TUNING SYSTEMS (TEMPERAMENTS)
There is no perfect keyboard tuning system. All tuning systems (temperaments) have their advantages and disadvantages. Modern tuning (used for pianos) uses equal temperament, in which the octave is divided into 12 semitones equally. But by doing so, none of the thirds are pure thirds. They are all wider (sharper) than a pure third. This is an unfortunate law of physics. Earlier tuning systems did not divide the octave equally, so some intervals were purer than others.
To use an analogy, let’s say that a “perfect” slice of pizza is one-sixth of a whole pie. Let’s also say that the pie needs to feed 8 people. If we divide the pie into 8 equal slices, each person will get the same amount, one eighth of the pie, which is less than the “perfect one sixth.” No one gets the perfect one-sixth. This is like equal temperament. But there are other (less equal) ways to divide the pie. If 2 people insist on a perfect one-sixth of the pie, the other 6 people are left with four sixths (two thirds) of the pie. If those six people now divide the rest of the pie equally amongst themselves, each of those six people will only get one-sixth of two thirds, which is one-ninth, smaller than the one-eighth had the pie been divided equally in the first place. And if 3 people get the perfect sixth, the remaining five would get even smaller slices.
Since early tuning systems did not attempt to divide the octave equally, some intervals sounded purer than in equal temperament, at the expense of other intervals which sounded worse than in equal temperament. So what was the advantage of these earlier tuning systems? Since early music did not modulate much, the music would contain mostly “good intervals” and avoid “bad intervals.” And these “good intervals’ sounded purer than in modern tuning.
During Elizabethan times, a commonly used temperament was mean tone temperament, in which the good sounding chords are C Major, D Major, Eb Major, E Major, F Major, G Major, A Major, and Bb Major; while the really horrible sounding chords are Db Major, Gb Major, Ab Major, and B Major. The music of that time simply did not include those “bad chords.”
Later tuning systems were closer to equal temperament. They were called “well temperaments.” J.S. Bach used such temperaments. The differences between chords were not as striking as in earlier temperaments. By the mid-nineteenth century, equal temperament became necessary because the music modulated so much that all chords needed to sound equally good. However, equal temperament is a compromise. None of the chords are perfect. Each chord is equally impure, like one-eight of the pizza pie. In earlier temperaments, some chords sound pure (like one-sixth of the pie) while other chords lose out and sound quite bad.
To hear an example of an early tuning system (mean-tone), listen to my recording of Sweelinck’s Mein junges Leben hat ein End, to appreciate the pureness of the chords.