A B Sinusoid D E F G 🎶 (part 1)

Music is an important part of our everyday life. It is with us when we drive, when we run, when we go out, when we chill out at home. It calms us, it relaxes us, it soothes us, it makes us happy, it affects our mood and our emotions. With their different sonorities thanks to different instruments, music is a complex combination of a very simple element, a note. What is a note if not a sound, a specific mechanical wave, a sinusoid with specific characteristics? How does a mathematical element that simple produce such a beautiful harmony? Let’s find out!

Mechanical wave

A mechanical wave is a wave propagating through a medium, solid or fluid [1]. Therefore, it can’t go through the void of space (contrary to electromagnetic waves [2]). Sound, earthquakes or ripples are mechanical waves. Only the first one has our interest here but let’s see what types of mechanical waves exist.

In transverse waves, oscillations are perpendicular to the direction of the wave. The membrane of a drum creates transverse waves when hit. S-waves of an earthquake are another example of such waves. Light has the same behaviour, but again, not a mechanical wave [2].

Contrary to transverse waves, longitudinal waves spread in the same direction of the wave. P-waves of an earthquake and, more importantly, sound are longitudinal waves.

The last type are surface waves which, as their name tells, propagate on the interface between two media, such as ripple spreading at the surface of water.

When the medium is limited in space and wavelength, we call it a stationary wave. A musical example of that is a vibrating wire like a piano wire or a guitar wire.

Note is sinusoid

Like said before, sound is a longitudinal wave. Usually, a sound is a combination of multiple components in one dimension that can be represented as a sinusoid, a periodic signal.

This signal has two characteristics: its intensity (measured in decibel dB) and its frequency (measured in hertz Hz). A period is a repeated pattern in a signal (also called a harmonic is acoustic) and the frequency is number of times this pattern is repeated in one second. A note is a signal with a determined frequency.

We need a reference to establish the whole set of notes. That reference is the universal A note, the A of the 3^rd octave, with a frequency of 440 Hz. We have a base note from which we can determine all others. For that we use the following equation:

A bit of explanation about this equation. f_n is the frequency of the note we want to calculate (in hertz Hz), f₀ is the frequency of our reference note (in hertz Hz), n is the number of semitone we want from the reference note and can be positive or negative depending if you want an upper or a lower note. n is divided by 12 to represent the twelve semitones in an octave. This means that if n is a multiple of 12 the outcome is the same note but at a different octave and its frequency will differ from a factor 2. With a few calculations we can have all the notes we need. [3] [4]

Now we know our solfège but we make notes with a music instrument. We will put mathematics aside to come back to physics.

Strings theory

First part: string instruments. When a string vibrates, it creates a mechanical wave which spreads along the string. When it reaches an extremity, it continues spreading along the string at the opposite direction. It is the stationary wave mentioned before. Still, we need to control the wave to create a note.

This stationary wave has a specific behaviour of another equation (thanks to the Buckingham theorem [5]):

Here comes the explanation. First, 𝜈, the frequency (in Hz), the parameter we are interested in. The other parameters are a constant 𝜅, the length of the string L (in m), the mechanical tension of the string (in kg.m.s^-2) and the lineic mass (mass per length unit) of the string 𝜇 (kg.m^-1).

It means that to change the frequency, we can change the length, the tension or the composition (materiel or thickness) of the string. A piano plays on the length only (requiring more strings) while a guitar plays on all of the parameters.

For the piano, every key is linked to a hammer hitting the corresponding string. One string for one key. Longer is the string hit, lower is the frequency, producing a deeper tone. In the other hand, shorter string equal higher frequency equal higher tone.

For the guitar (and string instruments played with a bow), all the strings have the same length but with different thickness and materials (nylon, steel). The tension is changeable thanks to the tuning pegs. A guitar having usually six strings, it would mean six notes but a guitar player can artificially change the length of the string by pressing the different frets.

There is still the question about the other instruments: wind instruments, percussion instruments. How do they produce sounds and music? I will first let you digest the equations and informations in this post and we will answer all this next month in the part 2. Meanwhile, try to hear the sinusoids when you play or hear string instruments.

References

[1] Giancoli, D. C. (2009) Physics for scientists & engineers with modern physics (4th ed.). Upper Saddle River, N.J.: Pearson Prentice Hall. ISBN-13 9780133892741

[2] All You Need Is Science, What is the colour of radio waves?

[3] Adolphe Danhauser, Théorie de la musique : Édition revue et corrigée par Henri Rabaud, Paris, Éditions Henry Lemoine, 1929, 128 p. (ISMN 979-0-2309-2226-5)

[4] Claude Abromont et Eugène de Montalembert, Guide de la théorie de la musique, Librairie Arthème Fayard et Éditions Henry Lemoine, coll. « Les indispensables de la musique », 2001, 608 p.  (ISBN 978-2-213-60977-5)

[5] Edgar Buckingham, “On physically similar systems. Illustrations of the use of dimensional equations”, Physical Review, vol. 4, n^o 4,‎ 1914, p. 345-376. https://doi.org/10.1103/PhysRev.4.345