Thursday, April 12, 2012

Loudness and Intensity - What's the difference?

What’s the Big Deal about Loudness and Intensity?

Language exists to communicate. If we don’t use the same meanings to the words, leaving out philosophical issues for now, communication is difficult; hence, a few definitions and words to help us in the discussions.

Many people confuse loudness and intensity and assume they are the same or similar enough that they can be treated interchangeably. This isn’t true. They are different and understanding why they are different and how to use them is one of the keys to understanding what we hear. This understanding will also help explain some of the reasons why different level compression methods sound so different.

What is Intensity?

Definition: Intensity is the external measured level of a sound.*

There are many measures of intensity. For now, we will use the common psychoacoustic (not mechanical engineering) definition of sound pressure level (SPL). SPL, usually measured in dB, is a measure of acoustic energy in the atmosphere and is defined as 20*log10 (actual pressure/reference pressure). There are assumptions built into SPL. Let’s keep it simple for now and ignore them. Note that there are several reference pressures used for SPL in different fields, we will use the one commonly used in psychoacoustics, not the ones used in mechanical engineering or sonar.

dB in electronics has a different kind of reference than dB in acoustics and psychoacoustics. In electronics, we do not know the meaning of 0dB in acoustic terms because we do not know the efficiency and frequency response of the rest of the system. (Electronics reminder: Remember, dB is always relative, so in electronics we must assign an arbitrary 0dB. Additionally, we don’t know, in general, how that corresponds to actual intensity in the atmosphere; we all have “volume” controls on our equipment. Speaking of which “volume” is neither horse nor mule, but rather a poorly defined attempt at one, the other, or both, depending on who contoured the control’s taper.) Therefore, electronics uses dBm, dBv, and other defined reference levels to provide a way to understand what the intensity of the analog of a signal is. There are assumptions built into these level measures as well, constant impedance, for instance. Again, let us keep it simple for now.

dB in digital terms is also different, in that it typically uses 0dB for a reference maximum level rather than a low or minimum level, and that levels in dB are therefore always negative.

What is Loudness?

Definition: Loudness is the internal, subjective experience of how loud a signal is.

The term loudness dates back at least to Fletcher, if not beyond. Loudness is not intensity.  Subjective experience, perceptual experience, and individual auditory periphery all affect perceived loudness. Loudness is not a measured level in the atmosphere.

Loudness and intensity can be mostly related by a complex calculation. Keep in mind, however, that every listener is a bit different, hearing injuries affect loudness in many ways, and intensity does not equal loudness. The relationship is complex, and could constitute an entire tutorial in and of itself.

In the worst cases, intensity is a very poor substitute for loudness, and vice versa.

What is Intensity For?

Intensity is an objective measure. It measures the actual fluctuations in air pressure in the atmosphere that constitute sound. Use intensity when you need to know the actual fluctuations in air pressure. For example, when you want to know how much noise your car makes, when you need to know how to achieve a certain sound pressure level in a concert hall, or how much force a sonic boom exerts.

Do not use intensity when you want to know how loud a sound is to a listener.

What is Loudness For?

Loudness is a perceptual, or sensation concept. It describes the experience the listener has. Use loudness when you are trying to estimate psychoacoustic parameters, want to know why somebody is shouting “turn that **** thing down!” and the intensity isn’t that high, want to know why somebody is shouting “turn up the sound” when the intensity is already excessive, or want to match loudness, not levels, across audio selections, either full-bandwidth or from remotes/phones.

An Occasionally Useful Approximation, the Power Law Relationship

What can we say about the relationship between loudness and intensity?

When the spectrum (frequency content) of a signal is unchanged, loudness is approximately proportional to the 1/3.5 power of the signal power, or the 1/1.75 power of amplitude. Let’s call this the power law relationship. For frequencies above the audible thresholds, this approximation works well. For certain special signals or signals that have a lot of energy just below or at the threshold of hearing this approximation does not work as well.

As discussed in Why We Hear What We Hear, Part 3, the cochlea has a mechanical filter that does frequency analysis. ERB’s are the bandwidths of these filters. This mechanical filter is a set of continuously distributed, heavily overlapping filters, not one set of 30-ish adjacent filters. Some points to keep in mind:

  • ERBs are a way to understand frequency in perceptual terms, and constitute the best current measure of the filter bandwidths in the cochlea.
  • ERB bandwidths are approximately 70Hz at low frequencies, and about ¼ octave at higher frequencies.
  • Signals in the same band convert intensity to loudness with the power law relationship. The loudness in a given band is referred to as the partial loudness corresponding to that frequency.
  • The loudness of a signal is the sum of the partial loudnesses.

This explains why two signals at frequencies reasonably removed from each other, each of which has ½ the energy of the original, will have 2 * (1/2)^(1/3.5) or 1.64 of the loudness of one of the signals presented at energy 1. Doubling the energy of a signal without changing the spectrum will increase the loudness by a factor of about 1.21. In contrast, if we double the energy of the signal by adding as much energy in at a frequency where energy was not initially present on the cochlea (i.e. well outside of an ERB), the loudness doubles. Doubling the loudness is roughly equal to increasing the intensity by 10dB without spectral modification.

