Kinds of Sound: Tone, Complex Tone, Noise and Bang

The four kinds of sound after DIN 1320 compared: waveform, line or continuous spectrum, periodicity, pitch and crest factor – with inharmonic tone mixtures, noise colours and a bang to listen to.

Explanation Tasks

Overview: waveform (top) and spectrum (bottom)

Click a kind of sound to select it. The cards use the current settings.

Waveform

Spectrum (level relative to the strongest component)

Features
Kind of sound
Periodic
Spectrum
Pitch
Duration
Wavelength (air, 20 °C)
Crest factor
Highest partial

Start runs the waveform: periodic signals move along as a rigid pattern, noise looks different in every frame.

Kind of sound

Selection

Tone Complex Noise Bang

Frequency f or fundamental f₀ Hz

Number of harmonics N Stretching B (0 = harmonic)

Kind of noise

white pink brown third octave

Centre frequency of the band Hz

Duration of the positive phase T₊

Examples

Keyboard: Space = play/pause, F = presentation mode, M = sound on/off

What is happening?

Four kinds of sound

Everything we hear is sound: small pressure fluctuations in the air. The German acoustics standard DIN 1320 distinguishes four basic forms. A pure tone (Ton) is a single sine oscillation. A complex tone (Klang) consists of several sine tones whose frequencies are integer multiples of a fundamental frequency. Noise (Geräusch) is an irregular signal without a fixed period. A bang (Knall) is a short, strong pressure pulse. In addition there is the tone mixture (Tongemisch): several tones whose frequencies are not in integer ratios.

The distinction follows two questions. Does the signal repeat regularly (periodic) or not? And does the spectrum consist of single lines, or is it continuous, spread over a whole range of frequencies? Pure and complex tones are periodic and have line spectra. Noise and bang are not periodic and have continuous spectra. The tone mixture lies in between: lines, but no period.

How to use the module: at the top, the “Overview” shows all four kinds side by side, each with waveform and spectrum. Clicking a card selects it. Below, the selected kind is shown large: the “Waveform”, the “Spectrum” and the “Features” with periodicity, kind of spectrum, pitch, duration and crest factor. The speaker icon in the title row plays the signal.

Pure tone: a single sine oscillation

Select “Tone”. The waveform is a sine curve that repeats exactly after the period T = 1/f. At 220 Hz that is about 4.5 ms; the module marks one period with a bracket. The spectrum contains a single line at 220 Hz. There is nothing more to it: frequency, amplitude and phase describe a pure tone completely.

Pure tones are rare in nature. A tuning fork comes close, as does a whistled note or the test tone of a sine generator. In the studio the sine tone is the most important test signal: 1 kHz for setting levels, a sine sweep for frequency responses, 50 Hz or 60 Hz as hum when something is badly grounded.

A note on language: in everyday speech the note of a piano or a violin is also called a “tone”. Acoustically it is a complex tone. The module follows the technical terms.

The features also show the wavelength λ = c/f, the distance between two pressure maxima in space. Here c = 343 m/s is the speed of sound in air at 20 °C. At 220 Hz that is 1.56 m, at 20 Hz just over 17 m and at 20 kHz only 1.7 cm. For a complex tone it applies to the fundamental, for third-octave noise roughly to the centre frequency. Broadband noise and the bang have no single wavelength, because they contain all frequencies at once.

The crest factor is the ratio of the peak value to the RMS value. For a sine it is √2 ≈ 1.41, i.e. 3 dB. That is the smallest value in this module and serves as the reference for the other kinds of sound.

Complex tone: fundamental and overtones

Select “Complex”. Now several partials sound at f₀, 2·f₀, 3·f₀ and so on, here with the amplitudes 1, 1/2, 1/3 … as in a sawtooth. In the spectrum the lines are equally spaced by f₀. On the logarithmic frequency axis they crowd together towards the top, because each octave spans twice as many hertz as the one below.

The waveform is more complicated than for the pure tone, but it still repeats exactly after T = 1/f₀. That is no coincidence: each partial k·f₀ oscillates exactly k times in this time and is back at its start. That is why a complex tone has a clear pitch, the pitch of its fundamental, even though it contains many overtones.

The strength of the individual partials determines the timbre. Change the “Number of harmonics N” and listen: with one partial a pure tone is left, with 16 it sounds bright and sharp. The pitch stays the same. Almost all musical instruments with a definite pitch produce complex tones: strings, wind instruments and the voice singing a vowel. How partials shape the waveform is shown in detail in the module “Harmonic Series”.

