Instead of searching for "a" color or "a" brand maybe you should try searching for a professional colorist. a.k.a. Color Consultant, Color Designer, and don't forget about color tinters who work behind the counter.

You have a very fun challenge for the right person. If you can find the right color pro, they will help you.

Otherwise, I'd simply suggest methodically moving from one brand to another; it's really just a matter of trial and error. Because what I perceive to be the color of a ball of string is probably different from what you're envisioning.

This won't be helpful if you're looking for low-mid-priced easy-to-find paint, but Farrow & Ball actually has a color called String 8.

Here is a link that might be useful: String 8 on House Beautiful slide #10