Popular music and mathematics

SUMMARY: Popular music can be analyzed and enhanced by mathematical techniques, and to some degree, the popularity of music can be predicted mathematically. 

The interaction between mathematics and popular music goes far beyond the popularity of numbers in song titles, like Tennessee Ernie Ford’s “16 Tons” or 2gether’s “U + Me = Us (Calculus).” Mathematics is fundamental to musical theory and composition. The twentieth-century subgenres math rock and mathcore are perhaps the most explicitly mathematical compositions, but there are also songs about mathematics concepts. These are usually intended to be humorous or educational, such as “That’s Mathematics” by mathematician and musician Thomas Lehrer. Mathematics is also increasingly important to recording and analyzing popular music, including its potential effects on learning. Experimental electronic artist Jamal Moss, founder of record label Mathematics, noted: “Mathematics is the body of sound knowledge centered on such concepts as quantity, music, structure, space, and change—and also the academic discipline that studies them.” 

Mathematics in popular music reflects society’s often polarized opinions on mathematics. For example, Jimmy Buffett’s song “Math Suks” expressed the singer’s feelings about the difficulty of mathematical concepts like fractions, algebra, and geometry. Other singers and groups embraced mathematics, like the Texas indie rock band named “I Love Math.” Mathematics is often found in album cover art. British band Coldplay’s 2005 X & Y album featured a cover with colored blocks that spell out “X and Y” in the binary code developed in 1870 by Emile Baudot for use with telegraph systems. Coldplay’s lead guitarist Jonny Buckland studied astronomy and mathematics at University College London. Some artists have been criticized for incorrectly using mathematics. Pink Floyd’s very popular 1973 album Dark Side of the Moon features cover art showing a prism and spectrum. It was correct in depicting some facts, like violet light refracting the most and red the least, but other aspects were not accurate, such as the relative dispersion of the different colors. Mariah Carey’s 2009 album titled E=MC2 borrowed from Albert Einstein’s well-known theory of relativity. 

Avant-garde composer Iannis Xenakis and post-rock subgenres math rock and mathcore are prominent examples of popular music that rely heavily on mathematics. Xenakis was one of the most significant avant-garde composers of the twentieth century and a grandfather of modern electronic music. His work incorporated mathematical models, such as probability theory, stochastic processes, group theory, set theory, game theory, and Markov chains. He developed algorithms to produce computer-generated music using probability theory and stochastic functions in the 1960s. In his 1966 cello solo “Nomos Alpha,” he divided the twenty-four sections of the piece into two layers. The first layer, consisting of every section not divisible by four, was determined by the twenty-four orientation-preserving elements of the octahedral group, while the second layer was a more traditional structure. The work was compared to a musical kaleidoscope, and its structure likened to a fractal. 

In the 1990s, post-rock like Slint’s Spiderland became a dominant genre in experimental rock. Critic Simon Reynolds coined the term “math rock” to describe music that “uses rock instrumentation for non-rock purposes, using guitars as facilitators of timbre and textures rather than riffs and power chords.” Math rock bands began to explore the use of dramatically alternating dynamic shifts and unusual time signatures and dissonance, and songs tended to avoid the verse-chorus-verse structure of pop songs. Mathcore developed largely independently of math rock, growing out of hardcore punk and extreme metal, with a huge debt to hardcore pioneers Black Flag. 

Mathematics Songs

In the early twenty-first century, the website "M A S S I V E: Math And Science Song Information, Viewable Everywhere" was part of the National Science Digital Library and contained over 2,800 mathematical and scientific songs. Popular YouTube songs include mathematical raps and parodies, such as “I Will Derive.” Hard ‘n Phirm’s song “Pi” rose in popularity because of the 2005 music video by award-winning director Keith Schofield. Some songs help students learn mathematics concepts like multiplication. Other songs showcase the mathematicians who loved to sing. The Klein Four Group is a Northwestern University a cappella group who sing about undergraduate and graduate level mathematics. They are most known for their song “Finite Simple Group (of Order Two).” 

Self-proclaimed “mathemusician” Lawrence Lesser wrote educational songs to increase mathematics awareness. Educators often incorporate mathematics songs into their classrooms to enhance student learning of specific concepts, and many students use music of various kinds to help them focus while they study mathematical concepts. However, these effects have not yet been definitively supported nor refuted. One study that investigated using jingles to teach statistics concepts found students who sang several jingles versus reading aloud definitions for the same concepts performed better as a group on a follow-up test. On the other hand, a study that compared classical, popular, and no music to enhance learning found that the students in the three groups performed no differently on a mathematics placement test. This matched findings regarding the effect of music on other academic areas. 

