**Resources**

The word ‘decibel’ creeps into studio speak constantly. It slips into conversations with the producer, the salesperson, and even the drummer

But do we really know what it means? It’s a much misunderstood quantity. In fact—and I’m not making this up—there is a movement to remove the decibel from audio and acoustics dialogs. Some scientists feel it’s just too confusing. So let’s look under the hood and see just how confusing it is.

Equation (1) absolutely defines the decibel:

dB = 10 x log_{10} ( Power A / Power B) (Eqn. 1)

The English translation of that equation goes something like “the decibel is ten times the logarithm of the ratio of two powers.” (Here, power is a physical quantity—the amount of energy delivered in a given time. Our most frequently used unit of power is the Watt, as seen in the ratings of power amps and speakers.)

A scientist can grasp this equation right away—or at least most of them claim they can. For those of us who can make practical use of the dB every day, it’s gonna take an entire article to truly absorb the power and meaning of the decibel. But as we’ll see, the benefits of this understanding are considerable.

**What it’s all about**

The equation for the decibel has two features built-in. First is the logarithm, or log, which we’ll discuss shortly. But it is important first to understand that a log function is part of the equation to make the math easier. Once we tackle its meaning technically, we’ll see that applying the log function to our ratio of powers makes the numbers much smaller and easier to deal with.

The second key element of the decibel equation is the stuff within parentheses: one power divided by another, a simple ratio of the two. The decibel equation uses a ratio so as to be consistent with the human perception of power and related quantities. After all, we are trying to put a number on our musical waveform. For that number to be useful, it must have some connection to our own internal perception of the sound.

These two features—the log and the ratio—have made the decibel a versatile and useful way to express the amplitude of our musical waveforms. Let’s look more closely at them now.

**Log on**

The log represents nothing more than a reshuffling of how we express numbers. It’ll be a little hairy, but so you don’t get discouraged I’ll give you the punchline up front: all a log does for us is to let us write really, really big numbers as really, really little ones, without losing their meaning. Okay?

The two following equations are both true and say very nearly the same thing:

10y = X (Eqn. 2)

log_{10}X = y (Eqn. 3)

Equation (2) is relatively straightforward. Ten raised to the power y gives the result X. To raise X to the power of y means to multiply X by X, y times. For example, 10 raised to the power of 3 gives you 10 x 10 x 10, or 1000. (Note that this kind of “power” is a rule of math; right now we’re not talking about Watts. Sorry for the confusion, but the mathematicians and the engineers didn’t consult very carefully before they named this stuff.)

The logarithm enables us to undo the calculation mathematically. If you start with the answer from above, 1000, the log function lets you get back to 3. Taking the log of 1000 gives us 3. (If you’re paying attention, you’ve probably guessed that the little “10” on the log symbol is to tell us that we’re working in powers of ten—ten raised to some power or another. There are other kinds of logs, some of which are used in circuit design, but they won’t affect our discussion here.)

Said another way, equation (3) answers the question “What power of ten will give us this number?” To take the log of 1000 is to ask “What power of ten gives us 1000?” The answer is 3. 103 = 1000, so log_{10}(1000) = 3.

What power of 10 will give us 1,000,000 (1 million)? If you’ve got a calculator, you can confirm that the log_{10}(1,000,000) = 6. Ten raised to the sixth power (10 x 10 x 10 x 10 x 10 x 10) gives us 1 million, as equation (2) would describe it.

Now we can calculate the power of 10, which gives us 100 trillion: log_{10}(100,000,000,000,000) = 14.

If you look again, you might see a simple way to do these logarithms without a calculator! All you have to remember is to “count the zeroes.” 1000 is a one followed by three zeroes, so it’s ten to the third power, and its logarithm is 3. The log of 1 million is 6, and 1 million has six zeroes. If you write out 100 trillion with all its zeroes, you’ll count 14... and the log of 100 trillion is 14.

And as hinted above, this is the motivation for using logarithms in audio: they make big numbers—potentially very big numbers—much smaller. 100,000,000,000,000 becomes 14. It converts U.S. defense budgets into football scores.

There’s an interesting twist; follow along in Figure 1. The log function will calculate the power of ten needed to give you a certain number—any number. So while the log_{10}(100) = 2 and the log_{10}(1,000) = 3, we can also find values in between. For example, the log_{10}(631) is about 2.8. In other words, 102.8 = 631.

To know this we need the help of a calculator, a computer, a slide rule, a class geek, or some tables full of logarithm answers—don’t expect to do these in your head.

