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12th root Heterodynes



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

For example, when two out-of-tune notes are played together, a slow vibration or beating is heard.  The frequency of the beats is equal to the difference in frequency between the two notes.

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




Just Intonation



In a Just musical scale, the frequency of each semitone is related to the root (tonic) frequency by a small, whole-number ratio.  For example, a musical fifth is 3/2 of the root frequency while a musical fourth is 4/3 of the root.

In other words, the 11 semitones above the root have no rational relationship to any of the other semitones, only to the root.  If you try to play in a key that starts on a note other than the root, several of the note's semitones will be significantly off the Just ratios and the shifted scale will sound noticeably out of tune.

It's interesting to note that when you play a chord in a Just scale, heterodyning produces frequencies that are already present in or suggested by the chord, with pleasing results.

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

Of course, the disadvantage of a Just scale is that it only plays in tune in one key.




Equal Temperament



The Equal-tempered scale divides each octave into 12 notes equally proportioned in frequency.  Each note is the 12th root of two (1.059463) times higher in frequency than the previous note.

This divvies up the Just intonation errors more equally among the semitones.  No matter which note you pick as the root, an equally tempered instrument will sound okay.

Of course, the heterodynes of two equally tempered notes aren't perfectly just, but then neither are the notes themselves!  There's a certain consistency to equally tempered instruments, like the guitar and keyboard, that makes them sound all right.

The 12th root of two is used because multiplying any frequency by that number twelve times perfectly doubles the frequency, producing a note that's exactly one octave higher.





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