How to tell a client that we're currently ALSO working with other clients without them getting upset

We haven't missed any deadlines, but the person is sending an email asking "How are things progressing?" every 4 hours or so. Sometimes though, we're working on another project and in the last 4 hours didn't work on their project at all.

From previous clients, I've learned that saying you've been working on another project only brings irritation on their part. Is there a nice way to them we've got other clients to worry about as well?

You don't need to tell them anything about other clients. That is commercial information anyway, so not appropriate to share.

However, calling every four hours does seem excessive. Maybe tell them you'll email over a daily progress report instead, as taking the calls is causing too many interruptions. Best idea is to find out why they feel the need to call so often.

You could use social engineering to change the client's behaviors. Setup a weekly or bi-daily update schedule where you communicate recent progress to the customer.

You certainly can not have customers emailing you every four hours. I don't believe its your client's business to know how many projects you are working on, how many clients you have, or who you're working for right now, as long as you meeting the specifications of any contracts that you have.

From previous clients, I've learned that saying you've been working on
another project only brings irritation on their part. Is there a nice
way to them we've got other clients to worry about as well?

Handle it from the other side. This is not about other clients - it is about calling every 4 hours. Would the client call his lawyer every 4 hours? No.

Explain the client yuo gladly invoice him for an hour every time he calls and interrupts the work stream of the people. Meeetings are ok - every 4 hours is a nuisance.

Give each client an exact time and date when you will send the next status report.
Then if they still call you, I've sometimes said: "I'll be working on that part of the project from 3pm to 5pm today".
Both these actions tell them you are organized, actually working on their project, and also subtly tells them they are not paying for 100% of your time.

If, after that, the client is still calling you for a status report it means they want to hurry you up, but they don't feel comfortable telling you directly. Perhaps their client/boss is calling them every hour. Yep, it could be they are just nice. If you think this, try and shuffle things around to move their project along more quickly. Then for the next project offer them two pricing choices: the normal one, and a premium price for an earlier delivery.

If you don't want to be treated like an employee, act like a business!