15.
François Gaspard, Boris Wilmot & Ohme (BE)

Sine

Texts by Raoul Sommeillier

” Music is a science which must have determined rules. These rules must be drawn from a principle which should be evident, and this principle cannot be known without the help of mathematics. I must confess that in spite of all the experience I have acquired in music by practicing it for a fairly long period, it is nevertheless only with the help of mathematics that my ideas became disentangled, and that light has succeeded to a certain darkness of which I was not aware before. “

Jean-Philippe Rameau (1683-1764), French composer and music theorist.

What’s a sound? Sinewaves

Music affects our feelings and our mood. It has a strong impact on the cultural, social and economical life of our society: music moves the world. Music is a highly interdisciplinary topic: it involves acoustics, audiology, auditorial physiology, psychoacoustics (thus acoustics and psychology), cognitive science, mechanics, electronics, informatics, engineering, and so on. Mathematics provides language, structure and calculus to science and engineering, and pervades many of the above disciplines. In particular mathematics is at the basis of the theory of sound and of the construction of musical scales.

In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. It can be described, modeled and even created using mathematical objects, especially with a famous mathematical function you may already heard about: the sine.

Music is a combination of many sounds, and therefore a combination of many mathematical objects!

Mathematics allows us to synthesize new types of sounds and to digitize the way music can be produced.

The musical signal is characterized by:

Frequency. Usually, one singles out the frequency of the fundamental component of a sound. This is the greatest common divisor of the frequencies of its partial tones.
Duration. This is often expressed in terms of relative duration; for instance, the musical score indicates the ratio between the duration of a tone and that of the other tones.
Intensity. The acoustic power carried by the acoustic wave. For each pure tones, this is proportional to the square of the wave amplitude.
Timbre (also called the color of the sound). This reflects how the acoustic power is distributed among pure tones. It depends on the source of the sound.

These acoustic elements are combined with musical features such as melody, harmony, tonality, rhythm, meter, texture and form.

These basic musical elements are tones (or notes). To be precise, the former term denotes a sound characterized by precise frequency and duration, the second one denotes the corresponding symbol on the staff (on a sheet of music).

The difference between these two meanings may be subtle; we will come back to this later. From a less linguistic and more musical point of view, pure tone is defined as a sinusoidal wave, that is, the real or imaginary part of a function like \(A \times sound(rt)\) with \(A > 0\) and \(r ∈ R\). Sound produced either by the human voice or a musical instrument is a superposition of pure tones. However, a real sound may also include inharmonic components.