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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 frequency of the beats equals the difference in frequency between the two sounded notes.

Since a musical chord is a mixture of frequencies, one might wonder why unintended beating doesn't ruin the sound of a chord.  The answer lies in the musical scale.




In a Just musical scale, semitone frequencies relate to the fundamental frequency (also called the tonic) by small, whole number ratios.  A musical fifth, for example, vibrates at 3/2 of the tonic while a musical fourth vibrates at 4/3 of the tonic.

This means that, numerically, the semitones aren't evenly spaced.  If you try to move the tonic to a different note of the scale, the semitones will no longer have just ratios and the scale will sound noticeably out of tune.

The advantage of a Just scale is that chordal beating (heterodyning) produces harmonic frequencies that are already in or suggested by the original notes, with pleasing results.

A 440Hz tonic, for example, 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 (3/2 ratio above 440Hz).

The disadvantage of a Just scale is that it only plays in tune in one key.




In the Equal-tempered scale, each note is the 12th root of two (1.059463) times higher in frequency than the previous note.  This spreads out the scaling errors among the key signatures so that playing the instrument in any key sounds okay.

Of course, the heterodynes of these equally-tempered notes are no longer perfect harmonics but then neither are the notes themselves!  In spite of all this, keyboards and guitars, both equally tempered instruments, usually sound okay.

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


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