Pitch, Loudness, and Intensity of Sound
Pitch is how high or low a sound seems and is determined by its frequency, loudness is how strong a sound seems and is related to its amplitude and intensity, and sound intensity is the power per unit area, I = P ÷ A, often expressed as a sound level in decibels, β = 10 log10(I ÷ I0) with I0 = 10−12 W/m2. In brief: higher frequency means higher pitch; larger amplitude means louder sound. Every increase of 10 dB means ten times the intensity, and +3 dB roughly doubles it. For a point source, intensity falls with the inverse square of distance.
We describe sounds as high or low, loud or soft, and these perceptions correspond to measurable properties. This guide explains pitch, loudness, intensity and the decibel scale, compares the quantities in a table, works through calculations of sound levels and the inverse-square law, and corrects common mistakes.
What determines pitch?
The pitch of a pure tone is determined by its frequency. Doubling the frequency raises the pitch by one octave: the A above middle C is 440 Hz, and the A one octave higher is 880 Hz. Musical notes are related by frequency ratios (a perfect fifth is 3:2). Most real sounds contain a fundamental frequency and harmonics; the mixture of harmonics gives the timbre (quality), which allows us to tell a violin from a flute playing the same note at the same loudness. On an oscilloscope, pitch appears as the number of waves per unit time, and timbre as the shape of the wave.
What determines loudness?
Loudness is the subjective impression of the strength of a sound. It increases with the amplitude of the sound wave, and the energy carried by the wave increases with the square of the amplitude. The intensity I is the sound power passing through a unit area perpendicular to the direction of travel: I = P ÷ A (W/m2). For a point source radiating equally in all directions, the power spreads over a sphere of area 4πr2, so I = P ÷ 4πr2: doubling the distance reduces the intensity to a quarter. The ear also responds differently at different frequencies, being most sensitive between about 2 and 5 kHz. Loudness depends on the frequency as well as the intensity.
What is the decibel scale?
The range of intensities that the ear can detect is huge, from the threshold of hearing (I0 = 10−12 W/m2 at 1 kHz) to the threshold of pain (about 1 W/m2), a factor of 1012. A logarithmic scale is used: the sound intensity level β = 10 log10(I ÷ I0), in decibels (dB). On this scale the threshold of hearing is 0 dB. A tenfold increase in intensity adds 10 dB, a hundredfold increase adds 20 dB, and doubling the intensity adds about 3 dB. Sounds above about 85 dB over long periods can damage hearing, and a sound that seems twice as loud corresponds to roughly +10 dB.
Sound levels of common sounds
| Sound | Level (dB) | Intensity (W/m2) |
|---|---|---|
| Threshold of hearing | 0 | 10−12 |
| Whisper | 30 | 10−9 |
| Normal conversation | 60 | 10−6 |
| Busy traffic | 80 | 10−4 |
| Rock concert | 110 | 10−1 |
| Threshold of pain | about 120 | 1 |
Worked example: intensity and decibels
- Level from intensity. A sound has an intensity of 10−6 W/m2. β = 10 log10(10−6 ÷ 10−12) = 10 × 6 = 60 dB.
- Two sources. Two identical sources each give 60 dB at a point. The intensities add: 2 × 10−6 W/m2, so β = 10 log10(2 × 106) = 63 dB (an increase of 3 dB, not 120 dB).
- Inverse-square law. A small loudspeaker radiates 0.10 W of sound evenly. At 2.0 m: I = 0.10 ÷ (4π × 2.02) = 0.10 ÷ 50.3 = 2.0 × 10−3 W/m2, so β = 10 log10(2.0 × 10−3 ÷ 10−12) = 10 × 9.3 = 93 dB.
- Doubling the distance. At 4.0 m the intensity is one quarter (5.0 × 10−4 W/m2), a drop of 10 log104 = 6.0 dB, so β = 87 dB.
- Pitch. A note of 330 Hz and another of 660 Hz differ by one octave, since the frequency ratio is 2:1.
How was sound level defined?
The bel was named after Alexander Graham Bell by Bell Telephone Laboratories in the 1920s to measure signal loss in telephone cables, and the decibel (one-tenth of a bel) became the standard unit of sound level. Harvey Fletcher and Wilden Munson published equal-loudness contours in 1933, showing the dependence of loudness on frequency, and the phon and sone were introduced to describe loudness.
Which mistakes are common?
Mistake 1: Confusing pitch with loudness
Pitch depends on frequency, and loudness on amplitude and intensity.
Mistake 2: Adding decibels directly
Two 60 dB sources give 63 dB, not 120 dB. Add the intensities, then convert.
Mistake 3: Thinking that a higher-pitched sound is louder
Pitch and loudness are independent properties.
Mistake 4: Assuming that the decibel scale is linear
Each 10 dB step is a factor of 10 in intensity.