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Journal of Humanistic Mathematics Volume 4 | Issue 1 January 2014 An Introduction to Fourier Analysis with Applications to Music Nathan Lenssen Claremont McKenna College, Deanna Needell Claremont McKenna College, Follow this and additional works at: Part of the Music Theory Commons, and the Num
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A$$% $: +>://&+$+.&$.'/ +/4/1/5  An Introduction to Fourier Analysiswith Applications to Music Nathan Lenssen Claremont McKenna College, Claremont, CA Deanna Needell Claremont McKenna College, Claremont, CA Synopsis In our modern world, we are often faced with problems in which a traditionallyanalog signal is discretized to enable computer analysis. A fundamental tool usedby mathematicians, engineers, and scientists in this context is the discrete Fouriertransform (DFT), which allows us to analyze individual frequency componentsof digital signals. In this paper we develop the discrete Fourier transform frombasic calculus, providing the reader with the setup to understand how the DFTcan be used to analyze a musical signal for chord structure. By investigating theDFT alongside an application in music processing, we gain an appreciation forthe mathematics utilized in digital signal processing. 1. Introduction Music is a highly structured system with an exciting potential for analysis.The vast majority of Western music is dictated by specific rules for time, beat,rhythm, pitch, and harmony. These rules and the patterns they create enticemathematicians, statisticians, and engineers to develop algorithms that canquickly analyze and describe elements of songs.In this paper we discuss the problem of   chord detection  , where one wishesto identify played chords within a music file. With the ability to quicklydetermine the harmonic structure of a song, we can build massive databases Journal of Humanistic Mathematics  Vol 4, No 1, January 2014  Nathan Lenssen and Deanna Needell 73  which would be prime for statistical analysis. A human performing such atask must be highly versed in music theory and would likely take hours tocomplete the annotation of one song, but an average computer can alreadyperform such a task with reasonable accuracy in a matter of seconds [17].Here we explore the mathematics underlying such a program and demon-strate how we can use such tools to directly analyze audio files.In Section 2 we provide the reader with a brief introduction to musictheory and motivate the need for mathematical analysis in chord detection.Section 3 contains an introduction to the mathematics necessary to derivethe discrete Fourier transform, which is included in Section 4. We concludewith a description of the Fast Fourier Transform and an example of its usein chord detection in Section 5. 2. Introduction to Music Theory We begin with some musical terminology and definitions. 2.1. Pitches and Scales  We define a  pitch   as the human perception of a sound wave at a specificfrequency. For instance, the tuning note for a symphony orchestra is A4which has a standardized frequency of 440Hz. In the notation A4, A indicatesthe  chroma   or quality of the note while 4 describes the octave or height. A scale   is a sequence of pitches with a specific spacing in frequency. As we followthe pitches of a scale from bottom to top, we start and end on the same noteone octave apart (e.g., from C3 to C4). Pitches an octave apart sound similarto the human ear because a one-octave increase corresponds to a doubling inthe frequency of the sound wave. Western music uses the chromatic scale inwhich each of the twelve chroma are ordered over an octave. These twelvenotes are spaced almost perfectly logarithmically over the octave. We canuse a recursive sequence [13] to describe the chromatic scale: P  i  = 2 1 / 12 P  i − 1 , where  P  i  denotes the frequency of one pitch, and  P  i − 1  the frequency of theprevious. We can hear the chromatic scale by striking every white and blackkey of a piano in order up an octave or visualize it by a scale such as that  74 An Introduction to Fourier Analysis with Applications to Music  Figure 1  A chromatic scale beginning and ending at C. There are thir-teen notes because C is played both at the top and the bottom. Imageis from  Wikipedia , , accessed on March 10, 2013. depicted in Figure 1. Note that pitches use the names A through G, alongwith sharp (  ) and flat (  ) symbols; see [12] for more on musical symbols.By using the chromatic scale as a tool, we can construct every otherscale in Western music through the use of intervals. An  interval   refers to achange in position along the twelve notes of the chromatic scale. We definethe interval of a  half step  or H as a change in one pitch along the chromaticscale. Two half step intervals makes a  whole step  and is denoted by W. Wealso use interval of three half steps known as an  augmented second   denotedby A. Scales are defined by a sequence of these intervals with the conditionthat the total sum of steps must equal 12. This guarantees that we startand end on the same chroma known as the root R. There are four prevalentscales in Western music: major, minor, augmented, and diminished. Theintervals that describe these scales can be found in Table 1. The table is used by selecting any starting note as the root and then using the intervalsto construct the remaining notes. For instance, a C minor (Cm) scale is C,D, E  , F, G, A, B  , C. Scale “Color” Defining Steps Chromatic RHHHHHHHHHHHHRMajor RWWHWWWHR(Natural) Minor RWHWWHWWRDiminished RHWHWHWHWRAugmented RAHAHAHR Table 1  Interval construction of the four core scales with the chromatic scalefor reference. Note that the intervals apply for when ascending in pitch only.When determining the descending scale, the order of intervals is reversed.
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