Why Do Musical Instruments Sound Different From One Another? (PART - 2)

“Emotions are the soul of music” with the quote of Goldsmth, let us continue to learn the concept of overtones.

From part 1, you learnt the basics of standing waves, harmonics, and overtones to know why musical instruments sound different from one another.

The fundamental or 1st harmonic is the basic standing wave. Higher-order harmonics like 2nd, 3rd, 4th and so on are the 1st, 2nd, 3rd overtones and so on, respectively. Physicists compared the harmonics with the wavelength.

A wavelength is nothing but a wave having three nodes or three antinodes. This wavelength measurement helps the physicists to discover the higher-order harmonics or the overtones.

Click here to read why do musical instruments sound different from one another? (PART - 1)

Physicists Express Harmonics In Terms of Wavelength

First Harmonic:

For instance, on a string with two ends, you notice that the fundamental cover exactly half a wavelength.

A full wavelength of the wave would cover two peaks, a crest and a trough. But the fundamental spans exactly one peak, which is half the wavelength.

So for the fundamental of a string with two fixed ends, the length of the string is equal to half a wavelength.

• • • Second Harmonic:

The second simplest standing wave you can have on a string with two fixed ends has 3 nodes. One at the end and one in the middle, plus 2 antinodes in between the nodes. It’s called the 2nd harmonic, and the string holds exactly one wavelength.

• • • Third Harmonic:

Similarly, the 3rd harmonic looks like it have 4 nodes and 3 antinodes and the string holds 3/2 wavelengths.

The Maths That Makes Musical Instruments Work

For a standing wave on a length of string, the number of wavelengths that fit on the string is equal to the number of harmonics divided by 2.

Musicians care about most of the frequency. A wave’s velocity only depends on its medium. For all harmonics, a standing wave’s frequency will equal to its velocity, divided by its wavelength.

For the fundamental with two fixed ends, the wavelength is twice the string’s length.

So the frequency of that fundamental standing wave, known as the fundamental frequency and is equal to the velocity, divided by twice the length of the string.

The frequency of the 2nd harmonic - standing wave with 3 nodes and 2 antinodes will be equal to the velocity, divided by the length of the string, which is twice the fundamental frequency.

The frequency of the 3rd harmonic with its 4 nodes and 3 antinodes and the frequency of the 3rd harmonics, with its 4 nodes and 3 antinodes, will be equal to 3 times the fundamental frequency.

The frequency of the standing wave with two fixed ends will just be equal to the number of harmonics, times the fundamental frequency. The number of harmonics is equal to the number you multiply by the fundamental frequency, to get the frequency of the harmonic.

This math makes musical instruments work.

Working of a Standing Wave on Piano, Guitar & Flute

Two fixed Ends

• • • Piano:

When you press down a key on a piano, you make a hammer strike a string, creating standing waves in that string.

Depending on the string’s mass, length and tension, every string in the piano gets tuned corresponds to a note, so it gives the fundamental frequency. For example, middle C has a frequency of 261.6 Hz.

• • • Guitars:

Guitars get tuned so that the fundamental frequencies of their strings correspond to set notes.

 

When you press down the strings in certain places in a guitar, you change the length of the active part of strings so that its fundamental frequency corresponds to a distinct note.

So for a standing wave with two fixed ends, we can relate wavelength, frequency, velocity, the length of the string and the number of harmonics.

Two Open Ends and One Closed End

• • Flutes/Pipes

• • • Two Open Ends:

The same thing is possible for a standing wave with two loose ends in an open pipe, for example, in a flute. A standing wave in a pipe with two open ends is kind of the opposite of the wave with two fixed ends. Instead of having a node at each end, it has an antinode at each end.

So, the fundamental standing wave for a pipe with two open ends will have two antinodes and one node in the middle of the wave. Then the 2nd harmonic will have 3 antinodes and 2 nodes and so on. But each harmonic still covers the same number of wavelengths.

As we know that the fundamental wave for a string with two fixed ends covers half of a wavelength, the fundamental wave for a pipe with two open ends also covers half of a wavelength. That half is just in a different section of the wave.

Like a string with two fixed ends, the second harmonic for a pipe with two open ends covers a full wavelength. In the pipe, the wave starts & ends with a peak instead of a node. So the equation for wavelength & frequency for a standing wave with two open ends will be the same as with two fixed ends.

• • • Closed At One End:

A pipe with one closed end and one open end works a little differently. These kinds of pipes are in the instruments, like pan flutes, where you blow across the top of a closed pipe to make music.

Here standing waves need a separate set of equations, for two reasons:

• The closed end of the pipe will be a node because the air molecules aren’t oscillating there.

• The open end will be an antinode because that’s where there is a peak in the oscillations.

So the simplest wave you can make in this pipe is a stretch from one node to one peak. But that is only a quarter of a wavelength in the pipe. Before, with both the string fixed at both ends and an open pipe, the fundamental spanned half a wavelength. But a pan flute pipe only covers a quarter of a wavelength.

Because the frequency of each harmonic is equal to the number of harmonic times the fundamental frequency. But for a pipe that is closed on one end, you can’t double the fundamental frequency or quadruple it or multiply it by any even number.

Because its results would need a node on both ends, which is impossible. So, a pipe that is closed on one end can’t have even-numbered harmonics.

Result From The Physics Of The Standing Waves

All of this helps to explain why musical instruments sound different from one another, even when they are playing the same note.

When you play a note, you are creating the fundamental wave with some other harmonics that are the overtones. For each instrument, different harmonics will have different amplitudes and therefore create a louder sound.

But because of the physics of the standing waves, instruments that have pipes with one closed end won’t create the even-numbered harmonics at all. That’s why a ‘C’ on the flute sounds so different from a ‘C’ on the bassoon.

Learn Music Online - Stay Home - Stay Safe

If you are a music lover, then learn music from your home. Meet our experts who teach piano, electronic keyboard, guitar online with our creative step-by-step structured curriculum. Get our additional course “Voice Culture” for free when you sign up for an Online Music Classroom membership! 

Check out the video tutorial, sheet music, midi, karaoke and bundle to buy from Music Library!

We provide you with an opportunity to buy musical instruments from Musician Product on Amazon. 

Join now at classroom.goldsmth.com

To Learn Music – Anytime – Anywhere

What is your opinion about this blog? Tell us in the comments below