The science of sound: vibration, pitch and tone colour

Sound is something shaking

Every sound starts with something vibrating, moving back and forth very fast: a guitar string, a drum skin, the air inside a flute, or the vocal folds in your throat. The vibration pushes on the air next to it, that air pushes on the next bit of air, and a wave of tiny squeezes and stretches spreads out, like ripples on a pond.

When the wave reaches your ear, it makes your eardrum shake in the same pattern, and your brain hears a sound. In air at 20 °C the wave travels about 343 metres every second; in water it is more than four times faster. In outer space there is no air to carry it, so sound cannot travel at all.

Try it: rest your fingers gently on the front of your throat and hum. The buzz you feel is the vibration that makes your voice.

Pitch: how many times a second

How fast something vibrates decides how high or low it sounds. The number of vibrations per second is called the frequency, measured in hertz (Hz), a unit named after the scientist Heinrich Hertz. 440 vibrations a second is the A4 you met in the note names lesson; 110 a second is A2, two octaves lower.

Faster means higher and slower means lower. The lowest key on a piano, A0, is 27.5 Hz, and the highest, C8, is about 4,186 Hz. A young ear can hear roughly 20 to 20,000 Hz; the top limit usually drops with age, and many adults hear up to about 16,000 Hz.

• 110 → 880 HzFour As at 110, 220, 440 and 880 vibrations a second, as pure sine tones.

An octave is double

Listen to the four As again. Each one vibrates exactly twice as fast as the one before. That is the secret of the octave from the note names lesson: two notes an octave apart have frequencies in the ratio 2 to 1, so they blend so well that they sound like the same note, one high and one low.

Other simple ratios sound smooth too. When two frequencies are in the ratio 3 to 2, such as 220 Hz and 330 Hz, you hear a perfect fifth. Roughly speaking, the simpler the ratio, the more often the two waves line up, and the calmer they sound together.

• 2 : 1 (octave)220 Hz and 440 Hz, first one at a time, then together.
• 3 : 2 (fifth)220 Hz and 330 Hz, first one at a time, then together.

Why A = 440

To play together, musicians must agree on how high one note is. Today most of the world tunes the A above middle C to 440 Hz. It was not always like this: for centuries pitch changed from place to place, often following the local organ. In 1859 France passed a law setting A at 435 Hz.

In 1939 an international conference in London recommended 440 Hz. The International Organization for Standardization (ISO) adopted it as a standard in 1955 and confirmed it in 1975 as ISO 16. Even so, not everyone uses exactly 440: many orchestras tune a little higher, somewhere between about 440 and 444 Hz.

• 440 vs 435 HzToday's A, then the French A of 1859, then both together.

Overtones: why a piano and a guitar sound different

A real instrument does not vibrate at just one speed. A string vibrates as a whole and, at the same time, in halves, thirds, quarters and so on. So a note at 110 Hz also contains quieter sounds at 220, 330, 440, 550 Hz and up: whole-number multiples of the lowest one. This family is called the harmonic series, and the higher members are overtones. We hear the lowest one, the fundamental, as the pitch.

Each instrument has its own mix of loud and soft overtones, and its own way of starting and fading away. That mix is its tone colour, or timbre. It is why a piano and a guitar playing the same A at the same loudness still sound different. A sine wave has no overtones at all, which is why it sounds so plain and smooth.

The overtone ladder below plays the first eight members of the harmonic series of 110 Hz. A few of them miss the piano keys a little: the 7th is about a third of a semitone lower than the G on the keyboard.

• Overtone ladderThe first eight harmonics of 110 Hz, one at a time.
• Build a toneStart with 220 Hz, then add 440, 660 and 880 Hz: the sound gets fuller.
• Sine waveA4 as a pure sine wave, with no overtones.
• PianoThe same A4 on the piano.

Equal temperament: twelve equal steps

If you tune every interval to the simplest ratios, some keys sound lovely and others sound badly out of tune. The usual answer today is twelve-tone equal temperament: the octave is divided into twelve exactly equal semitones. Going up one semitone multiplies the frequency by the twelfth root of 2, about 1.0595. Do it twelve times and you land on exactly double: the octave.

The gain is that every key works equally well, so music can move to any key. The cost is that most intervals are a tiny bit away from the simple ratios. Musicians measure these tiny gaps in cents: one semitone is 100 cents. An equal-tempered fifth is only about 2 cents narrower than a pure 3:2 fifth, but an equal-tempered major third is about 14 cents wider than a pure 5:4 third. Singers and string players, who can adjust every note, often lean toward the pure versions. How clearly you hear the difference varies from person to person.

Equal temperament has been the main tuning in Western music since the 20th century. Before that, several other tunings were in use, and keyboard players moved toward equal temperament only gradually.

• Equal-tempered thirdC and E as on a modern piano: listen for a slight shimmer.
• Pure third (5:4)The same third with E lowered by 14 cents: smoother and calmer.

Beats: the wah-wah of tuning

When two notes are very close but not the same, their waves slide in and out of step. When they line up the sound gets louder, and when they cancel it gets quieter, so you hear a regular wah-wah-wah. These pulses are called beats, and the number of beats per second equals the difference between the two frequencies: 440 Hz and 442 Hz give 2 beats a second.

Beats are a tuner's best friend. As you bring a string closer to the right note, the beats slow down, and when they disappear the two notes are in tune. Piano tuners even count beats to set intervals. If the notes move far apart, the beats become too fast to count and turn into a rough, buzzy sound.

• 440 + 442 HzTwo beats every second.
• 440 + 445 HzFive beats every second: a faster wah-wah.
• Tuning inThe second note moves closer to 440 Hz until the beats stop.

Try it: play the same note on two instruments, or sing along with a long held note, and listen for beats. Adjust slowly until they slow down and vanish.

Keep exploring

• Note namesMusic uses only seven letter names, over and over, like the days of the week. Find them on a piano and learn what sharp and flat mean.
• IntervalsAn interval is the distance between two notes, like counting the stairs between two people on a staircase. Hear every interval and learn how to count it.
• ScalesA scale is a set of notes climbing step by step, built from a simple recipe. Pick any starting note and hear major, minor, pentatonic, blues and more.