Question

A violinist is tuning her instrument to concert A (440 Hz). She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat frequency of 3 Hz, which increases to 4 Hz when she tightens her violin string slightly. (a) What was the frequency of the note played by her violin when she heard the 3-Hz beats? (b) To get her violin perfectly tuned to concert A, should she tighten or loosen her string from what it was when she heard the 3-Hz beats?

Short answer

The frequency was 443 Hz. She should loosen the string to tune it correctly.

Step-by-step solution

Understand Beat Frequency

When two sound waves of slightly different frequencies are played together, they produce a beat frequency, which is equal to the absolute difference in their frequencies. Given that the violinist hears a 3 Hz beat frequency when she first plays the violin, this means the frequency of her note is either 3 Hz above or 3 Hz below the concert A frequency of 440 Hz.

Initial Frequency Calculation

Let's denote the frequency of the violin as \(f_v\). To find \(f_v\), note that the beat frequency of 3 Hz implies: \(|f_v - 440| = 3\). This results in two possible frequencies: \(f_v = 440 + 3 = 443\text{ Hz}\) or \(f_v = 440 - 3 = 437\text{ Hz}\).

Determine the Effect of Tightening the String

When the violinist tightens her string slightly, she observes the beat frequency increases to 4 Hz. If tightening increases the beat frequency, the original frequency \(f_v\) must be greater than 440 Hz. If \(f_v\) were less than 440 Hz, tightening would decrease the beat frequency. Thus, the original frequency was 443 Hz.

Determine Adjustment Needed for Perfect Tuning

Since the original frequency \(f_v\) was 443 Hz and tuning requires the frequency to match 440 Hz, the violinist needs to decrease her frequency. Therefore, to achieve perfect tuning to concert A, she should loosen her violin string.

Key concepts

Violin Tuning

Tuning a violin involves adjusting the tension of the strings to match specific frequencies. Typically, musicians aim to align their instrument's notes with standard pitches used in concerts or performances. Concert A, known as A4, is usually set at 440 Hz. This tuning process ensures harmony when playing with other instruments or recorded tracks.
In this context, the violinist is trying to match her A string to the concert pitch of 440 Hz. She uses a reference tone, like one generated electronically, to hear any mismatch in frequency between the reference note and her violin. If the note from the violin isn't accurately tuned, it will create a beat frequency with the reference tone. Recognizing and adjusting these beats plays a crucial role in the tuning process.

Frequency Adjustment

Frequency adjustment in musical tuning involves altering the tension of the strings, which in turn changes the vibration speed and the pitch produced.
When you tighten a string, it vibrates faster, increasing the frequency. Conversely, loosening the string slows down the vibrations, lowering the frequency. This is crucial when fine-tuning instruments to match standard notes precisely.
In the violin problem, the musician examines the beat frequency to decide whether to tighten or loosen the string. She initially hears a 3 Hz difference from the concert pitch, meaning her string is slightly off. Tightening the string increases the beat frequency to 4 Hz, indicating that her original note was higher than 440 Hz. Therefore, to decrease the frequency to the desired 440 Hz, she needs to loosen her string.

Sound Waves

Sound waves are vibrations that travel through the air, and our ears perceive these as sound. In music, they are essential in producing different pitches. The frequency of a sound wave is what determines the pitch we hear.
Higher frequencies produce higher-pitched sounds, while lower frequencies result in lower pitches. When two sound waves of slightly different frequencies intersect, they create a phenomenon called beat frequency, manifesting as periodic increases and decreases in volume. This helps musicians like the violinist know when two notes are not exactly in tune.
In the exercise, the sound wave from the violin interacts with the pure electronic tone, creating beats that guide the adjustments needed to perfectly tune the instrument.