OK, so after re-reading the previous post that I created ("A Primer on Decibels") I thought about editing it to add some details that I very much skimmed over. However, that thread is already long enough, so to that end, I'll post here some additional and insightful information.

Although I spoke to the method by wich SPL is calculated, as well as how to determine the pressure associated with a given SPL value, what I really glossed over was the change in pressure versus the change in dB SPL, and the immense scale of pressures we are talking about.

So to that end, I created the table below. The first column shows the SPL in dB, but the actual pressure, as well as the change (from one value to the next) is also shown. This is pretty telling as to just how much range in pressure is involved, and why the decibel is so useful.

[dB SPL]{Pmeas} /Change/
[0] {2.0E-5} /0.0/
[10] {6.32E-5} /3.16/
[20] {2.0E-4} /3.16/
[30] {6.32E-4} /3.16/
[40] {2.0E-3} /3.16/
[50] {6.32E-3} /3.16/
[60] {2.0E-2} /3.16/
[70] {6.32E-2} /3.16/
[73] {8.93E-2} /1.41/
[76] {1.26E-1} /1.41/
[80] {2.0E-1} /1.58/
[90] {0.632} /3.16/
[100] {2.0} /3.16/
[110] {6.32} /3.16/
[120] {20.0} /3.16/
[130] {63.2} /3.16/
[140] {200.0} /3.16/

Note what's interesting here... if you look at the SPL (shown in column 1) and then look at the change (i.e. from 0 dB to 10 dB, 10 dB to 20 dB, and so on) you will see that there is a pattern emerging. Here's where it gets kind of interesting...

So, from the prior post we talked about a 10 dB change in SPL being perceived as twice / half as loud. However...note that the change in pressure corresponds to a factor of 3.16, and not 2.0. That is, a change in measured pressure of a factor of 3.16 equates to a 10 dB change in SPL and a halving / doubling of the perceived intensity of the sound.

I threw in some smaller increments in the list to illustrate the change in pressure relative to the previous value - the 3 dB increments are those values.

Now let's talk about the range of pressures. Remember that the reference pressure, which equates to 0 dB SPL is 20 micro-pascal. That is, 20 millionths of a Pascal. That's a pretty freaking small number. Now look at the pressure that corresponds to 100 dB SPL, which is 2 Pascal. Thus, for 100 dB SPL the measured pressure is 100,000 times as great as is the pressure at 0 dB. That's a huge, huge range...and this is why the logarithm becomes so useful; logs are often used wherever the range of values is very, very large.

Let's take it a step further...look at the pressure at 140 dB SPL - we see 63.2 Pascal. Hmmm...a pattern is emerging here. Look at the measured pressure values. For now, focus solely on the pressures associated with the 10 dB steps. Do you see the pattern of "2" and "632" ? So, if you can keep but a few numbers in your head (2 and 632) you pretty much can estimate the pressure as you march up the SPL scale - the digits do not change - only their powers of 10 change. The pattern repeats over and over and over.

Additionally, for the 3 dB steps, note how each 3 dB change is a factor of 1.41. 'Sound familiar? It's the scale factor that is related to the half-power (that is, - 3 dB, or equivalently, SQRT2/2) point. Now, if you look at the change between 70, and 73, as well as 73 and 76, you'll see that both are a factor of 1.41. What happens when you multiply 1.41 x 1.41? You get 2.0. This...should kind of make sense, because 1.41 is equal to the square root of 2.0, thus, multiplying them should give you the original value - that is, 2.0. Now, if you look at a span of 6 dB a factor-of-two change in SPL (or voltage and current) results, that is, each 3 dB change is a factor of 1.41, so two of them taken together (i.e. going from 70 to 76 dB - a net change of 6 dB) should reflect a doubling of pressure. If you take this further, you'll see that a factor-of-four change would equate to 12 dB.

Now, the Human ear. Wow. A very, very complex subject, and it has more than a little bearing on this discussion and what we perceive. This post isn't the right place for it, but I will leave you with this: Human hearing is non-linear. When I say that, I don't mean non-uniform frequency response. Instead, what I mean is that the frequency response of the Human ear depends upon the intensity of a sound incident upon that ear. That is, the higher the intensity of the sound, the more our ears tend towards (but never achieve) 'flat' frequency response. This is the reason why you hear less bass at lower listening levels (trust me, if you were to measure it, you will see that the bass is still there...it's just harder for you to hear) and not coincidentally the impetus for the invention of the "Loudness" circuit found in some many '60's, 70's and 80's era gear (and can still be found in gear to this day). Thus, a properly designed loudness circuit adds a lot of bass at very low volume levels, and as you turn up the volume, the amount of bass addedd is progressively diminished - at least, this is how they are supposed to be designed, and there are several variable that enter into just how useful and successful this approach can be.

Mark

Hey Old School

You really took us to school on this subject. I enjoy reading your posts since I actually learn something from each one. I like your analysis on the low frequencies and the human ear as it responds to linear amplitude. The ear is not linear.

Thanks. I do indeed like writing about this stuff...