
Wave phenomena such as reflection, interference, and superposition are responsible for many acoustical and musical phenomena. Echoes and reverberation occur when sound waves are reflected. Although each musical note on a piano keyboard is associated with a particular fundamental frequency, most musical instruments create a wide spectrum of harmonics, which are integer multiples of the fundamental frequency. Musical intervals, such as the octave, correspond to simple integer ratios of frequencies. Interference of waves with two different frequencies produces beats and determines whether two or more notes sound pleasant when played together.

echo, phase, beats, harmonic
 Review problems and questions 

 This depiction of a piano keyboard gives the letter names for notes in the Cmajor scale, along with their frequencies in hertz. Intervals such as the fourth, the fifth, and the octave refer to ratios of frequencies. For example, to go one octave to the right means doubling the frequency.
 What is the frequency ratio corresponding to the interval known as a fifth? (Express your answer as a ratio of integers and in decimal format.)
 What is the frequency ratio corresponding to the interval known as a fourth? (Express your answer as a ratio of integers and in decimal format.)
 What is the product of the two ratios you just computed?
 State a general relationship between the fourth, the fifth, and the octave.

 A fifth corresponds to a frequency ratio of 3/2, or 1.5. This can be seen by dividing 396 Hz (the fundamental frequency of the G_{4} note) by 264 Hz (C_{4}).
 A fourth corresponds to a frequency ratio of 4/3, or 1.33. This can be seen by dividing 176 Hz (the fundamental frequency of the F_{3} note) by 132 Hz (C_{3}).
 The product of these two ratios is 2.
 A fifth, followed by a fourth, equals an octave. Note that a fourth encompasses four white keys (implying that you move three white keys up or down). A fifth, in turn, encompasses five white keys, implying a move of four. Three plus four equals seven, and moving up by seven white keys changes the note by one octave (bringing you from C_{4} to C_{5}, say). One might therefore say “a fourth plus a fifth equals an octave.”


Beats are heard when two tones of different frequencies occur. The beat frequency equals the difference of the two original frequencies.
 What is the beat frequency if C_{4} and C_{5} are played at the same time?
 Does that beat frequency correspond to one of the marked notes? If so, which?
 What is the beat frequency if C_{4} and G_{4} are played simultaneously?
 Does that beat frequency correspond to one of the marked notes? If so, which?
 What beat frequency results if C_{5} and F_{5} are played together?
 Does that beat frequency correspond to one of the marked notes? If so, which?
 The fourth, the fifth, and the octave are among the most pleasantsounding and important intervals in Western music. Can you speculate why?

 The beat frequency is 264 Hz if C_{4} and C_{5} are played at the same time.
 This corresponds to C_{4} (one of the notes being played in the first place).
 The beat frequency is 132 Hz if C_{4} and G_{4} are played simultaneously.
 This corresponds to C_{3} (one octave below one of the notes being played in the first place).
 The beat frequency is 176 Hz if C_{5} and F_{5} are played together.
 This corresponds to F_{3} (two octaves below one of the notes being played in the first place).
 All of these intervals generate beats that harmonize with one of the notes being played.
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