Tone mixture: when the partials do not fit

The slider “Stretching B” shifts the partials according to the model of a stiff string: f_k = k · f₀ · √(1 + B · k²). With B = 0 everything is harmonic. Even small values push the upper partials noticeably sharp, and the features switch to “Tone mixture”.

The spectrum still consists of sharp lines, but the waveform no longer repeats. Start the animation: the harmonic complex tone moves through the picture as a rigid pattern, the tone mixture keeps changing its shape. There is no common period any more, and the pitch becomes ambiguous. You hear several pitches or a floating, metallic sound.

Examples are bells, cymbals, gongs and mallet instruments. The low strings of a piano are also slightly stretched, because the thick steel wire is stiff. Piano tuners compensate by tuning octaves slightly wider than 2 : 1 (stretch tuning). The preset “Low piano string” shows a small amount of stretching, “Bell” a large one.

The boundary to the complex tone is fluid. With very little stretching the ear still hears a clear pitch, although strictly speaking the signal is no longer periodic.

Noise: randomness and a continuous spectrum

Select “Noise”. The module generates random noise, the simplest case of a noise-like sound. The waveform has no pattern. Start the animation: every frame looks different, nothing repeats. The spectrum no longer has lines but a continuous area. All frequencies are present at the same time, with randomly changing phase.

The “Kind of noise” sets the distribution over frequency. White noise has the same power in every hertz. Its spectrum is flat, but it sounds very bright and hissing, because a high octave spans many more hertz than a low one. Pink noise falls by 3 dB per octave and so has the same power in every octave; it sounds balanced, like surf or a waterfall. Brown noise falls by 6 dB per octave and sounds dull, like distant rumbling.

The third-octave band is noise limited to one third of an octave around the centre frequency. It has a vague pitch: you hear roughly where it lies, but it is not a tone. Sibilants such as “sh” and “s” are band-limited noises of this kind.

The crest factor is different too: noise has occasional high peaks, typically about four to four and a half times the RMS value (12 to 13 dB). Real noises such as wind, applause, traffic or a fan are more complicated but have the same features: not periodic and with a continuous spectrum.

Bang: short, strong and broadband

Select “Bang”. The model is the Friedlander wave, the classic shape of a blast wave after an explosion: p(t) = (1 − t/T₊) · e^(−t/T₊). Here p is the sound pressure, t the time since the bang and T₊ the duration of the positive phase. The pressure jumps instantly to its peak, falls to zero at T₊ and then turns into a smaller negative phase. The module marks T₊ in the waveform. For listening, the bang repeats every two seconds.

The spectrum is continuous as for noise, but smooth instead of random, because the bang is a single, fixed event. Its maximum lies at f = 1/(2π · T₊). Above it the spectrum falls by 6 dB per octave; that is caused by the jump at the start. Change T₊: a short bang (below 1 ms, like a clap, a whip crack or a shot) reaches far into the treble and sounds bright and sharp. A long one (about 20 ms, like distant thunder or a far-away blast) is dull and low.

The difference from noise is the duration: a bang is a single short event, noise continues. Typical of a bang is its huge crest factor: the peak is much higher than the RMS value averaged over the time until the next bang (here 2 s). That is why sluggish level meters show a bang far too low, although its peak level can damage hearing.

Limits of the model

All signals are idealised. Pure and complex tones run without attack or decay, and the partials have fixed amplitudes 1/k. Real instruments change their spectrum all the time, and the attack in particular (a short, noise-like part) is important for recognising an instrument. A real complex tone therefore almost always contains some noise as well.

For listening, tone, complex tone and noise have the same RMS value, so that you compare timbres and not levels. They still do not sound exactly equally loud, because hearing is less sensitive to low and very high frequencies (see the module “Decibels and phons”). The bang is played with a fixed peak value. A limiter protects ears and speakers, so a bang from a loudspeaker never sounds like the real thing. A real gunshot reaches peak levels far above 140 dB SPL.

The spectrum shows the level relative to the strongest component. Lines and continuous spectra are not directly comparable: a line holds all its power at one frequency, a continuous spectrum spreads it out (measured as a power density). The noise spectrum is a measurement averaged over many segments (Welch's method) and therefore slightly uneven. The spectrum of the bang is the exactly computed Fourier transform of the pulse.

Tasks

Practice questions

Questions from studio, stage and mastering – to think about, discuss and check with the module.

1. A tuner shows the note of a guitar immediately, but with a hi-hat or snare drum only jumps around wildly. Why?

Show answer

A tuner looks for the period of the signal, or the fundamental of a series of harmonic lines. A guitar string produces a complex tone: periodic, with a clear fundamental. Hi-hat and snare wires produce mainly noise with a continuous spectrum and no period. The tuner finds no stable value. The shell resonances of a drum can be tuned once the snares are off, because they form lines again. The partials of a drumhead are inharmonic, though (a tone mixture).
2. In a theatre production a blank gun is fired on stage. The console's RMS meter shows only a moderate level. Why is caution still needed?

