I believe that it already does handle vertical and horizontal "frequencies". It just might be hard to see with examples you've tried. In the way I defined the lissajous function, the w argument is basically the ratio of the vertical and horizontal "frequencies". There are some subtleties here, because if it's not rational, then the curve doesn't "close", and if you don't extend your tmax far enough, you might not be getting enough "cycles" of the curve to see all of the "bumps". Also, the view angle can make it easier or harder to see.

For example, using my original definition, execute this:

ParametricPlot3D[
 Lissajous3D[1, 1, 4/3, Pi/3][t, s], {s, 0, .1}, {t, 0, 6 Pi}, 
 ViewPoint -> Front, ViewProjection -> "Orthographic", 
 ColorFunction -> (White &), MeshStyle -> None, 
 PlotPoints -> {20, 60}, 
 Lighting -> {{"Point", Blue, {0, -5, 0}}, {"Point", 
    Green, {3, -5, 0}}}, Axes -> False]

Then execute this:

ParametricPlot3D[
 Lissajous3D[1, 1, 3/4, Pi/2][t, s], {s, 0, .1}, {t, 0, 8 Pi}, 
 ViewPoint -> Front, ViewProjection -> "Orthographic", 
 ColorFunction -> (White &), MeshStyle -> None, 
 PlotPoints -> {20, 60}, 
 Lighting -> {{"Point", Blue, {0, -5, 0}}, {"Point", 
    Green, {3, -5, 0}}}, Axes -> False]

So, what you could do in your Manipulate is have a slider for vertical frequency and another for horizontal frequency (use new variable names for these--e.g. hf, vf-- and get rid of w), and then in the Lissajous3D expression, replace w with vf/hf.

Here's an approach.

• Make the Lissajous curve wrap around a cylinder (I'm assuming this will give the 3D shape you want, but if you're wanting something else, like an over-under knot type pattern, then this won't work--but something else will :) )
• Use ParametricPlot3D
• Use two parameters in the plot to create a ribbon
• Use lighting or other styling to enhance the 3D effect.

Here's an example output:

Here is the code for the ParametricPlot3D:

ParametricPlot3D[
 Lissajous3D[1, 1, 3, Pi/3][t, s],
 {s, 0, .1}, {t, 0, 2 Pi},
 ViewPoint -> Front,
 ViewProjection -> "Orthographic",
 ColorFunction -> (White &),
 MeshStyle -> None,
 PlotPoints -> {20, 60},
 Lighting -> {{"Point", Blue, {0, -5, 0}}, {"Point", 
    Green, {3, -5, 0}}},
 Axes -> False]

You can see the options I've used, and hopefully that will get you started. The documentation for each option should help you figure out how to get your preferred appearance.

Here's the definition for the Lissajous3D function:

Lissajous3D[cx_, cy_, w_, d_][t_, s_] := {cx Sin[t w + d], cx Cos[t w + d], cy Sin[t] + Rescale[cx Cos[t w + d], {-1, 1}, {s, .5 s}]}

The x and z coordinates plot the 2d lissajous, and the y coordinate the front/back distance as if it was wrapping around a cylinder. This may not be the exact 3d shape you want, but hopefully it'll get you started. You can switch the coordinates to change the orientation.