How can I tie a tight knot?
As you know, the vibration of the chime is what causes the sound. Within that vibration are peaks and troughs of vibration, but there are also nodes where the vibration is barely, if at all, noticeable. If you don't tie down the chime at exactly one of these nodes, you'll attenuate your sound and cause buzzing anyway.

Instead of tying your chimes, think about laying them in a way that makes them act like you want without the problems you don't want. This, believe it or not, has been figured out by makers of xylophones and marimba.

For example, a xylophones has lots of tuned steel slats that are struck by mallets, just like you want to do with chimes. They do bounce a little, but that's addressed by having a hole in one or both ends and a pin loosely in the hole preventing it from moving in an undesirable way. These slats are also resting on felt or other cloth risers at their nodes to prevent buzzing and damage from striking too hard &amp; bounces. (I've left the image large so the cloth strips can be easily seen.)



Also, most chimes for music are held in place by either loose cords or a rod. Granted, these are also generally vertically mounted, but it's something to think about, since you are going the opposite direction in that you want to tie them securely down.



I'm not recommending a product or website, but if you look at how someone else makes something similar, you'll notice that the chimes appear to not be held down at all. It would be interesting to look at this from the ends of the chimes to see if there's a cord running across the tubes, but just hidden. The supports here look like rubber to prevent buzzing.

https://www.hayneedle.com/product/20inwoodstockpipedreamwindchime.cfm

On the other hand, a single chime seems to be supported and held down by the same attachment device. It's a bad pic, but the support looks like a rubber O-ring, also presumably to prevent buzzing.

https://www.hayneedle.com/product/woodstockzenergywindchimesingle.cfm