What is Q?
Post date: Oct 24, 2013 7:24:05 PM
After reading Geoff Martin's post about the definition of Q, I became curious about Q values in other equalizers, so I measured the frequency responses of a few parametric equalizers. My goal is to understand their characteristics better so I can use them more effectively. I'm also interested in Q values from a technical ear training point of view. If I train myself to know what various Q values sound like, can the training for Q be applied to other equalizers?
How much variation in Q (or definition of bandwidth) exists among various EQs? More than I expected.
Let's get started with the frequency response plots of four parametric equalizer plugins:
- Apple Logic Channel EQ (unlinked Gain-Q)
- Avid Pro Tools Channel Strip EQ
- Avid Pro Tools EQ3
- iZotope OZONE 5 EQ ("analog" setting)
- Fc = 1 kHz
- Gain = +/-12 dB (first plot) and +/-6 dB (second plot)
- Q = 1.0
What information can we glean these plots?
- Not surprisingly perhaps, there are differences among the plugins, but the plugin differences vary from plot to plot, despite the same Q settings. The differences are presumably due to different definitions of Q and maybe some scaling factors.
- Pro Tools EQ3 and Logic Channel EQ share the same curve in the first figure (+/-12 dB).
- Pro Tools EQ3 and iZotope OZONE share the same curve in the second figure (+/-6 dB).
In the tables below, I've measured Q values for various gain settings on these equalizer plugins. The Q column on the left represents the setting I entered in the plugin. The other Q columns are values I measured for each plugin, based on either the half-gain bandwidth or -3 dB bandwidth. I include more information about the two Logic Channel EQ Gain-Q settings below.
Assumption: Q = (center frequency) / (bandwidth)
* Plugins do not allow gains at these values.
** Half-gain Q.
* Plugins do not allow gains at these values.
** Half-gain Q.
The Pro Tools Channel Strip EQ could be defining bandwidth as the difference between frequencies at the half-gain point, if only because the half-gain bandwidth stays roughly constant across a range of gains. But the half-gain Q values are not that close to the Q setting on the plugin. The -3 dB Q values are closer to the Q setting in some cases but they vary widely across the range of gains. After doing the measurements for the table, I went back for an additional measurement with gain at +16 dB. As it turns out the measured -3 dB Q at +16 dB match the Q settings exactly for Q = 1.0 and Q = 2.0. So perhaps there is some sort of proportional scaling factor, with Q defined by the -3 dB point.
I'm not sure how to interpret the measured Q values for the Logic Channel EQ with strong linked Gain-Q. The -3 dB Q values (not shown) for these two plugins are even further from the Q settings. There may be a Q scaling factor but I'm not sure why the measured Q would be so different from the Q setting.
The Logic Channel EQ with unlinked Gain-Q and Pro Tools EQ3 appear to be using the -3 dB definition of bandwidth.
The iZotope OZONE 5 EQ appears to be using Robert Bristow-Johnson's half-gain bandwidth definition. The OZONE 5 curves also line up precisely with the curves from Cycling '74's MaxMSP filtergraph~ object connected to a biquad~ object.
Here are screenshots of each of the plugins:
Logic Channel EQ:
Pro Tools Channel Strip:
Pro Tools EQ3:
iZotope OZONE 5:
Now on to more specifics about the Logic plugin.
Logic Pro X - Channel EQ
The Logic Pro Channel EQ has options that are not commonly found on other equalizers. One is the "link button", shown by a little chain icon next to Q. In their user manual, Apple says that the link button "turns on Gain-Q coupling, which automatically adjusts the Q (bandwidth) when you raise or lower the gain on any EQ band, to preserve the perceived bandwidth of the bell curve". To my ears at least, it sounds the opposite: the unlinked Gain-Q setting seems to preserve the perceived bandwidth. But I can understand why one might prefer the linked mode, even if the measured Q does not match the Q setting.
