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Overtones and harmonics - aren't these the same (apart from counting - 1st overtone=2nd harmonic etc)?
Almost, but not quite. The stiffness of the string causes the overtones to depart slightly from the harmonic series. The higher the pitch (overtone number) the sharper it is relative to the harmonic.
With its wide range of pitch, the piano exhibits this "inharmonicity" to the extent that tuners stretch the octaves. This makes the fundamental of a higher pitch match the sharp overtone of a lower pitch.
Overtones and harmonics - aren't these the same (apart from counting - 1st overtone=2nd harmonic etc)?
This is from Wikipedia:
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental (including the fundamental which is 1 times itself.)
So yes, harmonics are overtones, but not all overtones are homonics, only whole number partials (overtones). Or at least that is the way I understand it.