Chapter 1: The Interplay of Math and Music

Mathematics and music may seem to occupy distinct spaces in our society, yet they are intrinsically linked in ways that are often overlooked. Beneath the art of music lies a precise science, where mathematical concepts such as sequences, progressions, and averages come into play. This remarkable relationship between these two fields has deep historical roots and continues to inspire curiosity and inquiry. Many renowned mathematicians have also been passionate musicians, a connection dating back to ancient Greek civilizations, where significant relationships between music and math were first recognized. Various tuning systems emerged over time, ultimately leading to the widespread adoption of octaves, which align with theories of consonance and dissonance.

Moreover, the concept of the Golden Ratio is integral to musical composition. Composers frequently engage with mathematics through counting, identifying patterns, and applying geometric principles. When music is intertwined with math, the latter can shed its rigid connotations, revealing a more creative side. In fact, mathematical simulations can be structured in a way that suggests music acts as a fundamental component of mathematical functions, implementable through advanced programming languages like C.

Mathematics is often perceived as an abstract and impersonal discipline, filled with numbers and computations that can intimidate many. In contrast, music evokes emotions and is woven into the fabric of our daily lives. Almost everyone has participated in creating music at some level, whether by singing or playing an instrument. This stark contrast raises questions about the motivations behind exploring the connections between these seemingly disparate domains. While elements such as rhythm and pitch illustrate the mathematical aspects of music, finding musicality within mathematics is less intuitive. The organization and predictability of math can feel at odds with the fluidity of artistic expression.

However, upon deeper examination, it becomes clear that math and music are not as mutually exclusive as commonly believed. Instead, they share numerous connections that may explain the mutual appreciation between musicians and mathematicians.

Section 1.1: Fibonacci and the Golden Section in Music

One particularly intriguing aspect of mathematical principles in musical compositions is the presence of Fibonacci numbers and the Golden Section. The Fibonacci sequence, named after the medieval mathematician Leonardo de Pisa, begins with two ones, with each subsequent number formed by summing the previous two (1, 1, 2, 3, 5, 8, 13, 21, 34…). A key feature of this sequence is that its ratios converge to a constant known as the Golden Ratio (approximately 0.618).

Geometrically, the Golden Section refers to the division of a line into two unequal segments such that the ratio of the whole to the larger segment is equal to the ratio of the larger segment to the smaller one. This aesthetically pleasing ratio finds applications in art, especially in painting and photography, where key elements often align with the Golden proportion. Interestingly, studies have shown that this principle is also prevalent in musical compositions, where Fibonacci ratios can dictate rhythmic variations and melodic structures. In essence, the arithmetic progressions in music correspond to geometric progressions in mathematics, establishing a logarithmic relationship between the two.

The first video titled "A Hard Day's Math: the connections between mathematics and music" by Jason I. Brown delves into the intricate ties between these two fields, highlighting how math underpins musical theory and composition.

Subsection 1.1.1: Simulating Music through Mathematical Functions

As we explore the relationship between music and math, it's essential to consider how mathematical functions can embody musical characteristics. If we can attribute musical qualities to mathematical functions, we can define a set of rules to facilitate this connection. A suitable platform for illustrating these relationships could be a programming language like C.

Mathematical functions possess a typical range of values that can be expressed within specific limits. For example, considering a scale of three octaves, each consisting of 12 notes, yields a total of 36 notes. This framework allows us to associate musical notes with numerical values: SA corresponds to 1, RE to 2, and so on. We can examine three types of functions: logarithmic, polynomial, and trigonometric, to see how they relate to musical structures.

The polynomial function's positive values can range up to 32,767 (the maximum integer value in C). By segmenting this range into three parts: 1 to 36, 37 to 1296, and 1297 to 32,767, we can convert values from the higher ranges to the first. Negative values can also be transformed into positive ones, broadening their musical potential.

Logarithmic functions can yield a maximum value of approximately 10.39, which can be scaled similarly to polynomial functions. In contrast, trigonometric functions are periodic, necessitating a defined base for zero values. For instance, we can set this base at 18, with negative values ranging from 1 to 18 and positive values spanning from 18 to 36.

Conclusion: Bridging Two Worlds

While mathematical patterns in sound, harmony, and composition provide insights into the relationship between math and music, they do not fully explain the enduring fascination mathematicians have with music. Being a mathematician does not solely mean finding numbers everywhere; there exists a deeper connection that transcends mere calculations. Although both fields remain distinct, recognizing their interconnections could enrich mathematical education, freeing it from its often rigid reputation. By illustrating the artistic side of mathematics, we can reshape public perceptions, fostering a greater appreciation for its essence and universality. However, achieving this understanding will likely take time and effort.

This paper was co-authored by Rohan Jain and myself, presented at the Tech Fest at Manipal Institute of Technology, KA in 2002. Additionally, I developed a C program to demonstrate the concepts discussed above.