The Measurement of Loudness

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A stack of some of my older audio equipment. Yes it is very dusty.

I'm not a sound engineer of any sort, but I enjoy music and have been blessed with decent hearing acuity, so I tend to pay attention to noises going on around me. Now, I know what you're thinking! Surely, a measurement nerd interested in sound would have bought a cheap SPL meter off the internet and this is what this post is about.<br>And you'd be wrong! Hah! Because this post goes a bit further off the deep end because SPL and the dB(a) scales that we commonly associate with "sound volume" always confused me when I tried to understand them. Like with how measuring color is really difficult because it's at the intersection of a physical measurement and human perception (see: How the heck does one measure color?), sound is just as messy because it's again physical measurements (sound pressure) mediated by the human auditory system.<br>To measure loudness, we're going to have to go back a bit in history.<br>First came the decibel<br>As with a lot of audio signal technology, the history of loudness starts with the telephone systems and Bell Labs. Once you had a service that had to send electrical signals over long wires, you needed a language and units to easily discuss how much signal was coming through.<br>The decibel unit (dB) was adopted in the 1920s to replace an older unit of "miles of standard cable (MSC)" which represented the loss of power over a mile long piece of standard telephone cable. You could easily imagine a telephone tech at the end of a line taking measurements and saying that a line sounds like it had traveled through 15 miles of cable.<br>But once telephone systems got more advanced and it was possible to boost signals for longer transmission, engineers needed a way to both add and subtract signal. So they developed the decibel system because power and perception is more in line with logarithmic scales. It is a logarithm (base 10) of the ratio between two values (one being a reference point), 10*log_10​(P1​/P2​), such that 3 dB difference between the two meant a doubling in power, so a 10 deci-bel difference (a.k.a. one Bel), is a full power of ten.<br>Incidentally, 0 dB SPL, is referenced against the sound pressure level of 20 micropascals, which is about the threshold of hearing.<br>In practice, it meant that phone engineers could listen to their reference tone from the central office (which was apparently set to 1000 hz because it's a round number in the target audio hearing range), then they listened to a closer reference signal and used calibrated attenuators (a bunch of resistors) to compare how much the signal degraded versus the reference tone.<br>But humans don't hear decibels<br>Pretty quickly, it was obvious that the human perception of loudness wasn't the same as the decibel unit's measurement of pure electrical power. The dB scale would predict something should sound twice as loud and humans would say it wasn't when it deviated from the reference 1000 hz tone that everything was tuned to. Depending on the frequency, it'd be perceived as softer our louder, but rarely "the same".<br>So more researchers at Bell Labs came up with another paper in "Loudness, It's Definition, Measurement, and Calculation" Fletcher and Munson, The Bell System Technical Journal, 1933. Here, Fletcher and Munson essentially tackled the problem of measuring how humans perceived the loudness of tones.<br>To jump to the conclusion, the two researchers developed a new unit called the phon . It is a unit of perceived loudness where 40 phon, by definition, is set to be equal to 40 dB SPL at 1000 hz. For all 1000hz tones, phon is the same as decibels of sound pressure. It is also a logarithmic scale and so if our hearing was magically the same at all frequencies, the phon and db SPL would be equal.<br>But phon and db SPL become different values when we're talking about different frequencies. The two scales are related by what's called the "equal-loudness contour" which is a set of curves that show how much db SPL is needed to sound the same as the reference phon. See the diagram below.<br>Equal-loudness contour chart from WikipediaAs you can see, each line essentially maps a level of sound pressure that "sounds equally loud". We are pretty bad at hearing low frequencies so it takes a LOT of SPL at the low end, and it sweeps downward as our hearing gets better at picking up sounds. Then the curves eventually sweep upward as we lose ability to hear extremely high values. Also note that the curves get flatter as we change from from threshold sound levels to extremely loud ones.<br>Most of the paper talks about the technical implications of measuring and then using the equal-loudness contour that was originally mapped by Fletcher and Munson. Meanwhile, what's of more interest to me was how they measured their initial contour (which isn't on the contour chart above, but was the predecessor of the ISO blue line). The methods of...