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12th root Heterodyne



When two different vibration frequencies occur simultaneously, sum and difference frequencies called heterodynes appear.  In acoustics, heterodynes are called beats.

For example, when two slightly out-of-tune notes are played together, a slow vibration or beating is heard.  The frequency of these 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, each semitone frequency is related to the root (or tonic) frequency by a small, whole-number ratio.  For example, the musical fifth ratio is 3:2 and the musical fourth ratio is 4:3.  The major-third ratio is 5:4 and the minor-third ratio is 6:5.

This means that none of the instrument's semitone frequencies have any rational relationship to any of the other semitones, only to the root.

If you start a musical scale on a note other than the root, several of the semitones will differ significantly from the just ratios,  The shifted scale will sound noticeably out of tune.



Just Heterodyning


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, each higher in frequency than the last note by a factor of the 12th root of two (1.059463).

When you apply this rule 12 times, you double the frequency, which is one octave higher.

Equal-tempered instruments like the guitar and keyboard sound okay no matter which note on its scale you use as the root.

The irrationality of the scale isn't too noticeable because the most grievous dissonances of shifting a Just scale are spread out among all the semitones.



Tempered Heterodyning


When two notes are played together on an equal-tempered instrument, the resulting heterodynes can be just as unjust as the notes themselves!

But there's a certain consistency to the scale that makes it all sound good.





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