ABSTRACT
In this chapter we discuss how mathematics can be used to model the tones we hear in music. The principal mathematical tool we will use is called a spectrogram. Spectrograms allow us to create visual representations of the structure of music relative to both time and frequency. We begin the chapter by showing how various musical tones appear in spectrograms. This will lay the groundwork for our use of spectrograms to analyze music in the next chapter. The remainder of the chapter describes the mathematics used to create spectrograms. This mathematics involves the trigonometric functions, sine and cosine. The sine and cosine functions provide a basic model for pure tones. By taking finite sums of sine and cosine functions, we will have a basic model for tones from musical instruments. After laying the foundations of analyzing music in a time-frequency manner in this chapter, we will apply time-frequency analysis to a wide variety of music in the next chapter.
