65edo: Difference between revisions
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=65 tone equal temperament= | |||
'''65edo''' divides the [[Octave|octave]] into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the [[schisma|schisma]], 32805/32768, the [[sensipent_comma|sensipent comma]], 78732/78125, and the [[Wuerschmidt_comma|wuerschmidt comma]]. In the [[7-limit|7-limit]], there are two different maps; the first is <65 103 151 182|, [[tempering_out|tempering out]] 126/125, 245/243 and 686/675, so that 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo supports garibaldi temperament. In both cases, the tuning privileges the [[5-limit|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 [[wuerschmidt_temperament|wuerschmidt temperament]] (wurschmidt and worschmidt) these two mappings provide. | |||
65edo approximates the intervals [[3/2|3/2]], [[5/4|5/4]], [[11/8|11/8]] and [[19/16|19/16]] well, so that it does a good job representing the 2.3.5.11.19 [[just_intonation_subgroup|just intonation subgroup]]. To this one may want to add 13/8 and 17/16, giving the [[19-limit|19-limit]] no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit [[k*N_subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as [[130edo|130edo]]. | 65edo approximates the intervals [[3/2|3/2]], [[5/4|5/4]], [[11/8|11/8]] and [[19/16|19/16]] well, so that it does a good job representing the 2.3.5.11.19 [[just_intonation_subgroup|just intonation subgroup]]. To this one may want to add 13/8 and 17/16, giving the [[19-limit|19-limit]] no-sevens subgroup 2.3.5.11.13.17.19. Also of interest is the 19-limit [[k*N_subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as [[130edo|130edo]]. | ||
Revision as of 09:57, 1 January 2019
65 tone equal temperament
65edo divides the octave into 65 equal parts of 18.4615 cents each. It can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the wuerschmidt 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 65edo supports sensi temperament, and the second is <65 103 151 183|, tempering out 225/224, 3125/3097, 4000/3969 and 5120/5103, so that 65edo 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 wuerschmidt temperament (wurschmidt and worschmidt) these two mappings provide.
65edo approximates the intervals 3/2, 5/4, 11/8 and 19/16 well, so that it does a good job representing the 2.3.5.11.19 just intonation subgroup. To this one may want to add 13/8 and 17/16, giving the 19-limit no-sevens subgroup 2.3.5.11.13.17.19. 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 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.
Intervals
| Degree | Size (Cents) |
|---|---|
| 0 | 0.0000 |
| 1 | 18.4615 |
| 2 | 36.9231 |
| 3 | 55.3846 |
| 4 | 73.8462 |
| 5 | 92.3077 |
| 6 | 110.7692 |
| 7 | 129.2308 |
| 8 | 147.6923 |
| 9 | 166.1538 |
| 10 | 184.6154 |
| 11 | 203.0769 |
| 12 | 221.5385 |
| 13 | 240.0000 |
| 14 | 258.4615 |
| 15 | 276.9231 |
| 16 | 295.3846 |
| 17 | 313.8462 |
| 18 | 332.3077 |
| 19 | 350.7692 |
| 20 | 369.2308 |
| 21 | 387.6923 |
| 22 | 406.1538 |
| 23 | 424.6154 |
| 24 | 443.0769 |
| 25 | 461.5385 |
| 26 | 480.0000 |
| 27 | 498.4615 |
| 28 | 516.9231 |
| 29 | 535.3846 |
| 30 | 553.8462 |
| 31 | 572.3077 |
| 32 | 590.7692 |
| 33 | 609.2308 |
| 34 | 627.6923 |
| 35 | 646.1538 |
| 36 | 664.6154 |
| 37 | 683.0769 |
| 38 | 701.5385 |
| 39 | 720.0000 |
| 40 | 738.4615 |
| 41 | 756.9231 |
| 42 | 775.3846 |
| 43 | 793.8462 |
| 44 | 812.3077 |
| 45 | 830.7692 |
| 46 | 849.2308 |
| 47 | 867.6923 |
| 48 | 886.1538 |
| 49 | 904.6154 |
| 50 | 923.0769 |
| 51 | 941.5385 |
| 52 | 960.0000 |
| 53 | 978.4615 |
| 54 | 996.9231 |
| 55 | 1015.3846 |
| 56 | 1033.8462 |
| 57 | 1052.3077 |
| 58 | 1070.7692 |
| 59 | 1089.2308 |
| 60 | 1107.6923 |
| 61 | 1126.1538 |
| 62 | 1144.6154 |
| 63 | 1163.0769 |
| 64 | 1181.5385 |
| 65 | 1200.0000 |