Tobias Kunze wrote:

> well, depends on what you mean by "subharmonics".  The--original,
> I believe--meaning of the term stems from music theory (another
> sad chapter in the long history of failures in this field) and
> refers to an inverted series of overtones _below_ the fundamental,
> that is at 1/n f.  Clearly, the fact that such a spectrum is still
> harmonic as long as n has a finite limit did escape the theorists
> who came up with it.  Besides, no such spectrum exists in "nature",
> which puts them in the same category as unicorns.

I wouldn't go so far as to claim that this spectrum does not exist in nature, as I could find a
number of Gyuto monks that would disagree with you on this (and remember, every computer music
composer is required BY LAW to sample Tibetan monks at least once during his/her career).
Subharmonics can be found in a number of instruments, including wind instruments and the human
voice. The most common modes of oscillation are an octave below the pitch of the signal, and an
octave and a fifth below the pitch. In most cases, these subharmonics only appear during "noisy"
moments, and jump between the various modes. However, with the proper training, musicians can learn
how to sustain one of these subharmonic modes. In the voice, for instance, I have found that
generating the sound known as "vocal fry" while singing a note at the same time can result in an
octave drop in the perceived pitch of the tone.

Do these "subharmonics" exist as actual pitches? I don't think so, in the sense of appearing as a
sine wave below the main pitch in an FFT analysis. However, a look at the waveform of a signal that
exhibits an octave subharmonic will show something interesting: the waveform repeats on a level that
corresponds, not to every period (of the original pitch), but to every other period. In other words,
there is a regular repeating pattern, at the period of the subharmonic, that modulates the shape and
amplitude of the original signal. There is a distinct pattern that corresponds to the original
pitch, but every odd cycle (or even, depending where you start counting from ;) will display a
different shape, or a different amplitude.

As far as electronic instruments, subharmonics have been utilized since Oskar Sala's Trautonium
incorporated them around 1948 or so. In this form, the subharmonics are generated by some sort of
flip-flop, where N changes of state in the input signal results in a change of state of the output
signal. Subharmonic generators can usually be found in analog synths as suboctave generators,
outputting a signal 1 or 2 octaves below the original. The Blacet Frequency Divider uses subharmonic
generation to output signals ranging from 1/2 to 1/10 the original frequency.