PUSHING simply means giving extra development to compensate as much as is possible for underexposure.
Sometimes it is like a good tool to use, like a Sawzall or a Backhoe.
Here is what happens: the longer you develop a negative, the more contrast it gets. AND what CONTRAST means is How Many Steps in the print do you have between Black and White.
Photographic prints normally have 10 steps: Black, White, and 8 tones of gray. Think of it as an octave in music.
A one stop push (400>800) gives you a 7 tone scale. A two stop push, a 6 tone scale, and so on. White stays in the same place, and black walks closer as you climb the scale.
If you do it just right, you can get Black, White, and 3 shades of Gray from TMY, and THAT is EI 6400. I guess that if you do it just right, that is like having 5 notes on a guitar, and if you do it right you can get a C#, and you can rock down to Electric Avenue. Just bring your groove. We don't need 8 or 10 or 12 tones.
The trick is that the harder you push, the less room you have for mistakes. If you underexpose 1 stop on a Push, you don't get 5 notes, you get 4, and maybe you can compose in those 4 notes, maybe not.
A good musician doesn't need ALL the notes, you just need the right ones. And when you push, you are pulling keys off the piano, or taking frets off the guitar. Be sure you figure out what you want or you end up with junk.
PUSHING takes practice, and is a cr*ppy way to salvage once in a lifetime pictures.
If you have to use TMY, try 16 minutes in Xtol 1+1, agitate it every minute, at 68•. Remember, when you go close to the edge, watch your step.