-\providecommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
-
-% The Cartesian product of two things.
-\providecommand*{\cartprod}[2]{ {#1}\times{#2} }
-
-% The Cartesian product of three things.
-\providecommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
-
-% The direct sum of two things.
-\providecommand*{\directsum}[2]{ {#1}\oplus{#2} }
+\newcommand*{\triple}[3]{ \left({#1},{#2},{#3}\right) }
+
+% A four-tuple of things.
+\newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
+
+% A five-tuple of things.
+\newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
+
+% A six-tuple of things.
+\newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
+
+% A seven-tuple of things.
+\newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
+
+% A free-form tuple of things. Useful for when the exact number is not
+% known, such as when \ldots will be stuck in the middle of the list,
+% and when you don't want to think in column-vector terms, e.g. with
+% elements of an abstract Cartesian product space.
+\newcommand*{\tuple}[1]{ \left({#1}\right) }
+
+% The "least common multiple of" function. Takes a nonempty set of
+% things that can be multiplied and ordered as its argument. Name
+% chosen for synergy with \gcd, which *does* exist already.
+\newcommand*{\lcm}[1]{ \operatorname{lcm}\of{{#1}} }
+\ifdefined\newglossaryentry
+ \newglossaryentry{lcm}{
+ name={\ensuremath{\lcm{X}}},
+ description={the least common multiple of the elements of $X$},
+ sort=l
+ }
+\fi