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Computing the TOP tuning using Miller commas

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

3/9/2007 1:36:07 PM

(1) First get a list of the (r+1)-prime comms of the rank r temperament.

(2) Find the comma c with the maximum value for cents(c)/cents(th(c)),
where th(c) is the product of the numerator and the denominator of c.

(3) For each prime p, define ep = up/log2(p)-1, where "up" is an
indeterminate.

(4) Taking fp to be +1 if p is in the numerator of c, -1 if p is in the
denominator, and 0 if neither, form the set of equations {fq*eq -
fr*er} for each pair of primes q and r. Actually, you don't need the
whole list, just for q<r will do.

(5) To this set, add the equation <u2 u3 ... up| ki>, where ki is an
element of a comma basis for the temperament.

(6) Solve the resulting linear system for u2, u3, ... up. This is the
TOP tuning.

The method works in general, but sadly cannot be adapted to NOT tuning,
because NOT has r primes, not r+1, with equal relative errors, and
those do not correspond to commas.