Hellenistic roots
Pythagoras, the ancient Greek mathematician, made significant contributions in many fields including politics, cosmology, and philosophy. One of his lesser-known interests was music. He developed music theory and the understanding of harmony.
Pythagoras was born in 570 BCE on the Greek island of Samos. One of the founders of Western mathematics, his work with numbers and geometric shapes is still taught in schools today.
Pythagoras discovered the mathematical relationships underlying musical intervals. He noticed that the sounds produced by plucking strings of varying lengths on a musical instrument followed a pattern. By plucking a string that was half the length of another, he observed that the resulting note was eight scale degrees higher – one octave. This observation led to the concept of octave equivalence, where notes with frequencies in a 2:1 ratio are perceived as being harmonically related.
He further discovered that the division of a string into specific proportions created measurable intervals, such as the perfect fifth (a 3:2 ratio) and the perfect fourth (a 4:3 ratio). These findings laid the foundation for Western music theory, and Western tuning systems, as they revealed the fundamental mathematical relationships that underly musical harmony. (Fifths and octaves are the intervals or gaps between notes. A jump from a C to a G moves the scale higher by five degrees).
Plato, a student of Pythagoras’ followers, was deeply inspired by these mathematical approaches to music. In his work The Republic, Plato discussed the importance of music in education and its power to shape the character of individuals and society.
Baroque
There have been numerous academic studies on the relationship between mathematics and the music of Johann Sebastian Bach (1685 – 1750). Researchers have identified various mathematical patterns, such as symmetry, proportion, and numerical sequences within Bach’s music, suggesting a deliberate integration of mathematical principles in his compositions.
Bach’s ability to write in highly structured forms highlights a systematic approach that mirrors arithmetic logic. The Well-Tempered Clavier, for instance, explores all major and minor keys, showing a geometric exploration of harmony.
A Bach chorale refers to a harmonised version of a Lutheran hymn tune, typically written for four voices: soprano, alto, tenor, and bass. Bach composed hundreds of chorales, often as part of larger cantatas and passions, but many stand independently. The leading voices are crafted with smooth melodic motion and attention to harmonic progression.
Chorales exhibit structured, repeating patterns that lend themselves to analysis as dynamic systems, offering insights into stability and change in musical forms. Bach chorales are not just beautiful examples of music but also rich, structured data sets that illuminate the interplay between art and mathematics.
Bach’s use of counterpoint, where multiple melodies are woven together in precise, interdependent relationships, is an illuminating example of how the mathematical concept of symmetry is reflected in how these melodies interact—moving in contrary motion or mirroring one another.
The fugue, a genre in which Bach excelled, showcases his mathematical rigour. The fugue is a musical composition where a theme is repeated or imitated. It is transformed through systematic processes, akin to solving an equation with multiple variables.
In Bach’s hands, mathematics and music become an intertwined marriage of technical mastery and emotional richness. This is what makes his music enduringly powerful, bridging the worlds of abstract numbers and human experience.
Minimalism
In the modern era, Music in Fifths (1969) by Philip Glass is a direct example of Pythagoras’s ratios. The title refers to the intervals of perfect fifths that dominate the composition. Glass builds melody and harmony almost entirely using these intervals.
Glass employs what he calls ‘additive structure’, where phrases gradually expand or contract by adding or subtracting a single note or beat. The steady, unrelenting tempo and incremental changes create a hypnotic, meditative effect. If the initial phrase has 4 notes, the next iteration might have 5 notes, then 6 notes, etc., forming a predictable growth pattern. This is conceptually similar to fractals or iterative functions in maths, where simple inputs generate intricate outcomes.
Glass’s use of systematic processes mirrors algorithms in computer science. The incremental process embodies the mathematical concept of progression. Musicologist K. Robert Schwarz, in Minimalists, discusses Glass’s use of ‘reductive logic,’ where arithmetic drives the evolution of his pieces.
The concept of polyrhythms involves layering rhythms with different time signatures (e.g., 3:2 or 4:3). These directly correspond to mathematical ratios. For example, Steve Reich’s Clapping Music is based on phasing patterns, where one rhythm gradually shifts out of sync with another. This creates a mathematically predictable pattern of alignment and misalignment. Reich describes the mathematical basis of phasing in his book Writings on Music.
Fibonacci patterns
Studies have shown that some composers incorporate the Golden Ratio, a mathematical proportion often found in nature and art, into their compositions, lending support to the music’s appeal and beauty.
The ratio is derived from the Fibonacci sequence of numbers, when the next term is determined by adding the previous two (0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on). The pattern is seen throughout nature – petal patterns, veins on a leaf, branches of trees, and the circular designs on shells.
Scholars like David Yearsley in Bach and the Meanings of Counterpoint argue that Bach’s work reflects mathematical rigour, but there is no direct proof of Fibonacci sequences in his music.
Composers such as Claude Debussy (1862 – 1918) and Béla Bartók (1881 – 1945) did directly use the golden ratio in their music. Bartók’s Music for Strings, Percussion, and Celesta has structured sections based on Fibonacci numbers, such as the climax at the 0.618 point of the piece, approximating the golden ratio. Ernö Lendvai’s book, Béla Bartók: An Analysis of his Music, provides detailed analysis of Fibonacci structures.
Iannis Xenakis (1922 – 2001) was one of the most innovative composers of the twentieth century. He applied mathematical processes and algorithmic influences to his compositions. His piece Metastaseis uses transformations like the Fibonacci sequence and probability theory to determine the placement of notes. Xenakis wrote about his methods in Formalised Music: Thought and Mathematics in Composition.