If we take the same amount of energy, and spread it out from a tone into 2, 3, 4, or more ERBs, the loudness will increase as the energy is spread out, as long as all of the signal spectrum remains above the absolute threshold of hearing.

Understanding loudness behavior in regard to bandwidth is critical. Here are some examples as a reminder.

  • If we double the energy of a sine wave, its loudness rises by approximately 2^(1/3.5).
  • If we make a new signal by adding a second sine wave with the same energy to the first example at a frequency removed from the first sine wave (in terms of ERBs), the loudness doubles.
  • The ratio of loudness for these two signals, which have the same intensity, is 1.21 vs. 2.

Here’s a graphical example. In this example, the vertical axis is the relative loudness, with the single-band loudness set to 1 for simplicity. The curve shows the relative loudness when the same amount of energy is split over n bands, from 1 to 25. The numbers for over 15 bands are probably an overestimate, but that’s signal dependent.

relative loudness vs number of bands

As we can see in this graph, when the bandwidth of a signal grows, even though the energy of the signal remains the same, the loudness will grow. When the number of bands involved becomes large enough that adjacent bands become involved (as it must above 14 or 15 bands),the data in this plot is somewhat of an overstatement. In general, when energy is spread into an adjacent band, the effect on loudness will be somewhat less than when energy is added to a band distant in frequency from the original signal.

Two other things to observe about this example, first the range of loudness corresponding to a given signal energy can vary by about a factor of 10, corresponding to an increase in energy (for an unchanged spectrum) of a factor of about 3000. Clearly, increasing bandwidth is a powerful method for increasing loudness without increasing signal energy. Second, the graph is only an approximation. It is approximate because

  • Effects due to different listeners will change it
  • Effects regarding absolute threshold will change it
  • The power law is merely approximate in the first place
  • The distribution of energy (adjacent bands or far away) will affect the outcome

None the less, the point holds that loudness is not always very strongly correlated to intensity.

In summary, dB SPL is a measure of the intensity of a signal. Loudness can be discussed in dB equivalent or related terms, but in general, dB is not a measure of loudness, and should not be presented as such. What dB equivalent means is that the loudness of a signal is equal to the loudness of some other, unchanging reference signal at the specified SPL for the reference signal.

dB does not measure loudness, but there are some commonly cited approximations, such as that doubling loudness  without changing spectrum requires a gain increase of 10dB or so, as established by a set of experiments done by Fletcher, after Bell’s original experiments. Fletcher’s work has since been confirmed over and over by many other experimenters. The design of Fletcher’s experiments, Stevens’ experiments, and their successors are complex and interesting but beyond the scope of this article.

There is a Loudness Button on my Machine

There is a “Loudness” button on many receivers, amplifiers, and other audio equipment. Because of the well-known reduced sensitivity of the ear at low frequencies, some manufacturers choose to add a bass boost, sometimes variable from the front panel, and call it a “loudness” control. This bass boost is linear and time-invariant.  The best we can say about Loudness controls is that they make the sound louder. They cannot fully compensate for changes in sensation level without being signal dependent and time varying.

Unfortunately, we can boost something with a fixed curve only if the signal has a fixed level and a fixed spectrum, and music is neither. Processing for loudness restoration is not linear time invariant (time invariant frequency shaping), it’s not even close to being linear time invariant.

Take a look at this graph of loudness level contours versus intensity levels. The crowding of signals at low frequencies shows the effects of loudness growth at low frequencies that does not scale to dB. A real loudness restorer that worked for different intensities of presentation levels would have to be time-varying and signal dependent. It’s not that easy.

(Image from

Common Questions and Moving Forward

 Be careful to use the terms loudness and intensity as distinct terms and to bear in mind how loosely the two are related. When we use nonlinear processing such as level compression that spreads frequency content, we must realize that the loudness may rise faster than expected.

Does that mean that there is such a thing as loudness “enhancement”? Well, leaving aside the metaphysical question of exactly what an enhancement is, yes. If a nonlinearity spreads the spectrum of a signal, it is likely to become louder. Some examples of loudness “enhancement” would include

  • LP distortion grows with level. That means that as level grows, the signal bandwidth (including the distortion) increases. An increase in intensity is over-represented by the increase in loudness.
    • This can create an illusion of “more dynamic range”.
    • It can also be very annoying.
  • Tape distortion grows with level. It behaves different than LP’s, but to the same result, at usual saturation levels.

What about “Make it LOUD” sorts of processing, such as used in certain kinds of level compression processing for radio broadcast or some overly loud CDs? Oh, yes, they work. They certainly do “make it louder”, indeed.

  • It is a bit of an art to make a signal have a peak to RMS ratio similar to that of a sine wave.
  • They spread the spectrum very broadly.
  • You can create your own opinion about how they sound.

* You will notice that I say that I am treating intensity informally above. This is deliberate, intensity does have a formal definition in acostics, and a somewhat different meaning in psychoacoustics, at least in some quarters. In the case of hearing, the eardrum responds to pressure, and the head interacts with the volume velocities of the air around it to convert some amount of volume velocity to pressure at higher frequencies. This being how head related transfer functions (HRTF's) come into being, but that's another article for another day. More specifically, in this article, intensity refers to the actual differential pressure across the ear drum, that being what the head, pinna, etc, convert an acoustic sound field into.

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