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You've Got the Power

September 1, 2004

Although the terms power, amplitude, and loudness are often used interchangeably, there are subtle differences that distinguish them from each other. Our perception of loudness is an example of an emergent property, that is, something that emerges from a number of factors in combination and that can sometimes be more than the sum of its parts. Entire books have been written on these subjects, but here we'll take a quick look at the concepts and explain why knowing about them can help us produce music (see the sidebar “Further Reading” for a list of other resources).

POWER UP

When you throw a rock into a lake, a force is exerted on the water, and a series of waves ripple out from the point of impact. The same thing happens with sound. Two hands clapping exert a force on the air, which produces an expanding spherical wave, like a balloon. The wave has a certain amount of power behind it that is felt by the air molecules it passes over. Power, which is expressed in watts, describes the oomph behind a sound wave (more precisely, a force that can move a certain mass a certain distance in a certain amount of time). Air molecules don't have much mass, so it doesn't take much power to set them into motion. Fortunately, our ears are very sensitive: a tone at 3 kHz is audible when the eardrum moves one-tenth the diameter of a hydrogen molecule.

Now think about where you're standing when you hear the sound. As the wave expands, its total power remains constant. But just as the rubber of a balloon gets thinner as it gets inflated, the energy of the sound event gets spread over a larger surface. The farther you are from the sound event, the lower the percentage of the total wave passes over you. The wattage per unit area decreases, and the sound sounds softer. Intensity, measured in watts per square meter, represents the power level at the location of the listener.

FIG. 1: This figure shows a two-dimensional slice of a sound wave expanding from a vibrating tuning fork.

FIG. 1: This figure shows a two-dimensional slice of a sound wave expanding from a vibrating tuning fork.

FOLLOW THE BOUNCING BALL

We've been talking about the sound wave, the cause of the sound event. Now let's talk about the effect it has on the air molecules. Like little superballs, air molecules bounce back and forth when the wave passes over them, bunching up (compression) and spreading apart (rarefaction), until the energy that pushed them initially is dissipated (see Fig. 1). With more power pushing them, they move farther apart before bouncing back.

Air molecules oscillating in this way are an example of a vibrating system. Our world is full of things that vibrate back and forth: tides, springs, even pendulums and electricity. With alternating current, electrons are oscillating back and forth in a wire. Amplitude refers to how far a vibrating object moves when it oscillates (or the degree of displacement from the equilibrium position). With tides, we might talk about how many feet up the shore the water is reaching. With pendulums, we could describe the angle of the swing. With air molecules, we talk about pressure levels, that is, how much their density changes. Sound-pressure levels (SPLs) are measured in pascals, named for the 17th century French scientist Blaise Pascal.

Intensity and amplitude are proportional to each other: if you get more of one, you get more of the other. But how much more of each? Here's a basic principle of vibrating systems: the amplitude level is proportional to the power level squared. This means that if you reduce the amplitude by one-half, you get one-quarter the power. Subsequently, if you double the amplitude, you get four times the power.

FIG. 2: An oscilloscope view illustrates a sound wave as pressure changes in time. High points of the wave correspond to high ­molecular density (increased air pressure), and low points of the wave correspond to low molecular density (reduced air pressure).

FIG. 2: An oscilloscope view illustrates a sound wave as pressure changes in time. High points of the wave correspond to high ­molecular density (increased air pressure), and low points of the wave correspond to low molecular density (reduced air pressure).

Sound amplitude levels are easier to visualize than sound power levels. We see them when we use an oscilloscope. The rising and falling line represents changing SPLs (see Fig. 2). More precisely, sound amplitude levels are mapped to alternating current amplitude levels that, in turn, are displayed on the screen. The same thing happens with audio, except that amplitude levels drive a loudspeaker rather than generating an image.

Do amplitude levels determine loudness levels? Not exactly. The ear registers volume as the average intensity over time. You may get an instantaneous amplitude spike but not hear it because the average intensity remains unchanged.

