I should add to my initial post that the blade shape and stiffness (material property of the metal) are the most important factors, followed by how much damping is present both in the system (especially the handle) and in how it is held (boundary conditions in technical speak). To get a sweet ring the blade will have to be shaped such that it has low natural frequencies, or resonant frequencies, well defined in a frequency range that sounds sweet. It then needs to have low enough damping that it will continue to vibrate like a bell. HOWEVER, none of this matters much if you can't find a way to make the thing vibrate at those sweet resonant frequencies. That is why you'd probably find that it rings sweet after contact with certain materials,e.g. its sheath or a certain stone, while not so much with others, e.g. you pant leg. Higher frequency vibrations tend to damp out much faster even if the damping is low, so short and fat blades will probably not ring as sweet or as long, since their resonant frequencies are higher.
Btw, pinching the knife at the balance point helps because that point probably coincides with the first resonant frequency nodal point, which is a point that doesn't move when the knife vibrates at that frequency. By holding it there you greatly decrease the effective damping that your hand introduces, allowing it to vibrate for longer. :thumbup::thumbup: