Ideal tuning

If musicians could produce any desired pitch at any time, what tuning
would they use? That obviously depends on the piece in question.
Although many would answer "Just Intonation" to this question, there are
many examples where JI is not ideal. The 6/9 chord is one recently
discussed example; another is the chord 0 9 13 18 in 22tET, where the
ambiguity between the 3-limit and 7-limit is much like that of the
3-limit and 5-limit in the former chord. But let us leave these
ambiguous harmonies aside for the moment.

Wandering tonics are another typical example used against JI renditions
of many common-practice pieces. However, one can keep each individual
chord in JI while distributing the comma over all the melodic intervals
-- essentially, tune the melodic sequence roots in meantone temperament
and the chord over each root in JI. Since the just noticeable difference
for melody is much greater than that for harmony, this is clearly a
better-sounding (though difficult to perform) solution than straight
meantone. The truly optimal solution (if "badness" is quadratic in
tuning error, and harmonic tuning errors are weighted several times more
than melodic ones) would be to shade the JI chords ever-so-slightly
toward meantone, so as to reduce the melodic shifts further while
introducing a negligible amount of roughness into the chords.

In general, melodies operate within certain pitch sets and auxiliaries
to these sets. Again, the allowable mistuning for these melodic pitch
sets is greater than that for chords. One can envision a computer
program which then optimizes the tuning of a piece by minimizing some
overall "badness" function of these mistunings.

Even the ambiguous chords above can be handled in such a scheme; each
consonant harmonic interval is associated with one and only one just
ratio, and an overall error function is minimized. For example, in the
6/9 chord, there are four perfect fifths, two minor thirds, and one
major third, each of which should approach simple 5-limit ratios. If we
weight all 7 errors equally, and find the least-squared-error solution,
one constructs the chord using perfect fifths of 696.3 cents. If one
weights the errors in proportion to the odd limit, one constructs the
chord using perfect fifths of 695.9 cents. If one weights the errors in
inverse proportion to the odd limit, one constructs the chord using
perfect fifths of 697.2 cents. In the other chord, there are three 3/2s,
two 7/4s, and one 7/6. The optimal perfect fifths: Equal-weighting --
711.5 cents; Odd-Limit-weighting -- 712.8 cents;
Inverse-Odd-Limit-weighting -- 707.7 cents. In any case, the melodic
context will typically cause all the pitches to be altered slightly, but
probably not very much, since melodic tuning errors are less important
than harmonic ones.

(A word on those weighting schemes -- the Odd-Limit-weighting concerns
the ability of the intervals to evoke certain simple ratios rather than
more complex ones, while the Inverse-Odd-Limit-weighting may better
represent the overall level of consonance. Where both factors are
considered important, Equal-weighting may be a good compromise.