Monday, April 8, 2013

The tragedy of music, part two

In part one, we discussed Pythagorean tuning and its failings.  Despite this crucial issue, Pythagorean tuning was highly influential on musical theory.

In the 1500's, a variant of Pythagorean tuning called quarter-comma meantone became popular.  The "comma" in this name is not the Pythagorean comma we saw last time, but an entirely different comma.

A major third is an interval spanning three staff positions and four semitones.  It is not considered "perfect" in the same way as the perfect fifth, but is still regarded as a consonant interval, at least in theory.  Under Pythagorean tuning, the major third was a rather dissonant 81:64.  Quarter-comma meantone flattened this to a nicer 5:4, at the expense of a more dissonant perfect fifth.

Interestingly, we now have irrational intervals: the perfect fifth is the fourth root of 5.  This way, if we move up by four perfect fifths, and down by two octaves, we end up at 5:4, the justly-intoned major third.

This sort of trade-off is debatable, of course, but it was an explicit design goal of quarter-comma meantone; the flatter fifth was viewed as an acceptable price for a just third.

Now, suppose we start at middle C, as we did last time, and move up a major third.  We arrive at middle E at a ratio of 5:4.  Next we move up again to G♯, at a ratio of 25:16.  Finally, we get to C5, at a ratio of 125:64.  This is not an octave, but unlike in Pythagorean tuning, the interval is too flat rather than too sharp.  This means there's a gap between the octaves, unlike the overlapping octaves of Pythagorean tuning.

Like in Pythagorean tuning, one of the fifths must span this gap.  That fifth is sharpened by 128:125, a much larger interval than the Pythagorean comma.  It sounds extremely dissonant, to the point that it became known as the "wolf fifth" because it sounds like a wolf howling at the moon.  This moniker is sometimes also applied to the diminished sixth produced by Pythagorean tuning under the equivalent problem, but note that quarter-comma meantone is much worse in this regard.

Updated: In part three, we discuss modern equal temperament.