The average amplitude level is tricky to talk about because the amplitude level rises and falls above zero. A simple averaging of a periodic wave amounts to zero because the positive part winds up canceling out the negative part. Instead of average amplitude levels, we refer to a wave's rms (root mean squared) value. The wave is squared so it's entirely positive, and the average of the squared value is taken. For a sine wave oscillating between values of ±1, the rms value is 0.707.

We can also compare peak amplitude and instantaneous amplitude. Amplitude level changes constantly, and the rms value represents the extreme peak of motion in each direction. But when we make a digital recording of a sound, we store samples, which are instantaneous amplitude measurements obtained at the sampling rate. As stated above, instantaneous changes in amplitude may not register as changes in loudness if the average intensity is not affected. But a series of instantaneous amplitude measurements allow a digital-to-analog converter to “connect the dots” and create a good approximation of an audio wave. The perceived loudness of the wave is not represented by any one sample, but rather by the average amplitude level (squared) over time.

NAME THAT LEVEL

In the early 1920s, researchers at Bell Labs set out to devise a useful unit for describing sound power levels, because sound wattage has a couple of problems. First, air molecules are never completely inactive. There is always some degree of interaction among them. Thus, an absolute measurement is impossible. Without a concrete anchor in the case of sound power levels, comparative measurement is necessary. The Bell researchers created such an anchor by determining an average hearing threshold. People came into the labs, heard sine tones at 1 kHz at various power levels, and were asked “Can you hear it now? Can you hear it now?” Once they found an average threshold, they used it as a basis for comparison by dividing any measured power level by the threshold power level. This ratio is the basis of the decibel (dB) scale.

Another problem is that there is a huge range of audible power levels. To compress it into something more useful, they decided to express them on a logarithmic scale, like the Richter scale, which measures the strength of earthquakes. Simply put, this means that equal increments along the decibel scale represent a doubling of wattage levels. An increase of 3 dB doubles a given power level, while a decrease of 3 dB halves the power level.

The decibel unit is used to describe all three measurements: power, intensity, and amplitude. However, due to the mathematics of logarithms and the squared relationship of amplitude to power, an amplitude level that is doubled or halved represents an increase or decrease of 6 dB SPL. An instrument playing pianissimo is generally in the neighborhood of 40 dB SPL, while an orchestra playing at fortissimo is somewhere around 90 dB SPL, a change of about a millionfold. Things start to get painful at about 120 dB. When we deal with reverberation, the reverb time is the time it takes a sound to drop to -60 dB SPL, about one-thousandth the original level, which is effectively silence.

AUDITORY IDIOSYNCRASIES

If loudness were equal to acoustic intensity, things would be straightforward. But it's not that simple. Acoustic intensity is one thing; what the auditory system interprets as loudness can be another. (Our perception of sound is the basis for the field of psychoacoustics.)

For example, not all frequencies sound equally loud when played at equal power levels. In particular, lower frequencies need to be played at considerably greater power levels.

Imagine a musician playing an instrument at a given power level. Now imagine another musician joining in, playing at the same power level. The result is a doubling of the music's total power level. If they're playing different pitches, the result will be double the volume level. But if they happen to be playing the same pitch, it won't sound much louder than when there was just one player. As a rule of thumb, it takes eight musicians to produce an apparent doubling of volume. (Report this to a mathematician, and he or she will observe that the volume level is equal to the cube root of the intensity level.) That can give you problems if you're recording an ensemble. If the musicians go back and forth between unison and separate parts, the apparent volume level will change even though the meters show the same level.

Attack times also contribute to volume. Play a short synthesized tone repeatedly. Keep the amplitude level constant, but start with a slow attack and gradually shorten it until you have a sound that resembles a plucked string. Listeners will say that the sounds are getting louder.

