Is there any data in Tolkien (books/letters/etc..) to explain why there were precisely 9 Men rings and 7 Dwarf rings and 3 Elf rings? (the latter assumes that Sauron planned the nmbernumber of Elvish rings to be 3 before he was thwarted by Elves making the rings themselves)
Was it because of the amount of tribes/kings he needed to deal with? Some magical/power limitations? Magical properties of numbers 9 and 7 (ala 7 Horcruxes)? A whim of Sauron's?
UPDATE: Looks like 3 and 7 are explained in the answers (3 elf lords and supposedly 7 Dwarf houses, though nobody provided canon proof).
What was NOT explained was why there were 9 Man rings?
If 9 was a meaningful amount, why?
If it was merely "19-7-3" leftover, then why were there 19 total non-One rings made?