Loudness, Decibels, and Equal-Loudness Contours

Learning Objectives¶

By the end of this chapter, you should be able to:

• Distinguish sound pressure, sound intensity, and loudness, and state which are physical and which is perceptual.

• Explain why the enormous range of audible intensities makes a logarithmic scale the natural one.

• Calculate sound intensity level and sound pressure level in decibels from intensity or pressure, and convert back.

• Explain why a factor of ten in intensity is 10 dB and a factor of two is 3 dB, and use those two facts to do decibel arithmetic without a calculator.

• Combine the levels of two or more incoherent sources, and explain why doubling the number of violins does not double the loudness.

• Distinguish the decibel as a ratio from the decibel as an absolute level referred to a stated reference, and use dB SPL correctly.

• Read the equal-loudness contours, define the phon, and explain why bass disappears first when music is played quietly.

• Define the sone, use the rule that ten phons doubles the loudness, and explain why a 60-piece orchestra is not sixty times as loud as one instrument.

• Explain masking and the critical band, and give a musical consequence of each.

Introduction¶

Turn the volume knob halfway down. Is the music half as loud?

Nobody thinks so, and the reason is the subject of this chapter. Between the sound pressure at your eardrum and the loudness you experience lie two transformations, and neither is a simple proportion.

The first is enormous range. The quietest audible sound and the loudest tolerable one differ in pressure by a factor of about a million, and in energy by a factor of about a million million. No linear scale copes with that, and the decibel exists to tame it.

The second is frequency dependence. The ear is far more sensitive at some frequencies than at others, and, the awkward part, how much more depends on how loud the sound is. A bass note and a mid-range note that match in loudness when played loudly will not match when played quietly. Music therefore sounds thin at low volume, and it is a fact about you rather than about your loudspeakers.

This chapter separates three things that ordinary speech runs together:

Only the first is physics in the sense of Chapters 1 to 5. The other two are properties of a listener, and getting from the first to the third is what the chapter does.

7.1Pressure, Intensity, and the Range of Hearing¶

Two Physical Measures¶

Two physical quantities describe the strength of a sound, and both are used.

Sound pressure $p$ is the amplitude of the pressure variation, in pascals. It is what a microphone and an eardrum actually respond to.

Intensity $I$ is the power per unit area carried by the wave, in watts per square meter. It is related to pressure by

where $\rho v$ is the acoustic impedance of the medium, about 413 kg m⁻² s⁻¹ for air at room temperature.

The essential relation is the one Chapter 1 established: intensity goes as the square of pressure. Double the pressure and you quadruple the intensity. Every factor-of-two confusion in acoustics comes from losing track of this.

Twelve Orders of Magnitude¶

The numbers are worth writing out.

The pressure range is 10⁶ and the intensity range is 10¹². For comparison, the total acoustic power radiated by a full symphony orchestra playing as loudly as it can is somewhere around one watt, less than a small light bulb. The ear is an extraordinarily sensitive instrument working with extraordinarily small quantities.

A linear scale is hopeless here. If one millimeter on a graph represented the threshold of hearing, the threshold of pain would be a thousand kilometers away.

The Reference Values¶

The values in the first row of that table are not accidents; they are definitions, chosen because they are close to the average threshold of hearing at 1 kHz:

Every decibel figure in this book is referred to one of these, and they are consistent with each other: a pressure of $p_0$ in air carries an intensity of very nearly $I_0$.

7.2The Decibel¶

A Logarithmic Scale¶

The decibel compresses a huge range into a manageable one by taking a logarithm. For a ratio of intensities,

Because intensity goes as pressure squared, and $\log(p^2) = 2\log(p)$, the same level in terms of pressure is

The factor is 10 for power quantities and 20 for amplitude quantities. They give the same answer for the same physical sound; the difference is entirely in which quantity you started from. Using 10 where you need 20 halves every answer, and it is the most common error in the subject.

dB SPL and Other References¶

A decibel on its own is a ratio and says nothing absolute. “The passage is 6 dB louder” is meaningful. “The passage is 6 dB” is not, until a reference is stated.

dB SPL means referred to $p_0 = 20\ \mu$Pa, and is the one used throughout this book for absolute levels. Others exist, dBu and dBV in audio electronics, dBFS in digital audio (Chapter 15), and mixing them up produces errors of tens of decibels.

