Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz. c. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

Key Concepts

Beats Frequency

Beats frequency occurs when two sound waves of slightly different frequencies interfere with each other, creating a fluctuating sound intensity. The number of beats per second is equal to the absolute difference between the frequencies of the two waves. In this case, if the tuner hears four beats per second, it indicates that the frequency of the E string is either 4 Hz above or below the reference frequency of 659 Hz.

Frequency and Tension Relationship

The frequency of a vibrating string is directly related to its tension, length, and mass per unit length. Increasing the tension of a string raises its frequency, while decreasing tension lowers it. This relationship is described by the formula f = (1/2L)√(T/μ), where f is frequency, L is length, T is tension, and μ is mass per unit length. Understanding this concept is crucial for the tuner to adjust the E string to the correct pitch.

Harmonics and Overtones

Harmonics are the integer multiples of a fundamental frequency produced by vibrating strings or air columns. When a string vibrates, it produces a fundamental frequency along with overtones, which are higher frequencies that contribute to the overall sound quality. In piano tuning, the tuner listens for the harmonics of the strings to ensure they align correctly, which is essential for achieving a harmonious sound when the notes are played together.