I have an unconventional method that you might consider... :o) And it works!
Consider your 1 year old's scream to be akin to a tantrum. He might not be screaming because he isn't getting his way (or at least not all the time), but he IS doing it because it is an attention-getter. He likes getting a reaction....ANY reaction from the people around him. It doesn't matter if it is a positive or negative reaction, it's still a reaction. I'm sure the look on your face when he screams is enough for him..."Look!...Mommy's eyes bulge out and she clenches her jaw and makes a funny face! Wheeee!"
So...how do you stop this problem in its tracks? Easy. You have to make the reaction to his scream a whole lot less interesting.
This is the same technique that has been handed down in my family for generations...but it is usually used for tantrums. It usually only takes one time for the child to get the point...but it may take as many as three. Stick to this and I'll bet you'll stop this problem...
Take a small cup (I like Dixie cups) and put about 2-3 inches of water in it. Put it in the fridge and let it get nice and cold. Reeeeaallly cold (but don't use ice!). Make sure it is easily accessable.
When your adorable little one starts in on his series of shrieks, go to the fridge and get that cup of cold water. Hide it.
The most important point here is that he should NEVER see it.
Put the cup behind your back and walk up to him confidently. When his face is screwed up in the middle of a scream, dash that water right into it. (Don't worry that he might choke...he won't. He's breathing out in order to scream.)
Hide that cup behind your back and never let him see it. I suggest you use a Dixie cup, since you can immediately crumple it up.
This will stop the scream immediately and he'll probably freak out a little bit and start crying. Don't coddle him too much, and don't talk to him (and don't feel too bad...especially since this is going to stop the problem!)....just go and get a towel and clean him up. If you feel like you have to say something, say "Oh NO! What happened? Were you screaming? Uh oh!" Give him a hug and send him on his way....but act like the whole thing never happened. Don't explain....let his little mind work it all through. It won't take long for his logic to figure out that screaming equals a cold surprise...
A couple of added points: For this to work you can NEVER threaten. He shouldn't know where it comes from or why...Never say "If you don't stop screaming, I'm going to get the water" or, "If you scream, I'll do this!" The consequence should not even be related to you (this will save you a guilt trip, too). Never talk about the water or the screaming in front of him. He should never know that you had anything to do with it.
I hope this works out for you...even though it's an odd method, it is tried and tested...