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undertone series
In music, the undertone 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. The overtone series being based on harmonic division, the undertone series is based on arithmetic division.〔Nattiez, Jean-Jacques (1990) (). ''Music and Discourse: Toward a Semiology of Music'' (''Musicologie générale et sémiologue''). Translated by Carolyn Abbate (1990). Princeton, N.J.: Princeton University Press. p. 202. ISBN 0-691-02714-5. Undertone series shown on E.〕
==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 doubled in the opposite ratios, the undertones series is produced. Similarly, on a wind instrument, if the holes are equally spaced, each successive hole covered will produce the next note in the undertone series.
In addition, José Sotorrio showed that undertones could be made through the use of a simple oscillator such as a tuning fork. If that oscillator is gently forced to vibrate against a sheet of paper "it will naturally make contact at various audible modes of vibration."〔Sotorrio, José A (2002). ''Tone Spectra -and the Natural Elements of Music''. (1st Ed) Spectral Music, 2002.〕 Sotorrio explained that since the tuning fork produces a sine tone, it will normally vibrate at the fundamental frequency (e.g. 440 Hz), but "momentarily", it will make contact only at every other oscillation (220 Hz), or at every third oscillation (147 Hz), and so on. This produces audible "subharmonic spectra", (e.g. below A@440 Hz (1/1), A@220 Hz (1/2), D@147 Hz(1/3), A@110 Hz(1/4), F@88 Hz(1/5), and so on). Sotorrio claims it is possible to sustain these "subspectra" using a sine wave generator through a speaker cone making contact with a flexible (flappable) surface, and also on string instruments "through skillful manipulation of the bow", but that this rarely sustains noticeably beyond the "sub-octave or twelfth".〔
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