
Heterodyne
Problems?


Heterodyning

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 outoftune 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, wholenumber ratio.
For example, the musical fifth ratio is
3:2 and the musical fourth ratio is 4:3. The majorthird ratio is
5:4 and the minorthird 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 suboctave 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 equaltempered 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.
Equaltempered 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 equaltempered 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.



