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. a.What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

Verified step by step guidance

1. Identify the fundamental frequencies for notes A and E, which are 440 Hz and 659 Hz respectively.
2. Calculate the third harmonic of the A note by multiplying the fundamental frequency of A (440 Hz) by 3. This gives the frequency of the third harmonic of A.
3. Calculate the second harmonic of the E note by multiplying the fundamental frequency of E (659 Hz) by 2. This gives the frequency of the second harmonic of E.
4. Find the frequency difference between the third harmonic of the A note and the second harmonic of the E note by subtracting the frequency of the second harmonic of E from the frequency of the third harmonic of A.
5. The result from the previous step gives the frequency difference between the third harmonic of the A note and the second harmonic of the E note.

Key Concepts

Harmonics

Harmonics are integer multiples of a fundamental frequency produced by vibrating strings or air columns. For example, the first harmonic (fundamental frequency) is the lowest frequency, while the second harmonic is twice that frequency, and the third harmonic is three times the fundamental frequency. Understanding harmonics is essential for analyzing how different frequencies interact in musical instruments.

Frequency

Frequency is the number of cycles of a periodic wave that occur in one second, measured in Hertz (Hz). In music, different notes correspond to specific frequencies; for instance, the note A has a frequency of 440 Hz. The frequency of a note determines its pitch, and understanding frequency is crucial for comparing different musical notes.

Beats

Beats occur when two sound waves of slightly different frequencies interfere with each other, resulting in a fluctuating sound intensity. The beat frequency is equal to the absolute difference between the two frequencies. This phenomenon is important in tuning instruments, as it helps tuners identify discrepancies between the frequencies of different strings.