65edo
| ← 64edo | 65edo | 66edo → |
65 equal divisions of the octave (abbreviated 65edo or 65ed2), also called 65-tone equal temperament (65tet) or 65 equal temperament (65et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 65 equal parts of about 18.5 ¢ each. Each step represents a frequency ratio of 21/65, or the 65th root of 2.
Theory
65et can be characterized as the temperament which tempers out 32805/32768 (schisma), 78732/78125 (sensipent comma), 393216/390625 (würschmidt comma), and [-13 17 -6⟩ (graviton). In the 7-limit, there are two different maps; the first is ⟨65 103 151 182] (65), tempering out 126/125, 245/243 and 686/675, so that it supports sensi, and the second is ⟨65 103 151 183] (65d), tempering out 225/224, 3125/3087, 4000/3969 and 5120/5103, so that it supports garibaldi. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit würschmidt temperament (wurschmidt and worschmidt) these two mappings provide.
65edo approximates the intervals 3/2, 5/4, 11/8, 19/16, 23/16, 31/16 and 47/32 well, so that it does a good job representing the 2.3.5.11.19.23.31.47 just intonation subgroup. To this one may want to add 17/16, 29/16 and 43/32, giving the 47-limit no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of schismic/nestoria that focuses on the very primes that 53edo neglects and that instead elegantly connects primes 7, 13, 37 and 41 to nestoria. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the zeta edo 130edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.42 | +1.38 | -8.83 | +2.53 | +8.70 | +5.81 | -2.13 | -0.58 | +4.27 | -0.42 | +7.12 | -4.45 | +5.41 | -0.89 |
| Relative (%) | +0.0 | -2.3 | +7.5 | -47.8 | +13.7 | +47.1 | +31.5 | -11.5 | -3.2 | +23.1 | -2.3 | +38.6 | -24.1 | +29.3 | -4.8 | |
| Steps (reduced) |
65 (0) |
103 (38) |
151 (21) |
182 (52) |
225 (30) |
241 (46) |
266 (6) |
276 (16) |
294 (34) |
316 (56) |
322 (62) |
339 (14) |
348 (23) |
353 (28) |
361 (36) | |
Subsets and supersets
65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Andrew Heathwaite's composition Rubble: a Xenuke Unfolded.
130edo, which doubles its, corrects its approximation to harmonics 7 and 13.
Intervals
| # | Cents | Approximate Ratios * | Ups and Downs Notation | |
|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | D |
| 1 | 18.46 | 81/80, 88/87, 93/92, 94/93, 95/94, 96/95, 100/99, 121/120, 115/114, 116/115, 125/124 | ^1 | ^D |
| 2 | 36.92 | 45/44, 46/45, 47/46, 48/47, 55/54, 128/125 | ^^1 | ^^D |
| 3 | 55.38 | 30/29, 31/30, 32/31, 33/32, 34/33 | vvm2 | vvEb |
| 4 | 73.85 | 23/22, 24/23, 25/24, 47/45 | vm2 | vEb |
| 5 | 92.31 | 18/17, 19/18, 20/19, 58/55, 135/128, 256/243 | m2 | Eb |
| 6 | 110.77 | 16/15, 17/16, 33/31 | A1/^m2 | D#/^Eb |
| 7 | 129.23 | 14/13, 27/25, 55/51 | v~2 | ^^Eb |
| 8 | 147.69 | 12/11, 25/23 | ~2 | vvvE |
| 9 | 166.15 | 11/10, 32/29 | ^~2 | vvE |
| 10 | 184.62 | 10/9, 19/17 | vM2 | vE |
| 11 | 203.08 | 9/8, 64/57 | M2 | E |
| 12 | 221.54 | 17/15, 25/22, 33/29, 58/51 | ^M2 | ^E |
| 13 | 240.00 | 23/20, 31/27, 38/33, 54/47, 55/48 | ^^M2 | ^^E |
| 14 | 258.46 | 22/19, 29/25, 36/31, 64/55 | vvm3 | vvF |
| 15 | 276.92 | 20/17, 27/23, 34/29, 75/64 | vm3 | vF |
| 16 | 295.38 | 19/16, 32/27 | m3 | F |
| 17 | 313.85 | 6/5, 55/46 | ^m3 | ^F |
| 18 | 332.31 | 23/19, 40/33 | v~3 | ^^F |
| 19 | 350.77 | 11/9, 27/22, 38/31 | ~3 | ^^^F |
| 20 | 369.23 | 26/21, 47/38, 68/55 | ^~3 | vvF# |