Timbre also plays a role. Play some sine tones, then play some complex tones. Listeners will tell you that the complex tones sound louder, even though the intensity level remains constant.

If you're synthesizing your own instruments, you can make instruments sound louder without raising their amplitude level. Add some higher harmonics to give your sound some added presence, or shorten its attack to make it “pop out” of the mix.

FIG. 3: The curve that a compressor uses to control a signal''s ­dynamic level is called a transfer function. Below the threshold, the output level matches the input level. Above the threshold, the output level is mapped to a value below the input level. Extreme compression is also called limiting.

FIG. 3: The curve that a compressor uses to control a signal''s ­dynamic level is called a transfer function. Below the threshold, the output level matches the input level. Above the threshold, the output level is mapped to a value below the input level. Extreme compression is also called limiting.

THE ENVELOPE, PLEASE

The auditory system is selective about what it considers to be loud. We can also be selective about how loud our recordings are if we use a compressor, which reduces a signal's dynamic range (the difference between the lowest and highest levels) according to a transfer function (see Fig. 3). In addition to setting the threshold level and the slope of the transfer function, compressors feature adjustable attack and release times — that is, how long it takes the compression to kick on once the input level exceeds the threshold, and how long it takes the compression to stop after the input falls below the threshold. A long attack time can blur an instrument's transient, and a long release time can extend an instrument's release if the instrument's natural amplitude level falls below the compression threshold faster than the release time takes to activate.

Compression/limiting is useful for evening out an inconsistent signal, such as a player moving about and not maintaining a consistent distance from a microphone. Or it can be used for special effects if you use long attack and release times to manipulate envelope shapes. Lengthening the attack blurs the initial transient of each note. This can be useful if peaking occurs just at the note onsets, but you don't want to lose the track's punch in the overall sound by lowering its amplitude.

Let's say you're mixing a bass and a kick drum. Both of these instruments are played on strong beats. Since they are both bottom-dwellers frequency-wise, their combined sound can blur into an indefinite muddiness. How can you ensure that each retains its “personality” in a mix? One way is to put separate compressors on each. At the same time you can give the bass a quick attack time, maintaining its transient, but then applying an aggressive compression ratio following the attack. This maintains each bass note's juiciness without letting its sustains muddy the mix. Alternatively, you can lengthen the attack on the bass compressor and shorten the attack of the kick drum's compressor, allowing the kick drum to give a strong attack to each note, and letting the bass sustain it.

USE THE FORCE

The decibel measurement is helpful, but only to a point. Other measurement scales have been suggested over the years, but all have their problems. The bottom line is that there is no definitive scale for perceived loudness. But knowing a little about the properties that compose volume, we are better able to make sure that people who listen to our music hear it at the volume level we want them to.


Mark Ballora is assistant professor of music technology at Penn State University, where people rarely complain about the volume of his music.

FURTHER READING

This has been a “quick-start” introduction to loudness and its implications. Here are some resources for anyone wishing to delve into the subject a bit further.

Acoustic Power and Our Friend the Decibel

Essentials of Music Technology, by Mark Ballora (Prentice Hall, 2003)

An Introduction to the Psychology of Hearing, by Brian C.J. Moore (Harcourt Brace Jovanovich, 2004)

Listening: An Introduction to the Perception of Auditory Events, by Stephen Handel (MIT Press, 1989)

The Science of Musical Sounds, by Johan Sundberg (Academic Press, Inc., 1991)

Compression and Other Effects

The Computer Music Tutorial, by Curtis Roads (MIT Press, 1996)

Harmony Central: Effects Explained http://harmony-central.com/Effects/effects-explained.html

An Introduction to the Creation of Electronic Music, by Samuel Pellman
(Wadsworth Publishing, 1994)

The Audio System and Its Idiosyncrasies

Auditory Scene Analysis, by Albert S. Bregman (MIT Press, 1991)

The Psychology of Music, by Diana Deutsch (Academic Press, 1982)

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