From what I can see, Gain-Q coupling produces enormous differences in the definition of Q. For example, here is a screenshot of a +24 dB boost at 1 kHz, and Q = 2.0 in linked mode (note the blue rectangular chain link button in the lower left corner):
When I unlink the Gain-Q, the displayed setting is Q = 21.0, even though the frequency response curve does not change. Note that I did not specifically change the Q, it automatically changed from 2.00 to 21.0 when I unlinked Gain-Q.
Based on my own experience with equalizers, this particular setting sounds and looks more like a Q of 21 than 2, probably because I'm used to the classic definition of Q (e.g., bandwidth is determined by the frequencies -3 dB down from the peak). When I measure the bandwidth using the -3 dB down bandwidth, I get Q = 21.3.
Under the "extended parameters area" (accessible by clicking on the arrow in the bottom left corner), there is another parameter setting that affects the Q in linked mode called "Gain-Q Couple Strength", which has six options: light, medium, strong, asym light, asym medium, and asym strong. The plugin loads with Gain-Q linked in asymmetrical strong mode.
Here is a table showing Q values of unlinked and linked Gain-Q:
The Q values were taken from the Channel EQ interface itself, they were not measured. I started in unlinked mode, chose a Q of 2.0 for each gain, and then turned on linked mode and went through each of the six linked options and wrote down the Q value that appeared for each option. Each row represents the same frequency response curve (thus the reason for the color coding), despite the variation in Q values shown by the interface.
If I link Gain-Q and change the gain level, the Q setting will stay constant. If I unlink Gain-Q, the Q setting changes, inversely proportional to the table above.
For the symmetrical links, there are four different Q values for one curve. Here are the frequency response curves for unlinked and linked Gain-Q with:
fc = 1 kHz
gain = +/-24 dB
Q = 2.0
(I omitted symmetrical cuts in the plot because they mirror the boosts.)
The Channel EQ does display the frequency response in the interface, but I admit I understand this equalizer's functionality much better now than I did before doing the measurements.
With asymmetrical boost/cut equalizers we can't undo a boost with an equal but opposite cut at the same frequency and Q, like we can with a symmetrical boost/cut equalizer. For example, here is 1 kHz boosted by 24 dB and cut by 24 dB, both with Q = 1.0, on the same Logic Channel EQ plugin:
If I happen to have the boost and cut applied on the same plugin, at least it will show the resulting frequency response:
However, I found out that I can flatten the frequency response by setting the Q of the cut to 0.35 when the boost Q is 1.0, for +/-24 dB at 1 kHz. If I switch to unlinked Gain-Q, the Q values for the boost and cut both change to 10.0.
Equalizers give different frequency response curves for the same Q settings. Why? One reason might be that designers want to differentiate their products from those of their competitors. Can we duplicate the measured frequency response of one equalizer with another, even if the numbers on the interfaces do not match? Yes, according to the folks at Algorithmix, who say that: "The truth is that with a properly designed, fully parametric analytic PEQ [parametric equalizer], every amplitude and phase characteristic of any other equalizer setup can be recreated." They are essentially saying that we really only need one fully parametric equalizer, but we might choose one model over another because we like the way bandwidth scales, or remains constant, with gain changes.
Some analog equalizers intentionally vary bandwidth with gain, where large boosts/cuts have higher Qs (narrower bandwidth) and small boosts/cuts have smaller Qs (wider bandwidth). API's 550 and 560 equalizers are examples of this design. API calls the characteristic "proportional Q" and there is no independent control of Q. George Massenburg's 8200 analog EQ, on the other hand, offers constant shape reciprocal filter curves, with completely independent control of frequency, gain, and Q. Are software companies trying to mimic these design philosophies to give their equalizers an interesting gain-Q proportionality or "feel"?
Regardless of the measured and displayed Q values, we can get used to a specific equalizer's characteristics and learn the way it changes an audio signal, with reference to its displayed settings. But if we're switching from one brand of EQ to another, our expectations of Q and its relationship to gain may not match the interface settings. It doesn't really matter how we define bandwidth, but it is helpful to know what definitions equalizers are using.
All measurements were done using FuzzMeasure.