Thanks for the in-depth explanation, I think I get it now. Although I have to check up on these programmable AND gates that I never saw before.
I made a BD-version of your counter unit that can count to 6, 10 and 12 and outputs in 4-bit binary. It is not quite as versatile as yours, but it's smaller and perhaps easier to remember how it works. Picture (with explanation below):
The input is at the bottom of the picture. The pulses go to a BD n-counter which outputs a pulse after n pulses (to the next counter unit for hours, etc.). They also go through a BD 2-counter and directly to the first BD switcher. Thus the first switcher will switch every pulse and the other switchers every 2nd pulse.
EDIT: There's an error in the picture - there should not be redstone from the 2-counter to the n-counter. The n-counter should get its input from the first input. I modified the circuit and messed that up, sorry. Though when counting to even numbers you could wire it up like that and just count to n/2.
What the switcher BDs contain depend on which number the unit counts to, but are combinations of conductive blocks and non-conductive things, in this case minecarts. Let a conductive block be represented by 1 and a non-conductive by 0. Then, for a 12-counter, the switchers contain the following sequences from bottom to top:
1, 0
0, 1, 0, 1, 0, 1
0, 0, 1, 1, 0, 0
0, 0, 0, 0, 1, 1
For the 10- and 6-counters you just remove 1 or 3 blocks from the three last sequences. Note that the 1's have to be different kinds of conductive blocks, or they get stacked together and everything breaks down.
What happens is that when a BD-switcher switches to a conductive block (1) the BD above the relevant output bit gets power and places a conductive block which lets a current pass through. When the switcher goes to 0, the block is retracted and the output shuts off. You could also use sticky pistons with a conductive block for this.
The output at the 5th iteration looks like this:
You have to modify the setup to count to odd numbers, and I think you probably cannot count to some of those at all. But since you only need the 6, 10 and 12 anyway it works fine.
I think that using a combination of turntables and BD's it could be made even smaller.
Also note: I think your setup is really smart, I just wanted to show an alternative now that we figured out some new BD techniques. :)
