Chapter 6 Further Reading: Overtones & the Harmonic Series

Foundational Texts

Helmholtz, Hermann von. On the Sensations of Tone as a Physiological Basis for the Theory of Music (1863; English translation by Alexander Ellis, 1875) The foundational scientific text on acoustics and music perception. Helmholtz's chapters on combination tones, timbre, and the relationship between consonance and the harmonic series remain essential reading. The Ellis translation includes extensive notes that update Helmholtz's findings to the mid-19th century.

Rameau, Jean-Philippe. Treatise on Harmony (1722; English translation by Philip Gossett, 1971) Rameau's systematic derivation of harmonic theory from the harmonic series. Essential primary source for understanding how Western music theory connected physics and aesthetics. Gossett's translation includes helpful commentary.

Mersenne, Marin. Harmonie Universelle (1636) The foundational text of quantitative acoustics. Available in partial translation. Mersenne's descriptions of string vibration, his measurements of absolute pitch, and his early account of the harmonic series as a physical phenomenon are documented here.

Accessible Books for the Curious Reader

Fauvel, John, Raymond Flood, and Robin Wilson (eds.). Music and Mathematics: From Pythagoras to Fractals (Oxford University Press, 2003) An accessible collection of essays exploring the mathematical relationships in music, including extended discussions of the harmonic series, Pythagorean tuning, and the connections between music and mathematics.

Levitin, Daniel J. This Is Your Brain on Music (Dutton, 2006) An accessible account of the neuroscience and psychology of music perception, with clear explanations of harmonic series, timbre, and consonance from a cognitive science perspective.

Roederer, Juan G. The Physics and Psychophysics of Music: An Introduction (4th ed., Springer, 2008) A rigorous but accessible textbook covering the acoustic physics of musical sound, the physiology of the auditory system, and psychoacoustics. Excellent chapters on harmonics, timbre, and combination tones.

Sethares, William A. Tuning, Timbre, Spectrum, Scale (2nd ed., Springer, 2005) A deep exploration of the relationship between inharmonic timbres and non-Western tuning systems. Shows how the harmonic series assumption underlies Western tuning and how different spectral structures lead naturally to different musical scales.

On Pythagorean Tuning and Historical Acoustics

Godwin, Joscelyn. The Harmony of the Spheres: The Pythagorean Tradition in Music (Inner Traditions, 1993) A history of the Pythagorean tradition in music, tracing how integer ratio theory influenced Western music from ancient Greece through the Renaissance and beyond.

Lindley, Mark. Lutes, Viols and Temperaments (Cambridge University Press, 1984) A specialized but rewarding study of historical tuning practices and how temperament decisions affected Renaissance and Baroque music. Shows concretely how physics and culture interacted in shaping musical sound.

Barbour, J. Murray. Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951; reprint Dover, 2004) The standard reference work on Western tuning history. Covers Pythagorean tuning, meantone systems, well temperament, and equal temperament in mathematical detail.

On Inharmonicity

Fletcher, Harvey, E. Donnell Blackham, and Richard Stratton. "Quality of Piano Tones." Journal of the Acoustical Society of America 34:6 (1962): 749–761. The classic study of piano string inharmonicity and its effect on the characteristic sound of the piano. Documents the stretched octave phenomenon quantitatively.

Conklin, H. A. "Design and Tone in the Mechanoacoustic Piano." Journal of the Acoustical Society of America 99:6 (1996): 3286–3296. A detailed analysis of how piano design decisions (string length, stiffness, bridge coupling) interact to produce inharmonicity and the characteristic piano sound.

On the Voice and Formants

Sundberg, Johan. The Science of the Singing Voice (Northern Illinois University Press, 1987) The definitive scientific account of vocal acoustics, covering formants, the singer's formant, voice registers, and the interaction between vocal fold vibration and vocal tract resonance.

Ladefoged, Peter, and Ian Maddieson. The Sounds of the World's Languages (Blackwell, 1996) A comprehensive phonetic survey covering vowel formant patterns across languages, with spectrogram illustrations.

On Cross-Cultural Perspectives

Nettl, Bruno. The Study of Ethnomusicology: Thirty-Three Discussions (3rd ed., University of Illinois Press, 2015) An authoritative introduction to ethnomusicology that includes nuanced discussions of what is and is not universal in musical perception, including the question of octave equivalence.

Jacoby, Nori, et al. "Universal and Non-Universal Features of Musical Pitch Perception Revealed by Singing." Current Biology 29:19 (2019): 3229–3243. A landmark cross-cultural study examining rhythmic and pitch perception across diverse musical traditions, using a method (iterated reproduction) that allows quantitative comparison. Relevant to the question of whether harmonic series perception is universal.

Online Resources

PhET Interactive Simulations (University of Colorado Boulder) — "Wave on a String" https://phet.colorado.edu/en/simulations/wave-on-a-string Interactive simulation demonstrating standing waves, nodes, and harmonic modes.

Acoustics of Musical Instruments — National Museum of Scotland Online exhibition with audio examples demonstrating the harmonic series in different instruments.

Overtone singing demonstrations — YouTube Search for recordings of Tuvan throat singing (khoomei), Mongolian overtone singing, or Tibetan Buddhist chanting to hear the harmonic series exploited as a direct musical resource.

Praat: Doing Phonetics by Computer (free software) www.fon.hum.uva.nl/praat/ Professional-grade spectrogram and acoustic analysis software used by linguists and vocal scientists. Allows you to produce spectrograms of your own voice.