I'm not an expert, but I have a resource for you. The American Wood Council has an online calculator. http://www.awc.org/calculators/span/calc/timbercalcstyle.asp
You'll need to know or approximate a number of variables for input into the calculator: the species and wood grade you can get from your supplier - I used No 2 Spruce, as that was readily available here in Atlanta; your spacing (presumably 16"); and the deflection limit. There are a few other inputs, which I think are self-explanatory.
The difficult ones (requires math) are the dead load (you won't need live load) and deflection limit. Dead load is the weight of the drywall and fixtures or other structures hanging form the joists. 5/8 drywall weighs about 2.2 pounds per square foot (psf). You can add up the total dead load you expect, though it's probably less than the minimum the calculator will allow, so it's building in some extra allowance for you. Deflection limit is given as a fraction (L/240 or L/480 etc). The larger the denominator, the smaller the deflection you are specifying. You may want to play with the calculator to get an idea for how much is too much deflection. The L component of that fraction is the designed span - 23ft in your case. 23ft/240 is just under 0.1 feet. If you were to build to that deflection, the center of the ceiling should be within that height of the edges - around 1 inch.
Plugging some numbers into the calculator, making some assumptions on your behalf, it looks to me like you can make it, depending on your tolerance for deflection. I'm a little surprised, to be honest. I had imagined a span that long would need an engineered support solution.
Naturally, I am not an engineer or architect, but I think I have a fair grasp on this topic. I used this same approach in my build
, but haven't gotten to a point where I can check the validity of my calculations.