Show answer

A bang has an extremely high crest factor: the peak lasts only milliseconds, and the time-averaged RMS value stays small. Sluggish meters (RMS, VU, also measurements with “slow” time weighting) therefore grossly underestimate the peak level. For hearing, the peak level is what matters. Occupational noise regulations (e.g. the EU noise directive 2003/10/EC) set separate exposure action values of 135 and 137 dB(C) peak sound pressure level. A blank fired close to the ear can clearly exceed them. The same applies to microphones and the console: the peak overloads them although the RMS meter looks harmless.
3. Pink noise, rarely white noise, is commonly used to tune a PA or as a reference while mixing. What speaks for pink?

Show answer

Pink noise has the same power in every octave and every third octave. A third-octave analyser therefore shows it as a straight line, and deviations of the system stand out at once. This also suits music, whose long-term spectrum falls off towards the top in a similar way. White noise appears in a third-octave analyser rising by 3 dB per octave. It sounds harsh and puts a lot of power into the tweeters, which can endanger the drivers at high level. As a noise with a continuous spectrum, pink noise excites all frequencies at once, whereas a sine sweep excites one frequency after another.

Tasks

Predict first, then try it out, then open the solution. “Load starting point” sets the matching initial values.

1. From pure tone to complex tone

A 220 Hz tone is set. Turn on the sound and then switch to “Complex”. What changes in the waveform, in the spectrum and in what you hear? Does the period stay the same?

Load starting point

Show solution

The waveform becomes more jagged, and the spectrum gains lines at 440, 660, 880 Hz and so on. The sound becomes brighter and fuller. The period stays at T = 1/220 Hz ≈ 4.5 ms, so the pitch stays the same: the overtones change the timbre, not the pitch.
2. When does a complex tone become a tone mixture?

A complex tone with 220 Hz and 8 partials is set. Slowly increase the stretching B. When do you hear a difference? Start the animation and compare the waveform with B = 0.

Load starting point

Show solution

Mathematically, even the smallest B turns the complex tone into a tone mixture: the partials are no longer integer multiples, and the signal is no longer periodic. Audibly it usually takes about B = 0.001; the 8th partial is then about 3 % (roughly half a semitone) sharp. With a large B it sounds metallic and bell-like. In the animation the waveform with B = 0 moves as a rigid pattern, the tone mixture keeps changing its shape.
3. White, pink, brown

White noise is set. Listen and switch to pink and brown. How do the spectrum and the sound change? Why does white noise sound so bright although its spectrum is flat?

Load starting point

Show solution

White noise is flat (equal power per hertz), pink falls by 3 dB per octave (equal power per octave), brown by 6 dB per octave. White sounds hissing, pink balanced like surf, brown dull. Hearing divides frequencies roughly into octaves or thirds. The octave from 10 to 20 kHz spans 10,000 Hz, the one from 100 to 200 Hz only 100 Hz. With white noise the top octave therefore holds a hundred times as much power as the low one.
4. Noise with a pitch?

Third-octave noise around 1 kHz is set. Do you hear a pitch? Compare with a 1 kHz tone (kind “Tone”, frequency 1000 Hz), then move the centre frequency.

Load starting point

Show solution

Yes, but a vague one: you hear that the noise lies roughly where the 1 kHz tone is, and it rises with the centre frequency. There is no clear pitch as for the tone, because the spectrum is an area rather than a line and the signal is not periodic. The narrower the band, the more tonal the noise sounds. We tell sibilants such as “s” and “sh” apart by the position of such bands.
5. Short and long bang

A very short bang (T₊ = 0.3 ms) is set. Where is the maximum of the spectrum? Lengthen T₊ to 20 ms. What happens to the spectrum, the sound and the crest factor?

Load starting point

Show solution

The maximum lies at f = 1/(2π · T₊), about 530 Hz for 0.3 ms. Above it the spectrum falls by 6 dB per octave, so it reaches far into the treble, and the bang sounds sharp like a whip crack. At 20 ms the maximum lies at about 8 Hz, the treble is much weaker and it sounds dull like distant thunder. The crest factor drops, because the longer pulse contains more energy and the RMS value over the 2 s until the next bang rises. It still stays far above that of noise.

Settings

Sound

Level calibration (dB SPL)

MIDI keyboard

Kinds of Sound: Tone, Complex Tone, Noise and Bang