Music and Mathematics: A Pythagorean Perspective

Edel Sanders

Chair of Psychology Department at UNYP

I would give the children music, 

physics and philosophy, but the most important is music, for in the patterns of the arts are the keys to all learning.

Plato, c. 428-347 BC

The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.”  He can also be considered the “father of harmony,” given that his discovery of the overtone series and analyses of the acoustics and ratios involved in music have served as the foundation of harmony in western-hemisphere music composition ever since.  The Pythagorean, Quadrivium and Platonic classifications of mathematics were based on hierarchical dimensions, starting with arithmetic, then geometry, astronomy and finally music. 

Reportedly, Pythagoras experimented with the tones produced when plucking strings of different lengths.  He found that some specific ratios of string lengths created pleasing combinations (“harmonies”) and others did not.  Based on his careful observations, Pythagoras identified the physics of intervals, or distances between notes, that form the primary harmonic system which is still used today (Parker, 2009, pp. 3-5). 

Music is based on proportional relationships.  The mathematical structure of harmonic sound begins with a single naturally occurring tone, which contains within it a series of additional frequencies above its fundamental frequency (“overtones”), of which we are normally unaware on a conscious level.  Within this harmonic or overtone series, there is a mathematical relationship between the frequencies – they are specific integer multiples of each other.  For example, if the slowest frequency (the “fundamental”) were 100 Hz, then the overtones would be 2 x 100 (200 Hz), 3 x 100 (300 Hz) and so forth.  (The overtone series is often referred to as harmonics.)

Pythagoras observed several ratios of sound wave frequencies and the corresponding intervals between them, including 4:3 (known to musicians as the interval of a perfect fourth, or two pitches that are five semitones apart from each other) and 3:2 (a perfect fifth, seven semitones apart).  Note that pitch is the frequency or rate of vibration of a physical source such as a plucked string. 

The most prominent interval that Pythagoras observed highlights the universality of his findings.  The ratio of 2:1 is known as the octave (8 tones apart within a musical scale). 

When the frequency of one tone is twice the rate of another, the first tone is said to be an octave higher than the second tone, yet interestingly the tones are often perceived as being almost identical.  For example, a woman’s voice may fluctuate around 220 Hertz while a man’s voice is around 110 Hertz, approximately half the frequency of the woman’s.  However, if they sing together, it may sound as though they are singing the same melody together in unison, even though they are actually an octave apart.  This 2:1 ratio is so elemental to what humans consider to be music, that the octave is the basis of all musical systems that have been documented – despite the diversity of musical cultures around the world.  Moreover, this physical phenomenon is so fundamental that even non-human species such as monkeys and cats recognize it (Levitin, 2008, p. 31).

 The inherent properties of physics and mathematics within music, perceived so long ago by Pythagoras, may help to explain why many physicists and mathematicians are also musicians.  This point is illustrated by a quotation from Einstein:  “The theory of relativity occurred to me by intuition, and music was the driving force behind that intuition….My new discovery was the result of musical perception” (Suzuki, 1969, 90).

*Portions of this article are adapted from Edel Sander’s chapter in Musik i forskola och tidiga skolar (2015), a Swedish textbook for music educators.