The Secret Geometry of a Symphony: How Physics writes our Favorite Songs

Author: Saachi Khemchandani

Introduction

Have you ever heard about the Stradivari's miracle? Why is a 300-year-old violin still priced above all others? For centuries, musicians and historians have whispered about the “Stradivarius Miracle”. People might think that it is Stradivari's artistry that makes it so “magical”. Others may attribute this to the “magic” varnish. However, the real reason lies in the physics that was used in the violin's making.

A Stradivarius violin made in 1684 (from Capon, 2017). https://www.science.smith.edu/climatelit/stradivarius-violins/

While we often perceive listening to music as an emotional experience, it is fundamentally a physical phenomenon governed by air. Music isn't simply an art, it's a series of longitudinal waves that travel at about 343 meters per second at standard temperature. The music we listen to isn't just a piece of art but the perfect balance of density, resonance and wave interference. To understand why a 300-year-old instrument remains the standard, we must understand the mechanics of the vibrations.

The difference between noise and a masterpiece lies in the mathematical patterns hidden in the sounds. Musicians create music by organizing various sounds together. When we listen to music carefully, it can be noticed that each song has its own noticeable pattern. To further understand the concept of how a violin works and how a pattern in sound is created it is necessary to decode how a sound wave works.

Anatomy of a sound wave

To decode the nature of sound, one must understand how energy travels through a medium. In physics, a wave is described as a disturbance due to which energy is transferred from one point to another through a medium like water or air. Waves are typically of two types - transverse waves and longitudinal waves. While light waves are transverse waves, sound waves are longitudinal waves. These waves travel through a medium like the spruce wood of a violin or the surrounding air.  

Image displaying difference between noise and tone. (Source: Feature.fm blog)

The movement of sound is characterized by "push and pull" movements of the molecules. These movements are called compressions and rarefactions. When we think about characteristics of sound waves, we think about amplitude, intensity (how loud is it) and frequency (the pitch of the sound being produced). We mainly look at three fundamental properties namely wavelength, amplitude and frequency.

Longitudinal wave. (Source: https://www.geeksforgeeks.org/physics/longitudinal-waves/)

Just like we use instruments to measure a physical object, we use wavelength and amplitude to quantify the dimensions of a sound. The distance between two corresponding points on consecutive waves is the same and can be measured. This distance is called the wavelength. The amplitude is the height of the wave which is consistent and can be measured. The frequency of the wave is the number of cycles it completes in a second.

The relation between these properties is not random but bound by a mathematical equation. While listening to music the "tone" is often talked about. The tone mainly depends on two factors: the frequency of the wave and the amplitude. The frequency determines the pitch (degree of highness or lowness) of the tone. The amplitude affects the volume of a tone, higher the amplitude, the louder the volume. A mathematical connection between these properties is governed by the equation:

v = fλ

where v is the speed of sound (343 m/s), f is the frequency, and λ is the wavelength. This equation explains why, as a violinist moves their finger up the fingerboard (shortening the string and the wavelength), the frequency - and thus the pitch - must increase.

Timbre and Overtone

Even when two instruments play the exact same note, our brain can detect a "sonic fingerprint" that tells them apart. When we hear the same note being produced by a violin, guitar, piano or a human we can immediately differentiate between them. The quality of sound that distinguishes between these sounds is timbre (pronounced as tam-ber). Timber brings color, emotion, and individuality to musical sound. 

What we might consider a single note is actually a complex compilation of hidden layers. Overtones are hidden layers of sound that accompany every note and give each instrument or voice its distinctive color and identity. Timbre is determined by overtones. Without overtones all instruments would sound identical, thus stripping music of its emotional depth and identity. When a musical instrument produces a note, it doesn't emit a single frequency but a combination of one fundamental frequency and a series of overtones (also known as harmonics).

Timbre and various overtones of a sound. (Source: https://jews-harps.com/the-art-of-sound-coloration/)

However, this is not the simplest form of music. Music in its simplest form is monotonous - only composed of pure tones. Monotonic music is usually considered dull and lifeless like a 1990's ringtone. Real music on the other hand is polytonic - a mixture of pure tones played together in a manner that sounds harmonious. Sound with multiple frequencies - like that of an instrument or the human voice - would still be periodic but more complex than just a simple curve. The human voice and musical instruments produce sounds by vibration. What vibrates determines the type of instrument.

Classification of instruments. (Source: https://physics.info/music/)

Physics of the Violin body 

The violin acts as an amplifier, turning the almost silent vibration of a string into a room-filling performance. The study of the physics of the violin gives us a fascinating insight into how the instrument converts the player's intricate movements into musical sounds. The strings hardly make any noise by themselves; they are thin and slip through air barely making any noise. 

Every component of the instrument is precisely engineered so as to manipulate the air's pressure. The thicker, more massive strings vibrate more slowly, so the strings on the violin get thicker as you go down from the E to A to D to G strings. The tension is adjusted by tuning pegs: tighter gives higher pitch. The frequency of the sound being produced also depends on the length of the string that is free to vibrate. The body consists of the front and back plates, the sides and the air inside. All these serve to transmit the vibration of the bridge into the air around the instrument.

The bridge and body of the violin transmit some of the energy of vibration of the strings into sound in the air. The bridge itself is very effective at transmitting power to the body at frequencies of about 1 - 4 kHz, a range where the human ear is very sensitive.

Cross section at the bridge, seen from the tailpiece end. At low frequencies, the bridge oscillation approximates rotation around a point near the treble foot. (Source: https://newt.phys.unsw.edu.au/jw/violintro.html)

Human voice as an Instrument 

The most versatile instrument is not made of wood or metal, but of bones and muscles. The human voice is one of the most overtones rich instruments. With the human voice we can shape timbre through articulation, breath control, and resonance. Singers possess the specialized technique of amplifying specific overtones creating the illusion of two pitches being sung simultaneously.

The human vocal tract. (Source: https://www.phys.unsw.edu.au/jw/voice.html)

How do our ears work

Our biological hardware is the final stage in the journey from a vibrating string to a conscious thought. The way we hear music also depends on the way our biological hardware works. First, the sound waves are focused into the ear canal via the ear flap also known as the pinna. These waves impinge on the ear drum, causing it to vibrate. The three tiny bones of the middle ear - the hammer, anvil, and stirrup - transmit these vibrations to the oval window of the cochlea.

The cochlea is a fluid-filled spiral structure in the inner ear. Vibrations entering the cochlea create waves in the fluid, which move the basilar membrane, which holds ~30,000 hair cells in the organ of Corti. Sensitive hairs respond to the sound vibrations, and send signals to the brain via the auditory nerve. Brain finally processes audio as "music".

Inner anatomy of the human ear. (Source: Molecular Biology for the Auditory System, MIT OCW. Image courtesy: http://www.nasa.gov)

Conclusion

While the geometry and anatomy of the wave explain how sound moves, the "Stradivarius Miracle" may also be connected to historical climate conditions. During the late 17th century, Europe experienced a "Little Ice Age." The prolonged cold winters may have caused the spruce and maple trees of the era to grow slowly and uniformly, resulting in wood with a unique acoustic density and a higher modulus of elasticity (stiffness). In physics, the resonance of a body is heavily dependent on its density and stiffness. These specific wood properties allow the violin's front and back plates to vibrate with a "purity" that modern wood cannot replicate. When the bridge transfers energy to this high-density wood, it creates a specific harmonic profile - a "Stradivarius Timbre" - that perfectly balances the fundamental frequency with its overtones. While these unique wood properties may contribute to the characteristic harmonic profile associated with Stradivarius violins, they are only one part of a much larger story involving craftsmanship, design, aging, setup, and perception.

References

• Joe Wolfe. How does a violin work? UNSW. https://newt.phys.unsw.edu.au/jw/violintro.html

• Chia Madani. Overtones and Timbre. Kurdish Talent Academy. https://kurdishtalent.academy/post/364/overtones-and-timbre

• Mrittika Ghosh. The Physics Behind Music. https://www.researchgate.net/publication/369369732_The_Physics_Behind_Music

• Music and Noise. The Physics Hypertextbook. https://physics.info/music/

• Marren Vasquez. How A Violin Produces Music. The Masters Music School. https://tmms.co.uk/how-a-violin-produces-music/

• The Science of Sound. NASA. https://www3.nasa.gov/specials/Quesst/science-of-sound.html

• Steven Errede. The Physics of Music & Musical Instruments. UIUC. https://courses.physics.illinois.edu/phys406/sp2017/POM_Talks/2012_UIUC_POM_Talk/2012_UIUC_POM_Talk.pdf

About the author

Saachi Khemchandani is a high school student. She likes reading fiction novels. She recently completed her grade 10 and discovered her love for researching and writing articles, especially on scientific topics.