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Instrument Tuning and the Out-of-Tune Problems of the Past
Several hundred years ago, when musicians and musical theorists were experimenting with changing keys within a piece of music, a nasty problem kept bedevilling them. Whenever they tried to switch to a new key, their instruments sounded out of tune. Hellish out of tune.
The problem was the dang Pythagorean comma. As discussed in Chapter 4, if you tune an instrument using exact Pythagorean 3:2 frequency ratios, you end up with an octave that is slightly bigger than it ought to be. About a quarter of a tone too big.
For example, Middle C has a frequency of 261.6 Hz. So the frequency of the C above Middle C ought to be exactly double: 523.2 Hz.
But if you use exact Pythagorean fifths, you end up with C above Middle C having a frequency of 530.3 Hz. Noticeably too sharp.
If instead you tune in perfect 2:1 octaves, then the other notes derived from simple frequency ratios such as 3:2, 4:2, and so on, end up either too sharp or too flat.
What to do?