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Off Pitch: The Equally Tempered Scale's Advantages
The equal temperament solution works. Your brain accepts the small “pitch errors”—slight deviations from simple ratios—when they’re equally distributed over all 12 semitones. Since every semitone interval is exactly equal, you can construct diatonic scales using any of the 12 semitones as the tonic note, and the octave notes will always have a frequency ratio of exactly 2:1. Equal temperament makes something called modulation possible (coming up shortly).
Consequently, equal temperament has been the norm for about three centuries in Western music.
Equal temperament works only because the pitch errors are small—so small that your forgiving brain processes them as though they were simple frequency ratios.
When you try to tune a guitar or other stringed instrument using harmonics from string to string, it doesn’t quite work out because you’re not using equal temperament. That’s why the best tuning device is a digital tuner, with equally-tempered frequencies built into the electronics that are accurate to many decimal places.