Question

Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370 and 392 Hz. What beat frequencies are produced by this discordant combination?

Short answer

The beat frequencies produced by the discordant combination are 21 Hz, 22 Hz and 43 Hz.

Step-by-step solution

Identification of the given data

The give data is listed as below,

• The frequency of first key is, f₁ = 349 Hz
• The frequency of second key is, f₂ = 370 Hz
• The frequency of third key is, f₃ = 392 Hz

Definition of beat frequency

The beat frequency is defined as the difference between the frequency of two similar waves. If f₁and f₂are the frequencies of two similar waves then the beat frequency is expressed as follows,

\[{{\rm{f}}_{\rm{B}}}{\rm{ = }}\left| {{{\rm{f}}_{\rm{2}}} - {{\rm{f}}_{\rm{1}}}} \right|\]

Determination the beat frequencies produced by the discordant combination

Write the expression for the first pair to get the beat frequency.

\[{f_{B1}} = \left| {{f_1} - {f_2}} \right|\]

Substitute all the values in the above expression.

\[\begin{aligned}{f_{B1}} = \left| {{\rm{349}}\;{\rm{Hz}} - {\rm{370}}\;{\rm{Hz}}} \right|\\ = {\rm{21}}\,\;{\rm{Hz}}\end{aligned}\]

Write the expression for the second pair to get the beat frequency.

\[{f_{B2}} = \left| {{f_2} - {f_3}} \right|\]

Substitute all the values in the above expression.

\[\begin{aligned}{f_{B2}} = \left| {{\rm{370}}\;{\rm{Hz}} - {\rm{392}}\;{\rm{Hz}}} \right|\\ = {\rm{22}}\,\;{\rm{Hz}}\end{aligned}\]

Write the expression for the third pair to get the beat frequency.

\[{f_{B3}} = \left| {{f_3} - {f_1}} \right|\]

Substitute all the values in the above expression.

\[\begin{aligned}{f_{B3}} = \left| {{\rm{392}}\;{\rm{Hz}} - {\rm{349}}\;{\rm{Hz}}} \right|\\ = 43\,\;{\rm{Hz}}\end{aligned}\]

Thus,the beat frequencies produced by the discordant combination are 21 Hz, 22 Hz and 43 Hz.