source file: mills2.txt Date: Tue, 15 Jul 1997 00:04:33 +0200 Subject: harmonic metrics From: Carter Scholz I've been following the thread on harmonic spaces and metrics with interest, and although it's been a while since I've done any work in this area I'd like to comment briefly. I've found it somewhat fruitful to think of any given harmonic metric -- like any given tuning -- as one way of hearing among many ways. In fact I'm skeptical that there can be any such thing as a single metric (in the strict mathematical sense) for "consonance" or "concordance" or "harmonicity", because these concepts are so highly contextual. I don't believe that ANY metric can state categorically that a 15/1 is more or less "consonant" than a 5/3, or a 15/8 than a 16/15, or a root-position major triad than a second-inversion triad, or any such comparison, without carefully circumscribing its assertion by a precise definition of what "consonance" means in its context. (Tenney's _History of 'Consonance' and 'Dissonance'_ is a valuable look at how notions of "consonance" have changed over the centuries in European art music.) In designing a metric, there is an understandable tendency to shuttle between number and intuition, to say "my ears tell me..." about this quality, while also striving for quantitative objectivity. Which is NOT to say that the quality can't be measured, or that the design of metrics is therefore futile. But it does seem inevitable that any given metric will necessarily contain highly subjective decisions -- e.g. how to weight the various dimensions, whether or not it's octave invariant, whether the limit-system is based on primes or odd numbers or odd composites, et cetera -- decisions that carry with them theoretical assumptions about the nature of harmony and tonality, tending to radically simplify and somewhat falsify the processes by which we hear and interpret. I believe that a somewhat fuller and more useful description of harmonicity might be reached by using a GROUP of metrics, each descriptive of some particular aspect of pitch and harmonic space, instead of seeking a single "best" metric for a quality as multivalent and contextual as "consonance". Various metrics based on small-number-ratio theories of consonance have been discussed extensively on the list. I would like to see more discussion of what I consider "problem areas", the first two of them psychoacoustic, the third a more general agenda: 1) Critical bandwidth. Though there has been much discussion of octave invariance, there has been little of critical bandwidth, which requires that we bring absolute frequency, not just interval ratios, into any metric that claims psychoacoustic validity. (Sethares's extension of Plomp-Levelt does this, and takes steady-state timbre into account as well.) 2) Non-integers/approximations. Small-number-ratio metrics fail with irrationals. E.g. 300001/200000 is much more "dissonant" than 3/2; indeed, the closer the approximation, the more distant it's ranked by any small-number metric. The ear has a much more adaptable threshold. Some metric that describes near-misses could be quite useful. (Again, by incorporating critical bandwidth, Sethares's metric avoids this particular pitfall. Tenney alludes to, but does not incorporate, a "tolerance range" in his harmonic distance metric.) 3) Acculturation. A catch-all by which I mean any sort of bias or assumption or preference that may be hiding behind a veneer of objectivity. Octave-invariance has been discussed. Some discussion of where & how qualities become quantities might be helpful, e.g. the importance of 5-limit intervals in tertial harmony. Not that bias or preference could or should be banished, but it ought to be identified. Thus, a metric designed to describe the closeness of a pitch set to a harmonic series might reasonably rank a 7/1 closer than a 3/2. Et cetera. The goal might be to arrive at some set of simple metrics that do not confuse a multi-dimensional subjective quality like "consonance" with a single quantitative measurement. One could then choose the qualities that one deems important and use those metrics appropriate to the analytical or compositional task at hand.