So I thought that during a period of time the timbre and tone would change until the strings settle and keep the same tone for a long time?
As @el murdoque mentioned, it's about 2 weeks in my case.
When I was playing and using them (until I quit because of virus safety concerns), that would be two 4 hours gigs a weekend and maybe 2 to 5 hours of noodling between weekend gigs. So, during a two week period, maybe 20 to 25 hours.
But really, I only noticed the newness at home. Once the band kicked in, even after new ones were installed within days of the gig, it drowned out any newness factor. Considering a break-in period was never an issue with me.
And they'd last close to a year before I'd change. Only used the La Bella LTF-5A's. Short period of D'Addario Chromes, but changed 'em back to La Bella after a few months.
If your bass is 34inch scale, like mine, you'll probably get in-tune, distinct notes beyond the octave B with the .118 B string when properly intonated. Opened up a whole new playing area on my bass when I got away from .125's and larger for B strings. The roundwound set I use has a .115 B string.
No loss of bass goodness and increased in-tune range up the neck in those thinner B's.
The LTF-5A's are a good starting point for you, in my opinion. Hope you enjoy.