Sorry to disappoint all of you But I will be ON VACATION until August 3rd
In addition to the equal temperament described earlier, each piano has its own specific requirement for the piano technician to be familiarized with. Some would say, a piano has a "Soul"…
A piano string, when struck by the hammer, will produce a specific series of [overtone] depending of its thickness, stiffness, length and whether it is a ‘wound string’ (with a core made of one metal and overwinding of another). Also, age and accumulation of rust deposit on a string may affect its resonance.
Such differences in strings, from piano to piano and even for different registers within the same instrument, will then exhibit [inharmonicity]. This very inharmonicity is responsible for the ‘warmth’ property common to real piano tones. Therefore, octaves might be ‘stretched’ in a piano to a certain amount depending on the exact [inharmonicity] of the strings involved.