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Chapter 6: Problem 34

The highest musical note on the piano has a frequency of 4,186 \(\mathrm{Hz}\). Why would a tape of piano music sound terrible if played on a tape player that reproduces frequencies only up to \(5,000 \mathrm{~Hz} ?\)

Short Answer

Expert verified
Answer: The tape player would produce poor sound quality because it cannot reproduce any of the harmonics for the highest piano note due to its limited frequency range. Harmonics are essential for creating a rich and full sound, and without them, the music will sound dull and unnatural.

Step by step solution

01

Understanding Music Notes

A piano has 88 keys, with each key corresponding to a different pitched note. These notes create sound waves with various frequencies, which determine their pitch. The higher the frequency of a note, the higher its pitch. In this case, the highest music note on the piano has a frequency of 4,186 Hz.
02

Introduction to Harmonics

Every musical note has a fundamental frequency and a series of harmonic frequencies. Harmonic frequencies are integer multiples of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics will be 2f, 3f, 4f, and so on. The presence of these harmonics contributes to the richness and quality of the sound that we hear.
03

Frequencies Produced by the Highest Note on Piano

Now that we know the highest note on the piano has a fundamental frequency of 4,186 Hz, we can calculate its harmonics. Following the pattern described above, the first harmonic will be at 2(4,186) = 8,372 Hz, the second harmonic will be at 3(4,186) = 12,558 Hz, and so on. It is important to note that for a sound to be perceived as complete and rich, most of these harmonics should also be reproduced alongside the fundamental frequency.
04

Frequency Range of the Tape Player

In this scenario, the tape player can only reproduce frequencies up to 5,000 Hz. However, as we calculated in the previous step, even the first harmonic of the highest note of the piano is already above this limit (8,372 Hz). This means that the tape player will be unable to reproduce any of the harmonics for the highest piano note due to its limited frequency range.
05

Analyzing the Sound Quality

Since the tape player can only reproduce frequencies up to 5,000 Hz, it will only play the fundamental frequency of the highest musical note without any of its harmonics. This absence of harmonics will result in a dull, incomplete, and unnatural sound, making the piano music played on this tape player sound terrible.

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Most popular questions from this chapter

While a shock wave is being generated by a moving wave source, is the Doppler effect also occurring? Explain.

A conditioning drill consists of repeatedly running from one end of a basketball court to the other, turning around and running back. Sometimes the drill is changed and the runner turns around at half court, or perhaps at three-fourths of the length of the court. Describe how the number of round trips a runner can do each minute changes when the distance is changed and how this is related to a guitarist changing the note generated by a string by pressing a finger on it at some point.

Explain the process of echolocation. How is the Doppler effect sometimes incorporated?

Normal telephones do not transmit pure tones with frequencies below about \(300 \mathrm{~Hz}\). But a person whose speaking voice has a frequency of \(100 \mathrm{~Hz}\) can be heard and understood over the phone. Why is that?

A \(100-\mathrm{Hz}\) pure tone at a \(70-\mathrm{dB}\) sound level and a \(1,000-\mathrm{Hz}\) pure tone at the same sound level are heard separately. Do they sound equally loud? If not, which is louder, and why?
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