Question

What is the motivation for tuning “stretched octaves” on a piano? Be specific.

Answer 1

piano tuning is not simply perticular fixed set of pitches. it require an assesment of the vibration interacting among notes. a stretched string vibrates in different modes .

when piano string vibrates all the harmonic modes are exicited and the first harmonic is largest and the higher harmonic is vibrates integer multiple of first harmonic

in piano tuning stretched octaves are commonly encountered where the inharmonicity caused by the string thickness and tension make it necessary to widen every intervel very slightly

strecthed octaves are caused by the physics of standing wave in a strecthed wire. the pitch of each overtone is determined by the ratio of string restoring force divided by mass per unit length

piano string are made from high carbon steal which is stiff.

for a stretching tuning the octave is greter then a factor of 2

for a compressed tuning the octave is less then the factor of 2