the period was still proportional to the square root of the mass
1) This is true for an arbitrary nonlinear spring in the equation
F(x) = m (d/dt)^2 x (eq 1)
2) This is an example of a powerful technique for solving (and more
importantly, for understanding) differential equations. To apply the
technique to this problem, write
x = X x'
m = M m'
t = T t'
where capital X, M, and T can be thought of as the "length scale", "mass
scale", and "time scale" of the problem, and the primed variables x', m',
and t' can be thought of as the "reduced variables" or "dimensionless
variables". (Primes do not indicate differentiation.)
Then the equation of motion becomes
F(X x') = (M X / T^2) m' (d/dt') x' (eq 2)
Now, because the functional form of F is unknown, we can't play with X. In
contrast, we can change M to be anything we want if we simultaneously
change T so that (M / T^2) remains constant.
As the saying goes, "the same equations have the same solutions".
3) The symmetry exhibited by eq 1 is not a super-deep principle of
physics. If you have a force that depends on velocity (e.g. magnetism) or
a force that depends on mass (e.g. gravitation) then there won't be a
simple M/T^2 symmetry.
4) In addition the aforementioned multiplicative transformations, there are
other transformations that leave eq 1 invariant. Shifting t by an additive
constant is an obvious example.
5) The search for invariances (rescalings, shifts, etc.) in differential
equations has a long and glorious history. The Reynolds number is famous
example.
6) There is always considerable arbitrariness in choosing "the"
invariances. In eq 1 for example, there is no basis for saying that (M /
T^2) is a more fundamental invariance than (X M / T^2). If one is
invariant then the other is also; take your pick.