The interesting thing about butted is "double butted" means it tapers at one end, then tapers back at the other end of the tube. So double realy means the taper changes only twice. I think "triple butted" means it tapers inwards at 2 places, then tapers outwards once.
The main purpose of butting is that (a) it's designed to allow thin and springy tubes but also (b) it's designed to be thick at the ends for the joinery of the tubes. So, the thick ends of the tubes should have a long enough butt (typically 75-100mm or 3-4") so that the lug points do not dig into the ultra-thin part of the tubing. So an italian long-point lugged frame should probably have longer butts than a bocoma pro (short-point) lug set.
The benefits of quad-butting are imho doubtful. On some bikes like Miyata, I believe they put the downtube shifter bosses after the first (but not second) butt. Tearing at the shifter bosses is a very common downtube failure mode. So I can perhaps understand triple butting on a downtube, but I seriously doubt if it's needed when you could use low-temperature silver-solder for installing downtube shifter bosses, bottle bosses, and pump pegs, etc., to minimize heating and weakening of the tubes at the thinnest points.
- Don Gillies
San Diego, CA, USA