65edo: Difference between revisions

'''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 the [[zeta]] edo [[130edo|130edo]].

65edo contains [[13edo|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 [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded Rubble: a Xenuke Unfolded].