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.