Our speaker thinks of each name sounds "comfortable" when spoken.
By name we assume she means the names of all the things in of the world: field daisy, black bear, etc.
The word "comfortable" suggests familiarity. It suggests a bond, like the line about brotherhood and sisterhood. She wants to know the names (and the things themselves, probably) well enough for their names to feel comfortable on her tongue.
And using the word "music" elevates the idea of a name, and the act of naming (which is pretty much the basis of the art of poetry). But we think it also elevates the things that are named, which, presumably, is anything and everything in the world.
tending, as all music does, toward silence,
That naming, that comfortable music in the mouth, tends to turn into silence.
Anytime you say something, it's bookended by silence.
This seems like a gentle way of pointing again to death. There's not any music (or anything in the world) which does not tend toward silence (or death).
So, in a way, this is our speaker's re-imagining of the concept of life and death. Life is a music that eventually becomes silence. Envisioning it this way makes life sound beautiful (we all like music, right?) and death sound gentle.