Helix in a helix

Helix in a helix

I am trying to work out a "helix in a helix" mathematically. Intuitively I
think of this as a steel cable, which is made up of a number of smaller
steel cables all bound together in spiral. If I wanted to find the length
of one of the individual cables, it would be bound in a spiral in the
smaller cable and then those cables bound in a larger spiral cable. I know
that if I wanted to do a helix whose ends meet, I would use the
parametrization
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},b\sin(\omega{t})),t=0..2\pi$$
I've been trying to map out in my head how to, instead of curling the
helix, making the helix travel in the path of a helix. I've achieved it
partially with
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},t),t=0..\infty$$
But this doesn't keep the smaller helix in tact, and turns it into a sine
wave helix. I've also tried
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},tb\sin(\omega{t})),t=0..2\pi$$
But this gives me sort of a nautilus shape where the helix curls into
itself and increases in size and curls around into itself. What am I
missing?