A few weeks ago, I said that one day I might tell you what a decibel is. Guess what -- today is that day.

If you were listening last week, you already sort of know what decibel is: it's a tenth of a bel. Which leaves only two questions: what's a bel, and why do we talk about decibels all the time, instead of bels?

The main reason you hear about decibels more often than bels is similar to the reason why food packaging always incorrectly refers to kilocalories as simply "calories". Technically, that's no better than crossing out "kilometres" and writing "metres" on your car's odometer. But it is way more convenient. Counting calories in your diet is like counting pennies in your bank account; they're so small that you need to use big unweildy numbers to represent any significant amount of energy.

Calories are a measure of energy, by the way, and the word dates back to the old theory of "caloric." Caloric was an invisible substance that sprang into existence in order to transfer heat from a hotter body to a cooler body. These days, we are more inclined to talk about heat transfer in terms of conduction, convection, and radiation; but food energy is still measured in calories instead of the metric term, which would be joules.

Since calories are so small, nutritional information labels always talk about kilocalories. They just happen to call them calories, presumably because it's easier to say than kilocalories. And it takes up less space on the label. I heard once that the excuse for not labelling genetically modified food is that people won't buy it if it's labelled as genetically modified. Perhaps that's supposed to be the same sort of idea.

Anyway, I didn't come here to talk about calories, or kilocalories, or even decicalories, if you can imagine using coins worth a tenth of a penny -- I want to talk about bels, and decibels.

And of course, before I get too much further, we'll also be hearing some music.

While you're listening, you should keep in mind a couple of things. First, I dredged up this old topic because nobody has made any requests lately. Or perhaps you suggested something but I have forgotten what it was. So if you have a question that I might like to answer on a future episode of Mostly Mozart, call me in the studio at 352-3706, and ask me. If you don't catch the show where I give you the answer, and you happen to have access to the internet, you can look up all of my past shows on my web site: mozart.fiction.org.

The second thing to keep in mind is that a bel is a ten-fold increase in sound pressure, otherwise known as volume or loudness. A decibel, even though it's a tenth of a ten-fold increase in loudness, is not a one-fold increase in loudness. That's right, it's time to get out your slide rule, because we might be dealing with a few logarithms later on.

The first thing again, in case you weren't ready, is the phone number here at KCR's studio A: 352-3706. Ask me anything, and I'll do my best to give you the impression that I know the answer.

---

Comfort and Joy is happy to sponsor Mostly Mozart. Today's topic is bels. That's bel with one "L", by the way. A bel is a ten-fold increase in sound pressure. A decibel, or dB with a small "d" and a big "B", is one tenth of that -- but only if you know how logarithms work. It's definitely not a one-fold increase in sound pressure.

If you have one of those fancy stereos made by Sony or JVC, with labels all over the place boasting about DSP and AMS and one-bit DAC and 18 times oversampling, then the volume knob is probably labelled with a bunch of numbers. This seems to suggest that when you adjust the volume, you're judging the correct setting by the numbers, instead of just listening to the sound that comes out. Which is probably not how you do it, but hey, you never know. If you've got a really fancy stereo, you might even get to watch the numbers go up and down on a digital display panel, as you press the "louder" and "softer" buttons.

One thing you notice if you have one of these fancy stereos is that the numbers are all negative. Even if they don't have minus signs in front of them, you can tell they're supposed to be negative because 100 is really quiet, and 0 is probably loud enough to blow up your fancy amplifier.

The first important lesson in this, is that you can label a volume knob however you want, and people will still expect it to get louder when they turn it clockwise. You could probably label everyone's steering wheel to say "turn the wheel clockwise to direct the car to left", and they would just ignore it and steer properly anyway. Sometimes people are no fun at all. But hey, if you actually mix up the steering parts so the steering wheel does work backwards, well, they still won't read the label, but the result might at least be more fun.

Oops, the "off topic" light is flashing. I guess I should get back to bels.

Why does the volume knob have 0 decibels at the top? Well, let's see what happens if we put 0 decibels at the bottom, and count up from there. The only thing 0 decibels could mean is "no sound": when the volume is turned all the way down, no sound comes out. If you went up 10 decibels, or 1 bel, then you'd expect 10 times as much sound to come out. Unfortunately, 10 times 0 is 0, so you'd still get no sound. Hopefully you didn't spend all that money on your fancy stereo just so you could get no sound out of it.

