i'm sure lots of you have had this problem before, my chain tension is not the same all the way around my chainring. when it rotates, the chain will slacken, then tighten, and so on.. so my question is, is my chainring not completely round? and is there a way to tell so?

no chainring is completely round. it's physically impossible.

however, yours is probably farther from a true circle than a high quality new ring. the way you tell is if the chain tension changes with crank position as you described.

get a new chainring.

how bad is it? set the chain to be taut at the tight point (so there's like an inch of play or so in the chain). if the chain is loose enough at the loose point that you can take the chain off the cog or chainring, buy a new chainring. If it doesn't fall off, then i don't think it's a safety issue. You'll need to replace it with a more expensive one regardless if you want to get rid of the variance in slack

You'll be fine.

Your bottom bracket may be a bent. Watch your chainring from above as you spin the cranks, and see if it moves sideways during the rotation.

Hard to fathom bending a BB axle. Much easier to bang up a chainring from sloshing the bike on the ground.

There's always Sheldon Brown's method for centering the chainring. Only helps if the out of round condition is not too severe.
https://www.sheldonbrown.com/fixed.html#fing
Scroll down to the centering chainwheels heading.

It happened on my 30-year-old SS. I don't know how, it was like that when I got it. But it was bad enough that the chain was taut in one position but could still slip off while I was riding.

Try this before you start throwing money at the problem. Even if you have one of the physically-impossible-perfectly-round rings, you'll have variations in chain tension if it isn't centered properly. It has always worked for me (with a lot of patience) and can compensate for an out of round ring very well.

so is the sheldon brown method saying loosen the chainring bolts, then rotate the cranks, find the tightest spot in the chain, tighten the bolts at that spot, then secure the wheel?

What he's suggesting you do is loosen and re-tighten the chainring bolts just enough so that they don't move around on their own. Find the tight spot in the chain, then adjust the chainring by tapping it lightly until the chain in that spot loosens up. Find the next tight spot and do the same. Continue to do this until tension is acceptable at all points, then torque your chainring bolts.

The SB method only works if there is some play between the chainring nuts (inner female part) and the holes in the crank spider and chainring. More often than not the chainring is so badly out of round that methods such as this are futile, and the only solution is to replace the chainring with a better one. The unbranded stock chainring on the Lasco cranks that came with my Moto Messenger was so bad, that even with chain adjusted with only about 1/8" play at its tightest position (should be more like 1/2" play), the chain could be removed from the chainring at its loosest position, which is dangerous. At the time I replaced it with another chainring, which completely eliminated the problem. The replacement chainring was a steel Sugino RD2 part, so we're not talking anything exotic here. I have since replaced the crankset with an Andel unit, which is nearly perfect with virtually no discernible variation in chain slack as the cranks are turned. Another benefit of better quality chainrings is that typically the teeth are machined more precisely, resulting in smoother and quieter chain engagement.

Maybe it is biopace?

*ducks*

I haven't attempted Biopace on a FG/SS yet but I think my commuter MTB will get it soon. The stock riveted chainrings are horrible, and then I'll be able to say I have a FG/SS vertical dropout MTB with Biopace

I bought my polo bike with a biopace chainring. It's such a sketchy feeling ride without a derailleur, you really can't keep a fluid pedal motion because the chain binds at its tightest point, and goes completely slack at the loose. You mash, spin, mash, spin, smack your knee on the stem when your chain slips off, fix it, no bueno.

So yeah, get a chain tensioner.

wait... did you just get free biopace?

I have a friend who sheared off the the drive side spindle of his phil wood bb starting from a stop... so it can be done...

Pay the shipping and you can have two more from me.

Your chainring spider is also probably not perfectly round. With some time you should be able to find a better set up. Rotate your chainring around the spider and find the best spot, with the least play. I have been able to make up quite a bit of slack with this method.