65edt: Difference between revisions
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Revision as of 17:35, 20 March 2025
| ← 64edt | 65edt | 66edt → |
(convergent)
65 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 65edt or 65ed3), is a nonoctave tuning system that divides the interval of 3/1 into 65 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 31/65, or the 65th root of 3.
Theory
65edt is almost identical to 41edo, but with the 3/1 rather than the 2/1 being just. The octave is about 0.3053 cents compressed. Like 41edo, 65edt is consistent to the 16-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.3 | +0.0 | -0.6 | -6.5 | -0.3 | -3.8 | -0.9 | +0.0 | -6.8 | +3.7 | -0.6 |
| Relative (%) | -1.0 | +0.0 | -2.1 | -22.3 | -1.0 | -13.1 | -3.1 | +0.0 | -23.4 | +12.7 | -2.1 | |
| Steps (reduced) |
41 (41) |
65 (0) |
82 (17) |
95 (30) |
106 (41) |
115 (50) |
123 (58) |
130 (0) |
136 (6) |
142 (12) |
147 (17) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.1 | -4.1 | -6.5 | -1.2 | +10.9 | -0.3 | -6.1 | -7.1 | -3.8 | +3.4 | +14.2 | -0.9 |
| Relative (%) | +24.3 | -14.1 | -22.3 | -4.2 | +37.1 | -1.0 | -20.9 | -24.4 | -13.1 | +11.7 | +48.7 | -3.1 | |
| Steps (reduced) |
152 (22) |
156 (26) |
160 (30) |
164 (34) |
168 (38) |
171 (41) |
174 (44) |
177 (47) |
180 (50) |
183 (53) |
186 (56) |
188 (58) | |
Subsets and supersets
Since 65 factors into primes as 5 × 13, 65edt contains 5edt and 13edt as subset edts.
Intervals
| # | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 1/1 |
| 1 | 29.3 | 20.0 | 49/48, 50/49, 64/63, 81/80 |
| 2 | 58.5 | 40.0 | 25/24, 28/27, 33/32, 36/35 |
| 3 | 87.8 | 60.0 | 19/18, 20/19, 21/20, 22/21 |
| 4 | 117.0 | 80.0 | 14/13, 15/14, 16/15 |
| 5 | 146.3 | 100.0 | 12/11, 13/12 |
| 6 | 175.6 | 120.0 | 10/9, 11/10, 21/19 |
| 7 | 204.8 | 140.0 | 9/8 |
| 8 | 234.1 | 160.0 | 8/7, 15/13 |
| 9 | 263.3 | 180.0 | 7/6, 22/19 |
| 10 | 292.6 | 200.0 | 13/11, 19/16, 32/27 |
| 11 | 321.9 | 220.0 | 6/5 |
| 12 | 351.1 | 240.0 | 11/9, 16/13 |
| 13 | 380.4 | 260.0 | 5/4, 26/21 |
| 14 | 409.7 | 280.0 | 14/11, 19/15, 24/19 |
| 15 | 438.9 | 300.0 | 9/7, 32/25 |
| 16 | 468.2 | 320.0 | 21/16, 13/10 |
| 17 | 497.4 | 340.0 | 4/3 |
| 18 | 526.7 | 360.0 | 15/11, 19/14, 27/20 |
| 19 | 556.0 | 380.0 | 11/8, 18/13, 26/19 |
| 20 | 585.2 | 400.0 | 7/5, 45/32 |
| 21 | 614.5 | 420.0 | 10/7, 64/45 |
| 22 | 643.7 | 440.0 | 13/9, 16/11, 19/13 |
| 23 | 673.0 | 460.0 | 22/15, 28/19, 40/27 |
| 24 | 702.3 | 480.0 | 3/2 |
| 25 | 731.5 | 500.0 | 20/13, 32/21 |
| 26 | 760.8 | 520.0 | 14/9, 25/16 |
| 27 | 790.0 | 540.0 | 11/7, 19/12, 30/19 |
| 28 | 819.3 | 560.0 | 8/5, 21/13 |
| 29 | 848.6 | 580.0 | 13/8, 18/11 |
| 30 | 877.8 | 600.0 | 5/3 |
| 31 | 907.1 | 620.0 | 22/13, 27/16, 32/19 |
| 32 | 936.3 | 640.0 | 12/7, 19/11 |
| 33 | 965.6 | 660.0 | 7/4, 26/15 |
| 34 | 994.9 | 680.0 | 16/9 |
| 35 | 1024.1 | 700.0 | 9/5 |
| 36 | 1053.4 | 720.0 | 11/6 |
| 37 | 1082.7 | 740.0 | 13/7, 15/8 |
| 38 | 1111.9 | 760.0 | 19/10, 21/11 |
| 39 | 1141.2 | 780.0 | 27/14, 35/18 |
| 40 | 1170.4 | 800.0 | 49/25, 55/28, 63/32 |
| 41 | 1199.7 | 820.0 | 2/1 |
| 42 | 1229.0 | 840.0 | 45/22, 49/24, 55/27, 81/40 |
| 43 | 1258.2 | 860.0 | 25/12, 33/16 |
| 44 | 1287.5 | 880.0 | 19/9, 21/10 |
| 45 | 1316.7 | 900.0 | 15/7 |
| 46 | 1346.0 | 920.0 | 13/6 |
| 47 | 1375.3 | 940.0 | 11/5 |
| 48 | 1404.5 | 960.0 | 9/4 |
| 49 | 1433.8 | 980.0 | 16/7 |
| 50 | 1463.0 | 1000.0 | 7/3 |
| 51 | 1492.3 | 1020.0 | 19/8 |
| 52 | 1521.6 | 1040.0 | 12/5 |
| 53 | 1550.8 | 1060.0 | 22/9, 27/11 |
| 54 | 1580.1 | 1080.0 | 5/2 |
| 55 | 1609.3 | 1100.0 | 28/11, 33/13 |
| 56 | 1638.6 | 1120.0 | 18/7 |
| 57 | 1667.9 | 1140.0 | 21/8 |
| 58 | 1697.1 | 1160.0 | 8/3 |
| 59 | 1726.4 | 1180.0 | 19/7 |
| 60 | 1755.7 | 1200.0 | 11/4 |
| 61 | 1784.9 | 1220.0 | 14/5 |
| 62 | 1814.2 | 1240.0 | 20/7 |
| 63 | 1843.4 | 1260.0 | 26/9 |
| 64 | 1872.7 | 1280.0 | 44/15 |
| 65 | 1902.0 | 1300.0 | 3/1 |