Audio Processing

While music production techniques have allowed a certain amount of alteration and error correction by adjusting the relative levels and balance of the recorded elements, twenty-first century software capabilities progressed to the point where lower-quality vocals could be processed to professional-sounding quality. 

The software package most associated with this was Auto-Tunereleased in 1997 and developed by Exxon engineer Harold “Dr. Andy” Hildebrand. Hildebrand applied seismic data interpretation methods to the analysis and modification of musical pitch. Auto-Tune was an enhancement of existing phase vocoder technology, which used short-time Fourier transformsnamed after mathematician Joseph Fourierto convert time domain representations of sound into time-frequency representations that could be modified before being converted back. Extreme changes could leave tell-tale artifacts in recordings, in the form of a warble like a degenerating audiocassette tape. Audio processing became standard in many pop albums and on television shows, such as Glee. Some well-established singers regularly use Auto-Tune for both albums and in live performances. Other musicians have refused to do so out of fear that it would change the sound enough to make them unrecognizable. 

In the twenty-first century, Platinum Blue and Music Intelligence Solutions specialize in mathematically predicting hit songs, while services like iTunes and Spotify create suggested playlists or make recommendations. Platinum Blue CEO Mike McCready explained that he and others discovered mathematical patterns in hit songs while trying to build an automated recommendation platform. The algorithm his company used was based on roughly thirty song traits that were quantified mathematically, such as melody, harmony, beat, tempo, and rhythm. These traits were analyzed for patterns, resulting in groups of songs that were ranked according to probability of success. Hit songs tended to have identifiable similarities, but falling into a particular category was not a guarantee of success. For example, lyrics were an influential song component that were not reliably quantifiable, and aggressive marketing could have an effect not captured by the algorithm. McCready noted, “We figured out that having these optimal mathematical patterns seemed to be a necessary, but not sufficient, condition for having a hit song.” 

Music and Artificial Intelligence

In the 2020s, mathematics in the form of artificial intelligence (AI) revolutionized music production in profound and nontraditional ways. While innovation is a natural and celebrated aspect of music, the prospect of machine-originated artistry has many grappling to understand how it fits into a historical understanding of music. Questions abound whether only human-derived creations can be considered “true” music. Similar to AI’s impact in other areas, matters of ethics seeped to the forefront. For example, AI has the ability to learn, mimic, and reproduce the creative artistry of humans in ways akin to counterfeit productions. Many are calling for ethical guidelines to assist in the development of AI before the technology veers into uncontrollable pathways.

In March 2024, one of the first attempts to establish legal boundaries on AI-generated music occurred in Tennessee. In deference to Elvis Presley (1935–77), one of the first American music artists to attain mass popularity, Governor Bill Lee announced the passage of the ELVIS Act (Ensuring Likeness Voice and Image Security). The act sought to prevent AI users from morphing the voices of human artists into AI-generated creations. These phony productions are made possible by human programmers first collecting large volumes of the musical works of actual artists. AI software analyzes the musical content for patterns that can then be replicated in fraudulent productions.

Bibliography

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Hight, Jewly. “AI Music Isn’t Going Away. Here Are 4 Big Questions about What’s Next.” NPR, 25 Apr. 2024. Accessed 3 Oct. 2024.

Lesser, Lawrence. “Sum of Songs: Making Mathematics Less Monotone!” Mathematics Teacher, vol. 93, no. 5, 2000, pubs.nctm.org/view/journals/mt/93/5/article-p372.xml. Accessed 3 Oct. 2024.

Lynch, Peter. "Tom Lehrer: A Comical, Musical, Mathematical Genius." The Irish Times, 17 Sept. 2018, www.irishtimes.com/news/science/tom-lehrer-a-comical-musical-mathematical-genius-1.3632032. Accessed 3 Oct. 2024.

Roeloffs, Mary Whitfill. “Artists Slam AI Developers for Using Music without Permission in Letter Signed by Kacey Musgraves, Billie Eilish and More.” Forbes, 2 Apr. 2024, www.forbes.com/sites/maryroeloffs/2024/04/02/artists-slam-ai-developers-for-using-music-without-permission-in-letter-signed-by-kacey-musgraves-billie-eilish-and-more/?sh=6bca8ea33d97. Accessed 3 Oct. 2024.

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