We have seen that musical tones can be modelled mathematically by sinusoidal signals. If y...

We have seen that musical tones can be modelled mathematically by sinusoidal signals. If you read music or play the piano, you know that the piano keyboard is divided into octaves, with the tones in each octave being twice the frequency of the corresponding tones in the next lower octave. To calibrate the frequency scale, the reference tone is the A above middle C, which is usually called A-440, since its frequency is 440 Hz. Each octave contains 12 tones, and the ratio between the frequencies of successive tones is constant. Thus, the ratio must be 21/12. Since middle C is nine tones below A-440, its frequency is approximately (440)2–9/12 ≈ 261.6 Hz. The names of the tones (notes) of the octave starting with middle C and ending with high C are:

(a) Make a table of the frequencies of the tones of the octave beginning with middle C, assuming that the A above middle C is tuned to 440 Hz.

(b) The above notes on a piano are numbered 40 through 52. If n denotes the note number and f denotes the frequency of the corresponding tone, give a formula for the frequency of the tone as a function of the note number.

(c) A chord is a combination of musical notes sounded simultaneously. A triad is a three-note chord. The D-major chord is composed of the tones of D, F#, and A sounded simultaneously. From the set of corresponding frequencies determined in (a), make a sketch of the essential features of the spectrum of the D-major chord assuming that each note is realized by a pure sinusoidal tone. Do not specify the complex phasors precisely.