Question

Question: (II) What is the beat frequency if middle C (262 Hz) and C^# (277 Hz) are played together? What if each is played two octaves lower (each frequency reduced by a factor of 4)?

Short answer

If middle C and C^# are played together, the beat frequency is \(15\;{\rm{Hz}}\) , and if each is played two octaves lower then, the sound wave of the frequency will not be audible.

Step-by-step solution

Determination of the audible capacity

To check the audible capacity of sound waves, the frequency range of sound waves should lie between the human audible range, i.e.,20-20,000 Hz.

Given information

Given data:

The frequency of middle C is \({f_1} = 262\;{\rm{Hz}}\).

The frequency of middle C^# is \({f_2} = 277\;{\rm{Hz}}\).

Evaluation of the beat frequency and audible capacity

The beat frequency can be calculated as:

\(\begin{array}{c}{f_B} = \left| {{f_2} - {f_1}} \right|\\{f_B} = \left| {277\;{\rm{Hz}} - 262\;{\rm{Hz}}} \right|\\{f_B} = 15\;{\rm{Hz}}\end{array}\)

It is given that each frequency is reduced by a factor of \(4\), the beat frequency will be reduced by the same amount, and the new beat frequency can be given as:

\({f'_B} = \frac{{{f_B}}}{4}\) … (1)

After substituting the calculated value of \({f_B}\) in equation (1), you get:

\(\begin{array}{c}{{f'}_B} = \frac{{15\;{\rm{Hz}}}}{4}\\{{f'}_B} = 3.75\;{\rm{Hz}}\end{array}\)

As the frequency of human audible range lies between \(20 - 20,000\;{\rm{Hz}}\), the above-calculated frequency is below the \(20\;{\rm{Hz}}\) frequency so, the sound wave of this frequency is not audible.

Hence, if middle C and C^# are played together, the beat frequency is \(15\;{\rm{Hz}}\) , and if each is played two octaves lower then, the sound wave of the frequency will not be audible.