Introduction
People passing through the engineering quad on the Texas A&M University College Station campus will undoubtedly notice an art installation by Olafur Eliasson. The polished steel structure presents a type of optical illusion: despite being only a collection of cubes, composed of sharp edges and defined vertices, the sculpture appears as a sphere, a seamless figure with no definite beginning or end. The two ideas seem incompatible—the former represents that which is definite and the latter, the infinite—and yet when these cubes are superimposed so as to leave cutouts of negative space in between, viewers perceive the resulting shape to be a sphere. This work of art, like many others by Eliasson, draws inspiration from the principles of mathematics.
As the previous example illustrates, there exists the potential for a deeper study of the relationships between mathematics and the arts. One could argue that this is particularly so where music is concerned. The twentieth century was characterized in part by the growth of Contemporary and Expressionist styles in music, allowing for greater compositional development through the use of atonality, a nontraditional musical structure (i.e. the composition does not use any identifiable scale form and is often comprised of notes that “clash,” or are too close in frequency to exhibit waveform harmonics). Compositions based in atonality—regarded as “the first significant advance in this trend toward serialism in music”^{1}—employed all twelve chromatic notes of western classical music, which were each assigned a value of one through twelve at the composer’s discretion. The numbers were then randomly ordered by the roll of a twelvesided die, and music would be composed using the newly generated series of notes. No pitch could be repeated until all twelve tones had already been chosen once.^{2} This method created a form of music composition based solely on mathematics, specifically on discrete mathematics, which “[deals] with objects that can assume only distinct, separated values.”^{3}
The connection established between mathematics and music through atonality invites us to think about ways in which we might expand the relationship to relevant physics and engineering equations in order to turn them into musical compositions. Specifically, we turn from equations that operate under discrete mathematics to those under the jurisdiction of continuous mathematics such as recurring decimals, fractions, and irrational numbers to discover if there is a real relationship between continuous math and music. Binomial Rhapsody, a project under the ENGR[X] program at Texas A&M University, was the avenue by which I was able to marry the technical aspects of mathematics with the creative aspect of music and more concretely pinpoint the influence mathematics has on musical composition. The project began as an openended challenge proposed by Professor Shayla Rivera to the students of the engineering college: compose music from one (or more) of the 40 mathematical and scientific equations printed on the bricks of the walkway around the Zachry Engineering Education Complex. To do this, it was necessary to form a process by which I could translate the numerical output from an equation model of a continuous function into musical notes which, when assigned whole number values from one to twelve, would represent a set of discrete numbers. 
I WAS ABLE TO MARRY THE TECHNICAL ASPECTS OF MATHEMATICS WITH THE CREATIVE ASPECT OF MUSIC 
Mathematical Methods and Composition Techniques
I selected an equation that only had two variables contributing to a single output: F = m*a, or Newton’s Second Law of Motion. The purpose of picking an equation with three variables was to set one of the variables as a constant and manipulate a second variable to obtain a singular, unique output with each manipulation. Fewer variables in the chosen equation allow for greater compositional freedom, as only those notes representing few numeric variables need to be played at once. After choosing the equation, I initially set the acceleration to be a constant 1 m/s2, but realized that as I varied the value of mass, our output (the force variable) would be the same as our input (the mass variable), and our equation would simply generate a chromatic scale^{4}) when I assigned individual notes numeric values. To avoid the dull predictability of the chromatic scale, I decided to set the acceleration to an arbitrary value of 22 m/s2, a nod to the expected graduation year of all members of my performance ensemble. I then multiplied this value with the mass, which I varied from 1–12.
Using Google Sheets, I multiplied each combination of numbers defined by my process to obtain an output for the force (Figure 1). Because there are only twelve distinct musical notes in Western classical music (C, C#, D, D#, E, F, F#, G, G#, A, A#, and B), I then converted each of the numbers to a basetwelve system so that each output would correspond to one unique note. During the conversion, I retained the values of both the whole number quotient (the number multiplied by 12, the product of which was subtracted from the original numerical output for force) and the remainder (what was leftover from the whole number division); this was done to form chords. If both the remainder and the quotient had not been kept, the output would only be a single number, therefore yielding only a single note to play, resulting in a lack of harmony and only a solo melodic line. To avoid this problem and instead generate harmonic musical context, I kept both the remainder with the whole number quotient for nearly all calculations, and for the quotients that did not have remainders I used 12, the divisor, as part of the final note output. For cases in which the whole number quotient (regardless of whether there was a remainder) was divisible by 12, I divided the quotient by 12 and kept all notes obtained from this operation and its resulting remainders. While this process generated more musical output (because there were more numbers available), its primary purpose was to break down large numbers into numbers that I could translate into notes, ranging only from 1–12. I assigned each note a value (Figure 2), notes are labeled in red and orange and the assigned numerical values are labeled in green and blue) and chose to begin the labeling system at C because “middle C” is the approximate center of the eightyeight key piano, and the C Major scale—played on only the white keys—is typically the first scale learned because of its simplicity. To more plainly illustrate how the process worked, take the output of 264, or 22*12. Dividing 264 by 12 gives us 22, but 22 is beyond the scope of our 1–12 range of notes, so we again divide 22 by 12 to get 1 with a remainder of 10. This means that we have drawn the notes C (from the quotient 1), and A (from the remainder 10).
After performing these numerical conversions and assigning numbers to the notes, I used my knowledge of music theory to arrange the notes into chords (for example, after getting C, A, and B as the output, I added the note E to make an “a minor” chord). This was done for all rows of output (Figure 1). To improve the flow of the piece, I selected chords that had interval relationships of perfect fifths or perfect fourths between them, meaning there were respectively five and seven half steps between the starting note and the ending note. From a physics standpoint, a perfect 5th is an interval in which the higher and lower pitches have a frequency ratio of 3:2, and a perfect fourth is one in which the higher and lower pitches have a frequency ratio of 4:3.^{5} These ratios mean that the human ear perceives the two pitches to be consonant, or having a pleasant sound due to a lack of destructive interference in their soundwaves. Utilizing these interval relationships during chord transitions and between movements causes the human ear to naturally anticipate the next note, in turn making the piece sound smoother. While mathematics allowed me to obtain the general structure of the piece pitchwise, I personally decided upon the articulation (how notes were played), rhythm, dynamics, and all other components of musical development. I composed a theme (a melodic idea) for each section of the piece (Figure 3) based on the notes in the chords of that section – the first violin played these themes most frequently because of its tonal prominence, but these themes were passed from instrument to instrument. The piece begins and ends in the same manner, with just the piano playing a light melody over the chords of movements one and three, to give the piece aural symmetry.
WHILE MATHEMATICS ALLOWED ME TO OBTAIN THE GENERAL STRUCTURE OF THE PIECE PITCHWISE, I PERSONALLY DECIDED UPON THE ARTICULATION
Performances and Future ProjectsThe piece was originally performed following a brief presentation that detailed the composition process, given during the grand finale of Texas A&M EngineeringWeek, a celebration of engineering careers with activities designed to increase understanding of and interest in Science, Technology, Engineering and Math (STEM) fields. The ensemble performed again as the finale of the Engineering Project Showcase on April 26th, 2019. Their project was featured in the “OnCampus Aggies” section of the Fall 2019 edition of Texas A&M Foundation Spirit magazine. A full recording of the piece is accessible on YouTube (Figure 4). Binomial Rhapsody is an ongoing program sponsored by the Texas A&M College of Engineering through the ENGR[X] program. It is in its inaugural year and is open to any engineering major with musical or technical skills and a passion for innovation as it pertains to music. 
Impact
This project adds to a growing collection of musical pieces composed through mathematical means. Although much of music theory is underlined by mathematical relationships and patterns, it did not develop solely as a function of mathematical principles. Rather, music theory developed as a set of rules by which composers could create consistently beautiful pieces of music. Wherever there is a pattern, there is usually math, but not all patterns have been fully explored mathematically. By composing pieces grounded in mathematics, which are then arranged to sound more harmonic, we can reverse the process by which music is usually created, and thereby attain a deeper understanding of the compositional process. Additionally, by synthesizing components of mathbased music (our output from the equations) with components of traditional composition (composing a singable melody, which the first violin plays), we practice a positive form of humanalgorithmic interaction; our creative minds interact with a set formula. Today, the word “algorithm” has a negative connotation when spoken of in association to platforms and applications such as Facebook, YouTube, and Twitter, because there exists an assumption that a computerrun formula dehumanizes our interactions with both creative content and other human beings. However, by utilizing a formula and setting up an algorithmic process for only the base of a musical composition, we enable creativity and set a model for a brighter, more original future in which automating a process does not necessarily mean dehumanizing it.
While the atonality in music was explored in depth by composers such as Arnold Schoenberg in the 1900’s, it was done mainly through randomization and not set formulae. While the rules of twelvetone composition ensured that there was no true “pitchcenter” and allowed for a complete break from the previous norms of Western classical music, the technique based in rolling a twelvesided die could be difficult to work with as the randomization did not lead to aurally appealing compositions. Merging the technicalities of serialism in music and the freedom of “organic” music composition allows for a different, diverse soundscape.
The very existence of a project such as Binomial Rhapsody is indicative of the changing needs of the world at large: there is need for artistic creativity in the world of STEM. By strengthening the link between music and mathematics, we encourage future generations to invest in the arts despite declining funding and focus. While the fine arts do receive significant support in some areas of the United States, few students enjoy such privilege, causing “students who are economically disadvantaged to not get the enrichment experiences of affluent students.”^{6} Neglecting the arts portends a failure to develop the part of society that most visibly represents humanity and its creativity. Binomial Rhapsody enables those in STEM fields to develop their passion for music. By developing the creative skill set of a musician and the analytical skill set of an engineer, future engineers may be more likely to progress their efforts beyond the boundaries of what physically exists in our world today – it is only by dreaming that we make something possible.
This is the main reason why Binomial Rhapsody is so important: by practicing our ability to physically conceptualize the abstract, whether it be in music or in mathematics, we practice the very creativity that fueled the world’s best innovators, tested the bounds of the conventional, and challenged accepted ways of thinking. It is this creativity with which we continue to catapult ourselves into a modernity where we connect with thousands at the touch of a button and literally defy gravity. By making an effort to adopt a more abstract way of thinking, technological and scientific innovators develop creative solutions that can engender change and benefit the world at large.
WE CONTINUE TO CATAPULT OURSELVES INTO A MODERNITY WHERE WE CONNECT WITH THOUSANDS AT THE TOUCH OF A BUTTON 
AcknowledgmentsSpecial thanks to my ensemble members Joshua Tia, Christine Park, and Alexis Hou. You all make the composition process so enjoyable, and your flexibility allowed me to test so many different musical ideas in such a short span of time. Thank you to Professor Shayla Rivera, who came up with the challenge, who connected me to so many opportunities, and is always there when I need her guidance. And thank you to R. F. Walsh and Andrea Meier, who took the time to proofread the manuscript (multiple times!) and fix the details I overlooked. This paper is dedicated to my mother; the hours you spent watching me practice, critiquing my solos and compositions, and driving me to and from piano lessons, voice lessons, and choir clinics, have made me who I am today. 
References

Ritika Bhattacharjee ‘22Ritika Bhattacharjee ‘22 is a chemical engineering major with a minor in performance studies, specializing in music, from Katy, Texas. She is involved in numerous interdisciplinary projects and programs including the ENGR[X] program, Maroon and White Leadership Fellows, a biomedical engineering research lab (BioInSyst), and she is the student conductor of Ingeniare, the engineering chorus on campus. She hopes to continue work that merges opposing disciplines in order to inspire a more interconnected approach to conventional ideas. Vertical Divider
