Here is one technique that people tend to shy away from. They seem complicated and you have to do some manipulating the reed for them to be 100% accurate. But not if you have a fingering that will do it for you. Here are the steps I take to try and find the right combination to produce a particular quarter-tone, or microtone. There are a few places to find these fingerings (I have posted one of my fingering charts  and some published resources before but I will also compile a list and post them later today or tomorrow).

1. Find a fingering that produces the entire microtone, quart-tone.

2. Find a fingering that get you close to the pitch you need, and then a slight manipulation of your embouchure.

3. A large manipulation to your embouchure.

Always try to go the easies/least complicated path first, it is not going to help you perform these techniques in a piece if you have to change a lot of what you normally do. How often would you actually use them?! Please experiment and try them out in context, it does make a BIG difference when you apply them to music.

Here is a short video of showing a couple of quarter-tones. I will produce the quarter-tones illustrated in the picture (1. E 1/4 sharp, 2. A 1/4 sharp, 3. and G 1/4 sharp).

Have fun!