantone]] as the number of divisions of the 5th harmonic is multiple of 4. | ||
=== Harmonics === | === Harmonics === | ||
Revision as of 12:37, 24 March 2025
| ← 71ed5 | 72ed5 | 73ed5 → |
(semiconvergent)
72 equal divisions of the 5th harmonic (abbreviated 72ed5) is a nonoctave tuning system that divides the interval of 5/1 into 72 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 51/72, or the 72nd root of 5.
Theory
72ed5 is related to 31edo, but with the 5/1 rather than the 2/1 being just. The octave is slightly compressed (about 0.3372 cents). Like 31edo, 72ed5 is consistent through the 12-integer-limit, but it has a flat tendency, with prime harmonics 2, 3, 7, and 11 all tuned flat. It supports meantone as the number of divisions of the 5th harmonic is multiple of 4.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.3 | -5.7 | -0.7 | +0.0 | -6.1 | -2.0 | -1.0 | -11.4 | -0.3 | -10.5 | -6.4 |
| Relative (%) | -0.9 | -14.8 | -1.7 | +0.0 | -15.6 | -5.2 | -2.6 | -29.5 | -0.9 | -27.3 | -16.5 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (0) |
80 (8) |
87 (15) |
93 (21) |
98 (26) |
103 (31) |
107 (35) |
111 (39) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.8 | -2.4 | -5.7 | -1.3 | +9.8 | -11.8 | +10.7 | -0.7 | -7.7 | -10.9 | -10.4 | -6.7 |
| Relative (%) | +25.4 | -6.1 | -14.8 | -3.5 | +25.3 | -30.4 | +27.7 | -1.7 | -20.0 | -28.1 | -27.0 | -17.4 | |
| Steps (reduced) |
115 (43) |
118 (46) |
121 (49) |
124 (52) |
127 (55) |
129 (57) |
132 (60) |
134 (62) |
136 (64) |
138 (66) |
140 (68) |
142 (70) | |
Subsets and supersets
72 is a largely composite number. Since it factors into primes as 23 × 32, 72ed5 has subset ed5's 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 38.7 | 33/32, 36/35, 49/48, 50/49, 64/63 |
| 2 | 77.4 | 21/20, 22/21, 25/24, 28/27 |
| 3 | 116.1 | 14/13, 15/14, 16/15 |
| 4 | 154.8 | 12/11, 13/12 |
| 5 | 193.5 | 9/8, 10/9 |
| 6 | 232.2 | 8/7 |
| 7 | 270.9 | 7/6 |
| 8 | 309.6 | 6/5 |
| 9 | 348.3 | 11/9, 16/13 |
| 10 | 387.0 | 5/4 |
| 11 | 425.7 | 9/7, 14/11 |
| 12 | 464.4 | 13/10, 17/13, 21/16 |
| 13 | 503.1 | 4/3 |
| 14 | 541.8 | 11/8, 18/13, 26/19 |
| 15 | 580.5 | 7/5 |
| 16 | 619.2 | 10/7 |
| 17 | 657.9 | 16/11, 19/13, 22/15 |
| 18 | 696.6 | 3/2 |
| 19 | 735.3 | 20/13, 26/17, 32/21 |
| 20 | 774.0 | 11/7, 14/9 |
| 21 | 812.7 | 8/5 |
| 22 | 851.4 | 13/8, 18/11 |
| 23 | 890.1 | 5/3 |
| 24 | 928.8 | 12/7 |
| 25 | 967.5 | 7/4 |
| 26 | 1006.2 | 9/5 |
| 27 | 1044.9 | 11/6 |
| 28 | 1083.6 | 13/7, 15/8 |
| 29 | 1122.3 | 17/9, 19/10, 21/11 |
| 30 | 1161.0 | 35/18, 49/25, 63/32 |
| 31 | 1199.7 | 2/1 |
| 32 | 1238.4 | 33/16, 45/22, 49/24, 55/27 |
| 33 | 1277.1 | 21/10, 25/12 |
| 34 | 1315.8 | 15/7, 17/8, 19/9 |
| 35 | 1354.5 | 13/6 |
| 36 | 1393.2 | 9/4 |
| 37 | 1431.9 | 16/7 |
| 38 | 1470.6 | 7/3 |
| 39 | 1509.3 | 12/5 |
| 40 | 1548.0 | 22/9 |
| 41 | 1586.7 | 5/2 |
| 42 | 1625.3 | 18/7 |
| 43 | 1664.0 | 21/8 |
| 44 | 1702.7 | 8/3 |
| 45 | 1741.4 | 11/4 |
| 46 | 1780.1 | 14/5 |
| 47 | 1818.8 | 20/7 |
| 48 | 1857.5 | 26/9, 38/13 |
| 49 | 1896.2 | 3/1 |
| 50 | 1934.9 | 40/13 |
| 51 | 1973.6 | 25/8 |
| 52 | 2012.3 | 16/5 |
| 53 | 2051.0 | 13/4 |
| 54 | 2089.7 | 10/3 |
| 55 | 2128.4 | 24/7 |
| 56 | 2167.1 | 7/2 |
| 57 | 2205.8 | 18/5, 25/7 |
| 58 | 2244.5 | 11/3 |
| 59 | 2283.2 | 15/4 |
| 60 | 2321.9 | 27/7 |
| 61 | 2360.6 | 35/9, 63/16 |
| 62 | 2399.3 | 4/1 |
| 63 | 2438.0 | 33/8, 45/11, 49/12 |
| 64 | 2476.7 | 21/5, 25/6 |
| 65 | 2515.4 | 17/4 |
| 66 | 2554.1 | 22/5 |
| 67 | 2592.8 | 9/2 |
| 68 | 2631.5 | 32/7 |
| 69 | 2670.2 | 14/3 |
| 70 | 2708.9 | 19/4, 24/5 |
| 71 | 2747.6 | 44/9 |
| 72 | 2786.3 | 5/1 |