65edt: Difference between revisions
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== Theory == | == Theory == | ||
65edt is almost identical to [[41edo]], but with the | 65edt is almost identical to [[41edo]], but with the perfect twelfth rather than the [[2/1|octave]] being just. The octave is about 0.305 cents compressed. Like 41edo, 65edt is [[consistent]] to the [[integer limit|16-integer-limit]], and in comparison, it improves the intonation of primes 3, [[11/1|11]], [[13/1|13]], and [[17/1|17]] at the expense of less accurate intonations of 2, [[5/1|5]], [[7/1|7]], and [[19/1|19]], commending itself as a suitable tuning for [[13-limit|13-]] and [[17-limit]]-focused harmonies. | ||
=== Harmonics === | === Harmonics === | ||
| Line 223: | Line 223: | ||
| 1170.4 | | 1170.4 | ||
| 800.0 | | 800.0 | ||
| [[55/28]], [[63/32]] | | [[49/25]], [[55/28]], [[63/32]] | ||
|- | |- | ||
| 41 | | 41 | ||
| Line 233: | Line 233: | ||
| 1229.0 | | 1229.0 | ||
| 840.0 | | 840.0 | ||
| [[49/24]], [[81/40]] | | [[45/22]], [[49/24]], [[55/27]], [[81/40]] | ||
|- | |- | ||
| 43 | | 43 | ||
| Line 343: | Line 343: | ||
| 1872.7 | | 1872.7 | ||
| 1280.0 | | 1280.0 | ||
| [[ | | [[44/15]] | ||
|- | |- | ||
| 65 | | 65 | ||
| Line 356: | Line 356: | ||
* [[95ed5]] – relative ed5 | * [[95ed5]] – relative ed5 | ||
* [[106ed6]] – relative ed6 | * [[106ed6]] – relative ed6 | ||
* [[147ed12]] – relative ed12 | |||
* [[361ed448]] – close to the zeta-optimized tuning for 41edo | |||
[[Category:41edo]] | |||
Latest revision as of 13:08, 20 June 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 perfect twelfth rather than the octave being just. The octave is about 0.305 cents compressed. Like 41edo, 65edt is consistent to the 16-integer-limit, and in comparison, it improves the intonation of primes 3, 11, 13, and 17 at the expense of less accurate intonations of 2, 5, 7, and 19, commending itself as a suitable tuning for 13- and 17-limit-focused harmonies.
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 |