Pitch is a result of different frequencies. The faster the string vibrates, the higher the sound is. On a piano, the lowest note (A0) vibrates at 27.5 Hz, while the highest (C8) reaches 4186 Hz. That range is covered by 88 keys and about 230 strings, each making a different sound and vibration when played.
The frequency of a vibrating string depends on three factors:
Longer, looser and thicker strings. Some are wound with brass to increase mass and lower the frequency of vibration. Only one string needed, slow vibrations create a richer sound.
Shorter, tighter, and thinner strings. More than one string per note because the high notes are much softer. Doesn't change the pitch, but makes the sound fuller.
Higher frequencies require more energy to sustain because of their speed, so they fade out faster than lower ones. This causes higher notes to sound sharper, while low notes sound heavier and last longer.
Most of the waves created are transverse, except for some slight longitudinal stretching waves as the string expands and contracts during vibrations.
Hannah