<HTML>Michael G sez: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT= : 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">Yes, I agree. There is no likelihood of the exact science o= f sound and its harmonics being produced text-book fashion in the piano. The= principle  applies though - and I doubt it's far removed from the "tex= t book" version. What say you?</BLOCKQUOTE> Depends upon what you call far removed, but it's precisely that difference t= hat is important to piano tuners. You're right that the principle of coincid= ent partials explains where beats come from, but it's the  _departure_&= nbsp; from the principle which explains the compromises necessitated by inha= rmonic mismatches, and that is so vital to refined tuning. It's still in the= textbook; it's just on a different page. The other day I mentioned some phony figures to help illustrate this point. = You have a Steinway A, so here are some real measurements, from the temperam= ent area of a Steinway A. If A4 is tuned to 440 Hz., its 2nd partial vibrates at 880.92 its 4th partial is 1766.4 instead of 1760 (and its 8th partial is 3609.36, instead of 3520) If A3 is tuned to 220 Hz., its 2nd partial vibrates @ 440.48 instead of 440 its 4th partial is 882.6, instead of 880 its 8th partial is 1774.8, instead of 1760 This creates significant beating, even though the fundamental of the upper n= ote is twice that of the lower. You can see that at the 4:2 level, you've go= t 880.92 versus 882.6, (over a beat and a half per second). At the 8:4, it's= even worse. If A3 is then retuned so that its 2nd partial is also 440, its fundamental will be at 219.76, but the other coincident partials will co= me a little closer to matching. Fortunately, we do some of this automatically by ear, but it's still importa= nt to know when to go for smooth and when we are likely to have to squeeze i= t a little to get better results elsewhere in the piano. Bob Davis Stockton, California </HTML>