Psychology of Music (Tan et al., 2010): Chapter 2.1, The Physics of Sound

In this chapter, we look to the science of acoustics, which explores how sound is produced by a source, propagated through a medium, and received by an auditory system. More specifically, we are focusing on musical acoustics, which Hall (2001) defines as the study of how sound is produced by musical instruments, how architectural design affects sound transmission, and the perception of sound as music. As such, the current chapter explores 3 main areas, namely: the physical properties of sound waves, the acoustics of musical instruments, as well as the acoustics of music performance environments. In this blog, I'll talk about the first area and cover the other 2 in a separate post.

The physics of sound

How is sound produced? Sound can simply be thought of as the transmission of energy from a sound source to a receiver through a medium (usually, air). This process starts with the vibrations of a sound source, which agitate and compress nearby air molecules. These air molecules, now carrying energy from the sound source, then transfer this energy to other air molecules in their immediate vicinity. Importantly, this results in a series of compressions and expansions (i.e., rarefactions) of the air molecules, which can be graphically represented by the peaks and troughs of a sine wave (see Fig. 1), respectively. This oscillating (i.e., repeatedly moving in a cycle) wave is called a sound wave, which is the physical basis of what we know as sound.

Fig. 1 Sound as a sine wave (Tan et al., 2010)

One commonly taught property of sound waves is that they are longitudinal waves. What this means is that the movement of oscillation runs parallel to that of the entire sound wave. To illustrate (see Fig. 2), if a sound wave is moving along the horizontal axis, then the air molecules too are moving back and forth (i.e., oscillating) along the same axis. This is contrasted with transverse waves (e.g., light and other waves on the electromagnetic spectrum), where the oscillation is actually perpendicular to the direction the wave is travelling. So, if a light wave is moving from left to right, then it is oscillating up and down.

This means that in longitudinal waves (i.e., sound), peaks and valleys in the sine wave are a result of compressions (i.e., dense areas of air molecules gathering) and rarefactions (i.e., low-pressure regions), respectively. Meanwhile, for transverse waves, the global sine wave pattern is a result of the wave literally moving up and down to create peaks and valleys.

Fig. 2 Transverse vs. Longitudinal waves (Zohuri, 2021)

Properties of sine waves

We've been speaking a lot about sine waves. At this point, it's good to remember that sound waves aren't literally sine waves. Sine waves, like any mathematical construct, are simply abstract representations of the real, physical phenomenon (i.e., the sound wave). In other words, they are simply mathematical tools to better understand the real world. With that out of the way, let's familiarise ourselves with two key terms related to sine waves: frequency and amplitude.

Frequency (see Fig. 3) refers to the number of cycles that are completed by a part of the wave in one second. To calculate frequency, we first find the period, defined as the time (in seconds) it takes for one part of the wave to go from one peak to the next. Then, frequency, F, is basically the reciprocal of the period, P (i.e., F = 1 / P). For instance, if it takes 1 second for one part of the wave to go from peak to peak (or from one part of its cycle to the same part of that cycle), then P = 1 s, and F = 1 / 1 s = 1 Hz (hertz, read as cycles per second). Or, if the period is 0.5 s, then F = 1 / 0.5 s = 2 Hz.

That's quite math-y, but does frequency mean anything? Put into the context of sound, the frequency of a sound wave is responsible for the aural sensation of pitch. Here, a sound with a higher frequency gives a higher pitch, while one with a lower frequency gives a lower pitch. For example, middle A (A4) on a piano conventionally has a frequency of 440 Hz, with the same note an octave below and above having frequencies of 220 Hz and 880 Hz, respectively.

Moving on to amplitude (see Fig. 3), this parameter simply refers to the maximum displacement of the sine wave from its resting state. In other words, amplitude is the magnitude of displacement at the wave's peaks and valleys. Psychologically, amplitude is related to the loudness of the sound. Here, greater amplitudes are a result of more energy being propagated and consequently correspond to the perception of a louder sound.

