When I think of cadenzas, I think of a short virtuoso passage for a solo instrument at the end of a piece or movement. I remember when doing my flute exams the accepted practice was that any cadenzas for a baroque or classical concerto should be playable in 1 breath. Not everyone agrees with this today, but when writing my own cadenza for the Mozart flute concertos it was a useful constraint on duration.
We studied Mozart's 23rd piano concerto (K488) for A-level music. Unlike his wind concertos, Mozart actually wrote the cadenza for this concerto, although he most likely played the premier of the piece. It is believe he notated the cadenza because other performers wanted to emulate his virtuosity. In the first movement, the orchestra end on a Ic chord, leading into the piano cadenza, which eventually finishes with a V-I perfect cadence.
I was listening to the 1st Movement of Rachmaninov's 3rd piano concerto and the cadenza is almost a movement within itself. The advantage of the piano being it can add harmony and multiple melodic lines so you can prolong a cadenza; other instruments can only play 1 line so are limited in what they can achieve.
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