Mathematics and Music: The Place Where Numbers Meet Emotions

The relationship between mathematics and music is one of the oldest and most compelling discoveries in human history. Although at first glance one appears abstract and bound by precise rules while the other seems emotional and free, music is fundamentally built upon a mathematical order. Every melody, rhythm, and harmony that pleases the ear is a reflection of numbers ratios and patterns. In this article we will examine closely the historical origins scientific foundations and mathematical structures underlying the composition of music.

The Intersection of Mathematics and Music: Pythagoras’ Discovery

The first person to systematically investigate the relationship between mathematics and music was the ancient Greek philosopher Pythagoras. Pythagoras observed that the pitch of a sound produced by a taut string is directly related to the length of the string.

In his experiments he observed:

• When the string was divided into two equal parts the pitch rose by an octave
• When the string was divided into three parts the resulting sound formed a harmonious interval with the previous one

This discovery revealed that harmonious musical intervals are based on simple whole number ratios.

For example:

• 1/2 → Octave
• 2/3 → Perfect fifth
• 3/4 → Perfect fourth

Pythagoras’ work is regarded as the first scientific step toward explaining musical harmony through mathematical ratios.

^{Mathematics and Music (}Bogazici University Science Club⁾

The Mathematical Structure Underlying Music

The Nature of Sound and Frequency

Sound is physically a wave motion that vibrates at a specific frequency. This frequency determines whether a sound is high or low. Today the note A is universally accepted as having a frequency of 440 Hz. From a mathematical perspective sound is not a single vibration but rather the result of the combination of many sine waves with different frequencies and amplitudes.

In the 19th century the mathematician Jean Baptiste Joseph Fourier proved that complex sounds can be expressed as the sum of sinusoidal components. This work laid the foundation for many fields ranging from music to electronic sound processing.

Intervals Harmony and Timbre

In music the relationship between two pitches is called an interval and these intervals are directly expressed as frequency ratios. As Pythagoras demonstrated simple whole number ratios sound natural and harmonious to the ear.

Simple ratios:

• Create a consonant and balanced effect

Complex ratios:

• Produce a sense of dissonance

When a note is played not only its fundamental frequency is heard but also its harmonics multiples of that frequency. The characteristic sound of an instrument its timbre depends on the amplitude ratios of these harmonics. This is why a violin and a piano playing the same note sound different. This difference is entirely the result of a mathematical structure. Furthermore concepts in music theory such as the Pythagorean comma and the equal temperament system are also based on mathematical calculations.

Rhythm Sequences and Composition

Rhythm is formed by dividing time into specific ratios. The note values used in music are entirely based on mathematical time proportions:

• 4/4 → Four quarter notes per measure
• 3/4 → Waltz rhythm
• 9/8 → Aksak rhythm in Turkish music (2+2+2+3)

Scales and pitch sequences are also organized according to mathematical systems. Pythagoras created the first mathematical scale models using whole number ratios. In addition the Fibonacci sequence and the golden ratio appear as hidden structures in many compositions. In the Fibonacci sequence the ratio between consecutive terms approaches the golden ratio. This creates a natural balance in the flow of a piece.

It is known that in some compositions by composers such as Mozart and Debussy the climactic points occur at approximately the 61% mark of the work. This ratio is a musical manifestation of the golden ratio.

The Harmonious World of Mathematics and Music

The human brain perceives frequency relationships based on small whole number ratios as natural and harmonious. This is one of the fundamental reasons why music is a universal language. The renowned philosopher Leibniz expressed this truth in these words:

“Music is a hidden arithmetic exercise of the soul which does not know that it is counting.”^【1】

Mathematics is the invisible system that organizes sounds; music is the emotional embodiment of this order. The black and white keys of a piano generate infinite melodies while faithfully adhering to mathematical ratios.

In conclusion although mathematics and music appear as distinct fields they are two universal systems that speak the same language and complement each other.