A triple leaf partition has two air cavities rather than the typical one air cavity. Similarly, a quadruple leaf partition has three air cavities. Triple and quadruple leafs are most often created when constructing a new wall in front of an existing wall. The other common ways include installing resilient sound clips or resilient channel over existing drywall. This applies to both walls and ceilings alike. A leaf in a partition is a solid layer, like drywall or plywood.
The drawings above show single leaf (no air cavity), double leaf (like most assemblies), triple leaf, and quadruple leaf assemblies, both clipped and standard framing. It is necessary to remember that two layers of drywall directly against each other still only count as one leaf because there is no air cavity between the layers. In the sketch below we show again double through quadruple leaf walls, but this time they all have the same number of drywall layers.
It is not illogical to presume that the quadruple leaf wall on the far right would have the best sound isolation. After all, the sound has to go through drywall, air space, drywall, air space, drywall, air space, and then again drywall, often with insulation within each air space. In a double leaf wall, the sound has to make it through only one air space. While it makes sense that the quadruple or triple leaf assemblies are better than double leaf assemblies, it is, unfortunately, not the case.