Frequency and Pitch

The pitch of a sound (how high or low it is) depends on the frequency of the sound wave: how many vibrations happen per second. A low rumble has a low frequency, perhaps 50 vibrations per second. A high whistle has a high frequency, maybe 5,000 vibrations per second. Frequency is measured in hertz (Hz), named after the German physicist Heinrich Hertz. One hertz means one vibration per second. The relationship between frequency and pitch is at the heart of music, voice and many other parts of life.

  • PitchHow high or low a sound isDecided by frequency
  • UnitHertz (Hz)Vibrations per second
  • Human range20 Hz to 20 kHzDrops with age
  • Middle C261.6 HzCentre note on a piano
  • A above middle C440 HzStandard tuning note
  • Bat sonarUp to 200 kHz10x above human hearing

What is frequency?

Frequency is the number of complete waves (or vibrations) that pass a fixed point each second. If a guitar string vibrates back and forth 440 times in 1 second, the sound it makes has a frequency of 440 Hz, which sounds to your ears as the note A above middle C.

Higher frequency means more vibrations per second, which sounds higher pitched. Lower frequency sounds deeper.

The range of human hearing

Young people with healthy ears can usually hear sounds from about 20 Hz to 20,000 Hz (20 kHz):

  • 20 Hz: a deep rumble (felt as much as heard)
  • 100 Hz: a deep male voice
  • 500 Hz: a typical adult speaking voice
  • 1,000 Hz: a clear whistle
  • 5,000 Hz: very high notes on a piccolo
  • 15,000 Hz: the high pitch of an old TV set or a teenagers ringtone
  • 20,000 Hz: the upper limit (most adults stop hearing this in their teens)

As people get older, the highest frequencies become harder to hear (a normal effect called presbycusis). By age 30, many adults cannot hear above 16,000 Hz. By 50, the upper limit may be around 12,000 Hz. Loud noise damage (from concerts, headphones, machinery) can accelerate the loss.

Below and above human hearing

  • Infrasound: sounds below 20 Hz. Felt rather than heard. Whales, elephants and rhinos communicate using infrasound that can travel for kilometres.
  • Ultrasound: sounds above 20 kHz. Used by bats and dolphins for echolocation, and by humans in medical scanners and industrial cleaners. (See ultrasound.)
Fact Different animals hear different ranges. Dogs hear up to about 45 kHz (way above human range, which is why dog whistles work). Cats hear up to about 65 kHz. Bats can hear up to 200 kHz. At the other end, some elephants can hear infrasound vibrations below 10 Hz, sensing other elephants stomping the ground from kilometres away.

Pitch and music

Music depends on the careful choice of pitches. The notes of the musical scale are chosen so that their frequencies form simple mathematical ratios:

  • Octave: two notes whose frequencies are exactly a 2:1 ratio. Middle C is 261.6 Hz; the C above is 523.2 Hz, exactly double. Your ear hears them as the same note, just higher.
  • Perfect fifth: 3:2 ratio. C and G.
  • Perfect fourth: 4:3 ratio. C and F.
  • Major third: 5:4 ratio. C and E.

These simple ratios are why music sounds "right" to our ears. The discovery is usually credited to the Greek mathematician Pythagoras, who noticed that strings whose lengths formed simple ratios sounded harmonious together.

How instruments change pitch

  • String instruments: longer strings vibrate slower (lower pitch), shorter strings vibrate faster (higher pitch). Tighter strings vibrate faster (higher); looser ones slower. Thicker strings vibrate slower.
  • Wind instruments: longer air columns make lower pitches; shorter ones make higher pitches. That is why a tuba is huge (very low notes) and a piccolo is tiny (very high notes).
  • Drums: a bigger drum head vibrates slower (lower pitch). Tightening the drum head raises the pitch.
  • Voice: the vocal cords stretch and relax to change frequency, much like guitar strings being tuned.
Did you know? The note A above middle C has been the international standard tuning note for orchestras since the 1930s, fixed at exactly 440 Hz. Before this standard, different orchestras tuned to slightly different frequencies (440 Hz in London, 442 Hz in Berlin, 444 Hz in some American cities). A musician swapping orchestras might find their carefully tuned instrument suddenly slightly off.

The Doppler effect

You may have noticed that an ambulance siren sounds higher pitched as it comes towards you, then lower as it passes and moves away. This is the Doppler effect: the frequency of the sound waves reaching you changes depending on whether the source is moving towards you or away.

  • Moving towards you: each wave gets crammed in closer, so frequency goes up, pitch sounds higher.
  • Moving away: each wave gets stretched out, so frequency goes down, pitch sounds lower.

The same effect applies to light from distant galaxies, which is how astronomers discovered the universe is expanding. Galaxies moving away from us have their light shifted to lower frequencies (redder colours), called redshift.

Try this Take a rubber band, stretch it over the open top of a small box and pluck it. Listen to the pitch. Now stretch it tighter and pluck again: the pitch goes up. Loosen it and the pitch goes down. Now shorten the rubber band by pressing it down with a finger halfway along: the pitch goes up again because the vibrating part is shorter. You have just built a one-string instrument and explored exactly how stringed instruments control pitch.
Deeper dive: why a piano has 88 keys (and the maths behind it)

A standard modern piano has 88 keys, covering a range of just over 7 octaves: from the lowest note A0 (27.5 Hz) to the highest note C8 (4,186 Hz). Why exactly 88, and why does each octave contain 12 notes?

The 12 notes in an octave come from a Western musical tradition stretching back to the ancient Greeks. Pythagoras noticed that simple mathematical ratios of frequencies sound harmonious. By multiplying frequencies by the right ratios, musicians built up the familiar 7-note major and minor scales, plus 5 extra "in-between" notes called sharps and flats.

For centuries, musicians tuned each note using the simple ratios Pythagoras found. The problem is that the maths does not quite close up: going around the cycle of fifths produces notes that do not perfectly match notes reached by stacking octaves. This caused chords in some keys to sound off-pitch.

In the 1700s, musicians adopted equal temperament: each of the 12 notes in an octave is given a frequency exactly the 12th root of 2 times the one below. This squashes all the simple ratios slightly out of perfect tune, but lets you play in any key without retuning. The famous composer Bach wrote his Well-Tempered Clavier partly to celebrate this new flexibility.

By the 1880s, manufacturers had settled on 7 octaves plus a few extra notes as the standard piano. Below A0, notes start to sound like a buzz rather than a pitch; above C8, they sound like a faint ping. The 88-key range covers the practical limits of musical pitch for both ears and instruments. Even the most modern grand pianos still have exactly 88 keys today.

For more, see amplitude and volume and ultrasound.