How imperfect is sound? Part One.

by Gerard

Pierre Auguste Renoir – Gabrielle and Jean

Note or noise?

Take a heap of sand. Now, take one grain away, is it still a heap? Of course. But what if we continued to do this until there was only one grain of sand left? Would it be a heap? Our intuition tells us no, but this seems to be the logical conclusion of our reasoning.

This dilemma is commonly know as the Sorites Paradox. You might wonder why I’m talking about this in a post about sound. To answer that I first need to say a little about the nature of sound, specifically, the relationship between pitch and timbre.

A pitch is a single frequency, for example, the note A is generally placed at 440 hertz (cycles per a second). We perceive these cycles as sound. However, in all but the most controlled environments we will never hear a pure sine wave. In addition to the loudest (usually) and most perceptible frequency, known as the fundamental, there are a series of mathematically related overtones. It is the relative amplitudes (volume) of these overtones that give a sound its unique ‘colour’ usually referred to as timbre. For example, the same frequency played on a violin, will emphasise a different set of overtones to those that would be emphasised if it were played on a guitar.

Now, back to the paradox. In sounds such as those I have described about, the overtones themselves are sine waves (some overtones may have their own series of overtones, but this is not important for our purposes). Based on this fact, theoretically any sound can be broken down as the sum of a set of sine waves at varying amplitudes. Now we can rephrase the paradox in terms of sound.

Take a sine wave of a specific frequency. Add an overtone (another sine wave), is it still a note? Yes. Add another. Is it still a note? Of course. Eventually, we will have summed up so many sine waves that the sound we have constructed is equally loud across all frequencies. This is the very definition of noise.

When, then, does a note become noise? It seems there is no satisfactory answer, but the paradox in itself has some interesting implications.

It seems odd now, to me at least, to speak of ‘right notes’ and ‘wrong notes’. But what it also shows is that pitch and timbre, so rigorously separated ideas in western music, in reality are inextricable from each other. This correlation between pitch and harmony has been explored by some modern composers, such as Olivier Messiaen, who have used harmony to generate timbres, as well as other composers who have used the overtone series as the basis for their melodies.

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