So you want to make an album? (part 16)
KeithHandy posted in So You Want... on July 10th, 2007
To read the entire series, do what yo momma told you. Er, I mean, follow the link.
Installment 16: Math is loud (or “I got logarithm”… or something)
All the spiritual and psychological stuff I’ve been pontificating on, as valuable as it is, does not negate the fact that there are a few hard, cold math concepts you should know if you want to get the most out of recording. I’ll try to explain a couple of important points, in the most human terms possible. If you only vaguely understand this the first time through, refer back to it later, as it becomes more practical than theoretical.
To better understand two very basic aspects of sound — amplitude, and pitch — it’s helpful, if not essential, to know what a logarithm is. You may not need to know what goes on under the hood of your equipment, but you absolutely need to understand that human beings perceive amplitude and pitch in a logarithmic way, as opposed to a linear way.
A logarithmic scale is sort of like a distorted lens through which all multiplication has been “translated” into addition, and all division has been “translated” into subtraction. In other words, every time you multiply a real, actual value by a constant amount, you are adding a constant amount to its logarithm.
Hang in there. You’re gonna be okay.
We measure amplitude in “decibels“, abbreviated as “dB”. When we describe the difference between the loudest and quietest things we can record, that’s called “dynamic range”, and it’s measured in decibels. A typical dynamic range for a digital recording device might be 90 dB, for example. But what does that mean?
Amplitude is relative. If one sound is twice as loud as another sound in the same recording — that is, if the waveform appears twice as high on your screen — it will always be twice as loud, no matter how loud or soft you have your overall volume set to. So it’s not meaningful to say sound A is “some number of watts more” than sound B. It’s only meaningful to say it’s something times as loud, or some percent as loud. However, we can say sound A is some number of decibels more than sound B, because a decibel is a logarithmic (there’s that word again) unit. This means whenever you increase or decrease the amplitude of something by a certain number of decibels, you are actually multiplying and dividing the height of that waveform by a certain amount.
For recording purposes, it’s best to remember this rule of thumb: any time you double the height of the waveform on your screen, you’re increasing its amplitude by approximately six decibels. To help hammer this nail into your brain, you can use the following graphic aid/mnemonic device:
Likewise, any time you decrease it by six decibels, you’re approximately cutting it in half. Each time you go down six more decibels, you’re cutting it in half again. Theoretically, to go all the way down to absolute silence, you have to go down an infinite number of decibels. That’s why input meters say “negative infinity dB” at the bottom. Notice they also define “zero dB” as the loudest you can go without distorting.
If you accidentally make an exact copy of one of the tracks in your project (I’ve done this), and it plays back perfectly synchronized to the original track, its amplitude will double (since it is two copies of itself), and you will hear it as approximately six decibels louder than it should be. BUT — and this is where it gets a little weird — if you mix together two different sounds that are about the same volume, the mixed sound will only be about three decibels louder than either individual sound. That’s because if they’re not the same exact sound, their peaks and valleys won’t be happening at the exact same times, so they don’t do as much “damage” together. In a third scenario, if you mix a sound with an exact copy of itself, but invert (flip upside-down) the copy so that it’s a mirror image of the original (the peaks become valleys and vice-versa), it will cancel itself out completely, resulting in silence (”negative infinity dB”).
A geekier way of rephrasing the above paragraph is this: if you mix two identical waves together that are perfectly in phase, the result will be 6 dB louder. If they are ninety degrees out of phase, meaning if you slide one of them back in time just enough to be 1/4 of a wave cycle late, the mixed result will be only 3 dB louder. (How this sort of relates to the second example in the above paragraph: if sounds are mixed together without any attempt to correlate them, as will be the case with any real-world sounds, instruments, and voices, they might as well be about 90 degrees out of phase on average.) If two waves are exactly 180 degrees out of phase, meaning if you slide one wave back in time enough to be exactly 1/2 of a wave cycle late, so that it looks like an upside-down version of the first one, the mixed result will be silence.
This understanding of decibels will be helpful when you’re setting levels and mixing, especially since not all level controls are logarithmic. (Ever notice some volume knobs and sliders are hard to control at lower volumes, because they “jump” too much with the slightest nudge? Those are linear.) Likewise for meters. Likewise for curves and crossfades. But I mentioned pitch too, which will be particularly important when using equalizers and filters — how do logarithms relate to that?
Well, first of all, we don’t actually use logarithmic units for pitch, unless you’re talking about the language of music itself: that’s right, musical notes, the little black dots used by Mozart, are actually a logarithmic representation of pitch. Each time you go up an octave, you’re doubling the pitch, and each time you go down an octave, you cut the pitch in half. (Since the octave is divided up into twelve semitones, this means every time you go up a semitone, you’re multiplying the pitch by the twelfth root of two, which is a hair less than 1.06.) But octaves and semitones are musical concepts; all the equalizers and filters you will ever use show pitch in its plain old linear scale, “hertz“, meaning cycles per second. (Humans supposedly hear from 20 to 20,000 Hz, but realistically it might be more like 30 to 16,000.) BUT… you will notice something funny about the numbers on an equalizer. They will generally look something like this:
30 Hz - 60 Hz - 125 Hz - 250 Hz - 500 Hz - 1 KHz - 2 KHz - 4 KHz - 8 KHz - 16 KHz
So even though pitch is shown in linear units, it’s scaled logarithmically, so the lower frequencies appear to be more “spread out” on the left while the higher frequencies are more “squished together” on the right. This is because, again, we hear pitch logarithmically — meaning each time it doubles, we hear it as going up about the same amount. So the middle of our range of hearing isn’t 10,000 Hz — it’s more in that 500 to 1000 range. Incidentally, if you lost all your hearing between 10,000 and 20,000 Hz, you wouldn’t be losing “half your hearing”, you’d only be losing the top octave. Things would sound a little duller, but not horribly muffled.
This will be helpful to know as you play with any equalizers or filters. If you practice with them a little, you should eventually be able to make a good guess as to where in the spectrum an unpleasant “ringing” is, and be able to quickly zero in on it and filter it out, while barely doing any damage to the music itself. Wouldn’t it be nice to be able to do that?
Anyway, you’ve had enough meat for one day. Enjoy your pudding. Class dismissed!
