Everything is math: part 2

 

The Sacred Mathematics of Music: The History, Discovery and Importance of Numbers in Music

Music and mathematics may at first appear to be two entirely different subjects. One belongs to the world of creativity, emotion and artistic expression, while the other is associated with calculation, measurement and logical reasoning. 

Yet the two have been connected for thousands of years. Behind almost every piece of music lies a mathematical structure, from the vibration of a single musical note to the complicated rhythms of an orchestra or the precise relationships between musical scales.

Throughout history, numbers have played an important role in how people understand, create and interpret music. 

Ancient philosophers believed that numbers revealed the hidden structure of the universe and that musical harmony was evidence of a greater mathematical order. Some even believed that the same numerical relationships responsible for musical beauty governed the movements of the planets and the organisation of the cosmos.

The discovery that musical harmony could be expressed through simple numerical ratios was one of the most important developments in the history of music, mathematics and philosophy. It helped establish the foundations of musical theory and inspired religious, philosophical and mystical ideas that continue to influence music today.

The earliest relationship between numbers and music

Long before written musical notation existed, human beings were making music using their voices, hands and natural objects. Drums, flutes, rattles and other instruments were used in ceremonies, celebrations, religious observances and everyday life.

Although prehistoric people would not have described music mathematically, they would have recognised that certain sounds worked well together, while others sounded unpleasant or discordant.

The earliest musical instruments discovered by archaeologists include prehistoric flutes made from bird bones and mammoth ivory. Some examples date back approximately 40,000 years. Their existence demonstrates that early humans were capable of producing different pitches and organising sounds into musical patterns.

As civilisations developed, people began to explore the relationship between sound, measurement and proportion.

Ancient Mesopotamia provides some of the earliest evidence of sophisticated musical organisation. Archaeological discoveries, including musical instruments and cuneiform tablets, demonstrate that the Babylonians developed systems for organising musical pitches and tuning instruments.

One particularly important discovery is a Babylonian tablet known as the Hurrian Hymn No. 6, dating to approximately 1400 BCE. It contains musical instructions associated with a hymn to the goddess Nikkal. Although scholars continue to debate how precisely the notation should be interpreted, it provides remarkable evidence of early musical theory.

The ancient Egyptians also used music extensively in religious ceremonies, temple rituals, celebrations and funerary practices. Their instruments included harps, lutes, flutes, drums and sistrums.

Music was often associated with divine forces, particularly the goddess Hathor, who was connected with music, dance, joy and fertility.

Although these early cultures understood practical aspects of musical organisation, it was the ancient Greeks who developed particularly influential mathematical explanations of musical harmony.

Pythagoras and the discovery of musical ratios

Pythagoras (c. 570–495 BCE)

Pythagoras is traditionally credited with discovering the mathematical relationships between musical intervals. His teachings helped establish the idea that music could be understood through numbers.

One of the most famous stories in the history of music concerns Pythagoras and his investigation into the sounds produced by different lengths of vibrating strings.

According to later accounts, Pythagoras noticed that certain musical sounds appeared particularly harmonious when their string lengths were related by simple numerical proportions.

The story is sometimes told as an observation made in a blacksmith’s workshop, where different-sized hammers produced harmonious sounds. However, this version is generally regarded as legendary rather than historically reliable.

The more important principle is that the relationship between the length of a vibrating string and the pitch it produces can be measured mathematically.

Pythagoras is traditionally associated with experiments using a monochord, an instrument consisting of a single string stretched over a resonating body, with a movable bridge allowing the vibrating length to be changed.

These experiments demonstrated some remarkable relationships.

When a string is divided into two equal vibrating lengths, the shorter section produces a note one octave higher than the original.

When the string is divided in the proportion of 2:3, the resulting notes form a perfect fifth.

When divided in the proportion of 3:4, they produce a perfect fourth.

These relationships became fundamental to Western musical theory.

The important mathematical ratios

Musical interval	String-length ratio	Frequency ratio
Octave	2:1	2:1
Perfect fifth	3:2	3:2
Perfect fourth	4:3	4:3
Major third	5:4	5:4

The distinction between string length and frequency is important. A shorter string vibrates more quickly and produces a higher note. Consequently, the frequency ratios are the inverse of the corresponding string-length ratios.

For example, if a string produces a note at 220 vibrations per second, halving its vibrating length produces a note at approximately 440 vibrations per second.

This is an octave.

What made this discovery so remarkable was that musical harmony could be explained using relatively simple numbers.

Rather than being entirely subjective, certain musical relationships appeared to have a measurable physical foundation.

The importance of the number 2:1 and the octave

The octave is perhaps the most fundamental numerical relationship in music.

If a musical note has a frequency of 100 vibrations per second, the note one octave above it has a frequency of 200 vibrations per second.

Likewise:

• 110 Hz becomes 220 Hz.
• 220 Hz becomes 440 Hz.
• 440 Hz becomes 880 Hz.

Although the notes are different in pitch, they sound closely related. Most musical cultures recognise some form of octave equivalence, although the ways they divide and organise the octave differ.

The octave is also important because it provides the framework within which musical scales are organised.

The familiar musical notes

 repeat as they move higher or lower in pitch:

C, D, E, F, G, A, B, C.

The second C is an octave above the first.

This repeating structure allows instruments to cover a wide range of pitches while retaining a recognisable system of musical relationships.

The number two therefore has a particularly important position in music. It represents doubling, repetition and the relationship between different registers of sound.

The ancient Greek idea of the harmony of the spheres

For Pythagoras and his followers, the relationship between numbers and music extended far beyond musical instruments.

They believed that numerical relationships were fundamental to the structure of the universe itself.

This led to the philosophical concept known as the Harmony of the Spheres, sometimes called Musica Universalis, or the Music of the Universe.

According to this idea, the movements of celestial bodies followed mathematical proportions that could be compared with the intervals and harmonies found in music.

The planets, stars and heavenly bodies were imagined as participating in a vast cosmic harmony.

This did not necessarily mean that the planets literally produced audible musical notes. Rather, it was a philosophical understanding that the universe was organised according to mathematical relationships similar to those found in music.

The Greek philosopher Plato explored related ideas in his dialogue Timaeus, where he described the creation and organisation of the cosmos through numerical proportions and harmonic relationships.

Later philosophers and astronomers developed these ideas further.

The concept became particularly influential during the medieval period, when music was considered one of the mathematical disciplines through which the order of creation could be understood.

For those interested in the spiritual and mystical history of music, the Harmony of the Spheres is especially fascinating. It represents an early attempt to connect sound, mathematics, astronomy and the divine.

Music as one of the four mathematical arts

During classical antiquity and the medieval period, music was regarded as part of a group of four mathematical disciplines known as the Quadrivium.

These were:

• Arithmetic – the study of numbers themselves.
• Geometry – the study of shapes, proportions and spatial relationships.
• Music – the study of numerical relationships expressed through sound.
• Astronomy – the study of celestial movements and their mathematical patterns.

The four disciplines were considered interconnected.

Arithmetic provided the numbers, geometry explored proportions, music demonstrated how numerical relationships could be heard, and astronomy explored mathematical order in the heavens.

This was a very different way of understanding music from the modern idea that it is primarily an artistic subject.

Medieval scholars believed that studying music could reveal something about the underlying structure of reality.

The famous philosopher and mathematician Boethius (c. 477–524 CE) was particularly influential in preserving this tradition.

In his work De institutione musica, he described three forms of music:

• Musica mundana – the music of the universe, associated with cosmic order.
• Musica humana – the harmony of the human body and soul.
• Musica instrumentalis – music produced by voices and instruments.

These ideas helped establish the importance of music as an intellectual and philosophical discipline throughout medieval Europe.

Numbers and the development of musical scales

One of the most important ways numbers influence music is through the construction of musical scales.

A musical scale is an organised sequence of notes arranged according to pitch.

Different cultures developed different scales, and these were often based on mathematical relationships between frequencies.

The ancient Greeks developed several musical systems, including the diatonic, chromatic and enharmonic genera.

Their approach to musical organisation was based partly on the mathematical relationships between intervals.

The Pythagorean system was particularly influential in Western music.

It used the perfect fifth, with a frequency ratio of 3:2, as the basis for constructing a series of notes.

By repeatedly multiplying a frequency by 3/2 and adjusting the result into the same octave, a series of pitches can be created.

However, this system has a mathematical difficulty.

Twelve perfect fifths do not correspond exactly to seven octaves. The small difference between these two quantities is known as the Pythagorean comma.

This may seem like a minor mathematical problem, but it created considerable challenges for musicians who wanted to play in different keys using fixed-pitch instruments.

Over centuries, musicians and theorists developed different tuning systems to address this problem.

The number 7 and the seven musical notes

In Western music, the familiar seven-note diatonic scale consists of:

C – D – E – F – G – A – B

These seven notes form the basis of the major and natural minor scales, although other scales and musical traditions use different numbers of notes.

The seven-note system has ancient roots, particularly in Greek musical theory, although its development into the modern system took place over many centuries.

The notes are not equally spaced.

In a major scale, the pattern of whole tones and semitones is:

Tone – Tone – Semitone – Tone – Tone – Tone – Semitone

For example, the C major scale is:

C – D – E – F – G – A – B – C.

This arrangement creates the familiar sound of a major scale.

The number seven also has an interesting place in religious, mystical and philosophical traditions.

It is associated with the seven classical planets, the seven days of the week and various systems of spiritual symbolism.

It is tempting to assume that the seven notes of the Western scale were deliberately chosen because of these associations. However, there is no convincing historical evidence that the seven-note scale originated from the mystical symbolism of the number seven.

Nevertheless, later writers and esoteric traditions have frequently drawn comparisons between the seven musical notes and other groups of seven found in nature, religion and cosmology.

The number 12 and the musical octave

Another important number in music is twelve.

On a modern piano, an octave contains twelve semitones:

• Seven white-key notes.
• Five black-key notes.

Together, these form the twelve pitch classes of the familiar Western chromatic system.

The twelve-note system allows musicians to move between different keys and create an enormous variety of musical combinations.

The relationship between twelve and the octave is particularly important because modern Western music generally uses equal temperament.

In twelve-tone equal temperament, the octave is divided into twelve equal logarithmic steps.

Each semitone has a frequency ratio of:

2^{1/12}

This means that every semitone is approximately 1.05946 times the frequency of the note below it.

For example, the standard musical note A above middle C is commonly tuned to 440 Hz.

Moving up twelve semitones doubles the frequency to 880 Hz.

This mathematical system allows instruments to play in all twelve keys with consistent interval relationships.

It is not perfectly identical to the simpler whole-number ratios discovered through ancient string experiments, but it provides a practical compromise that makes modern Western keyboard music possible.

The discovery of frequency and vibration

The ancient Greeks understood that the length of a vibrating string affected its pitch, but the scientific explanation of sound developed much later.

During the scientific revolution, researchers began investigating the physical nature of sound in greater detail.

The French mathematician and philosopher Marin Mersenne (1588–1648) made important contributions to the study of vibrating strings.

His investigations helped establish the mathematical relationship between string length, tension, mass and frequency.

A string’s frequency depends on several factors:

• Its vibrating length.
• The tension applied to it.
• Its mass per unit of length.

This explains why tightening a guitar string raises its pitch, while loosening it lowers the pitch.

The development of acoustics eventually allowed scientists to understand that sound travels as waves through a medium such as air.

Frequency describes how many complete vibrations occur each second and is measured in hertz (Hz).

A low note has fewer vibrations per second, while a high note has more.

For example:

Musical note	Frequency
A2	110 Hz
A3	220 Hz
A4	440 Hz
A5	880 Hz

This is why numbers are so important in modern music production. Electronic instruments, synthesizers, recording equipment and digital audio software all rely on mathematical descriptions of sound.

Numbers and musical rhythm

Pitch is only one part of music. Numbers are equally important when it comes to rhythm.

Rhythm determines how music moves through time. It involves the organisation of beats, pauses, accents and repeated patterns.

Musical time is generally organised into groups of beats known as measures or bars.

The time signature tells musicians how these beats are grouped.

Some common examples include:

Time signature	Musical organisation	Typical character
2/4	Two beats per bar	Marches
3/4	Three beats per bar	Waltzes
4/4	Four beats per bar	Much popular music
6/8	Six eighth-note beats, often felt in two groups of three	Folk and dance music

The number three is particularly important in the waltz, where the rhythm follows a repeating pattern of one-two-three.

The number four is extremely common in modern popular music, rock, blues and many forms of dance music.

The number six is often associated with compound rhythms, where beats are divided into groups of three.

These numerical patterns influence how music feels.

A piece in three-four time often has a flowing, circular quality, while four-four time can create a more straightforward and regular movement.

Of course, musical rhythm is not restricted to these patterns. Many cultures use complex rhythms involving five, seven, nine, eleven or other groupings.

Indian classical music, for example, uses rhythmic cycles known as tala. These can contain different numbers of beats and may involve elaborate patterns of subdivision and repetition.

African musical traditions also contain sophisticated polyrhythms, where several contrasting rhythmic patterns occur simultaneously.

Fibonacci numbers and the golden ratio in music

One of the more intriguing connections between mathematics and music involves the Fibonacci sequence and the golden ratio.

The Fibonacci sequence is a series of numbers in which each number is produced by adding the two previous numbers.

It begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

As the numbers become larger, the ratio between successive numbers approaches the golden ratio, approximately 1.618.

The golden ratio has attracted considerable interest in art, architecture, nature and mathematics.

Some writers and music enthusiasts have suggested that composers have used Fibonacci numbers and golden-ratio proportions when structuring their compositions.

For example, a composer might place a major musical climax approximately 62% of the way through a piece, reflecting the mathematical proportions associated with the golden ratio.

There are often claims that famous compositions by composers such as Mozart, Beethoven and Debussy contain deliberate golden-ratio structures.

However, these claims need to be treated cautiously. Some examples depend on selective measurement or interpretation, and there is not always evidence that the composers consciously intended to use the golden ratio.

Nevertheless, mathematical patterns can certainly be used as compositional tools.

A modern composer might deliberately use Fibonacci numbers to organise the length of musical sections, the number of notes in a phrase or the placement of important musical events.

The number 432 and the mystical interpretation of music

Another number that frequently appears in discussions about music, spirituality and sound is 432.

Some people believe that music tuned to A = 432 Hz has special spiritual, healing or cosmic qualities.

The argument is sometimes connected with ancient mathematical systems, the proportions of the universe and the supposed natural vibration of the Earth.

There are also claims that 432 Hz is more harmonious with the human body than the more widely used modern concert pitch of A = 440 Hz.

Historically, musical tuning standards have varied considerably. There was no single universal concert pitch before modern standardisation.

Today, A = 440 Hz is widely used as the international reference pitch, although orchestras and musicians sometimes tune to different standards.

Music tuned to A = 432 Hz is slightly lower in pitch than music tuned to A = 440 Hz.

There is currently no convincing scientific evidence that 432 Hz has unique healing or spiritual properties compared with other tuning standards.

However, this does not prevent people from finding it personally relaxing or meaningful.

From a historical and esoteric perspective, the fascination with 432 reflects a much older belief that numbers, sound and the natural world are connected.

It is worth remembering that ancient Pythagorean ideas about numerical harmony were philosophical and mathematical observations, not evidence that a particular modern tuning frequency has supernatural powers.

Music, mathematics and the development of instruments

Numbers have also influenced the physical construction of musical instruments.

The design of stringed instruments depends on mathematical proportions.

The length and tension of strings determine the pitches they produce, while the position of frets on a guitar must be carefully calculated to create the correct intervals.

The construction of a piano is another excellent example.

A modern piano contains approximately 88 keys, arranged across a range of seven octaves plus a few additional notes.

Its strings vary in length, thickness and tension to produce different pitches.

Wind instruments also depend on mathematical relationships.

The length of the air column inside a flute, trumpet or pipe determines its resonant frequencies. By changing the effective length of that air column, the musician can produce different notes.

In electronic music, mathematics becomes even more important.

Synthesisers generate sound through electronic oscillations, while digital recording systems represent sound as numerical information.

Modern music production involves calculations relating to:

• Frequency and pitch.
• Tempo and beats per minute.
• Sound waves and amplitude.
• Digital sampling rates.
• Audio compression.
• Harmonic frequencies and overtones.

Without mathematics, much of modern music technology would simply not exist

The spiritual and mystical significance of musical numbers

Throughout history, people have regarded music as something more than entertainment.

Music has been used to communicate with deities, accompany religious ceremonies, create sacred atmospheres and support meditation.

Ancient Greeks believed that musical harmony reflected the order of the cosmos. Medieval Christian thinkers associated musical proportions with divine creation. In Hindu traditions, sacred sound has an important place in spiritual philosophy, including the concept of Nada Brahma, often translated as the universe is sound.

In many religious traditions, particular numbers have acquired symbolic associations with music.

For example:

• One can represent unity, the original source or the fundamental tone.
• Two represents duality and the relationship between two musical pitches.
• Three can represent triads, the basic three-note structures of many Western chords, as well as spiritual trinities.
• Four is associated with four-beat rhythms and, symbolically, the four elements in some philosophical traditions.
• Five appears in the five-note pentatonic scale, found in many musical cultures.
• Seven is associated with the seven notes of the Western diatonic scale.
• Twelve represents the twelve semitones of the modern Western chromatic octave.

Some of these are direct musical relationships, while others are symbolic interpretations made by later philosophical and spiritual traditions.

The pentatonic scale is particularly interesting. It uses five notes within an octave and is found in many musical cultures across the world, including Chinese, African, Celtic and other traditional musical systems.

Its simplicity makes it especially versatile, and it can create melodies that sound harmonious without requiring the more complicated structures of some seven-note scales.

For modern Pagans and practitioners of witchcraft, the relationship between numbers, sound and ritual can provide an interesting area of exploration.

Drumming, chanting, repetitive singing and the use of bells can all create structured sound patterns within ceremonies. Repeated rhythms can help establish a particular atmosphere and support concentration or meditation.

Numbers can be incorporated into these practices through repetitions, rhythmic cycles and the symbolic significance attached to particular intervals or patterns.

These are matters of spiritual interpretation and personal practice rather than scientifically established magical effects.

Why numbers remain important in music today

Today, mathematics is involved in virtually every aspect of music.

Composers use numerical relationships to create structure, musicians use mathematical timing to maintain rhythm, and instrument makers rely on precise measurements to produce accurate pitches.

Even the simplest melody contains mathematical relationships.

When we hear two notes an octave apart, we are hearing a frequency relationship of 2:1. When we hear a perfect fifth, we encounter a relationship close to 3:2. When we listen to a regular rhythm, we are experiencing the organisation of time through numerical patterns.

Computer-generated music makes this relationship particularly obvious. Digital instruments use mathematical calculations to generate pitches, rhythms, harmonies and sound effects.

Yet mathematics does not remove the emotional qualities of music.

A mathematical explanation can tell us why two notes have a particular frequency relationship, but it cannot completely explain why a melody makes someone feel joyful, nostalgic, peaceful or sad.

Music is both a physical phenomenon and a deeply human experience.

The ancient Pythagorean idea that music reveals something about the hidden order of the universe may never be scientifically proven in its mystical form. Nevertheless, the discovery that sound itself follows mathematical laws remains one of humanity’s most extraordinary intellectual achievements.

In that sense, every musical note is more than simply a sound. It is a vibration, a measurable frequency and part of a mathematical relationship that has fascinated humanity for thousands of years.

And perhaps that is why music has so often been regarded as something sacred: it allows us to hear mathematics rather than merely see it. 

Further fascinating reading: Three particularly interesting subjects to explore alongside this article are the Pythagorean Harmony of the Spheres, the history of sacred music and the relationship between musical frequencies and the seven-note scale. Together, they provide a deeper understanding of why music has been associated with mathematics, spirituality and the structure of the universe for so many centuries.