**Under pressure**

The smallest sound pressure that humans can hear (rounding off a little) is about 20 micropascals. (Scientists express pressure in the unit of the Pascal, but we don’t use it directly in audio. Why not? Read on.) Let’s compare that figure to some other pressure levels.

Listening to a conversation at normal levels might occur with amplitudes of around 20,000 micropascals. We could reasonably monitor our pop mix at about 630,000 micropascals. We might occasionally crank it to more than 6,000,000 micropascals. Even at this level the neighbors aren’t complaining, the drummer wants it louder, and if we don’t monitor this loud for long, we probably aren’t at much risk of hearing damage yet.

Pain starts happening at about 63,000,000 micropascals. The difference between detection and pain in the human experience of air pressure is many millions of micropascals.

You can see the problem with these numbers: they are too big to be useful in the studio. “Yeah, let me push the snare up about 84, pull the strings down about 6,117, and see if the mix sits right at 1,792,000.” Fuhgeddaboudit.

The decibel helps us out here. As we noted above, the log function exhibits the following helpful property:

log_{10}(BIG) = small (Eqn. 4)

The log of a big number results in a much smaller number. And since we clever humans can hear across a vast range of amplitudes, this math trick comes in handy. So it’s a fundamental part of the decibel equation: dB are logarithms, so we can express huge ranges of pressures with very small numbers.

**Ratio-nale**

Also built into the decibel equation is a ratio. Mathematically, a ratio compares two numbers.

For example, I come from a small town in Texas where the tallest building is ten stories. I moved to Boston with its 50+ story towers. “Those buildings are at least five times taller than any we got back home,” I said.

That’s using the ratio. 50 divided by 10 equals 5; the ratio of Boston-like towers to small town buildings is about 50 / 10, or 5. This simple mathematical feature is put into the decibel equation because our senses seem to work in this way.

We’ll talk about loudness in a moment or two, but first let’s consider pitch, where this behavior is much easier for us to hear instinctively. Musical harmony is built on ratios. The octave, for example, represents a doubling of pitch, a ratio of 2 to 1.

The orchestra tunes (sometimes) to A440. Also identified as A4, this is the first A above middle C. A440 is a musical note whose fundamental frequency is exactly 440 Hertz. Double that frequency to 880 Hertz and you’ve gone up an octave. To go down an octave, reverse the ratio (1 to 2, or mathematically 1 / 2). That is, cut the frequency in half. The note A an octave below A440 has a frequency of 220 Hertz.

All the musical intervals represent ratios. It gets messy as the intervals get smaller, and the ratios depend on the type of tuning used. For example, a perfect fifth is a ratio of 3 to 2 (mathematically 3 / 2, or 1.5) and a perfect fourth is a ratio of 4 to 3 (4 / 3 = 1.3333). This is in just intonation; as we’ll see below, the numbers are tweaked a little for equal temperament. But no matter what the tuning, it is the ratio that rules.

As I mentioned in an early ‘Nuts & Bolts’ column, it doesn’t make musical sense to think in terms of the actual number of Hertz between two notes. You won’t get hired back as an arranger for asking the trumpet player to play 217 Hertz above the tenor sax. But if you request a minor third up, or an octave up, then the trumpeter can oblige.

Looking at Figure 2, let’s identify these fundamental pitches mathematically, not musically. Start two Es below middle C, labeled E2. It has a fundamental frequency of 82.5 Hertz—a meaningless observation for a musician, but an important one for an audio engineer. Go up an octave. There, the pitch is exactly twice the starting pitch. That’s the very definition of an octave.

One octave up leads us to E3, with a fundamental pitch of about 165 Hertz (2 * 82.5). Next octave up is E4, the first E above middle C. E4 has a fundamental pitch of about 330 Hertz. E5 yet another octave above has a pitch of about 660 Hertz.

Four pitches. One musical value. They all sound very nearly the same, musically speaking. In harmony, all E notes perform very nearly identical functions.

And the difference between them is musically obvious: octaves. No matter which note with the pitch of E natural you begin with, if you go up in pitch to the next E natural you’ve progressed by the interval of an octave.

But there’s a subtle illusion going on here. Using Figure 2, watch as the absolute numbers fail and the ratio takes over. The ‘distance’ (as measured in Hertz) from E2 to E3 is 82.5 Hertz. And this 82.5 Hertz difference has meaning to our hearing system; it’s an octave.

