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Science and Math for Audio Humans – Your Friend, The Logarithm

by Danny Maland

I apologize for the title, but after a string of rather straightforward article names it was time for something with a bit more zing.

Oh, and here's the disclaimer: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's just flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.

For the last couple of topics, I've been semi-carefully avoiding the discussion of sound intensity in the quantitative units that we usually use. The reason for this is that I positively hate it when somebody effectively says, "You're not going to understand what this means just yet, but I'm going to fling this term/ concept/ whatever around blithely for ages until I explain it sometime in the future." You sit there, trying to understand what's being put across to you, and there's this 800-pound gorilla in the room - some key piece of information that is obviously important, but a piece that you don't have the tools to deal with yet.

When we're talking about sound, the quantitative descriptions of frequency and wavelength are pretty straightforward and familiar. Hertz is nothing more exotic than wave cycles per second, and wavelength is any measure of linear distance that you might prefer. Counting is something we do every day, and simple distance is similarly common. When it comes to pressure in air, though, most of us don't have the same sort of intuitive experience. I'm going to guess that you've never heard an exchange like this:

“Dude, the guitar player's amp was creating pressure waves of 20 Pa RMS (Pascals Root Mean Square), and we were 100 feet away!”

“Seriously, Dude? He had to have it so loud that you were seeing 0.15 mmHg (millimeters of mercury) at that distance? Wow.”

Air pressure units just aren't something that we're “calibrated” in. Even voltage is still a little unwieldy:

“It's okay, we've got level to spare. All those inputs together are peaking at 6.15 V (volts), and the processor can handle a little over 12.27 V. No problem.”

Beyond us not having an intuitive “feel” for what the measurements mean, there's also this whole problem of the range that the standard measurements cover. For instance, my “loud guitar player” example used 20 Pa RMS, but an undamaged human hearing system is capable of detecting sound pressure waves of only 0.00002 Pa RMS. (One atmosphere is 101, 325 Pa, and that is considered the upper limit of an undistorted sound pressure wave in air. That is, the rarefaction, or decompression cycle of the wave just reaches vacuum. Suddenly, that guitar rig isn't quite so impressive...)

  • Figure 2 shows you a comparison of exponential and logarithmic functions. The top graph is an exponential curve, and the bottom graph is a logarithmic curve.

As an aside, if you want to generate these graphs, you can go to wolphramalpha.com and drop in the following inputs, one at a time:
x^10 x= 0 to 1 y = 0 to 1
10 log base 10 of x (x=0 to 10) y=0 to 1.1

So, that's neat, but how does it help us make a compact measuring stick?

If you look at the exponential curve, you can see that it “accelerates” as the x-axis (horizontal) values rise. For each “unit” increase on the x-axis, the corresponding space required for the graph on y the y-axis (vertical) increases. This is most apparent if you consider 0.6 to 0.8, and 0.8 to 1. The “spreading” of the space required on the y-axis is dramatic. The logarithmic curve, being the inverse case, “decelerates” as the x-axis progresses. This being so, the corresponding y-axis space for the graph becomes more and more compact for each unit increase in x.

Yes, I realize that the last paragraph was incredibly abstract. The bottom line is that logarithms, and the use of logarithmic scales, let us compact and view very wide ranges of measurement values in convenient and comprehensible ways. This doesn't only apply to pressure or voltage, by the way. Frequency response charts benefit greatly from using logarithmic scales, because human hearing has a range of about 19,980 “whole number” frequencies. Try to represent that in a linear fashion, and you get something like Figure 3. The very-critical-for-human-hearing range of 1 kHz to 4 kHz, and the range below that (where lots of very important fundamental frequencies reside, like “Concert A” at 440 Hz) get smashed over to one side. The red highlight accents that range.

So, what if we used a logarithmic scale? One where each unit of horizontal space in the graph represented one octave? (An octave is a doubling of frequency, and is a fundamental “unit” in both music and sound.) If we choose this kind of representation, we get Figure 4.

The most critical range is nicely spread out, making the whole thing much easier to read. The highest couple of octaves are compressed horizontally, which means that much less space is wasted on representing 10,000 whole number frequencies at the top of the scale.

So, what's the purpose of this part of the discussion? Well, this whole bit is the foundation of understanding the decibel, which (besides the octave) is probably THE most important “measuring stick” in audio.