Question

If a violin string is tuned to a certain note, by what factor must the tension in the string be increased if it is to emit a note of double the original frequency (that is, a note one octave higher in pitch)?

Short answer

The tension in the violin string must be increased by a factor of 4 to emit a note of double the original frequency.

Step-by-step solution

Understand the problem

In essence, the exercise is asking to find out how much the tension in a violin string must be increased if the emitted note has to be doubled. By understanding that doubling the frequency increases the pitch by an octave, the problem nicely lies within the realm of understanding waves and vibrations.

Apply the formula for frequency

The formula for the frequency (\(f\)) of a vibrating string under tension is given by: \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \(f\) is the frequency, \(L\) is the length of the string, \(T\) is the tension, and \(\mu\) is the linear mass density. Since we're dealing with the same string in both cases, the length and density of the string remain constants. Therefore, if the frequency is doubled, the relationship becomes: \( 2f = \frac{1}{2L} \sqrt{\frac{T'}{\mu}} \), where \(T'\) is the new tension.

Rearrange and compute

By rearranging the equations, we can find an expression in terms of the new tension \(T'\). Squaring both equations and solving for \(T'\) yields: \(T' = T \times (\frac{2f}{f})^2 \). Since \(\frac{2f}{f} = 2\), we have: \(T' = 4T\).

Interpret the result

This result implies that the tension in the string must be increased by a factor of 4 for the frequency to be doubled (or for the note to be one octave higher). This makes intuitive sense because for the frequency to be double, a greater tension is needed to propagate the waves faster.

Key concepts

Violin String Tension

Violins have strings that vibrate to produce sound, and the tension of these strings plays a central role in defining the pitch of the sounds they emit. When you increase the tension, the string vibrates at a higher frequency and the pitch goes up.
To understand this, think of the violin string as a tightrope. A tightrope walker standing on a looser rope will cause it to sag and bow downward. Tightening the rope makes it straighter, allowing for faster movement under the walker's feet. Similarly, with a violin string, higher tension leads to tighter waves.
In practical terms, if you want your violin to sound one octave higher, you have to increase the tension in the strings. Specifically, you need to quadruple the tension. This quadrupling ensures that the frequency of the sound doubles, resulting in a pitch that is one octave higher.

• Higher tension in the string leads to higher pitch.
• Doubling the frequency requires quadrupling the tension.
• Tension affects how fast waves travel across the string.

Pitch and Octaves

Pitch is the perception of how high or low a sound feels to us, directly linked to the frequency of the sound waves.
When a violinist adjusts a string to produce a higher pitch, they are essentially increasing the frequency of the vibrations.
An octave is a specific musical interval, representing a doubling of frequency. When you move from one note to another one octave higher, the frequency of the sound is twice as high.

• Pitch is related to the frequency of the sound wave.
• An octave change involves doubling the sound frequency.
• Instruments must physically change string tension to achieve octave changes.

To achieve a one-octave increase in pitch on a string instrument, such as a violin, you must change the string tension significantly by increasing it fourfold.

Sound Waves in Strings

Sound waves on strings are fascinating to study as they reveal how music is physically produced through vibration. When a violin string vibrates, it creates waves that travel along the string and push against the surrounding air, producing the sound we hear.
The frequency of these waves determines the pitch. This frequency is influenced by several factors:

• The tension in the string: Higher tension leads to higher frequency.
• The length of the string: Shorter strings tend to have higher frequencies.
• The mass density of the string: Lighter strings vibrate more easily at higher frequencies.

The formula for this is reflected in \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \), where \(T\) is tension, \(L\) is length, and \(\mu\) is the linear mass density. Thus, to produce any particular frequency, such as one that is an octave higher, the tension must be adjusted accordingly. Increasing the tension makes waves travel faster, allowing for quicker vibration and higher pitches. Understanding these principles allows musicians to fine-tune their instruments accurately.