# Harmonics and smooth “timbral” instrumental sounds

Indeed, everyone involved in music has often heard of harmonic sounds and perhaps even learned how to produce them on their instruments, but without understanding what they were and why they sounded so different from “traditional” notes.

In this article, I will explain what a sound is and its characteristics, emphasizing harmonics and their fundamental function. The reader will understand what timbre is and why various instruments sound different. They will also acquire the basics of classical harmony, understanding why chords (in particular, major) are formed in a certain way rather than another.

### What is sound?

Imagine beating the surface of a drum. The constituent skin possesses a high level of elasticity. Therefore, the application of an impulsive force produces a vibration. In other words, the surface begins to move rhythmically from bottom to top until it has exhausted the energy you have transmitted at the moment of percussing it.

For physical reasons (which I omit, as they require knowledge of differential equations), “elemental” vibration has a sinusoidal structure (i.e., y = sin(x)), as shown in the following diagram:

Over time, the movement progresses from a negative minimum to a positive maximum. In reality, the shape is not correctly sinusoidal, but more precisely, sinusoidal with exponential damping. Nothing complicated! It just means that the movement becomes smaller and smaller until it stops when the energy runs out.

As it is easy to understand, this surface movement rhythmically pushes the air molecules together, generating what is referred to in the jargon as a “pressure wave.” When pushed forward, the closer molecules will impact the slightly more distant molecules, imparting a force that will cause them to move in turn. This process will repeat rapidly (in the air at 20 °C, the propagation speed is about 343 m/s), coming to stress your eardrums and thus to hear sound.

### What are harmonics?

In a sense, I have already answered this question, but allow me a quick flashback. Remember when I said that the physical problem required differential equations? Whether or not you know what they are, keep in mind that the solution of this equation is what is commonly called a wave function and has the following simplified form in one dimension:

A(x, t) takes the name “wave” because it is a function that varies and propagates in space and time. If you fix a point x, the function will divert precisely like the graph I showed: A(x) = F – cos(wt + d). Similarly, if you set a time instant, you will see that the function takes on a different value at each point in space. Despite this little math effort, which I invite you to associate with the image of sea waves, know that this is indeed a harmonic!

Harmonic sounds are nothing but pure sine waves at a specific frequency. At this point, you may wonder what is so special about it, so it is time to talk about instrumental timbre.

### The timbre and richness of harmonics

In the previous section, we understood that a harmonic is a sine wave. So, I ask, are there different possible harmonic sounds with the same frequency? Assuming that each instrument can produce pure harmonics, will an A at 440 Hz produced by a piano and that of a violin be distinguishable? As you can easily understand, the answer is negative. No characteristic elements can distinguish two sine waves at the same frequency (apart from the phase shift, which is a secondary element).

But then why is a C played on a classical guitar with nylon strings and the same C emitted by a clarinet so clearly distinguishable? The answer is straightforward: each instrument possesses a personal tonal range that distinguishes it (within certain limits) from others. All right, the reasoning seems correct, but exactly what is the stamp?

### Natural sounds are much richer than you think.

Imagine playing middle C on the piano. What do you think you are emitting? A single harmonic? No. What has been discovered is that the emission of a particular sound by means that are not excessively (and often deliberately, as in the case of the tuning fork) poor produces a sequence of harmonics, the structure of which is constant. In the case of C, one will have:

Of course, the starting note does not matter. What matters are the intervals so that we can observe that:

• The fundamental note is C (frequency = f)
• The first harmonic is C, one octave above the fundamental (frequency = 2f)
• The second harmonic is the right fifth (G) after the first octave (frequency = (3/2)2f)
• The third harmonic is still C, two octaves above the fundamental (frequency = 4f)
• The fourth harmonic is the major third (E) above the second octave (frequency = (5/4)4f)

This structure is regularly repeated for any instrument, but what changes is the amplitude and phase shift of each harmonic. In other words, an actual sound is the (infinite) sum of sinusoidal terms that become smaller and smaller as the frequency increases and whose amplitude characteristics vary according to the medium used to produce the sound.

As a result, the actual wave (due to the superposition of harmonics) will no longer be sinusoidal. Still, it will have a pattern, for example, similar to that shown in the following diagram, where I have considered an A at 440 Hz, followed by four harmonics and a noise term due to imperfections in the media:

The curve is much more jagged and irregular, but these irregularities allow for the warm, glassy, metallic, etc. sounds performers seek. If only harmonics were played, the result would be more or less similar to that of a whistle or hiss, an effect sometimes desired at particular times (as, for example, in a section of Capriccio no. 24 of Paganini), but certainly not for an entire composition.

### Spectral analysis to better understand the role of harmonics

A frequency (i.e., spectral) analysis of the sound produced by an instrument can be performed to realize the contribution of individual harmonics. In the case of the wave shown above, the result is as follows:

As can be seen, the diagram has a peak at the frequency of the fundamental (i.e., A at 440 Hz), followed by a subsequent peak of smaller amplitude for the octave and so on, until a spurious high-frequency peak due to noise (generally minimized in actual performance). The “shape” of these peaks (along with the phrase, which, in this case, for simplicity, has been omitted) determines the timbral characteristic of the instrument and, caught by the ear, allows one to immediately distinguish a banjo from a trombone or a violin from a timpani.

### Very brief digression of harmony

Those who have studied the basics of harmony know that the C major triad consists of a major third (C – E) followed by a minor third (E – G). Looking at the sequence of harmonics shown above, you might, if you have not already done so, realize why the intervals of unison, octave, and right fifth are called “perfect consonances” (they are the first 3 sounds in the sequence) and why the interval C – E is also a consonance.

The discussion is too long to be covered in one paragraph. Still, it is sufficient to realize that the major triad is the “child” of the sequence of harmonics emanating from the fundamental. In contrast, a fact that might surprise many readers, the minor triad (e.g., C – E♭ – G) is entirely artificial in that the minor third interval from the fundamental does not appear in the first harmonics)! This is also why minor tones have been looked upon negatively for quite some time.

### Conclusions

I hope the article has helped to clarify what harmonic sounds are and to understand their importance in timbral evaluation and in choosing an appropriate orchestral ensemble for particular circumstances. In future articles, I will explain how to achieve harmonic sounds on the classical guitar, and at that time, we will also see how the same instrument can generate a very varied tonal range. Should you have any questions or wish to propose topics for discussion, please get in touch with me via the form provided. I will be happy to respond to you and consider your proposals!

### For more on the physics of music

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Physics and Music: The Science of Musical Sound
• This foundational text is written for students who want to go beyond the perceptual stage of music to learn how musical sound is created and perceived
• It surveys a wide range of topics related to acoustics, beginning with a brief history of the art and science of music
• Succeeding chapters explore the general principles of sound, musical scales, the primary ways in which sound can be generated, the characteristics of instruments, the use of mechanical and electronic recording devices, hi-fi stereophonic and quadraphonic sound, the design of electronic musical instruments, and architectural acoustics
• Comprehensive yet accessible, Physics and Music includes over 300 diagrams, photographs, and tables
• Each chapter concludes with questions, problems, and projects, in addition to references for further study
Fundamentals of Musical Acoustics
• Landmark book by leading expert, hailed for its astonishingly clear, delightfully readable explication of everything acoustical important to music-making
• "Comprehensive
• rigorous

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Last update on 2024-07-29 / Affiliate links / Images from Amazon Product Advertising API