65edo
| ← 64edo | 65edo | 66edo → |
The 65 equal divisions of the octave (65edo), or 65(-tone) equal temperament (65tet, 65et) when viewed from a regular temperament perspective, divides the octave into 65 equal parts of about 18.5 cents each.
Theory
65et can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the würschmidt comma. In the 7-limit, there are two different maps; the first is ⟨65 103 151 182], tempering out 126/125, 245/243 and 686/675, so that it supports sensi temperament, 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 temperament. 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 and 31/16 well, so that it does a good job representing the 2.3.5.11.19.23.31 just intonation subgroup. To this one may want to add 17/16 and 29/16, giving the 31-limit no-7's no-13's subgroup 2.3.5.11.17.19.23.29.31. 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.
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 Rubble: a Xenuke Unfolded.
Prime harmonics
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Intervals
| Degree | Cents | Approximate Ratios * | Ups and Downs Notation | |
|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | D |
| 1 | 18.46 | 81/80, 100/99, 121/120 | ^1 | ^D |
| 2 | 36.92 | 45/44, 55/54, 128/125 | ^^1 | ^^D |
| 3 | 55.38 | 33/32 | vvm2 | vvEb |
| 4 | 73.85 | 25/24 | vm2 | vEb |
| 5 | 92.31 | 135/128, 256/243 | m2 | Eb |
| 6 | 110.77 | 16/15 | A1/^m2 | D#/^Eb |
| 7 | 129.23 | 14/13 | v~2 | ^^Eb |
| 8 | 147.69 | 12/11 | ~2 | vvvE |
| 9 | 166.15 | 11/10 | ^~2 | vvE |
| 10 | 184.62 | 10/9 | vM2 | vE |
| 11 | 203.08 | 9/8 | M2 | E |
| 12 | 221.54 | 25/22 | ^M2 | ^E |
| 13 | 240.00 | 55/48 | ^^M2 | ^^E |
| 14 | 258.46 | 64/55 | vvm3 | vvF |
| 15 | 276.92 | 75/64 | vm3 | vF |
| 16 | 295.38 | 32/27 | m3 | F |
| 17 | 313.85 | 6/5 | ^m3 | ^F |
| 18 | 332.31 | 40/33 | v~3 | ^^F |
| 19 | 350.77 | 11/9, 27/22 | ~3 | ^^^F |
| 20 | 369.23 | 26/21 | ^~3 | vvF# |
| 21 | 387.69 | 5/4 | vM3 | vF# |
| 22 | 406.15 | 81/64 | M3 | F# |
| 23 | 424.62 | 32/25 | ^M3 | ^F# |
| 24 | 443.08 | 128/99 | ^^M3 | ^^F# |
| 25 | 461.54 | 72/55 | vv4 | vvG |
| 26 | 480.00 | 33/25 | v4 | vG |
| 27 | 498.46 | 4/3 | P4 | G |
| 28 | 516.92 | 27/20 | ^4 | ^G |
| 29 | 535.38 | 15/11 | v~4 | ^^G |
| 30 | 553.85 | 11/8 | ~4 | ^^^G |
| 31 | 572.31 | 25/18 | ^~4/vd5 | vvG#/vAb |
| 32 | 590.77 | 45/32 | vA4/d5 | vG#/Ab |
| 33 | 609.23 | 64/45 | A4/^d5 | G#/^Ab |
| 34 | 627.69 | 36/25 | ^A4/v~5 | ^G#/^^Ab |
| 35 | 646.15 | 16/11 | ~5 | vvvA |
| 36 | 664.62 | 22/15 | ^~5 | vvA |
| 37 | 683.08 | 40/27 | v5 | vA |
| 38 | 701.54 | 3/2 | P5 | A |
| 39 | 720.00 | 50/33 | ^5 | ^A |
| 40 | 738.46 | 55/36 | ^^5 | ^^A |
| 41 | 756.92 | 99/64 | vvm6 | vvBb |
| 42 | 775.38 | 25/16 | vm6 | vBb |
| 43 | 793.85 | 128/81 | m6 | Bb |
| 44 | 812.31 | 8/5 | ^m6 | ^Bb |
| 45 | 830.77 | 21/13 | v~6 | ^^Bb |
| 46 | 849.23 | 18/11, 44/27 | ~6 | vvvB |
| 47 | 867.69 | 33/20 | ^~6 | vvB |
| 48 | 886.15 | 5/3 | vM6 | vB |
| 49 | 904.62 | 27/16 | M6 | B |
| 50 | 923.08 | 128/75 | ^M6 | ^B |
| 51 | 941.54 | 55/32 | ^^M6 | ^^B |
| 52 | 960.00 | 96/55 | vvm7 | vvC |
| 53 | 978.46 | 44/25 | vm7 | vC |
| 54 | 996.92 | 16/9 | m7 | C |
| 55 | 1015.38 | 9/5 | ^m7 | ^C |
| 56 | 1033.85 | 20/11 | v~7 | ^^C |
| 57 | 1052.31 | 11/6 | ~7 | ^^^C |
| 58 | 1070.77 | 13/7 | ^~7 | vvC# |
| 59 | 1089.23 | 15/8 | vM7 | vC# |
| 60 | 1107.69 | 243/128, 256/135 | M7 | C# |
| 61 | 1126.15 | 48/25 | ^M7 | ^C# |
| 62 | 1144.62 | 64/33 | ^^M7 | ^^C# |
| 63 | 1163.08 | 88/45, 108/55, 125/64 | vv8 | vvD |
| 64 | 1181.54 | 160/81, 99/50, 240/121 | v8 | vD |
| 65 | 1200.00 | 2/1 | P8 | D |
* based on treating 65edo as a 2.3.5.11.13/7 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 octave |
Generator (reduced) |
Cents (reduced) |
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 / photia |
| 1 | 28\65 | 516.92 | 27/20 | Gravity |
| 5 | 20\65 (6\65) |
369.23 (110.77) |
10125/8192 (16/15) |
Qintosec |
| 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) |
Trisedodge / countdown |