source file: m1421.txt Date: Tue, 19 May 1998 12:26:49 -0400 Subject: Ideal tuning From: "Paul H. Erlich" 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.