It had to happen. I could only see the same tiresome flawed argumentation being repeated so many times before I had to make a whole specific webpage about it, so here it is.

This basically explains the same things as my Volume Controls page, but then in a more concise way and specifically aimed at debunking the same incorrect linear reasoning that is often applied to all things related to sound.

It always goes as follows: someone is citing a *decibel* figure somewhere, nowadays YouTube videos are a common place where I encounter this scenario. Often it suffices that merely a decibel meter of some kind is visible in the video, showing a few measurements, for instance 80 dB and a bit later 85 dB. There is no need for the persons in the video to mention these figures or make any remark about them for the inevitable to happen: in one of the highest ranking comments, we will see someone proudly demonstrating that they believe to understand the decibel scale by making claims akin to the following:

- The decibel scale is a logarithmic scale, and 61 dB is ten times more intense than 60 dB.
- For every 3 decibel increase, its actually double the sound level or volume. So 115 to 129 (14db higher) is roughly 5 times louder than the original car horn.
- 60 is normal household conversation level. Singing loud is already 70 db plus which is 10 to the ten which is ten TRILLION times kinetic energy being produced.

All those claims are wrong in some way, some more than others, some have multiple mistakes combined. The errors stem from incomplete understanding of logarithms, and/or the properties of the human auditory system.

The *‘bel’* is a unit of measurement named after Graham Bell, used to compare signal or power levels of any kind, and is denoted with the symbol B. What is important is that it is a *relative* unit, that compares one measurement either to another or to a reference value. The bel is the 10-base *logarithm* of the *ratio* of two power values. In practice the bel proved to be too large for general use, therefore it is almost always used with a factor 1/10, yielding the *decibel,* denoted with the symbol dB. In other words, 1 B = 10 dB.

It is very important to note that the unit is **logarithmic.** The logarithm is the inverse function of the exponential function: if `x` = 10^{y}, then `y` = log(`x`). A logarithmic function turns multiplications into sums. The slope of the curve decreases with increasing `x`: ever larger input increments are needed to keep raising the function's response by the same step size.

In this text, “log(`x`)” denotes the 10-base logarithm of `x`. When for some reason you can only compute the natural logarithm ln(`x`), basic maths teaches us that log(`x`) = ln(`x`)/ln(10).

What this means in practice, is that each time the value being measured is *multiplied* by the same factor `F`, the same offset is *added* to the logarithm of the measurement: log(`x` · `F`) = log(`x`) + log(`F`). This is **always** the case, not just when it seems most convenient, and this is also where a lot of the reasoning about decibels takes a wrong turn because people keep thinking linearly while they should be thinking logarithmically.

The decibel is used to measure sound signal levels, both in an absolute way to denote how “loud” is a sound carried by pressure waves, as well as in a relative way to denote what is the amplitude or power of an electrical signal representing sound inside a machine. Professional audio equipment will have volume controls displaying dB values.

Why is it practical to use the decibel to measure the loudness or amplitude of sound? Because it matches well with how human hearing perceives sound loudness. **We do not perceive sound in a linear way.** Our ears also have an approximately logarithmic response to sound. It allows us to cope with an enormous dynamic range, from a silent whisper to a roaring jet engine and anything in between.

In my volume controls article, the focus was mostly on the relative scale because that's what volume controls are about, but in this article it's the other way round. For completeness, I'll start by explaining the relative scale anyhow. Theoretically, the absolute and relative scales cannot be readily interchanged. When taking a 90 dB(A) sound pressure wave and attenuating it by 20 dB, there is no guarantee at all that it will be perceived exactly as 70 dB(A), but in many cases it will be a reasonable approximation.

The *relative scale* is used for all kinds of physical quantities, and indicates the relative amplitude of one signal compared to another. The symbol is simply ‘dB’. The calculation of the dB value depends on whether one is working with amplitudes or power values:

- For a ratio
`P`between two**power**values, the formula is 10·log(`P`). - For the ratio
`R`between two**amplitudes,**the formula is 20·log(`R`).

The reason for the difference between these formulas, is that power is proportional to amplitude squared, and the logarithm of a value squared equals 2 times the logarithm of that value: log(`x`^{2}) = 2·log(`x`).

Here we are more interested in an **absolute scale** however, more specifically the *“sound pressure level” (SPL),* which is an estimate of how loud a certain sound is perceived by an average human listener. A decibel meter can be used to obtain such estimate. There are several different absolute scales, but the most widely used one is dB(A), where the ‘A’ denotes the weighting that is being applied during the measurement. *A-weighting* corresponds to the frequency response curve of an “average human” across the most important frequency range (it does not properly model deep bass sounds or high trebles). To determine the dB(A) value for a certain sound, the sound is filtered through a filter bank that approximates the A-weighting. Next, the 10-base logarithm of the power is taken and the result is multiplied by 10. This can be done across short or long time windows, depending on whether a sharp peak is to be measured or an average over a longer period.

By design, the dB(A) scale will reach a value of zero at a level that is perceived as pure silence, the so-called *hearing threshold.* In practice, people will generally already consider 30 dB(A) as ‘silence’ because that is about the background noise level in many present-day ‘silent’ environments. Going below 30 dB(A) typically requires seeking a desolate environment or employing sound dampening. Being in a 0 dB(A) environment is actually a weird experience.

If you now think that 0 dB(A) is actual perfect silence: *wrong.* There can still be sound waves at that level or below, it's just that humans can no longer perceive them. Anechoic chambers and high-end microphones can easily reach *negative* dB(A) values. True perfect silence would be *minus infinity* dB(A) but is pretty much impossible to achieve.