| 21 | 387.69 | 5/4, 64/51 | vM3 | vF# |
| 22 | 406.15 | 19/15, 24/19, 29/23, 34/27, 81/64 | M3 | F# |
| 23 | 424.62 | 23/18, 32/25 | ^M3 | ^F# |
| 24 | 443.08 | 22/17, 31/24, 40/31, 128/99 | ^^M3 | ^^F# |
| 25 | 461.54 | 30/23, 47/36, 72/55 | vv4 | vvG |
| 26 | 480.00 | 29/22, 33/25, 62/47 | v4 | vG |
| 27 | 498.46 | 4/3 | P4 | G |
| 28 | 516.92 | 23/17, 27/20, 31/23 | ^4 | ^G |
| 29 | 535.38 | 15/11, 34/25, 64/47 | v~4 | ^^G |
| 30 | 553.85 | 11/8, 40/29, 62/45 | ~4 | ^^^G |
| 31 | 572.31 | 25/18, 32/23 | ^~4/vd5 | vvG#/vAb |
| 32 | 590.77 | 24/17, 31/22, 38/27, 45/32 | vA4/d5 | vG#/Ab |
| 33 | 609.23 | 17/12, 27/19, 44/31, 64/45 | A4/^d5 | G#/^Ab |
| 34 | 627.69 | 36/25, 23/16 | ^A4/v~5 | ^G#/^^Ab |
| 35 | 646.15 | 16/11, 29/20, 45/31 | ~5 | vvvA |
| 36 | 664.62 | 22/15, 25/17, 47/32 | ^~5 | vvA |
| 37 | 683.08 | 34/23, 40/27, 46/31 | v5 | vA |
| 38 | 701.54 | 3/2 | P5 | A |
| 39 | 720.00 | 44/29, 50/33, 47/31 | ^5 | ^A |
| 40 | 738.46 | 23/15, 55/36, 72/47 | ^^5 | ^^A |
| 41 | 756.92 | 17/11, 48/31, 31/20, 99/64 | vvm6 | vvBb |
| 42 | 775.38 | 25/16, 36/23 | vm6 | vBb |
| 43 | 793.85 | 19/12, 27/17, 30/19, 46/29, 128/81 | m6 | Bb |
| 44 | 812.31 | 8/5, 51/32 | ^m6 | ^Bb |
| 45 | 830.77 | 21/13, 55/34, 76/47 | v~6 | ^^Bb |
| 46 | 849.23 | 18/11, 31/19, 44/27 | ~6 | vvvB |
| 47 | 867.69 | 33/20, 38/23 | ^~6 | vvB |
| 48 | 886.15 | 5/3, 92/55 | vM6 | vB |
| 49 | 904.62 | 27/16, 32/19 | M6 | B |
| 50 | 923.08 | 17/10, 29/17, 46/27, 128/75 | ^M6 | ^B |
| 51 | 941.54 | 19/11, 31/18, 50/29, 55/32 | ^^M6 | ^^B |
| 52 | 960.00 | 33/19, 40/23, 47/27, 54/31, 96/55 | vvm7 | vvC |
| 53 | 978.46 | 30/17, 44/25, 51/29, 58/33 | vm7 | vC |
| 54 | 996.92 | 16/9, 57/32 | m7 | C |
| 55 | 1015.38 | 9/5, 34/19 | ^m7 | ^C |
| 56 | 1033.85 | 20/11, 29/16 | v~7 | ^^C |
| 57 | 1052.31 | 11/6, 46/25 | ~7 | ^^^C |
| 58 | 1070.77 | 13/7, 50/27, 102/55 | ^~7 | vvC# |
| 59 | 1089.23 | 15/8, 32/17, 62/33 | vM7 | vC# |
| 60 | 1107.69 | 17/9, 19/10, 36/19, 55/29, 243/128, 256/135 | M7 | C# |
| 61 | 1126.15 | 23/12, 44/23, 48/25, 90/47 | ^M7 | ^C# |
| 62 | 1144.62 | 29/15, 31/16, 33/17, 60/31, 64/33 | ^^M7 | ^^C# |
| 63 | 1163.08 | 45/23, 47/24, 88/45, 92/47, 108/55, 125/64 | vv8 | vvD |
| 64 | 1181.54 | 87/55, 93/47, 95/48, 99/50, 115/58, 160/81, 184/93, 188/95, 228/115, 240/121, 248/125 | v8 | vD |
| 65 | 1200.00 | 2/1 | P8 | D |
* based on treating 65edo as a 2.3.5.11.13/7.17.19.23.29.31.47 subgroup temperament.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-103 65⟩ | [⟨65 103]] | +0.131 | 0.131 | 0.71 |
| 2.3.5 | 32805/32768, 78732/78125 | [⟨65 103 151]] | -0.110 | 0.358 | 1.94 |
| 2.3.5.11 | 243/242, 4000/3993, 5632/5625 | [⟨65 103 151 225]] | -0.266 | 0.410 | 2.22 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 3\65 | 55.38 | 33/32 | Escapade |
| 1 | 9\65 | 166.15 | 11/10 | Squirrel etc. |
| 1 | 12\65 | 221.54 | 25/22 | Hemisensi |
| 1 | 19\65 | 350.77 | 11/9 | Karadeniz |
| 1 | 21\65 | 387.69 | 5/4 | Würschmidt |
| 1 | 24\65 | 443.08 | 162/125 | Sensipent |
| 1 | 27\65 | 498.46 | 4/3 | Helmholtz / nestoria / photia |
| 1 | 28\65 | 516.92 | 27/20 | Larry |
| 5 | 20\65 (6\65) |
369.23 (110.77) |
99/80 (16/15) |
Quintosec |
| 5 | 27\65 (1\65) |
498.46 (18.46) |
4/3 (81/80) |
Pental |
| 5 | 30\65 (4\65) |
553.85 (73.85) |
11/8 (25/24) |
Countdown |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct