Sunday, March 11, 2007


I received a few emails about my post on Morton Feldman's Piano and String Quartet, and one of them concerned my use of the terms "tempered" and "untempered". While in everyday usage, "untempered" would suggest some unruliness or lack of control, in this musical context, nothing of the sort is intended.

In a musical temperament, conflicts between two or more ideal values for a given interval (or neighborhood of intervals) are resolved by compromising on a single value. In 12-tone equal temperament, a tuning system with some considerable historical weight and currency, the compromise is manifest in several ways: the pythagorean comma is "tempered out" so that the sum of 12 perfect fifths is an octave multiple of the initial tone; the syntonic tonic is tempered out so that the best major third is equivalent to four octave-reduced fifths; sharps and flats are "enharmonically equivalent; the octave divides equally into 2, 3, 4, 6, and 12 parts; and in practice, when a real, physical instrument is tuned up, some compromises are made to account for variations or biases in the instrument itself or in our perceptual apparatus (octaves, for example, may be subject to a register-dependent stretching or narrowing).

These compromises present both costs and benefits: the variance of intonation of intervals from an intonational ideal (which might be a form of just intonation) may be considered to be a cost, and unlimited transposition may be considered a benefit. An increase or reduction in the beating rate of an interval can be heard, in context, as either a cost or a benefit. The capacity for a given interval to represent more than one rational interpretation in a strategic, ambiguous, or even punning way, can be heard as either a cost or a benefit. (In fact, the principles and practices of western common-practice tonality are so closely connected to the characteristics of the prevailing intonational system (first meantone, and rather much later, via a wide range of subtly different unequal circulating temperaments, 12-tone equal temperament) that it is reasonable to wonder about the connection between the further history of that tonal tradition and whether the tuning system has potential to change further or tuning and tonal practice have settled together into some form of inertia.)

In his writing for piano, Feldman very strictly observed the properties of the temperament, assuming, for example, enharmonic equivalence, and not indicating double sharps or flats. However, in his later writing for instruments which were not restricted to the pitches of a fixed temperament, Feldman very carefully notated using all available varieties of accidentals, and spoke of this practice not in terms of any precise intonational system but rather in terms of "directionality". While Feldman indicated that this directionality was to be reflected in the intonation, he was rather unclear about precisely how this was to be done. If an d# and an eb (and possibly an fbb, as long as we're at it), for example, were to be intonationally distinct and "directional", which is the higher or lower pitch? Is d# lower than eb -- as in a meantone environment -- which would emphasize the fact that a d that being raised, or is d# higher than eb -- as in pythagorean tuning -- emphasizing what one might call a leading tone quality?

In the end, I suspect that Feldman was agnostic on this point; indeed, putting his tonal practice into a rigid, single system would have been an anathema for him. It was probably all the same to him whether sharps were lower than flats or vice versa as long as some consistent application of directionality was present, and the instruments which were not by nature restricted to the compromises of a temperament were not so restricted in his scores. It is precisely this practice which I have described as "untempered".

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