Principles of Sound and Music: Introduction

Black and white photo of a man at a keyboard with a complicated electronic console and an enormous horn behind him
Thaddeus Cahill playing his Telharmonium c. 1906

This is a series introducing the basic principles of sound and music, with applications in analog and digital sound design, modular synthesis, and other kinds of contemporary experimental music. We will start by covering basic concepts and work towards specific solutions, problems, and techniques. While some articles will build on others, the series will not be strictly sequential. Some topics I plan on covering: the physical nature of sound; basics of sound perception; ways to analyze the parts of timbre; mathematical representations of sound and how to read them; scales, harmony, and tuning systems; time domain and spectra; waveshaping and distortion; etc.

Each topic will be treated in depth, without any oversimplification or “hand waving”—incomplete reasoning which puts on a show of providing reasons but which doesn't really explain anything. If you've ever encountered presentations of music theory that read like a list of arbitrary rules and conclusions, and wondered things like “why do we use 12 tones? why are scales usually 7 notes?” Or if you've read brief introductions to synthesis and thought “wait, isn't there more to sound than brightness and envelopes?” rest assured that this series will be different.

As such, it will require a little patience and persistence, but no real background knowledge. For those who do have some background knowledge in these topics, note that there are some commonly repeated ideas about sound and music that are simply untrue, but many more ideas that come from a way of thinking about sound and music that, while valid, diverges significantly from mine. Some readers will agree with my takes, some will not, but I will try to be clear enough about what's at stake that each can decide for themselves. Indeed, the focus on experimental music demands that you question at least some of things presented here or in other sources.

General Approach

Approaching the science of music and sound from the point of view of the experimentalist means, first and foremost, understanding why we have arrived at these particular modes of analysis and guidelines for creation. When we really understand why, we start to see why and how things could be otherwise, and this insight drives our creative practices to new places. Science itself is constantly changing. In the course of time, we will find out that some of the things we now believe are false. But most of the things we believe now we'll believe in the future, too. We just won't believe them in the same way. Newton could analyze the orbits of all the planets according to a universal force of attraction, gravity. For Einstein, that force of gravity became the warping of space itself. But on an everyday scale, Einstein's theories give the same predictions as Newton's. Music theory changes in similar ways. For example, the seventh of the C major scale, once seen as ambiguously either B or B, now is unambiguously B, while the scale with B in the seventh is called a mixolydian scale. We were never really mistaken about what this seventh was or how it worked with other chords, but we organized our thoughts about this differently in the past. As musical experimentalists, we get to be part of this process of change, generating new pieces that don't follow along with our typical understandings of how music works, and thereby changing that understanding.

Unfortunately, many of the popular technical treatments of music and sound oversimplify things by glossing over controversies, the reasonings behind theories, and gaps in our knowledge. While many of these expositions are great and interesting in their own right, they are a problem for experimental music, where we want to know the weak points in our theories just as much, if not more, as the strong points.

Our analyses of sound, as well as the more technical analyses of music, have developed a vocabulary of mathematical symbols and terms, as well as conventional ways to write them, to communicate various properties of sounds, sound composition, and sound design. Just as the serious student of classical music will have to learn a lot about musical notation, the serious student of experimental electronic music will have to learn a bit about mathematical notation. In both cases, while it is perfectly possible to be an excellent musician without knowing any notation at all, it can be difficult to communicate with the wider community of musicians without knowing something about this notation. Nevertheless, mathematical notation is not commonly part of the training of most electronic musicians. As a consequence, I believe, there is not enough communication between the communities of musicians and the people designing the equipment with which they work. This leads to a lot of deliberate obfuscation in musical interfaces, as well as controls that confine new possibilities to familiar terrain, just to avoid having to explain a common mathematical concept. Sometimes this obfuscation is even presented as a virtue, as if muddledness and lack of control was our only weapon against rigid musical choices. I promise you that it is not.

If you are one of those people who encounter mathematics with some trepidation, relax. While understanding complex math can be beneficial for a musician (and I will include enough here so that those who want to go down that path can do so), there is not all that much you need to learn to be able to read and intuitively understand the mathematical expressions describing sounds. We do have to start somewhere, and for my purposes, that somewhere is high school algebra. This series assumes that you know very basic things about how to read an equation, like that letters stand in for values, that putting letters and numbers next to each other means multiply, that you treat whatever's inside parenthesis as if it were a single thing but that otherwise multiplication happens before addition, etc. If you get stuck or want to read out of order, a good place to look is the Glossary of Musical and Mathematical Terms.