You will also meet dB A, which is a measurement weighted by a filter approximating the ear’s sensitivity at moderate levels. Exposure limits are quoted in dB A precisely because a sound’s damage potential depends on where its energy lies, not just how much there is.

Decibel Arithmetic Without a Calculator¶

Two facts make most decibel problems mental arithmetic.

The second is approximate, $10\log_{10}2 = 3.01$, and the approximation is excellent.

From those two, by adding:

7.3Combining and Comparing Levels¶

Adding Incoherent Sources¶

Two violinists play the same note. How much louder than one?

The answer depends on whether their sounds are coherent, locked in a fixed phase relationship, or not.

Two loudspeakers fed from the same signal are coherent, and their pressures add: $2p$, which is 6 dB. Two violinists are not coherent: their frequencies and phases drift independently, so the interference at any point averages away over time and their intensities add instead: $2I$, which is 3 dB.

Real musicians are incoherent, so for $n$ identical independent sources,

The consequence for orchestration is sharp and slightly deflating. Sixty violins produce $10\log_{10}(60) = 18$ dB more than one. And since §7.5 will show that 10 phons is a doubling of loudness, 18 dB is under two doublings: sixty violins sound less than four times as loud as one. Orchestras have so many strings and so few trumpets for that reason: the section needs numbers merely to keep up.

Distance, Level, and the Six-Decibel Rule¶

Combining the inverse-square law of Chapter 2 with the definition of the decibel:

Doubling the distance therefore costs $20\log_{10}(2) = 6$ dB. Ten times the distance costs 20 dB.

This holds outdoors. Indoors it fails beyond the critical distance, for the reason Chapter 2 gave and Chapter 14 develops: most of what reaches a distant listener has bounced.

Levels of Real Musical Sounds¶

A rough table, for calibration. All at the listener’s position in a concert hall:

An orchestra’s usable dynamic range is therefore about 40–50 dB, which is far less than the 120 dB the ear can handle. The limit at the bottom is the noise of the hall and the audience; at the top it is what the instruments can produce.

Note also that dynamic markings are not absolute levels. A pianissimo in a Mahler symphony and a pianissimo in a string quartet differ by a good deal, and performers read the marking as an instruction about relative level and character rather than about decibels.

7.4Loudness Level: The Phon¶

Equal-Loudness Contours¶

Now the frequency dependence. The experiment is straightforward to describe: play a 1 kHz reference tone at a known level, then play a tone at some other frequency and ask the listener to adjust it until the two sound equally loud. Repeat across the spectrum, and across reference levels, and average over many listeners.

The result is the set of equal-loudness contours, standardized as ISO 226.

Three features of those curves carry most of the content.

The ear is most sensitive around 3–4 kHz. This is the ear-canal resonance of Chapter 6, and it is why that region is where speech intelligibility lives and where a singer must compete (Chapter 13).

Sensitivity falls sharply at low frequencies. At 50 Hz the threshold is about 45 dB SPL: a sound 45 dB above the 1 kHz threshold is only just audible.

The curves flatten as level rises. This is the important one, and it is the subject of the next subsection.

Reading the Contours¶

The phon is defined by those curves: a sound has a loudness level of $N$ phons if it is judged as loud as a 1 kHz tone at $N$ dB SPL.

Why Quiet Music Loses Its Bass¶

Because the contours are not parallel, the ear’s frequency response depends on level.

The consequence is familiar to everyone. Music played quietly sounds thin. Turning the volume down does not merely make everything smaller; it changes the balance, taking the bass first.

7.5Loudness Itself: The Sone¶

Doubling Loudness¶

The phon says when two sounds are equally loud. It does not say when one is twice as loud, and it cannot: phons are defined by a matching procedure, and matching says nothing about ratios.

To get ratios, ask listeners directly: adjust this sound until it is twice as loud as that one. The answer, robustly reproduced, is

The sone encodes this. One sone is defined as 40 phons, and

with $S$ in sones and $P$ in phons. So 40 phons is 1 sone, 50 phons is 2 sones, 60 phons is 4 sones.

Ten phons is a factor of ten in intensity. So a tenfold increase in intensity is only a doubling in loudness, which is the compression that makes the ear’s enormous range usable.

