Tuple: In logic and mathematics: Element made up of n things, with a first, second ... nth element, as in "(x_{1}, .., x_{n})"
Tuples are often named with their number of terms explicated: as in "ntuple". A pair is a twotuple.
It is not difficult to reduce tuples to pairs with suitable definitions or assumptions, for one can analyse a threetuple as a twotuple made up of an element and a tuple, and similarly for fourtuples (an element and a threetuple) etc.
This is the standard practice in set theory. Also, as is also common practice in setheory, one may take the element to be Ø (the empty set). This is one way to generate  a settheoretical equivalent of  the natural numbers.
Thus with "#(.)" = "the number of":
#(Ø) = 0
#({Ø}) = 1
#({Ø,{Ø}}) = 2
#({Ø,{Ø,{Ø}}) = 3 (etc.)
Those seeking even more refinement can use {} = Ø. And the percipient reader will have noticed that settheory thus enables one to generate natural numbers from nothing at all.
And also one may define an ordered twotuple (x_{a}, x_{b}) using sets:
(x_{a}, x_{b}) =df {{x}, {x,y}} & {x} = x_{a} & {x,y} = x_{b}
on the principle that in {{x},{x,y}} one can keep apart the two elements (one an unitset, one a pairset), and thus as it were pair off first and second element of the tuple involving x_{a} and x_{b} in that order.
This was independently thought of  in that order  by Wiener and Kuratowski. It showed that a special logic of relations as conceived in Principia Mathematica could be avoided, as relations could be analysed as tuples, and tuples as sets of the above kind.
In case one has proper classes, the above reduction of tuples to sets doesn't hold for proper classes. To have pairs of these too, Cech assumes special axioms for pairs in general, for both sets and proper classes. This then guarantees arbitrary ntuples of arbitrary things.
