I'd like to preface this by saying that there really is no right or wrong answer to this question. You can go online and find this topic on many audio forums and the pages of arguments will be endless. It really comes down to your preference, and I am going to try to explain why mine is almost always going to be Compression First.
First, when you Eq something, what are you doing? Well, you are finding a few bad frequencies and cutting them out, and then finding a few good ones and boosting them, correct? Now, when you boost frequency, you are adding energy and volume to the audio. This is an important concept to understand.
When you compress audio, you are working on the dynamic range of the waveform. The dynamic range, basically, is the most quiet the audio can get up to the loudest. When you compress, you are controlling that range by bringing down the peaks in volume to a more manageable level to work better with the average sound of the audio. The lower the threshold and the more compression added, the lower the peak level becomes.
Now, when you Eq first, you are adding level (volume) to the signal. Usually being a bass and/or a treble boost. The compressor is going to react to that boost and take some of it away in an effort to control the dynamics of the track. So you will lose some of your Eq'ed sound. You will then have to go in after the compressor and add another Eq to get the sound you had before the compressor did its job and controlled the output volume. To me, this is counter-productive to workflow. Why Eq twice when you can just do it all after the compressor has worked on the raw signal?
Having said that, I did previously say ALMOST always earlier in this article. I would like to put this theory to you as well.
When you are working with the raw, unprocessed signal, you have many frequencies that are making the track muddy, or mid heavy, or some other natural problem. If you compress the raw signal, your compressor will sometimes grab those and give you kind of a "false positive" of where the threshold should be set. If you Eq out a few of those naturally "honky" or "shrill" frequencies, you should be left with mostly "good" frequencies and the compressor will be able to work more efficiently. I have found this out myself by setting a compressor up on the raw track and then going back and adding some pre-Eq and watching my compressor input go down because of the lack of muddy mid frequencies. So cleaning up the mids in your audio BEFORE compression, I believe, is a decent workflow tool, because it allows you to compress a better, cleaner version of the audio and gives you a better base to work from when you add the "real" Eq after the compressor has done it's job containing the dynamics to a more even range.
This is how I understand the differences and this is the way that I teach others. Like I said before, there are many people out there who think BEFORE is the only way and many that think AFTER is the only way. I lean more towards the AFTER camp, but it's really up to your personal ear as to which you like best.
I hope after reading this you have a better understanding of the basics of using Eq with compression. Feel free to ask me your questions and I will try to respond as quickly as I can.
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Head Engineer, LIRS