Question

The middle-C key (key 52 ) on a piano corresponds to a fundamental frequency of about \(262 \mathrm{~Hz},\) and the sopranoC key (key 64) corresponds to a fundamental frequency of \(1046.5 \mathrm{~Hz}\). If the strings used for both keys are identical in density and length, determine the ratio of the tensions in the two strings.

Short answer

Answer: The ratio of tensions in the two strings is 1:16.

Step-by-step solution

Write the fundamental frequency formula for both keys

To find the ratio of the tensions of the two strings, first write the fundamental frequency formula for each of the keys (middle-C and soprano-C): Middle-C key: \(f_{1} = \frac{1}{2L} \sqrt{\frac{T_{1}}{\mu}}\) Soprano-C key: \(f_{2} = \frac{1}{2L} \sqrt{\frac{T_{2}}{\mu}}\)

Set up a proportion to compare the two strings

Since the two strings have the same density and length, the ratio of their fundamental frequencies should be proportional to the ratio of their tensions. We can write this proportion as: \(\frac{f_{1}}{f_{2}} = \frac{\frac{1}{2L} \sqrt{\frac{T_{1}}{\mu}}}{\frac{1}{2L} \sqrt{\frac{T_{2}}{\mu}}}\) Here, \(f_{1} = 262 \mathrm{~Hz}\) (middle-C frequency) and \(f_{2} = 1046.5 \mathrm{~Hz}\) (soprano-C frequency).

Simplify the proportion and solve for the tension ratio

Next, simplify the proportion: \(\frac{262}{1046.5} = \frac{\sqrt{\frac{T_{1}}{\mu}}}{\sqrt{\frac{T_{2}}{\mu}}}\) Square both sides of the equation to get rid of the square roots: \(\left(\frac{262}{1046.5}\right)^{2} = \frac{\frac{T_{1}}{\mu}}{\frac{T_{2}}{\mu}}\) Now, multiply both sides by \(T_{2}\) to isolate the ratio \(\frac{T_{1}}{T_{2}}\): \(\frac{T_{1}}{T_{2}} = \left(\frac{262}{1046.5}\right)^{2} \cdot 1\)

Calculate the tension ratio

Finally, calculate the tension ratio: \(\frac{T_{1}}{T_{2}} = \left(\frac{262}{1046.5}\right)^{2}\) \(\frac{T_{1}}{T_{2}} = \frac{1}{16}\) So, the ratio of the tensions in the two strings is \(1:16\).

Key concepts

Fundamental Frequency

The fundamental frequency is a crucial concept in the world of sound and vibration. It refers to the lowest frequency produced by any vibrating object, like a string or a column of air. The fundamental frequency determines the pitch of the sound, making it an essential element in music and acoustics.
To calculate the fundamental frequency of a string, we use the formula:

• \( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \)

  Where:

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

This equation shows how the fundamental frequency is influenced by string length, tension, and density. Higher tension results in higher frequencies, while greater density or length lowers the frequency. Understanding this relationship helps in designing and playing musical instruments like pianos.

Piano Physics

The physics of a piano involves the transformation of mechanical energy into sound energy. When a key on a piano is struck, it causes a hammer to hit a string inside the piano. This action makes the string vibrate, producing sound waves that we hear as music.
Pianos use different strings of varying lengths, tensions, and densities to produce a wide range of notes. For instance:

• Lower notes are produced by longer and thicker strings, which vibrate more slowly.
• Higher notes come from shorter, thinner strings, which vibrate at a faster rate.

Each string is tuned to a specific tension to achieve its intended pitch. The keys, hammers, and strings work together in harmony, guided by principles of physics, to create the piano's distinctive sound. Musicians leverage their understanding of these mechanics to master their instrument and produce beautiful music.

String Density

String density, or linear mass density, is a measure of how much mass a string has per unit length. It's denoted by the symbol \(\mu\) and is a fundamental property influencing a string's vibration characteristics.
The string density affects the sound produced by an instrument in several ways:

• Thicker, denser strings tend to produce lower-pitched sounds, as they have more mass per unit length, causing them to vibrate more slowly.
• Lighter, less dense strings vibrate more quickly, resulting in higher-pitched sounds.

The density of a string is as important as its length and tension when it comes to determining its fundamental frequency. For a given tension and length, increasing the density will lower the frequency of the sound produced.
In the study of a piano's middle-C and soprano-C keys, the density remains constant, allowing us to focus on tension differences to understand frequency changes.

Wave Equations

Wave equations are mathematical expressions that describe the behavior of waves as they travel through different mediums, like strings or air. In the context of stringed instruments like the piano, wave equations help us understand how strings vibrate to produce sound.
For a vibrating string, the wave equation is expressed as:\[\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}\]Here:

• \(y\) represents the displacement of the string at a point and time.
• \(v\) is the wave speed, dependent on the tension and density of the string.

Solving this wave equation gives insight into how waves propagate along the string, affecting its frequency and the sound emitted. Understanding these equations allows musicians and instrument makers to manipulate variables like string tension and length to achieve the desired tones.
In the problem of determining string tension ratios, wave equations facilitate the correlation between physical properties and sound frequencies, providing a mathematical framework for music acoustics.