This is for an invention I'm working on. I've figured out the mechanical bits, but I'm stuck on the electrical bits. I'm trying to figure out how to make a LED turn on and off in the following way: First, divide a second into 60 parts. The LED can be on for anywhere from zero of these 60 parts to 60 (all) of these 60 parts. A dial controls how many parts of 60 that the LED stays on - how long each flash is. Or you could also say that the LED is on for 'so many' parts of a second that's been divided by 60. I hope I'm making this clear. I've researched hertz, frequency, pulsed width modulation, the 555 IC, Triacs, Relays and on and on, but I can't seem to find the best way to do this.
I'm not expecting anyone to write up an entire schematic for me, but I have pages and pages in my notebook of different ways to go at this, so if anyone can just push me in the simplest and most feasible direction I would really appreciate it.
Thank you.

I don't have an answer for you, but to make sure others understand this correctly - if the dial is set to, say, 23, then the LED should be on for 23/60 of a second, then off for 37/60 of a second, then on again for 23/60, then off for 37/60, etc. ad infinitum - correct?

Yes, that is absolutely right.
Thank you for clarifying.