At the other end, the loudest level above which even an extremely brief non-repetitive exposure becomes unacceptable, is about 120 dB(A), the ‘pain threshold’—the name says it all. Higher levels are possible but will cause more pain and faster permanent hearing damage as the level rises. Permanent hearing damage can already occur below the pain threshold: it all depends on the loudness level and the duration of the exposure. The point at which a continuous exposure would start to become damaging is debatable, but the generally accepted levels are around 85 dB(A) although at that level we're talking hours of exposure before it becomes problematic. The duration required for sounds above this level to become damaging, shortens with increasing level.

There is a practical upper limit to sound pressure levels. Around certain values it becomes meaningless to keep using the dB(A) scale because humans would no longer perceive an increase in loudness, their ears would only be destroyed faster, and at some point they would actually *die* when reaching pressure levels that equate to explosions and utter destruction.

In between the extremes, here are a few examples of realistic SPLs for things most people can relate to. Mind that the **distance** to the sound source is important. Roughly speaking, the SPL will decrease by 6 dB with each doubling of the distance to the sound source, when ignoring any effects of reflections. Another important fact to consider is that **1 dB** is about the smallest difference that can be perceived by humans.

- A normal conversation between people 1 m apart: about 60 dB(A)
- Playing an instrument in a classical orchestra: about 94 dB(A)
- Jackhammer at 1 m: 100 dB(A)

By far the most common incorrect reasoning I keep seeing repeated, is the lack of understanding that just as the decibel scale itself, the human auditory system is *nonlinear* as well. Many sensory systems in the human body are nonlinear because linear measurements are in general totally impractical when it comes to dealing with the real world.

People will see some increase in decibel values being displayed, and they will then attempt to calculate the linear increase in sound pressure or power using their basic understanding of the exponential and logarithmic functions. Then they assume that we perceive this linear increase. This is **wrong,** just as a linear volume control is wrong and annoying to use.

There is no need to calculate the linear increase at all. The same logarithmic transform that was already used to compute the decibel values, is also a good approximation of how our ears will perceive the increase in loudness. One can simply consider the decibel values as such to get an idea of how much louder the sound will seem.

Yes, it is true that adding 10 dB means that the power of the sound itself is multiplied by a factor 10. But this does not mean we perceive the sound to be 10 times louder. We only perceive it as *having increased by 10 decibels.* Our ears are not linear and they do not care about the linear increase. So please, stop doing these pointless conversions of dB values to their linear counterparts. They are only important in very specific circumstances. Embarrassing oneself through half-arsed knowledge about the decibel scale is *not* one of those circumstances.

The conversion to linear values is sometimes claimed to be justified to be relevant for the effects of hearing damage, but this is also unwarranted. The relation between SPL and hearing damage is complicated and involves both a time aspect as well as the actual signal distribution over time, and even the frequencies of the sounds involved. The effect of all these factors is even more nonlinear than how amplitude or power relates to perceived loudness, hence throwing around 1-dimensional linear figures is utterly *meaningless* and the only thing one proves by doing it anyhow, is that one has no real idea what one is talking about.

Then there are the scaling factors that are often being confused, because many do not understand that comparing power values is not the same as comparing amplitude values. Occasionally we will also have someone confusing the bel and decibel, or who will just invent their own scale factors at leisure. As explained above, power is proportional to amplitude squared, therefore an extra factor 2 is required when calculating decibels based on the ratio of amplitudes as opposed to power values. Pressure is an amplitude value, hence so is SPL, sound *pressure* level. It takes an additional 6 dB to double the amplitude, while it only requires an additional 3 dB to double the power. But again, in many cases *nobody should care.* Just stick to the decibel values themselves.

Nowadays one can install decibel meter apps instead of buying an expensive dedicated decibel meter. Is this trustworthy? Taking the above explanation into consideration, it should be clear that as far as *comparing* decibel values goes, a smartphone with a good microphone will be quite usable. The logarithm is the saving grace here: even if the signal response of your particular microphone is a certain factor `F` stronger or weaker than the microphone the developer of the app was using, this factor will become a mere offset value log(`F`) in the decibel scale. Subtracting the dB measurements eliminates their common offset. So, if the actual difference between the sounds being measured is say 7 dB, then the smartphone should still roughly show a 7 dB difference as well, as long as the deviations in microphone responses aren't too drastic.

The problem lies within the *absolute* values. Even when one buys a dedicated decibel meter, it will also not be guaranteed to be accurate for measuring absolute dB(A) values unless it is *calibrated.* This involves mounting a device to its microphone that produces a well-controlled sound with known SPL, and then adjusting the meter until it displays that SPL value. In theory a smartphone app can also display an accurate absolute SPL value under several assumptions:

- The sensitivity of the particular smartphone's microphone is known by the decibel meter app,
- the app can either identify the microphone, or knows which smartphone type uses what microphone and all phones of that type use exactly the same microphone model;
- the variations between microphones is negligible, despite manufacturing differences of the microphones and differences caused by the soldering process, dust accumulating in the microphone, bumper case shape, etc.;
- the author of the app knows what they're doing and had access to a well-calibrated decibel meter to calibrate the app's response.

There is quite a lot that can go wrong in the above list, hence I wouldn't trust a smartphone when it comes to proving a point when it really matters. However, in general the error will be small enough for the occasional times when you need to have an idea of the actual decibel value, so don't go buy that expensive decibel meter and calibrator unless you really need to.

©2023 Alexander Thomas