Next, start at E3 and go up to E4. Again it’s an octave difference. However, measured in Hertz, it is a 165 Hertz difference. And again E4 to E5 is an octave, worth 330 Hertz. An octave equals 82.5 Hertz one moment, 165 Hertz another moment, and then 330 Hertz yet another time.

Conclusion: We can’t express an octave in Hertz unless we know the starting pitch. However, we can always express it as a ratio: 2 to 1.

The musical significance of the octave is well known, easier to experience than to describe. You may have also attached specific musical meaning to many (probably all) of the other intervals: the buzz of the perfect fifth, the warmth of the major third, the bittersweet mood of the minor seventh.

Such complex human feelings about the pitch differences between two notes reduce almost insultingly to some simple math. To go up an octave, multiply by 2. And the same principle applies to all the other intervals: to change by a certain musical amount, multiply by a specific mathematical amount.

Minor headache: aside from the octave, these numbers aren’t so neat in equal temperament, the most common form of tuning in pop music. To go up a perfect fifth, simply multiply by about 1.49830708. To go up a major third, multiply by the unwieldy (and rounded off) 1.25992105.

The numbers are rather unappealing, but the fundamental principle is straightforward. Don’t add a certain number of Hertz to go up by a certain musical amount. Instead, multiply by the appropriate ratio.

The idea of the ratio is built into our musical pitch labeling scheme. Notes are described on the familiar musical staff, and labeled with the familiar short, repeating alphabet from A to G.

Peek at the numbers and something peculiar is revealed. If we plot the musical staff using linear mathematics, in which all the lines and spaces of our traditional notation system are spaced an equal number of Hertz apart rather than simply an equal distance apart, we get the rather strange looking staff shown in figure 3.

The traditional notation scheme masks the actual quantities involved. But for good reason. The musical relevance of the notes is captured in our notation. The relationship between C and G is always the same: it’s a perfect fifth at any octave, at any location on the staff.

And so it is shown that way on paper. It isn’t musically important how many Hertz apart two notes are, but it is certainly important how many lines and spaces apart they are, as arrangers well know.

Physically, the piano presents the same illusion. Figure 4 shows a piano in which the physical location of the keys is determined by the number of Hertz between the notes. Unplayable. Unmusical.

Ratios are a part of music. On paper and on a keyboard we see that the ratio is a convenient way to take physical properties and rearrange them in a way that is consistent with their musical meaning.

**Amplitude ratio**

Now we can come back to the decibel once more, armed with the knowledge that this whole discussion of ratios works equally well for loudness as it does for pitch. That is, as we mentally ‘process’ musical pitch in a relative way (i.e. based on ratios), we similarly process amplitude.

So we need a way to quantify the amplitude of our musical signal that has musical meaning. The decibel, built in part on the ratio, accomplishes this.

Research has shown that in order to double the apparent loudness of a signal, the power must increase approximately tenfold. (Now we’re talking about power in Watts again, not powers of ten as in our discussion of logs.) Table 1 summarizes. Starting with a power of 1 watt and doubling the loudness, we end up at 10 watts. Doubling the loudness from 1 watt required an increase of 9 watts.

Now, at 10 watts, doubling the loudness requires the power go up ten times to a new value of about 100 watts. This doubling in loudness requires an increase of 90 watts. The next doubling, to 1,000 watts, comes courtesy of a 900 watt addition of power.

In all cases the perceptual impact was the same: it got roughly twice as loud. This is the power amplitude analogy of the octave. Here we are talking about loudness, not pitch. But just as our perception of pitch is driven by a ratio, so is our perception of power: mulitiply by 10 to double the apparent loudness.

The equation for the decibel therefore has a ratio built in. As table 1 reveals, the decibel difference between each of the power settings is always the same: 10 decibels. So the actual number of watts changes, depending on the initial power setting. But the amount of change required to double the loudness—to have the same perceptual impact on our listening systems—is always 10 dB.

Decibels are the amplitude equivalent of the musical pitch labeling scheme. They convert the physical quantity into a number that’s consistent with how we perceive that quantity.

You may already have a very specific idea in your mind about what a 6 dB increase in level sounds like. Or a 3dB decrease. You can probably start at any note and sing up an octave or down a fifth, etc. Music schools offer ear training to teach this ability for pitch. Audio schools offer audio ear training (ŕ la ‘Golden Ears’ by David Moulton) to accomplish the same thing in the amplitude domain.

