I have published this picture many times before, but I guess I can't repeat it often enough.
This is data that anyone who wants to design a speaker should really study and understand, as GAB so strongly advised, all those years ago. He will never be wrong about this.
Now note that many instruments have fundamentals below the 400 to 500 Hz range. Note that some have fundamentals above the crossover point to the mid of the speaker in question. The other thing to understand is that the acoustic power of an instrument is maximal at its fundamental, and progressively decreases from second, fourth etc.
I think you can see from this chart the the 400 to 500 Hz range is a really good place to cross from woofer to mid. If the mid is not really capable, use 2. You can also see that crossing to the tweeter at 4 to 5 KHz range is another sweet spot. Unfortunately too many mids suffer cone break up, with rise in output and beaming below that frequency.
That chart show why crossing to a 4.5" mid at 113 Hz is a really poor design choice.