Quad chainring (4x12)? Does anyone yet make an affordable quad chainring?  I would love to see a 4x12 with my staggered cog idea such that all 12 gears are shifted progressively using the 4 chainrings.  Each chainring would engage only 3 cogs.  I think the "double shifting" would be quite easy.  On the gear indicator (1 thru 12), every 3rd gear could be marked with a different color number to indicate to the rider that a "double" shift is required next.  Generally when the rider settles into a good pace on reasonably flat pavement, he/she will only be using 1 chainring and using the 3 gears for it to fine tune the cadence.  For example, if the chainrings are labelled A,B,C and D and each cog for each of those chainrings is labeled 1 thru 3, someone might be using C2 gear but shift to C1 for a slight headwind and C3 for a slight tailwind.

I think it is a good and interesting idea and would look cool too.

I played around in a spreadsheet and came up with these ratios:


20/40 = 0.5
20/33 = 0.61
20/27 = 0.74
28/31 = 0.9
28/26 = 1.09
28/21 = 1.33
36/22 = 1.64
36/18 = 2
36/15 = 2.4
44/15 = 2.93
44/13 = 3.38
44/11 = 4.00


Notice that the 15 cog repeats cuz the difference between the 36 and 44 chainrings is close enough to 20%.

Also notice the overall spread of the gears is a very wide 8 with a very low 1st gear (0.5) and a reasonably tall high gear (4.0).  That should cover almost any situation for the average rider.
You could do this for the front with a Mountain Tamer Quad (which is an adapter which turns a triple into a quad) and an appropriately chosen front derailleur. The chainline issues become more severe than they do with a triple. 

For the rear, you could use a 12 speed internal gear hub (Sachs Elan) or go up to 14 with a Rohloff Speedhub with a chain tensioner. 

The question is why would you want to do this (like all your questions)? With 3x9's or 2x11's there are more than enough gear combinations that its pointless for riders to have more. With more gears, things have to be machined more carefully and become less robust as well. You need a good gear combination, not the perfect combination (though Nuvinci makes a CVT for bicycles). Also, who in their right mind looks at the cassette when shifting? Nobody. On top of this, front derailleurs are crude and sloppy devices just by how they work -- with 4 chain rings, you can get one to work depending on the chainring spacing, but probably not well.

So, while you could run a 4x12, it would be utterly useless for anyone who wants to ride.