Fig. 3 Sine waves varying in frequency and amplitude (Tan et al., 2010)

Complex sounds

Sine waves are pretty neat. A little too neat, if you ask me. In reality, sine waves are a result of pure tones (i.e., a minority of sounds with only one frequency), with a majority of sounds consisting of multiple frequencies being combined in all sorts of ways. This simultaneous combination of frequencies is best described as a complex wave. The many different ways in which individual frequencies of varying amplitudes could combine lead to a rich array of sound qualities, or timbre, in which a complex wave can give rise to. Timbre is essentially the unique signature of a sound, and what allows you to differentiate between a piano and a trumpet even when they are playing the exact same melody.

The advantage sine waves have over complex waves is that they are easier to analyse. How might we analyse complex waves? To do this, we use a Fourier analysis, which can decompose any function (such as a complex wave) into a series of easy-to-analyse sine waves. This is highly convenient and useful for us, given that a complex wave is basically the sum of a number of sine waves of different frequencies (see Fig. 4). 

Fig. 4 A complex tone with its sinusoidal components (Tan et al., 2010)

The harmonic series

Once split into its component parts, we can start analysing the complex wave. The component sine wave with the lowest frequency is referred to as the fundamental frequency. This frequency informs the pitch of the sound. On top of the fundamental, we have a series of higher frequencies commonly referred to as harmonics or overtones (see Fig. 5).

Fig. 5 The harmonic (overtone) series (Tan et al., 2010)

A really cool thing about the fundamental and its harmonics is that their relationship can be elegantly mapped onto simpler integer ones. To illustrate this, starting from the fundamental (i.e., the 1st harmonic), the 2nd harmonic (or 1st overtone) has a frequency double that of the fundamental, the 3rd harmonic (or 2nd overtone) has a frequency triple that of the fundamental, and so on. For example, starting from F₁ (i.e., the fundamental), if F₁ has a frequency of 440 Hz, then F₂ = 880 Hz, F₃ = 1320 Hz, and so on.

And for the musically inducted, the harmonic series is a hint as to why some musical intervals and chords, such as a major chord (root - major third - perfect fifth), sound so natural, consonant, and pleasant. If you look at the first 5 harmonics of the harmonic series in Fig. 5, what you'll notice is that you spell out a major chord! Specially, F₁ (the fundamental) is the note C2, F₂ (the first harmonic) corresponds to C3 (the same note but just one octave higher), F₃ corresponds to G3 (a perfect fifth above C3), F₄ corresponds to C4 (same note as the fundamental, but just 2 octave up), and finally, we have F₅ which corresponds to the note E4 (a major third above C4). Again, this means that, when you play a single note, the first 5 harmonics (or fundamental + first 4 overtones) are simultaneously sounding out a major chord. How cool is that??

Other than having component waves of different frequencies, complex waves also consist of waves with different amplitudes. Graphically, this can be shown using a power spectrum analysis (see Fig. 6), which simply plots the amount of energy (a proxy for amplitude) that each frequency has. Overall, the take-home message here is that the different frequencies of varying amplitudes are what give each complex sound its unique aural characteristics!

Fig. 6 Example of a power spectrum analysis (Tan et al., 2010)

More on fundamentals and harmonics

Ever wonder why some sounds feel smooth and buttery, while others have a rougher edge to them? This has something to do with the relationship between fundamentals and their higher harmonics! When higher frequencies have simpler and integer relationships with the fundamental frequency, this gives rise to harmonic complex tones, or tones with smooth and melodious timbres. This changes when the relationship between the frequencies becomes messier and more complex. The result of this is anharmonic complex tones (see Fig. 7) with a rougher and noisier timbre. At best, they may be perceived as the raspy yet still melodious tone of a country singer. And at worst, what you might get is the overwhelming and annoying static from a radio!

Fig. 7 Fourier analysis of an anharmonic tone (Tan et al., 2010)

Concluding remarks

And so ends our journey exploring the physical properties of sound. There's still more in the chapter, though! In a follow-up post, I'll briefly cover the other parts of the chapter on the acoustics of musical instruments and venues. 

References

Hall, D. E. (2001). Musical Acoustics. Pacific Grove.

Tan, S., Pfordresher, P., & Harré, R. (2010). Psychology of Music: From Sound to Significance. http://ci.nii.ac.jp/ncid/BB01824497

Zohuri, B., & Moghaddam, M. (2021). Directed energy beam weapons the dawn of a new military age. Journal of Material Sciences & Manufacturing Research, 1–8. https://doi.org/10.47363/jmsmr/2021(2)120