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The most soaring vocal melody is in Johann Sebastian Bach's Mass in \(B\) minor. A portion of the score for the Credo section, number \(9,\) bars 25 to \(33,\) appears in Figure \(P 17.23\) The repeating syllable \(\mathrm{O}\) in the phrase "resurrectionem mortuorum" (the resurrection of the dead) is seamlessly passed from basses to tenors to altos to first sopranos, like a baton in a relay. Each voice carries the melody up in a run of an octave or more. Together they carry it from D below middle C to A above a tenor's high C. In concert pitch, these notes are now assigned frequencies of \(146.8 \mathrm{Hz}\) and \(880.0 \mathrm{Hz}\). (a) Find the wavelengths of the initial and final notes. (b) Assume that the choir sings the melody with a uniform sound level of \(75.0 \mathrm{dB} .\) Find the pressure amplitudes of the initial and final notes. (c) Find the displacement amplitudes of the initial and final notes. (d) What If? In Bach's time, before the invention of the tuning fork, frequencies were assigned to notes as a matter of immediate local convenience. Assume that the rising melody was sung starting from \(134.3 \mathrm{Hz}\) and ending at \(804.9 \mathrm{Hz}\). How would the answers to parts (a) through (c) change?

Step by step solution

01

Calculate the Initial and Final Wavelengths

We know that the velocity (v) of sound in air at room temperature is approximately \(343 m/s\). The relationship between velocity, frequency (f), and wavelength (位) is given by the formula: v = f * 位. Therefore, we can rearrange this formula to find the wavelength: 位 = v / f. Plugging in the given frequencies, we can find the wavelengths of the initial and final notes.

02

Calculate the Pressure Amplitudes

Sound level 饾惪 is related to pressure amplitude 饾憙 by \(饾惪=20 log(P/P0)\) where \(饾憙0=2脳10^(-5) Pa\) is the reference pressure amplitude (the quietest sound that the average human ear can hear). We can rearrange this equation to solve for 饾憙 using the given sound level of 75dB.

03

Calculate the Displacement Amplitudes

The pressure amplitude P and the displacement amplitude s are related through the formula \(P = s * 蟻 * v * 蠅\) where 蟻 is the density of air, approximately \(1.21 kg/m^3\) at room temperature; v is the speed of sound in air, and 蠅 is the angular frequency. We can rearrange this formula to find s.

04

Recalculate for Different Frequencies

Lastly, we repeat steps 1-3 now with the frequencies of \(134.3 Hz\) and \(804.9 Hz\). Then we can compare these results to the results from the original frequencies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavelength

When it comes to sound waves, the term "wavelength" is central in understanding how waves propagate through mediums like air. Wavelength, represented by the Greek letter \( \lambda \), is the distance between two successive crests or troughs in a wave.
For sound waves traveling through air, which is a common scenario in acoustics, the wavelength is linked to the speed of sound and the frequency of the wave. The formula to determine wavelength is:

• \( \lambda = \frac{v}{f} \)

Here, \( v \) is the speed of sound (approximately 343 m/s in air at room temperature), and \( f \) is the frequency of the wave.
So, by knowing the frequency of a sound note, such as those sung in a melody, we can calculate the wavelength, giving us insight into the spatial characteristics of sound.

Pressure Amplitude

Pressure amplitude is an essential concept when discussing the intensity or loudness of sound waves. It refers to the maximum change in pressure exerted by a sound wave in the surrounding air.
In the context of human perception, sound levels are often measured in decibels (dB), which express the pressure amplitude relative to a reference level. The formula to relate sound level \( L \) to pressure amplitude \( P \) is:

• \( L = 20 \log\left( \frac{P}{P_0} \right) \)

Here, \( P_0 = 2 \times 10^{-5} \) Pa is the reference pressure amplitude of the softest sound the average human ear can hear.
When singing at 75 dB, the pressure amplitude can be calculated by rearranging the formula above. Understanding pressure amplitude is crucial, especially in musical settings, as it helps us comprehend how intensely a note is perceived by our ears.

Displacement Amplitude

Displacement amplitude describes how far the particles in the medium through which a sound wave travels are moved from their rest position. This movement reflects part of the sound waves' energy.
Displacement amplitude \( s \) can be calculated using the relation:

• \( P = s \cdot \rho \cdot v \cdot \omega \)

Where \( P \) is the pressure amplitude, \( \rho \) is air density (typically around 1.21 kg/m鲁 at room temperature), \( v \) is the speed of sound, and \( \omega \) is the angular frequency, which is linked to frequency by \( \omega = 2 \pi f \).
Displacement amplitude provides insights into the energy transport and the physical movement of the sound wave, adding a layer of depth to our understanding of sound dynamics.

Frequency

Frequency deals with how often the sound wave oscillates as it travels through a medium. It is usually expressed in hertz (Hz) and fundamentally represents the number of wave cycles that pass a specific point in one second.
Musically, frequency defines the pitch of a note. Higher frequencies correspond to higher pitches, and vice versa.

• Frequency \( f \) can be mathematically defined by the inverse of the period of the wave.

A key part of the exercise involves identifying how frequency correlates with other elements like wavelength and amplitude. For example, given the constant speed of sound, frequency inversely affects wavelength: if the frequency increases, the wavelength decreases, provided the speed of sound remains the same. Thus, notes from a musical piece, like those in Bach's Mass in \( B \) minor, have unique frequencies that influence their perceived characteristics.