Optimal patent val: Difference between revisions

By tempering a JI scale using the ''N''-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [http://www.huygens-fokker.org/scala/ Scala] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution".

To limit the search range when finding the optimal patent val a useful observation is this: given ''N''-edo, and an odd prime ''q'' ≤ ''p'', if ''d'' is the absolute value in cents of the difference between the tuning of ''q'' given by the [[POTE tuning]] and the POTE tuning rounded to the nearest ''N''-edo value, then d < 600/''N'', from which it follows that N < 600/d. Likewise, if ''e'' is the absolute value of the error of ''q'' in the patent val tuning, then ''e'' < 600/''N'' and so ''N'' < 600/''e''. If ''N''-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/''N'', and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime ''q'' in the POTE tuning, is bounded by 1200/''N''. Hence, ''N'' < 1200/error(''q''). If now we take the minimum value for 1200/error(prime) for all the odd primes up to ''p'', we obtain an upper bound for ''N''.