**References available upon request**

A close look at the decibel equation reveals that it is a single number expression for two numbers. That is, 30 dB represents a comparison of one number to another. It doesn’t make sense to say that a power of, say, 1,000 watts equals 30 dB. To put 1,000 watts into the decibel equation we need a second wattage to put into the ratio. If we start with a reference of 1 watt, then we can calculate that 1,000 watts is 30 decibels higher than 1 watt (10*log(1,000 / 1) = 30).

So we’re a little stuck. The decibel is meaningless without referring to two numbers. If we have a power from an amp or a speaker, what are we comparing it to? Answer: a fixed, agreed-upon power level. A reference.

We are comparing a new amplitude to the current amplitude when we say something like “turn the snare up 3 dB.” If you were to resort to the equation, you’d put the current amplitude in the bottom of the ratio (the denominator), and the top of the ratio (the numerator) has the new, louder snare that you desire.

Of course, this equation is never used during a session. The faders on your computer screen or on your mixer are labeled in decibels already. Someone else already did the calculations for us.

If we aren’t comparing a signal to its current value, we’re comparing it to some reference value. If the reference is 1 watt, 10 watts is 10 dB above our reference and 100 watts is 20 dB above this reference. So the correct way to express decibels here is something like “100 watts is 20 decibels above our reference of 1 watt.” It gets tiring, always expressing a value in decibels above or below some reference value.

Here’s the time saver. If the reference is 1 watt, we can express it as dbW (“dee bee double you.”) The ‘W’ tacked on to the end identifies the reference as exactly 1 watt. Cool. This shortens our sentence to “100 watts is 20 dBW.” Done. The reference, which is required for the decibel statement to be meaningful, is attached to the dB abbreviation.

So while equation (1) is the equation for the decibel, other sub-equations exist, with different reference values. For example, sometimes 1 Watt is too big to be a useful reference power. When that happens, we can use the much smaller milliwatt (0.001 Watts) instead. If the power reference is one milliwatt, the suffix attached to dB is a lower case ‘m,’ for milli.

dBm = 10 x log_{10} (Power / 0.001 Watts ) (Eqn. 5)

And a little physics lets us leave the power domain and create expressions based on the quantities we see more often in the studio: sound pressure and voltage. Sound pressure decibel expressions use the threshold of hearing (20 micropascals) as the reference pressure, and we tack on the suffix “SPL” to express sound pressure level in terms that have perceptual meaning:

dBSPL = 20 x log_{10} (Pressure / 20 micropascals) (Eqn. 6)

Note that the equation changes a little. Instead of multiplying by 10, we multiply by 20. This is a result of the physical relationship between acoustic power and sound pressure. Likewise, we can use decibels to describe the voltage in our gear. But in the world of voltages, we’ve a few references to contend with:

dBu = 20 x log_{10} (Voltage / 0.775 Volts ) (Eqn. 7)

dBV = 20 x log_{10} (Voltage / 1 Volt ) (Eqn. 8)

It is a quirk of history that the unwieldy reference of 0.775 volts was chosen. Apply a standard one milliwatt of power across a load of 600 Ohms (which was a standard in another industry, not audio), and you’ll find a voltage of 0.775 V results.

Even as the idea of a 600 Ohm load lost its meaning to us in the modern professional audio industry, the quirky standard voltage remains. Someone, tired of the clumsiness of that number chose an easier to remember reference: 1 Volt. Good idea, messy result—now we have too many standards. Sort of misses the point of a “standard,” doesn’t it?

The output voltages specified in the back of the manual for your new multitrack recorder might be expressed in dbu, for example +4 dBu. Or it might show up in dBV, like -10 dBV. In both cases the manual is just telling you the voltage of the output, relative to some industry reference.

Thank goodness they are expressing it in decibels. Maybe one day in the future all the manufacturers will also use the same reference. And maybe one day in the future all slow drivers will get out of the left lane.

The fact is, one rarely—if ever—uses the decibel equation in any of its forms during the course of a session. But the hardware designers and software code jockeys who create the gear we use certainly do. And if we are to speak comfortably and accurately about decibels, it helps to know a little of the math that makes it possible.

Even if the math eludes or bores you, at least know that someone went to a lot of trouble to find a way for us to express the level of the signal in much the same way we express its pitch. The decibel offers a perceptually meaningful description of amplitude, one that our ears and brain can make sense of.

So don’t turn it up 2 volts, turn it up 12 dB!

*Alex Case plans to one day strike it rich and measure his bank account in dBDollars. Send donations via talkback@recordingmag.com.*