Monday, April 15, 2013

The tragedy of music: part three

In part one, we discussed some issues with Pythagorean tuning, and in part two we continued to quarter-comma meantone.  The logical conclusion is equal temperament, a widely used (though not quite universal) modern system.

Equal temperament is, in a way, simpler than any of the earlier systems.  Instead of building an octave out of some fixed ratio, we start with an octave and subdivide it.  The most natural way to do this is to make a semitone the twelfth root of two.  Note that this is simply "a semitone" rather than a chromatic or diatonic semitone; in equal temperament, those are the same thing.  Everything else can be built rather easily out of this fundamental unit, and we don't have overlapping or separated octaves.  This, in turn, means no wolf fifth.

Still, this fundamental unit can be unwieldy.  Writing out "the twelfth root of two" all the time is annoying, and the mathematical notation for it is rather ugly.  For this purpose, the so-called cent was invented.  There are 1200 cents in an octave; 100 make up a semitone.  It is a logarithmic unit.  Increasing a tone by 1200 cents means doubling its frequency.  700 cents make up a perfect fifth, and 400 cents are a major third.  The cent is a unit of relative measure; it is not meaningful to equate a single note to a given number of cents, except in relation to another note.  That "other note" is often middle A, which, for convenience, is typically tuned to 440 Hz.

This system is very practical, which accounts for its widespread use.  However, this is not a happy story with a happy ending; there is a problem.  The twelfth root of two is an ugly, irrational number, and all of the other intervals are powers of it.  There are no simple integer ratios at all.  While the cent may make the system look nice and clean, it is an artificial unit created to hide the irrational semitone.  In effect, we've taken the sourness of the wolf fifth and extended it over the whole octave, spreading thinly to mask the dissonance.  Practical this may be, but ideal it is not.

This leads me to a stark realization: the Platonist's ideal of "perfect yet unattainable" forms is inapplicable to music.  These problems are not of a physical nature.  Any musical system will suffer from them, except for the trivial system which has one note per octave.  There is a fundamental disconnect between equal temperament and just intonation.  We cannot have both, even in theory.  And that is the tragedy I've been talking about.