Masking¶

A sound that would be perfectly audible on its own can be rendered completely inaudible by another sound. This is masking, and Chapter 6 supplied the mechanism: a masker’s traveling wave excites a region of the basilar membrane, and anything whose own region falls inside that excitation is lost in it.

The upward spread of masking has direct musical consequences. A loud bass line buries a quiet high part far more than the reverse, so arrangers give the melody to a high instrument and a mix engineer who wants to hear the hi-hat turns the bass down rather than the hi-hat up.

It is also, remarkably, what makes MP3 possible. If a component of a signal is masked, a listener cannot hear it, so a codec can simply discard it and spend the saved bits elsewhere. Chapter 15 develops this.

The Critical Band¶

Masking is strongest when masker and masked are close in frequency, and the width of the region over which they interact strongly is the critical band.

It is roughly 100 Hz wide below 500 Hz, and roughly a third of an octave above that, which means it is approximately constant in musical terms, a little wider than a major third, across most of the range. That is not a coincidence: the critical band corresponds to a roughly constant distance along the basilar membrane, and Chapter 6 showed that the membrane’s map is logarithmic.

The critical band is the ear’s frequency-resolution limit, and it does a great deal of work in what follows:

• Two tones within a critical band interfere audibly and sound rough; two tones further apart do not. This is the physical basis of dissonance (Chapter 8).

• Noise masks a tone only if the noise lies within about a critical band of it.

• Loudness adds differently within and across critical bands. Spread the same energy across several bands and it sounds louder than if it is concentrated in one; so a chord is louder than a single note of the same total intensity, and a full orchestration carries further than a doubled unison.

7.6Summary¶

• Intensity goes as pressure squared. The decibel therefore has two forms: $10\log_{10}$ for intensity and power, $20\log_{10}$ for pressure and amplitude. They describe the same sound.

• Hearing spans 10¹² in intensity, from 10^-12 W/m² to about 1 W/m², referred to $p_0 = 20\ \mu$Pa. A decibel figure without a stated reference is a ratio, not a level.

• Two facts do most decibel arithmetic: a factor of ten is 10 dB, a factor of two is 3 dB (in intensity).

• Incoherent sources add in intensity, so $n$ of them give $10\log_{10}(n)$ dB. Sixty violins are 18 dB above one, less than four times as loud.

• Distance costs 6 dB per doubling outdoors, and much less than that indoors beyond the critical distance.

• Equal-loudness contours describe the ear’s frequency dependence. The phon is defined by matching against a 1 kHz tone, so phons and dB SPL agree at 1 kHz and nowhere else. Sensitivity peaks near 3–4 kHz and collapses at low frequencies.

• The contours flatten as level rises, so the ear’s frequency balance depends on volume. Quiet music loses its bass first.

• The sone measures loudness itself: 1 sone is 40 phons, and each 10 phons doubles it. A tenfold increase in intensity is a twofold increase in loudness.

• Masking renders an audible sound inaudible, spreading much further upward in frequency than downward. It is the basis of perceptual audio coding.

• The critical band is the ear’s resolution limit, roughly a third of an octave, hence roughly constant in musical terms. It underlies roughness, dissonance, and the way loudness sums across a spectrum.

7.7Check Your Understanding¶

Five short, auto-graded questions cycle within one compact activity, using five different response styles.

7.8Conceptual Questions¶

1. Explain why the decibel scale uses $10\log$ for intensity and $20\log$ for pressure, and why both give the same level for the same sound.

2. A student writes “the sound was 70 dB”. Explain what is missing from this statement and why it matters.

3. Explain why two loudspeakers fed from the same signal give +6 dB while two violinists playing the same note give +3 dB.

4. An orchestra has sixty violins and three trumpets. Use the addition rule to comment on whether this is a sensible balance.

5. Explain why music played quietly sounds thin, using the shape and spacing of the equal-loudness contours.

6. A 50 Hz tone and a 3 kHz tone are played at the same dB SPL. Which sounds louder, and roughly by how many phons at moderate levels?

7. Explain why a loud bass line masks a quiet cymbal much more than a quiet cymbal masks a loud bass line.

8. Explain how masking allows an audio codec to discard part of a signal without an audible change, and state what would go wrong if the codec’s model of masking were inaccurate.

7.9Problems¶