Music exists at the sublime confluence of mathematics and artistry, embodying a synergy that elevates both disciplines. The essence of music theory, particularly in the analysis of chords, mirrors an advanced form of modular arithmetic, showcasing a mathematical elegance. Musical scores serve as encoded scripts of melodies, awaiting decryption and performance. Particularly, Johann Sebastian Bach epitomizes this blend of mathematical intricacy and artistic beauty, standing as a paragon of the fusion between the two realms.
An enlightening post by Elise Cutts in Scientific American unveils how physicists have discerned mathematical patterns within Bach’s compositions, patterns that enrich the conveyance of information. Through a novel representation of musical scores as networks—comprising nodes (dots) interconnected by edges (lines)—this research quantifies the informational content across hundreds of Bach’s works.
By representing scores as simple networks of dots, called nodes, connected by lines, called edges, scientists quantified the information conveyed by hundreds of Bach’s compositions.
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So Kulkarni boiled down 337 Bach compositions into webs of interconnected nodes and calculated the information entropy of the resulting networks. In these networks, each note of the original score is a node, and each transition between notes is an edge. For example, if a piece included an E note followed by a C and a G played together, the node representing E would be connected to the nodes representing C and G.
Networks of note transitions in Bach’s music packed more of an informational punch than randomly generated networks of the same size—the result of greater variation in the networks’ nodal degrees, or the number of edges connected to each node. Additionally, the scientists uncovered variation in the information structure and content of Bach’s many compositional styles. Chorales, a type of hymn meant to be sung, yielded networks that were relatively sparse in information, though still more information-rich than randomly generated networks of the same size. Toccatas and preludes, musical styles that are often written for keyboard instruments such as the organ, harpsichord and piano, had higher information entropy.
The original paper Information content of note transitions in the music of J. S. Bach is out in Phys. Rev. Research’s on Feb. 6th, 2024. As stated in the paper, “our simple framework serves as a stepping stone for exploring further musical complexities, creativity, and questions therein”. By integrating these insights with emerging theories, we stand on the cusp of exploring musical creativity and complexity in unprecedented ways, opening new avenues for understanding the interplay between music, mathematics, and perhaps, information design.