Undertone series

In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones must be produced in unusual ways. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division.

Terminology The hybrid term subharmonic is used in music in a few different ways. In its pure sense, the term subharmonic refers strictly to any member of the subharmonic series (1⁄1, 1⁄2, 1⁄3, 1⁄4, etc.). When the subharmonic series is used to refer to frequency relationships, it is written with f representing some highest known reference frequency (f⁄1, f⁄2, f⁄3, f⁄4, etc.). As such, one way to define subharmonics is that they are "integral submultiples of the fundamental (driving) frequency". The complex tones of acoustic instruments do not produce partials that resemble the subharmonic series, unless they are played or designed to induce non-linearity. However, such tones can be produced artificially with audio software and electronics. Subharmonics can be contrasted with harmonics. While harmonics can "occur in any linear system", there are "only fairly restricted conditions" that will lead to the "nonlinear phenomenon known as subharmonic generation". In a second sense, subharmonic does not relate to the subharmonic series, but instead describes an instrumental technique for lowering the pitch of an acoustic instrument below what would be expected for the resonant frequency of that instrument, such as a violin string that is driven and damped by increased bow pressure to produce a fundamental frequency lower than the normal pitch of the same open string. The human voice can also be forced into a similar driven resonance, also called "undertone singing" (which similarly has nothing to do with the undertone series), to extend the range of the voice below what is normally available. However, the frequency relationships of the component partials of the tone produced by the acoustic instrument or voice played in such a way still resemble the harmonic series, not the subharmonic series. In this sense, subharmonic is a term created by reflection from the second sense of the term harmonic, which in that sense refers to an instrumental technique for making an instrument's pitch seem higher than normal by eliminating some lower partials by damping the resonator at the antinodes of vibration of those partials (such as placing a finger lightly on a string at certain locations). In a very loose third sense, subharmonic is sometimes used or misused to represent any frequency lower than some other known frequency or frequencies, no matter what the frequency relationship is between those frequencies and no matter the method of production.

Methods for producing an undertone series The overtone series can be produced physically in two ways – either by overblowing a wind instrument, or by dividing a monochord string. If a monochord string is lightly damped at the halfway point, then at 1⁄3, then 1⁄4, 1⁄5, etc., then the string will produce the overtone series, which includes the major triad. If instead, the length of the string is multiplied in the opposite ratios, the undertones series is produced. Vocal subharmonics or subharmonic singing is a vocal technique that lets singers produce notes below the fundamental and follows the undertone series. It can extend down from the regular vocal range an octave and further below when well controlled. It can be described as having a stable vocal fry-like sound. These pitches are produced by a combination of oscillations of turbulent airflow in the vocal tract. Coming from multiple sound sources such as the true and false vocal cords. Singers often describe it as feeling like stable points below regularly sung notes where it snaps or jumps specific intervals. This technique might also happen by accident when talking or singing in a fry voice. String quartets by composers George Crumb and Daniel James Wolf, as well as works by violinist and composer Mari Kimura, include undertones, "produced by bowing with great pressure to create pitches below the lowest open string on the instrument." These require string instrument players to bow with sufficient pressure that the strings vibrate in a manner causing the sound waves to modulate and demodulate by the instrument's resonating horn with frequencies corresponding to subharmonics. The tritare, a guitar with Y-shaped strings, cause subharmonics too. This can also be achieved by the extended technique of crossing two strings as some experimental jazz guitarists have developed. Also third bridge preparations on guitars cause timbres consisting of sets of high pitched overtones combined with a subharmonic resonant tone of the unplugged part of the string. Subharmonics can be produced by signal amplification through loudspeakers. They are also a common effect in both digital and analog signal processing. Octave effect processors synthesize a subharmonic tone at a fixed interval to the input. Subharmonic synthesizer systems used in audio production and mastering work on the same principle. By a similar token, analog synthesizers such as the Serge synthesizer and many modern Eurorack synthesizers can produce undertone series as a side effect of the solid state timing circuits (e.g. the 555 timer IC) in their envelope generators not being able to re-trigger until their cycle is complete. As an example, sending a clock of period N into an envelope generator where the sum of the rise and fall time is greater than 2 N and less than 3 N would result in an output waveform that tracks at 1⁄3 of the frequency of the input clock.

Comparison to the overtone series

Subharmonic frequencies are frequencies below the fundamental frequency of an oscillator in a ratio of 1/n, with n a positive integer. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz (1⁄2), ~146.6 Hz (1⁄3) and 110 Hz (1⁄4). Thus, they are a mirror image of the harmonic series, the overtone series.

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