Question

Stringed musical instruments, such as violins and guitars, use stretched strings. Explain (a) how tightening and loosening the strings tunes them to their designated tone pitch or frequency; (b) why the strings of lower tones are thicker or heavier; and (c) why notes of higher pitch or frequency are produced when the fingers are placed on the strings.

Short answer

Tightening strings increases pitch; thicker strings produce lower tones; shortening strings raises pitch.

Step-by-step solution

Understanding String Tension and Frequency

When you tighten a string, you increase its tension. According to the wave equation for a string, the frequency of the vibration increases with increased tension. Conversely, loosening the string decreases the tension and lowers the frequency. This is expressed in the formula \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \( f \) is the frequency, \( L \) is the length, \( T \) is the tension, and \( \mu \) is the linear mass density of the string.

Exploring String Thickness and Tone

Thicker strings have more mass per unit length (higher linear mass density \( \mu \)). Lower frequencies require either more tension or a denser string to maintain the same length. Due to the practical limits on tension, lower-tuned strings are heavier or thicker, allowing them to vibrate at lower frequencies and produce lower tones.

Shortening the String Length for Higher Pitch

When you place your fingers on a string, you effectively shorten its vibrating length \( L \). According to the wave frequency formula \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), reducing \( L \) increases the frequency of vibration, producing a note with a higher pitch or frequency. This is why pressing on the string results in a higher tone.

Key concepts

String Tension and Frequency

String tension plays a crucial role in determining the frequency of the sound produced by musical instruments like guitars and violins. When you tighten a string, you increase its tension. This, in turn, affects the frequency of vibration because it is directly related to tension. According to the formula \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where:

• \( f \) is the frequency,
• \( L \) is the length of the string,
• \( T \) is the tension, and
• \( \mu \) is the linear mass density.

A higher tension \( T \) increases the frequency, resulting in a higher pitch. Conversely, loosening the string reduces tension and hence the frequency, producing a lower pitch.
It’s like tuning a radio, except you’re adjusting how tightly the strings are stretched. Tuning pegs on instruments are used exactly for this purpose: to adjust the tension and hit the desired note perfectly.

String Thickness and Tone

The thickness of a string significantly impacts the tone it produces, especially seen in instruments like bass guitars and cellos. Strings of lower tones are typically thicker or heavier, meaning they have a higher linear mass density \( \mu \). Low frequency sounds require either more tension or a denser string. Given the practical constraints on tension, thicker strings are preferred.
Thicker strings vibrate more slowly, resulting in lower frequency tones. Imagine comparing a tight rope to a thicker cable; the cable will generally oscillate more slowly if stroked with the same force. This property is valuable in enabling a range of tonal options in a single instrument, allowing for rich, deep sounds.

Pitch and Frequency in Strings

On stringed instruments, altering the pitch is as simple as changing the effective vibrating length of the string. When you press down on a string using a finger, you shorten its length \( L \), which is a direct method to increase the frequency as per the formula \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \). By shortening \( L \), you increase \( f \). This results in a higher pitched sound. It's similar to how woodwind instruments work by covering holes to adjust their "length" and thus change pitch.
Understanding this relationship empowers musicians to produce any desired pitch by adjusting either the length, tension, or thickness of the strings, making playing versatile and expressive.