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Twelfth Root of Two     Heterodynes in Music



When two different vibrations are mixed together, sum and difference frequencies, called heterodynes, are created.  In acoustics, heterodynes are called beats.

For example, when two slightly detuned notes are sounded together, a slow vibration or beating is heard.  The beat frequency equals the difference in frequency between the two notes.

Since a musical chord is a mix of different frequencies, one might wonder why unwanted beats don't ruin the chord.  The answer lies in the musical scale.




In a Just scale, note frequencies relate to the Fundamental frequency (the tonic) by small, whole number ratios.  As a result, the semitones aren't evenly spaced.  If you try to move the tonic (the "I") to a different note, the intervals will be off and the transposed scale will be noticeably out of tune.

In Just scales, chordal beating (heterodyning) tends to produce harmonic notes already in or suggested by the original notes, with pleasing results.

For example, a 440Hz root beating with a 220Hz sub-octave generates a difference frequency of 220Hz (already present) and a sum frequency of 660Hz which is a musical 5th interval (3/2 ratio above 440Hz).

The disadvantage of a Just-scaled instrument is that it only plays in tune in one key.




In the Equal-tempered scale, each frequency is the twelfth root of two (~1.059463) times higher in frequency than the last.  In this way, the tuning issues are spread out amongst the key signatures, making all of them sound okay—it doesn't matter which note is the tonic.

And conveniently, after multiplying the tonic by the twelfth-root of two twelve times, its frequency is perfectly doubled, which just happens to be one octave.

Of course now the heterodynes aren't perfect but then neither are the note intervals themselves!  In spite of it all, keyboards and guitars, both equally tempered instruments, usually sound okay.

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