## MINI LESSON

DECIBELS

Composed by Jeremiah Sheets

Additional notes by Prof. Monty McGovern, PhD

I was recently speaking with a colleague while setting levels on camera and was asked a very common question: “What is a decibel?”

Just like any good technician, he was curious, but certainly not new. He was seasoned and educated and knew very many things. In some ways he was much more intelligent than me. He knew how to do all the technical things he was required to do in his position.

Yet his question, common as it may have been, stood out to me. Not by the words themselves, but by the profound way in which he said them:

“So… what is a ‘decibel’?”

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That emphasis reminded me that knowing about decibels and how to use them is not the same as understanding what a decibel is. Most technicians in the field learn the former, but don’t necessarily understand the latter.

With that, I challenged myself to come up with an explanation both simple and complex enough to satisfy anyone still reading this…

Here we go.

# V1

v1

In its simplest form that most everybody already understands: Decibels are loudness!

… while that definition isn’t technically wrong, it is wronger than v2, and perhaps a bit misleading.

# V2

v2

The decibel is a more manageable expression of what we consider loudness. Without which the figures used would be considerably more difficult to… express.

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Notice the words “what we consider”. In reality, “loudness” is not a physical force to be measured. What some consider loud, other’s might consider less loud, or maybe more loud. Loudness is defined entirely by the listener.

The force we actually want to measure is “sound pressure” and the unit we use to define it is the “pascal”. These units range from absurdly small to comparatively large. If for example we wanted to know the typical range of human hearing, a sound pressure of 0.00002 pascals (20 µPa) would be the quietest sound most healthy young humans can hear. At the other end, a pressure of around 60 pascals might be the loudest sound most healthy humans can hear before experiencing physical pain.

That doesn’t seem too difficult to handle at first glance, however consider this:

The threshold of hearing in pascals is 3,000,000 (three million) times less than the threshold of pain.

One extreme is 3,000,000 times louder than the other.

In other words:

Let that sink in for a moment...

Thankfully we don't actually perceive loudness in this way. Regardless, we clearly need a more manageable range of numbers. And I think we can come up with an even righter definition.

# V3

v3

The decibel is a logarithmic expression of a measurement of sound pressure level. That’s it.

You can stop here.

But if you’re a winner who likes to win, read on.

We’ll skip the math lesson. Honestly I don’t think I’m well enough equipped in the smarts department to actually give one, but let’s pretend it’s because I’m impatient and just stick to the relevant bits.

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Logarithms often have a convenient way of taking a large range of numbers and expressing them more manageably. For example: In a standard base-10 logarithm (the most common type), the difference between 1 and 10 is “1”. The difference between 10 and 100 is also “1”. The difference between 10 and 10000 is “3”, and using the formula to the right, the difference between 0.00002 and 60 is just under “130”. In case that’s not familiar, those first two numbers represent sound pressure and the last number represents the logarithmic ratio between those pressures in decibels.

### 20log

##### 10

#### P

#### P(ref)

Math notes from the Professor:

"The logarithm of a number n to the base b is the power a such that b^a equals n. Thus if b = 10 and one number n is 10000 times another one m, then the logarithm to the base 10 of n is 4 more than the logarithm of m: log n = log m+4. In this way you see that the difference between widely separated numbers can be expressed in a manageable scale."

- Monty McGovern, PhD

Professor of Mathematics at University of Washington

Because the logarithm is expressing the ratio between the two values, any reference can be chosen, and as long as the the difference in the ratio is the same, the resulting logarithmic value is also the same.

If we choose a static value to apply to 0.00002 and we call that a “reference”, then we can express any exact pressure in pascals as an exact value in dB SPL. That sounds useful… And so, that’s exactly what we do!

We apply the value “0dB SPL” to the threshold of hearing, which means the threshold of pain becomes around “130dB SPL”. Pick any number in pascals or in dB SPL and you can figure out the other…

0.006 pascals is near enough 50dB SPL. 100dB SPL is exactly 2 pascals. And so on.

# A little Cleanup

You might have noticed that when using the pascal as a reference above, the unit changed from “dB” to “dB SPL”. It’s important to note that decibels as logarithms don’t always have to refer to sound pressure level. In fact, they can be used to express Power (dBm), Voltage (dBV, dBv, or dBu), Digital Bit Levels (dBFS), or really any other reference unit where we need to express ratios with the specific formula. Even SPL can include weighted scales like dBA or dBC. When defining a number in decibels you must always include the reference when applicable.

I’d also like to point out that the thresholds of hearing and pain can vary significantly from person to person. In different texts you might find pain levels closer to 120dB SPL, or upwards of 140dB SPL. Those are very large differences, yet none of them are wrong! Some are just maybe wronger than others given a particular individual, or even a particular time of day.

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Finally you might have noticed the numeral "20" in front of the formula.for dB SPL.

# A Story

What's with all the "righter" and "wronger" text?

WRITE STORY ABOUT ASIMOV'S RELATIVITY OF WRONG - how it got me into science and math including chemistry and acoustics...

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Maybe link to "suggested reading" page?