What is A Subharmonic?

A subharmonic is a harmonic that is one or several harmonics below a given fundamental frequency. It is defined as an inverse of the harmonic (overtone) series, with a ratio of 1/n, where n is the harmonic^[1]. Given a fundamental frequency of 440 Hz (Middle-C), examples of its subharmonics include 220Hz (1/2), 146.67 Hz (1/3), 110 Hz (1/4), and so on.

Subharmonic Singing

Literature

Subharmonic singing is a vocal technique, more commonly used by bass singers, to produce a tone that is usually out of their lower vocal range, turning a regular bass singer into an "Oktavist",a singer whose range is an octave below a typical bass's vocal range^[2].

It is important to note that the term "subharmonic" in this sense is not a representation of the subharmonic series, but instead describes a phenomenon for when the resonant frequency being sung by an individual is lower than what is expected. However, the frequency relationships produced by a human voice utilizing this technique is still reminiscent of the harmonic series, not the subharmonic series.

In this context, "subharmonic" is used in a very loose sense as a reflection of other techniques utilizing the harmonic series, wherein a pitch is made to sound higher than usual by removing its lower partials.

How it Works

Subharmonic tones do not naturally occur in the voice, and so a common way to sing subharmonic-ally is through overtone singing, a technique where two frequencies are combined^[3]. The first subharmonic below a fundamental frequency is the inverse of the first harmonic, which is described with two frequencies interacting with a ratio of 3:2. This ratio also denotes the interval between a root note and its perfect fifth above it. The convergence of these two soundwaves amplifies every other crest of the root note, therefore bringing out the first subharmonic^[4].

Analysis

To analyze how the fundamental frequency interacts when its second harmonic is layered on top of it, I recorded the sinusoidal waves of the frequencies 110 Hz (A2), 164.81 (E3), and finally both of them played at the same time, which produced a frequency of 55 Hz (A1).

I did this analysis on Audacity and attached the images of the soundwaves below:

A2 Sinusoidal Wave

E3 Sinusoidal Wave

A2 + E3 Waves Producing A1 Frequency

I also cross-checked this analysis on Logic Pro as I could make use of its pitch detector feature (as displayed to the right of the screen). Also note that the pitch labeled in Logic Pro is an octave lower than intended. I played the exact same frequencies as shown in my analysis in Audacity from above and noted the timestamps below to show the interactions between the fundamental frequency and its second harmonic.

Logic Pro Analysis

0:02 - A0 is played

0:10 - A1 is played

0:18 - E2 is played

0:30 - A1 is played

0:32 - A1 and E2 played simultaneously, producing A0 tone