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Yes that is true Tarantella dance is connected with the Tarantula spider. It is said though, that Tarantella was performed as a therapy against the bite of the spider ☺!!
Another thing that a lot of people do not kow is that both the dance and the spider took their names from Taranto (Taras) a city in Southern Italy, which was created by the Spartans.
So I guess dancing is a good thing after all 🙂!
P.s.: The song you posted is a legendary one by beloved Leonard Cohen, which is beautiful to dance to!!!

I guess the debate will go on for eternity as to whether dance originates from, or cures, the bite of the lady spider.

Do tarantulas make webs? or are they hunting spiders like my beloved Wolf spider pets in the Himalayas?

Oops I do not know the answer to this. But would love to know 😉
I guess it doesn't have to do with dancing lol

No, it is more basic than that :-)