Igor Stravinsky once accurately described the close connection between mathematics and music: “Musical form is close to mathematics—not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationship.” Indeed, in musical cultures of different times and places, mathematics always underlies music in various ways. To illustrate this phenomenon, this article will focus on the mathematics involved in the overtone series (also known as the harmonics series) and examine its presence in musical cultures around the world.
Overtone series, which refers to a series of frequencies, suggests that each musical note heard is a combination of overtones produced simultaneously when it is played, instead of a single frequency. Burdette L. Green, who earned a Ph.D. at Graduate School of Ohio State University, provided an explanation: “...the complex wave forms emitted by most musical instruments result from a set of frequencies, each component of which is an integral multiple of the fundamental frequency” [1]. For example, suppose that the fundamental frequency (i.e. the lowest frequency of a periodic wave that determines the pitch) is 131 Hz, which is the frequency of C3 played on a string instrument with two fixed ends. The first overtone has a frequency of 262 Hz, which gives a note an octave above C3. By multiplying the fundamental frequency by 3, 4, 5, and 6, the frequency of 2nd, 3rd, 4th, and 5th overtones can be obtained, each corresponding with one note [2]. The list of overtones goes on infinitely with the interval (i.e. the distance between two tones) between two notes getting smaller and smaller.
When the note is played, the sound heard is a combination of its overtones, which forms a complex wave pattern. This results in a more complex and richer sound than what is heard when the string vibrates only at the fundamental frequency. As the number of overtones increases, each overtone gets softer, and therefore, human ears can only hear the blending sound of the fundamental frequency and the first several overtones. However, although C3 has always had this series of overtones, it has a different tone colour (i.e. timbre or perceived sound quality) when played on different instruments. This is because the relative loudness of individual overtones and the resulting combination of overtones vary among different instruments [3]. While overtones are usually drowned out by the fundamental frequency for most instruments, the overtone of an electrical bass, for example, sounds louder than the fundamental. Therefore, overtones play an essential role in determining the timbre of musical instruments.
The overtone series forms the basis of many musical cultures around the world, as a wide range of instruments, including strings, woodwinds, and even human voices, rely on overtone. For example, in Chinese Qin music, the harmonics of Qin (a traditional seven-string instrument) is called fan yin, which means “floating sounds.” It generates an ethereal sound and is believed to represent heaven, bringing “cosmic harmony” and “deep peacefulness” to listeners [4]. As one of the three timbres that Qin can produce, fan yin enhances the overall expressive effect of Qin, contrasting with the other two timbres when used together in monophonic Qin music. For instance, in Plum Blossoms in Three Movements, fan yin is used every time the theme enters, which contrasts with san yin (also known as “scattered sound” representing earth) that produces detached notes in the episodes [5]. This creates a cheerful atmosphere through the depiction of plum flowers blossoming in cold winds, symbolizing high moral integrity and noble character.
Harmonics played by string instruments are not only common in Chinese Qin music but are also evident in the tanpura of Indian music. The tanpura, a long-neck string instrument that is plucked, usually accompanies the Indian vocalist by playing continuous drones which are described as “melodic background” for the performance by Datta et al. [6]. Its overtone-rich sounds support the singing and make the vocalist sound more powerful by creating an underlying metallic, buzzing sound, giving Indian Classical music its unique style.
Apart from string instruments, woodwind instruments, such as the didgeridoo, a typical instrument in Australian aboriginal music, also exploit harmonic series. The didgeridoo, possibly the oldest instrument in the world with a history of 40,000 years, is an instrument made from about 1.3-meter long eucalyptus branches hollowed by nesting termites [7]. Similar to the Indian tanpura, it usually produces resonating, droning sounds to accompany chanting or singing. Played with vibrating lips and circular breathing, the didgeridoo can produce a fundamental note with rich harmonics. Further, it is often used to imitate animal sounds such as those of kangaroos, dingoes, and owls, since the connection with the land and their ancestors always remains essential in aboriginal culture. According to Robert Lawlor, “For the Aborigine, the observation of nature immediately requires a state of empathy, which leads to an imitative expression” [8].
Finally, the overtone series is used extensively in several aspects of singing in many musical cultures. An example of this use of the overtone series is Tuvan throat singing. Specifically, the performer not only sings a fundamental note but also produces its related overtones. Tuvan throat singing is usually sung in an open landscape, which carries the penetrating sound for a long distance. Surprisingly, it shares similarities with the Australian didgeridoo, because throat singing conjures natural imagery such as mountains, waterfalls, and grasslands, thereby allowing people to communicate with the cosmos and to achieve harmony between man and nature [9].
In conclusion, mathematics plays a fundamental role in producing music, as demonstrated by the reliance on overtone series in many musical cultures. While the use of the overtone series is widespread, the timbre created by the overtones of different types of instruments endow each musical culture with its own unique characteristics.
[1] Green, Burdette. "The Harmonic Series from Mersenne to Rameau : an historical study of circumstances leading to its recognition and application to music /." Doctoral dissertation, Ohio State University, 1970. Accessed August 31, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486653137096899.
[2] Suits, Bryan H. “Overtone Series.” pages.mtu.edu, 1998. Accessed August 31, 2021. https://pages.mtu.edu/~suits/overtone.html.
[3] Pouska, Andrew. “The Harmonic Series and Timbre.” StudyBass. Accessed August 31, 2021. https://www.studybass.com/gear/bass-tone-and-eq/the-harmonic-series-and-timbre/.
[4] Kouwenhoven, Frank. “Meaning and Structure ‐ the Case of Chinese qin(Zither) Music.” British Journal of Ethnomusicology 10, no. 1 (January 2001): 39–62. https://doi.org/10.1080/09681220108567309.
[5] Zi De Guqin Studio. “【古琴Guqin】《梅花三弄》Famous Traditional Chinese Music Depicting Plum Blossoms.” www.youtube.com, November 3, 2018. https://www.youtube.com/watch?v=MvxbfpuccXo&ab_channel=%E8%87%AA%E5%BE%97%E7%90%B4%E7%A4%BEZiDeGuqinStudio.
[6] Asoke Kumar Datta, Ranjan Sengupta, Kaushik Banerjee, and Dipak Ghosh. Acoustical Analysis of the Tanpura Indian Plucked String Instrument. Singapore Springer, 2019.
[7] Block, A. J. “What Is a Didgeridoo (the Droning Aboriginal Australian Wind Instrument)?” Didge Project, October 28, 2015. Accessed Aug 31, 2021. https://www.didgeproject.com/free-didgeridoo-lessons/what-is-a-didgeridoo/.
[8] Lawlor, Robert. Voices of the First Day : Awakening in the Aboriginal Dreamtime. Rochester, Vt., 1991.
[9] Vincenzi, Guilia. “Sounds of Nature: Throat Singers of Tuva - Pilot Guides - Travel, Explore, Learn.” Pilot Guides. Accessed August 31, 2021. https://www.pilotguides.com/articles/throat-singers-of-tuva/.