Let's try it the other way, with 0 at the top. The only thing 0 could mean at the top of the dial is "maximum volume," where your amplifier is amplifying as much as it possibly can. If you went down 10 decibels, or 1 bel, then you'd get one tenth of your maximum volume. If you went down another 10 dB, you'd get one hundredth of your maximum volume. Another 10 dB down, and you'd have a thousandth of your maximum volume. This is sounding much more useful. If you went all the way down to 100 dB below maximum volume, then no matter how loud your fancy amplifier is, one ten billionth of its maximum volume would be sufficiently close to "no sound" that you wouldn't notice it.

So that's why you've got negative numbers on your volume control. Maybe you didn't know that was a science question, but it is now.

I promised you some logarithms. Bels are logarithmic, which means that they represent ratios instead of amounts. Ratios don't usually have units; for example, if I am twice as tall as you, you could say the difference between my height and your height is 2. Two what? Not 2 metres; not 2 feet; not 2 anything, just "2". Bels are like that too. If the difference in sound level between a lawn mower and a jet engine is 1000, what do we have 1000 of? Not watts, not volts, not lawn mowers, nothing. Definitely not 1000 lawn mowers.

Adding and subtracting ratios like bels is different than adding and subtracting absolute measurements like metres. 2 metres plus 3 metres is 5 metres; but if you make something twice as loud, then three times as loud as that, then the result is 6 times as loud as when you started. You multiply because it doesn't make sense to add. If you make something one times as loud, and one times as loud again, and so on, you'll never get anywhere; you'll still end up one times as loud as you started. Hopefully you're getting the picture here, that you can't add ratios. You have to multiply them.

It is possible to convert ratios into something you can add, and that's where logarithms come in. If you've played with powers of 10, you've noticed that 10^2 times 10^3 is 10^5, and you calculate that by adding the 2 and the 3. That's really all there is to it. If you convert your ratios to powers of 10, then it is perfectly reasonable to multiply the ratios by adding the exponents. Only now you'll call the exponents "logarithms".

For example, let's say the lawn mower is 10 times as loud as my voice, and the jet engine is 1000 times as loud as that. If we convert those numbers to powers of 10, then the difference between my voice and the jet engine is 10^1 x 10^3, which is 10^4. And rather than saying the jet engine is 10,000 times as loud as my voice, I'll just say that it's 4 bels higher. Or perhaps it would be better to say 40 decibels, or 40 dB.

The phrase "40 dB" is much less unwieldy than "10,000 times as loud," and in practice that's the sort of ratio we usually talk about when it comes to sound pressure.

On this mixing board in the studio, I've got faders with 0 dB at the top, again meaning maximum volume, and -5, -10, -20 dB, and the usual label at the bottom that represents wishful thinking, -infinity dB. That's the only way to get no sound at all, which is appropriate because getting no sound at all is just as hard as anything else that's labelled infinity. Especially if you have young children.

Now you know what 10 dB and 0 dB mean, but what is 1 dB? Well, there's only one thing it can mean. Whatever ratio it represents, if you multiply that ratio by itself 10 times, it has to equal a ratio of 10, because that's how we defined 10 dB. In other words, 1 dB is the 10th root of 10, which is about 1.26. If you check that and multiply 1.26 by itself 10 times, sure enough, you will get about 10: 1 bel, or 10 decibels.

1.26 times as loud is not a good point of reference. But there is one really convenient checkpoint between 0 and 10 dB, and that is 3 dB. You'll notice that 3 dB is one of the most popular numbers of dB, and if you want to know why, just multiply 1.26 by itself 3 times. 1 dB plus 1 dB plus 1 dB is 3 dB; and so 1.26 x 1.26 x 1.26 is the ratio represented by 3 dB. Conveniently enough, the cube of 1.26 is pretty darn close to 2. So 3 dB is twice as loud.

I hope you got all that.

If you'd like to learn more about decibels, why not read the manual for your CD player. It will probably claim to have a dynamic range of 96 dB. Dynamic range is the difference in volume between the loudest and quietest noises that your CD player is able to make.

You can also ask a geologist about the Richter scale, which measures something almost identical to sound pressure. "5.0 on the Richter scale," as they say, is 10 times as big as a 4 Richter earthquake.

And finally, if you happen to have a cable splitter in your house, so you can watch a video upstairs and downstairs at the same time, then take a look at it. It will probably say something about 3 dB loss, which refers to the fact that each output only gets 1/2 the power of the input.

But you probably don't have a cable splitter, do you. Or a radar detector.