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Figure 1. Waves with Q of 7. Figure 2. Waves with Q of 20. Figure 3. Sound Designer II with a bandwidth of 200 Hz. Figure 4. Sound Designer II with a bandwidth of 67 Hz. Figure 5. Pro Tools with 1 octave bandwidth. Figure 6. Pro Tools with 1/3 octave bandwidth.
Figure 1. Waves with Q of 7.
Figure 2. Waves with Q of 20.
Figure 3. Sound Designer II with a bandwidth of 200 Hz.
Figure 4. Sound Designer II with a bandwidth of 67 Hz.
Figure 5. Pro Tools with 1 octave bandwidth.
Figure 6. Pro Tools with 1/3 octave bandwidth.

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Hertz, Q, and Octaves
All about the word
By Dave Moulton

While I was working on the eq clinics for Playback, it became apparent that we needed to explain the differences between the various ways that bandwidth is expressed in equalizers.

Definition: bandwidth refers to how much of the frequency range around a center frequency is altered when a peaking/cutting type equalizer is used. It is expressed in three different ways: as Q, in Hertz bandwidth, and in octave bandwidth. The relationship between these expressions is less than totally obvious.

Q

Q is a mathematical term that can be used to express the ratio between the center frequency of an equalizer and the bandwidth of the equalizer where the amplitude is down 3 dB on either side of the center frequency. If we have an equalizer set at 1 kHz, and at full boost it is 3 dB down at 995 Hz and 1005 Hz, it has a 10 Hz bandwidth, or 1000/10, for a Q of 100. A big number suggests a very narrow bandwidth and a small number suggests a very broad bandwidth. A Q of 1, for instance, for the above equalizer would have 3 dB down points at something like 600 Hz and 1600 Hz.

Bandwidth in Hertz

Bandwidth expressed in Hz is fairly obvious, but it is also misleading because the Q of a bandwidth expressed with a given number of Hertz will vary with the center frequency of the equalizer. 1000 Hz bandwidth means something dramatically different for an equalizer tuned to 500 Hz compared to one tuned to 5,000 Hz. This is because of the exponential nature of octaves and our perception of frequency.

Bandwidth in octaves

So the third way of expressing bandwidth, in terms of octaves, makes some sense. An octave is an octave (a 2:1 range in frequencies) regardless of where it falls in the spectrum. Got it?

A tale of three eqs

Now, lurking in my computer are the three different ways of expressing eq bandwidth. The Pro Tools software expresses bandwidth in octaves. Sound Designer II (SDII) expresses bandwidth in frequency. And the Waves Q10 equalizer plug-in expresses bandwidth in Q. It couldn’t be much more convenient for the purposes of this article!

So take a look at some graphic measurements from my TEF analyzer. These are 1/12th octave (there’s that term again, Mommy!) Real Time Analysis measurements (from 50 Hz to 12 kHz) of pink noise being sent through each of the three equalizers with 10 dB boosts at both 200 Hz and 2 kHz.

First let’s look at the Waves measurements. With a given Q of 7 (the Waves default bandwidth), we see a bandwidth shown of 130-270 Hz (about an octave) for the 200 Hz eq and a bandwidth shown of 1600-2700 Hz (slightly less than an octave) for the 2 kHz eq. The calculations, however, suggest a bandwidth of 28 Hz surrounding 200 and 285 Hz surrounding 2 kHz.

Obviously, the algorithm Waves is using to calculate Q is different from the one I learned in the good old analog synth days. This is of little concern, but a good cautionary example of why you should use your ears as well as arithmetic and trust your ears when the arithmetic doesn’t seem to work so well. Check out figure 1.

Now look at figure 2. The next illustration shows Waves with a given Q of 20 (calculated 20 Hz bandwidth at 200 and 200 Hz at 2 kHz). Now the measured response curves are much narrower, about 3/12ths of an octave each.

By comparison, take a look at the SDII parametric eq measurements. With a given bandwidth of 200 Hz, the 200 Hz boost is considerably broader, ranging from approximately 80 to 315 Hz (approximately two octaves). Meanwhile, at 2 kHz it is quite narrow (about 1/4 of an octave). Obviously, if you use such an equalizer, a little arithmetic is called for when you think about bandwidth. See figures 3 and 4.

Pro Tools expresses eq in octaves, and the pink noise measurement supports these settings pretty well, with the 1 octave setting yielding slightly less than an octave and the 1/3 octave setting yielding exactly 1/3 octave. Look at figures 5 and 6.

Now, none of these are better or worse from an audio quality standpoint, and this isn’t a product review. From an ergonomic standpoint, however, expression of bandwidth in Hz is probably the least convenient because the significance of the number changes as a function of other things.

The use of Q is subject to confusion over the mathematical definition of Q (which varies as a function of its application in various engineering contexts; for example, Q is also used to express the directionality of loudspeakers, and it is used as a working design concept in virtually any LC electrical network), and the correlation to music production applications is not obvious.

So my vote goes to the use of octaves bandwidth for this expression. It has an obvious and easy correlation to musical issues, and it is psychoacoustically constant across the spectrum.

But with a little practice, you can fairly easily move from one to another without too much p ain—if you’re willing to use a little gonzo arithmetic!

Dave Moulton is limited in bandwidth, but he responds to a variety of cues, especially when it hurts.

 




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