Subharmonic singing

Hi, I post this here in the hopes that people will read up on this technique and start writing music using it for soloists or specialized choirs. I have been practicing this technique for a while and while I am nowhere close to being able to do it reliably (in performance) I have gotten to the point where I can produce notes down to A1. By the way, I am a tenor with a really high voice, normally I cannot sing below F3 and when my voice is very warm I can only sing down to A3 with reasonably good tone.

How is this possible? It is due to a special quirk of the vocal cords and the way the cochlea converts vibrations into pitches. I do not know exactly what is happening in the vocal cords when using the subharmonic technique, but it essentially boils down to forcing the vocal cords to vibrate out of sync (when you sing in normal chest voice they vibrate at the same rate). By getting them to vibrate in the ratio of 3:2, i.e. 3 vibrations of one vocal cord for every 2 of the other, a tone an octave below the tone "being sung" is produced. This is due to the fact that the cochlea tries to interpret all compound sound waves (i.e. intervals) as belonging to the same harmonic series. If the intervals are tuned in such a way that their ratio is made up of small integers, the cochlea literally fills in the rest of the series. It is able to complete the calculations most effectively when the ratio is closest to 1:1, or for those tones whose relationship appears very early on in the overtone series - furthermore, the ratio of 3:2, the perfect fifth, is the most effective interval for this. It is worth mentioning that the new tone which is produced is the difference between the frequencies. For example, 300 Hz and 200 Hz, which is a perfect fifth (because the frequencies are in the ratio of 3:2), produce the tone of 100 Hz - literally 300-200=100, hence the name difference tone that is normally given to this phenomenon. These calculations happen between all intervals, harmonics and (I believe) also between the difference tones and "real" sound waves, but the tones which are most audible are those which are plausible fundamentals in an imaginary harmonic series. Difference tones have not been the subject of much study, despite being familiar to musicians since who knows when (it was known as far back as the early Baroque at the very least), so I do not know exactly why they are sometimes perceived as "real tones" and sometimes only as auxiliary tones like harmonics (a question which is driving me mad). Despite being claimed to be purely a psychoacoustic phenomenon, I believe they are measurable by special instruments and I can personally attest to the fact that the pressure of the vibrations is real (it is most noticeable when I experiment with difference tones on the viola, where producing difference tones approaching 0 Hz causes the bow to shake somewhat violently). By the way, this is a principle that organ manufacturers have used for hundreds of years to achieve "impossible tones" that require pipes much larger than are actually present. You can actually hear the separate pipes sometimes in old organs whose pipes are slowly deteriorating.

Back to subharmonic singing, you can force your vocal cords into a false resonance in many different ratios and as I said before the difference of the frequencies is what determines the resultant pitch. There is theoretically no limit to how deep you can sing, except that it becomes more challenging and requires finer vocal control as the ratios get smaller. Rest assured, if I as a hobbyist can extend my range by a fifth easily, an extension of at least an octave should be perfectly reasonable (although if writing for specific singers one should of course consult them first). Here is an example of what it sounds like:

For those of you who are more interested in string writing, this technique also exists on string instruments but it is much more difficult. I think it will be some time before it is part of the standard technique, but I fully believe it will be the hallmark of string writing in our century. There is a Japanese violinist who has mastered the technique and has written a few works using it: