The Maths of music - ENG

Design and translation: Stéphanie Jolicoeur

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The mathEMATICs of music

2. Pythagoras and proportions

1. It's all about waves...

3. Geometry, really?

5. The acoustics of the Maison symphonique

4. Music and brains

Properties of sound waves

The nature of sound

Music is the art of playing with sounds! On the physical plane, sounds correspond to waves which are themselves the result of vibrations: vibration of a string, vibration of the air in a pipe, etc.

Vibration generates a shaking of the air (the sound wave), which propagates, like throwing a pebble into a river. The air molecules are "jostling" and move and collide. Once the wave has passed, each particle returns to its rest position, that is to say the position it occupied before the passage of the wave. These tiny movements of colliding particles cause pressure changes in the air: thus sound is born.

Even if sounds form complex waves which are not perfectly sinusoidal, some notions of basic physics make it possible to describe the musical sonorities!

Timbre

Amplitude

Frequency

A young human can hear sounds ranging from 20 Hz to 20,000 Hz! Below and beyond, the sounds do exist, but are not perceptible to us.

The pitch of the note (low or high) is related to the frequency of the sound wave, i.e. the number of periods (or peaks) that the wave travels in one second. It is expressed in Hertz (Hz). A high-pitched sound has a high frequency: the crests of the wave are close together.

DID YOU KNOW...

Frequency

Amplitude

The height of the wave crest is called the amplitude; it corresponds to the sound volume expressed in decibels (Db). The higher the crest of the wave, the greater the amplitude of the wave, the higher the volume.

Noise is measured on a scale ranging from 0 to 130 decibels on the Decibel Scale. 0 dB represents the threshold of audibility and 130 the pain threshold. Most everyday sounds are between 30 and 90 dB.

did you know...

Timbre

If sound is a wave characterized by an amplitude and a frequency, it is also characterized by its shape : the timbre, or sound quality. By changing the shape of the wave, you can hear the same amplitude and frequency, but a different timbre.

Although being an auditory phenomenon, the timbre of a voice or a musical instrument is often described using a tactile, gustatory or visual vocabulary! A sound is therefore "clear", "round", "warm", “sour”, or even... “chocolate”!

did you know...

Pythagoras was a Greek philosopher who lived from 580 BC. to 495 BC. He was also a brilliant mathematician and founder of the Pythagorean Brotherhood, which formulated principles that influenced the thought of great thinkers like Plato and Aristotle, and contributed to the development of mathematics and Western rational philosophy.

660 Hz

440 Hz

Frequency 1

Frequency 2

A bit of theory… As early as Antiquity, Pythagoras managed to calculate the sound ratios of the most harmonious musical intervals. The interval between two notes is measured by calculating the frequency ratio between the highest note and the lowest note.

FIFTH

Sol

396 Hz

The fifth, which has a mathematical ratio of 3/2, is the most consonant interval after the octave. Example: this is the mathematical relationship between the frequencies of Do3 and Sol3, where Sol3 (396 Hz) = 3 / 2 of Do3 (264Hz).

264 Hz

132 Hz

The ratio between the two notes of an octave is 2:1. So the frequency of the higher note is twice the lower one.

OCTAVE

Do

Si

La

Sol

Fa

Mi

Re

Do

Example : the octave is the interval between two notes of the same name (low C and high C).

Johann Sebastian Bach is a German musician, organist and composer who lived from 1685 to 1750. He wrote hundreds of pieces for organs, harpsichord, orchestras, as well as masses. He remains one of the greatest composers of European culture.

The Toccata and Fugue in D minor by the great German composer Johann Sebastian Bach is certainly the most famous organ piece in the world. It was composed between 1703 and 1707 and is a remarkable illustration of how the art of musical writing can be explained by mathematics. The fugue is a musical form where different parts take up the same motif.

The Golden ratio

Translation and symmetry

In geometry, translation and symmetry are geometric transformations which make it possible to obtain an image figure from an initial figure following a "sliding" of x units horizontally and y units vertically or a vertical or horizontal "rollover".

Translation and symmetry

Bartók was fascinated by mathematics and by the shapes that can be found in nature, such as the spirals of snail shells or the rows of a pine cone. These two examples are forms of natural architecture illustrating one of the oldest numerical rules which has been called the Golden Rule.

The Fibonacci arithmetic sequence is obtained by adding a number to its immediate predecessor in the series: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5, etc., which gives 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on. The further you go in the series, the closer you get to the golden rule of dividing any number by its successor.

The Golden Ratio

Béla Bartók is a Hungarian composer and pianist, who lived from 1881 to 1945. A pioneer of ethnomusicology, he remained attached to his native country all his life, even after immigrating to the United States.

A team of Quebec researchers recently discovered that when we listen to music we like, our brain releases dopamine. It is a molecule associated in particular with pleasure that gives us a real hormonal “boost”.

Music stimulates and encourages physical exertion because it literally "turns on" all areas of our brain, not just the auditory area (area of ​​the brain responsible for interpreting sounds).

The emotions induced by the music translate into physical reactions such as shivers down the spine, goosebumps, rapid heartbeat or even blinking of the eyes, which are an indicator of the level of anxiety of the individual.

It takes barely 500 milliseconds for our brain to distinguish whether a music is happy or sad: this suggests that the human brain responds as quickly to music as to a life-threatening stimulus. And knowledge of music has nothing to do with it: when you ask a group of people (musicians or not) to group together pieces according to the emotion they arouse (sadness, joy, anger, nostalgia), they all form the same groups.

70% of the hall surfaces are covered with beech wood, due to its valuable acoustic properties. In some places the wood is polished and smooth, and in others it is less so; it is to modulate the sound propagation.

Inside the building, all potential sources of noise such as heating systems, elevators or toilets have been designed to prevent the propagation of sound.

The hall is enveloped in a structural shell that insulates it from all noise and vibrations coming from outside. The pillars of the room are placed on cushions, and it is lined with 175 cushions which reinforce sound insulation.

The auditorium is equipped with movable ceiling panels allowing the geometry of the place to be modified at will. This mechanism makes it possible to adapt the acoustics according to the type of event presented or even the musical genres: the adjustment of the levels of reverberation makes it possible, for example, to create intimate or grandiose atmospheres.

Watch this!

Ear practice

Timeline

Quiz

Maxime Goulet

Which Quebec composer composed a piece inspired by the motto of the Olympic Games?

Translation and symmetry

What are the two geometric motifs that we often hear in Bach's fugues?

Pythagoras and Fibonacci

Name two mathematicians who have a connection with music.

Listen to these excerpts from Also Sprach Zarathustra, composed by Richard Strauss in 1896. Do you hear a change in frequency, amplitude or timbre?

Ear practice!

2008

1936

1703-1707

1202

Do you recognize these dates?

580 BC

May the maths of music be with you!