Every culture anyone has studied makes music. A 2019 survey of ethnographic records from hundreds of societies, led by the psychologist Samuel Mehr, found song in every one of them, used for lullabies, healing, dancing and love. Music theory is the attempt to describe how that shared human art works: why some notes sound restful and others restless, why a beat makes us move, why a song feels finished. The German philosopher Gottfried Leibniz wrote in a letter of 1712 that music is "a hidden arithmetical exercise of the soul, which does not know that it is counting." This first level shows what is being counted.

What sound is

Sound is vibration traveling through a medium such as air, as waves of pressure. When a guitar string is plucked, it swings back and forth, pushing on the air beside it. Each push squeezes the air a little and each pull thins it out. These squeezed and thinned regions ripple outward until they reach your eardrum, which vibrates in step with them. (Only the parts of wave physics that matter for music are needed here.)

Pitch and frequency

Pitch is how high or low a sound seems, and it is set mainly by frequency, the number of vibrations per second. Frequency is measured in hertz: one hertz is one vibration per second. A string vibrating 220 times a second sounds lower than one vibrating 440 times a second. Healthy young human ears can detect roughly 20 to 20,000 vibrations per second.

One relationship matters more than any other. When you double the frequency of a note, you get the note an octave higher. Notes an octave apart sound so alike that people across many traditions give them the same name. Musicians also need a shared reference so that instruments agree. Most orchestras and bands today tune to an A at 440 vibrations per second, a standard agreed at an international conference in London in 1939, though some orchestras tune slightly higher and early music groups often tune lower.

The overtone series

A plucked string does not vibrate in only one way. It swings as a whole, but at the same time it also vibrates in halves, in thirds, in quarters and so on. Each of these smaller vibrations produces a fainter, higher tone. The lowest frequency, called the fundamental, is the pitch we hear and name, and the higher ones are called overtones or harmonics. Overtones fall at whole number multiples of the fundamental: two times, three times, four times and onward. This ladder of tones is called the overtone series or the harmonic series.

The series explains a great deal of music theory. The second harmonic is an octave above the fundamental. The third is a fifth above that, the fourth is two octaves up, and the fifth harmonic adds a major third. In other words, the basic notes of a major chord are already hidden inside a single plucked string. Ancient Greek writers told a story that Pythagoras discovered these number relationships by hearing hammers in a blacksmith's shop. The tale is almost certainly a legend, and hammers do not actually behave the way it claims, but the link between simple numbers and pleasing sounds that it describes is real.

Timbre: why a flute is not a violin

Play the same A on a flute, a violin and a trumpet, and you will still tell them apart instantly. That difference is called timbre, pronounced "TAM ber." Timbre is tone color: the quality that lets us tell two sounds apart even at the same pitch and loudness. In 1863 the German scientist Hermann von Helmholtz published a famous study showing that timbre depends largely on the mix of overtones in a sound and on how the sound begins and fades. A flute's tone is close to a pure fundamental with weak overtones, which is why it sounds smooth. A violin's bow drags the string in a way that makes many strong overtones, giving it a bright, rich edge. The start of a note, its attack, matters too: a piano's hammer strike and a bowed cello's gradual swell give the ear very different clues.

Worked example: finding a hidden note on a guitar

Take a guitar and pluck the thickest open string. You hear a low E, but your ear also receives its overtones. Now rest a fingertip very lightly on the string directly above the twelfth fret, exactly halfway along, without pressing it down, and pluck again. A soft, bell-like tone rings out an octave higher. Your finger has silenced the string's motion as a whole and left only the vibration in halves, which is the second harmonic.

Move your finger to the seventh fret, a third of the way along, and touch lightly again. Now you hear a note higher still: the third harmonic, a fifth above the octave. Touch at the fifth fret, a quarter of the way along, and you get two octaves above the open string. With nothing but one string and a finger, you have walked partway up the overtone series and heard, one at a time, the tones that were blended together in the ordinary plucked note.