Lastly, some might wonder at filling a series written for musical experimentalists with mathematics. Won't all this math make us more like robots and less like creative humans? Isn't mathematics part of an overly mechanistic worldview to which, as artists, we're trying to offer alternatives? Isn't mathematics something that we're absolutely sure about, so not really a place to experiment? Yes and no. It is my contention, here and in other areas of my work, that mathematics always involves choices. Mathematics really does capture some element of universality. That is, math expresses things that are true across all times, places, and ways of thinking, human or otherwise. But math isn't itself a universal truth; it expresses universal truths. And mathematics can only ever express universal truths according to a contingent and particular way of thinking.

As an example, the Ancient Greeks thought that numbers were things like \(1\), \(2\), \(3\), etc.—their concept of number didn't include things like \(1.5\), \(-5\), or \(0\). Sometimes this story is told as if the discovery of zero, decimal points, and negative numbers were just a matter of progress, something the Greeks would have eventually done if their civilization hadn't given way to all of those others which actually did come up with these concepts. But this is not the case. The Greeks were well aware of fractional measures which lay between the whole numbers, but to them these measures could only be expressed by two numbers, e.g. \(3/2\) (\(=1.5\)). After all, how could you possibly measure something other than to use some measuring device evenly divided into sections, and then count the sections? Things like \(\sqrt{2}\) were for the Greeks not, properly speaking, measurable. You could either use a measuring stick that could exactly measure the sides of the square, but not the diagonal, or a measuring stick that could exactly measure the diagonal, but not the sides (the diagonal is longer than the sides by a factor of \(\sqrt{2}\)). Even given a perfect world with infinite accuracy and resolution, you can never do both without some error. Today, we think about things like \(\sqrt{2}\) as numbers, just not numbers that we can write down in a finite amount of time (because they go on forever after the decimal point). That is, we say that \(\sqrt{2}\) is a number because we imagine counting an infinite number of little tick marks, and on this infinite measuring device we can put all the incompatible measuring devices of the Greeks. Certainly something has been gained by these changes in how we think about these problems. But has something also been lost? Quite possibly.

The way we are taught mathematics typically involves lots of hand waving and assertions without real explanations. Consequently, we aren't shown these choices, and mathematics looks like the antithesis of creativity, just a boring list of facts and mechanistic procedures. But as we learn more about the real reasons for things, math starts to look less like a rigid structure and more like a playful game, like a John Cage piece that creates wonderful things through random chance coupled with a few rules that we chose to give the piece a pleasing shape. As such, the mystery becomes not about how mathematics involves choices, but about how something so totally full of choices can nevertheless produce such clear knowledge that we can use it to smash atoms and send probes beyond the solar system. This realization that somehow a game of rearranging symbols on paper (mathematics) is able to show us immutable truths about the world is what Wittgenstein refers to as the “mystical feeling” in mathematics. There is, I think, a readily apparent affinity between this mystical feeling and the mysteries of music, where somehow the simple arrangement of sounds in time and space can lead to a sense of profundity and truth, despite (or even because of) the way these sounds lack the clear semantic content of language.

In this series, the choices we'll see expressed in mathematical formulae will be much more mundane. For example, if you think about sound as a one dimensional wave, it can be expressed as an amplitude that is a function of time. But is sound a one dimensional wave? Sort of. Mostly. Being able to push on those sort ofs and mostlys is what makes experimental music so interesting. Though I will certainly miss some of these, I'll do my best to draw attention to these sort ofs and mostlys as we learn about various topics.


I will try to post a new article here at least once every two weeks. Individual posts will more or less stay fixed, but the glossary will be a living document, updated as needed. To be notified when a new post goes up, follow @ns.instruments on Instagram or subscribe to our newsletter.


Other than this series, there are a great many wonderful educational resources out there. I am indebted to too many to list here, but I want to give a particular mention to Julius O. Smith's wonderful online books on digital signal processing. These are an invaluable resource for someone who wants to approach the design of signal processing algorithms, or just learn more about how the algorithms work, especially in the digital domain.

For more about the difference between Ancient Greek and modern concepts of number, see Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.

For a different but related take on the choices involved in mathematics and what this says about mathematical objects, see this Numberphile Video.

The phrase “mystical feeling” is from Ludwig Wittgenstein, Tractatus Logico-Philosophicus.

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