All I know is the dancing I did, and even now, 30 years later, I have the tempos engrained in my body. As I said there’s physics to it, and muscle memory. To check the “right speed” all I need is to dance it a bit.
Musicians can be all over the map, un-dance-ably slow or fast, because they’re not moving their whole body like a dancer is.
About “putting the tune under the dancer’s feet” as Mr Gumby so well says, there are players who are very good at watching the dancers and adjusting their speed to match.
I’m not one of them! I worry about getting the speed the way the dancer wants it.
Here’s a BPM horror story. I was hired to pipe for a group of dancers, their regular piper couldn’t make it. No worries because the teacher is always there to advise, right? Well for this one show the teacher couldn’t make it either.
So I attended their rehearsal, saw the dancers practice their show, and took metronome readings (they were practicing to recordings). As I noted the tempo of each dance I would ask the teacher “is this the right tempo? Is this how they will be doing it at the show?” and the teacher assured me that all the tempos they practiced were exactly how she wanted them done at the show.
So at the show itself I brought a set list with tempo markings, and played to a metronome. Each dance was done to the exact tempo they had rehearsed them to, the exact tempo the teacher told me to use.
The dancers were furious. I was never hired again.
Why? I can only assume that the dancers, at a show, would take dances slower or faster than they rehearsed them, at whim, and that their usual piper follows them. BTW their usual piper is extremely good at playing for dancers- it’s been his fulltime job for 30 years now.