Common applications might be a bassline sidechained to a kick drum, or hi hats and cymbals sidechained to a snare drum. In these examples, the purpose is to allow the kick or snare drum to punch through more easily while any clashing frequencies are temporarily reduced in volume. Other variations on this might involve using a "dummy" clip full of regularly spaced clicks aligned to your track's kick drum, where the main output is muted, allowing us to use this signal as a trigger for a sidechain compressor. This has the advantage of allowing us to continue to sidechain a signal even when the track's main kick drum isn't playing, a cool creative effect in certain circumstances.
In general, when you are using sidechaining as a mix tool, you should be looking for around 3dB in gain reduction, you can take this further (as well as playing with attack and release times) if you want to use this as an exaggerated, creative effect. Most people should be able to get the results that they want from using a sidechain compressor in this way. However i'd strongly encourage you to play around with the following.
1. Use Volume Automation Instead
Using volume automation sounds somewhat counter intuitive. The reason why we use compressors is to avoid having to pencil in tons of volume automation. However, it may be worth the effort as volume automation allows a lot more precise control, and you can really hear this in your overall mix. For example, you can reduce a signal slightly before a transient, therefore generating room so that the attack portion then cuts through as much as possible. If you use a gain plugin (such as "Utility" in Ableton) and automate the volume on this, it will save you having to redraw automation everytime you change the track's volume.
2. Use an Envelope Generator Instead
This is a variation on the point above, but works equally as well. There are many plugins available that will generate a repeating volume envelope on any given sound source. You see where I am going with this, all you need to do is draw a shape that reduces the volume of your target sound whenever a transient is playing. This gives you the same precise control as penciling in volume automation. As with volume automation you can also choose to have your sidechain effect running when there are no transients playing.
Check out LFO Tool by Xfer Records, and VolumeShaper by Cableguys for a couple of good envelope generator plugins, pictures below.
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