User:Overthink/41edo well temperament: Difference between revisions

A highly notable tuning system is [[41edo]], which is [[distinctly consistent]] and [[consistent to distance]] 2 in the 9-odd-limit, consistent to the [[15-odd-limit]], and is overall great for its size in all [[prime limit]]s up to [[13-limit|13]]. However, the accuracy of some intervals is debateable, so we will construct a well temperament in order to fix this issue.

Our well temperament will be based on a chain of fifths, and we will target harmonics [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. In 41edo, these prime harmonics are mapped as follows (corresponding to [[andromeda]] temperament if the fifths are all the same):

! Fifths down

Prime 5 is closest to the root on the circle of fifths; being only 8 fifths down due to 41edo tempering out the [[schisma]]. We build our well temperament down the chain of fifths to reach a more accurate prime 5, using slightly flat fifths of 701.792[[{{c}}]], or 1/12 schisma flat of just. We continue this chain of schismic fifths down until we reach -12 fifths, or an exact [[81/80]]. We then stack down 12 [[parapyth]] fifths of 704.002{{c}} to obtain accurate approximations of harmonics 7, 11, and 13. Finally, the remaining fifths are all 1/12-schisma flat, giving a total of 29 schismic fifths and 12 parapyth fifths.