MathLibArts

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1 Jamie Kasprovitz Math 151 A Survey of Mathematics Prof. Vazquez The Art of Mathematics in Liberal Arts At first glance, when I was faced with the question of how mathematics and liberal arts are connected, I immediately said to myself, "they're not". However, upon doing some research I've now conceded that I could not have been more wrong. Mathematics and liberal arts are often viewed as two separate and distinct subjects. Math is typically associated with science and technology, with a focus on logic, reasoning, and analysis. Liberal arts are comprised of the humanities, social sciences, and fine arts, and focus mainly on critical thinking and problem- solving skills. However, the relationship between math and liberal arts is more intertwined than many people realize, myself included. While these two fields of study may seem very different, they have several important connections. Math has had a significant impact on many specialties within liberal arts including history, music, philosophy, and literature. One way that mathematics and liberal arts are connected is through the learning and application of logic and deductive reasoning. These mathematical concepts teach how to analyze arguments and evaluate evidence. At its core, mathematics is about learning how to use logic and reasoning skills to solve problems. When you're solving a math problem, you're essentially using logic to identify the steps needed to arrive at a solution. This involves breaking down the

2 problem into smaller parts, identifying patterns, evaluating the order of steps to take, and using deductive reasoning to arrive at a solution. Mathematical concepts are necessary skills that are important to learn and apply in any educational or job setting, including jobs one would acquire using a degree in liberal arts. For the remainder of this essay, I will focus on the connection between mathematics and music. Mathematics and music have a relationship that dates back thousands of years. Musicians have used mathematical concepts to help them create and understand music. One of the primary ways that mathematics has impacted music is through harmony and chords. When musicians create harmony, they're combining different notes to create melodies. These notes are created by sound waves, which have different frequencies. The frequency of a sound wave determines the pitch of the note it produces. Musicians use math to create "chord progressions" that sound aesthetically pleasing. For example, many songs use chords based on the intervals of the perfect fifth and the major third, which correspond to integer ratios 3:2 and 5:4. This creates melodious chords that sound good together and have been used in famous musical pieces such as Beethoven's "Fifth Symphony". (Interval song chart) Mathematics also plays an important role in the rhythm and meter of music. Time signatures, such as 4/4 or 3/4 are used to organize music and establish a rhythm. Musicians can also use fractions and ratios to create more complex rhythms called polyrhythms or cross- rhythms, where two or more rhythms are combined to create a new sound. These kinds of rhythms are commonly heard in jazz and rock music, and music from other countries such as Africa and India. Additionally, math has also impacted music through the development of computer software and technology for creating and manipulating sound. Digital audio software uses algorithms to process sound waves and create new sounds. This technology has

3 revolutionized the way music is created and produced, making it easier for musicians to experiment with their sound and musical style. Another way that mathematics has affected music is with the creation of tuning systems. "A tuning system, or temperament, is a way to define individual pitches for music from the set of all possible high and low tones." (Wood) The Greek philosopher and mathematician Pythagoras (of the Pythagorean theorem fame) is credited with the development of the Pythagorean tuning system, which is based on the idea that musical harmony is directly related to mathematical ratios. Pythagoras believed that the relationships between musical notes could be understood in terms of ratios such as 2:1 for the octave, and 3:2 for the fifth. The Pythagorean tuning system has been widely used in western music for centuries. In addition to discovering the mathematical principles of tuning systems, Pythagoras also helped progress the study of harmonics and acoustics. Pythagoras was also considered to have discovered that musical intervals could be expressed as numerical ratios. By studying the consonance and dissonance produced by hammers of different sizes and weights when struck, Pythagoras discovered weight ratios. "Hammer number one was twice the weight of hammer two (ratio 2:1), which is the ratio of an octave (between the two musical pitches the frequency is exactly double). The third and fourth hammer had had the ratio to the first as 4:3 (perfect fourth), and 3:2 (perfect fifth), etc." (Popular Beethoven) While Pythagoras' theory regarding the hammers was proven eventually to be incorrect, it indeed was true when applied to strings with tension, such as on a piano. Pythagoras has also been rumored to be quite interested in the effect that music has on people's mental and emotional states, and made music a large part of the daily practice at his school.

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4 Finally, the mathematical concepts of the Fibonacci sequence (a series of numbers in which each number ( Fibonacci number ) is the sum of the two preceding numbers, also frequently found in nature) and the golden ratio have been used to analyze musical structures like the creation of melodies and the placement of chords. By using mathematical concepts to analyze music, musicians can gain new insight into the underlying patterns of musical compositions. In conclusion, the relationship between math and music is more harmonious than I had originally thought. While I don't believe that math will ever become my favorite subject, writing this essay honestly opened my eyes and helped me realize how much math has impacted every aspect of our lives for centuries and has perhaps helped me gain a new appreciation for the equations and sequences that help us to better understand the world around us. Citations : Lam Kai Shun, C. (n.d.). The Philosophical Implications of Set Theory in Infinity . Journal of Physical Mathematics. Retrieved February 23, 2023, from https://www.hilarispublisher.com/open-access/the-philosophical-implications-of-set-theory-in- infinity.pdf Interval song chart . Musicca. (n.d.). Retrieved February 22, 2023, from https://www.musicca.com/interval-song-chart Wood, S. (n.d.). Tuning systems . Music Crash Courses. Retrieved February 22, 2023, from https://musiccrashcourses.com/lessons/tuning_systems.html Anderson, G. H. (1983). Pythagoras and the Origin of Music Theory. Indiana Theory Review , 6 (3), 35–61. http://www.jstor.org/stable/24045969 Popular Beethoven. (n.d.). Retrieved February 22, 2023, from https://www.popularbeethoven.com/pythagoras-on-music/ Watkinson, J. (2020, September 14). Digital Audio: Part 3 - the mathematics of sampling . The Broadcast Bridge - Connecting IT to Broadcast. Retrieved February 22, 2023, from https://www.thebroadcastbridge.com/content/entry/15640/digital-audio-part-3

5 Mongoven, C. (2008, March 27). A STYLE OF MUSIC CHARACTERIZED BY FIBONACCI AND THE GOLDEN RATIO . Retrieved February 23, 2023, from https://www.researchgate.net/profile/Casey-Mongoven/publication/265261482_A_STYLE_OF_ MUSIC_CHARACTERIZED_BY_FIBONACCI_AND_THE_GOLDEN_RATIO/links/ 558d6a8208ae1e1f9bab138a/A-STYLE-OF-MUSIC-CHARACTERIZED-BY-FIBONACCI- AND-THE-GOLDEN-RATIO.pdf

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