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The undertone series is a series of notes that results from inverting the intervals of the overtone series. While the overtone series occurs naturally as a result of wave propagation and sound acoustics, musicologists such as Paul Hindemith consider the undertone series to be purely theoretical.
Hindemith, Paul (1945) . The Craft of Musical Composition. translated by Authur Mendel (revised ed.). New York: Associated Music Publishers. p. 78. “It seems to me repugnant to good sense to assume a force capable of producing such an inversion. … [The undertone series] can never have for music the same significance as the overtone series. … This “undertone series” has no influence on the color of the tone, and lacks the other natural advantages of the overtone series…”