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. 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 (, , , , etc.). When the subharmonic series is used to refer to frequency relationships, it is written with f representing some highest known reference frequency (, , , , 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.