<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META http-equiv=Content-Type content="text/html; = charset=iso-8859-1"> <META content="MSHTML 6.00.2800.1264" name=GENERATOR> <STYLE></STYLE> </HEAD> <BODY bgColor=#ffffff> <DIV><FONT face=Arial size=2>Hello Bob Davis</FONT></DIV> <DIV><FONT face=Arial size=2>Thanks for your input. I agree that the = frequencies produced by the model A aren't mathematically produced but sonically = with many reasons for their diversions. I'm very glad to get those figures of = yours if only to be able to listen out for them on my A. I will see = (hear) what I can hear, analyse what I hear (see) from you and maybe, just maybe, = there might be some conclusions to be drawn... But I doubt it! Which is why we do = what we do do (do do do) Dum dum  ;-)</FONT></DIV> <DIV><FONT face=Arial size=2>You'll notice I'm having a good session = with Isaac OLEG on this subject - a subject which is endless ...Yes? And I do = indeed do it by ear. ;)</FONT></DIV> <DIV><FONT face=Arial size=2>Regards</FONT></DIV> <DIV><FONT face=Arial size=2>Michael G (UK)</FONT></DIV> <DIV><FONT face=Arial size=2></FONT> </DIV> <DIV style="FONT: 10pt arial">----- Original Message ----- <DIV style="BACKGROUND: #e4e4e4; font-color: black"><B>From:</B> <A title=BobDavis88@aol.com = href="mailto:BobDavis88@aol.com">BobDavis88@aol.com</A> </DIV> <DIV><B>To:</B> <A title=pianotech@ptg.org href="mailto:pianotech@ptg.org">pianotech@ptg.org</A> </DIV> <DIV><B>Sent:</B> Tuesday, October 21, 2003 9:31 AM</DIV> <DIV><B>Subject:</B> Re: Fw: "The Invisible Tool"/beat rates - and where = they come from</DIV></DIV> <DIV><BR></DIV><FONT face=arial,helvetica><FONT lang=0 = face="Bookman Old Style" size=3 FAMILY="SERIF">Michael G sez:<BR><BR> <BLOCKQUOTE style="PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #0000ff 2px = solid; MARGIN-RIGHT: 0px" TYPE="CITE"></FONT><FONT lang=0 style="BACKGROUND-COLOR: #ffffff" = face=Arial color=#000000 size=2 FAMILY="SANSSERIF">Yes, I agree. There is = no likelihood of the exact science of 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 "text book" version. What say you?</BLOCKQUOTE><BR></FONT><FONT lang=0 style="BACKGROUND-COLOR: = #ffffff" face=Arial color=#000000 size=3 = FAMILY="SANSSERIF"><BR></FONT><FONT lang=0 style="BACKGROUND-COLOR: #ffffff" face="Bookman Old Style" = color=#000000 size=3 FAMILY="SERIF"><BR>Depends upon what you call far removed, but it's = precisely that difference that is important to piano tuners. You're right that the = principle of coincident partials explains where beats come from, but = it's the  _departure_  from the principle which explains the = compromises necessitated by inharmonic mismatches, and that is so vital to refined = tuning. It's still in the textbook; it's just on a different page.<BR><BR>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 temperament = area of a Steinway A.<BR><BR>If A4 is tuned to 440 Hz., <BR>its 2nd partial = vibrates at 880.92<BR>its 4th partial is 1766.4 instead of 1760<BR>(and its 8th = partial is 3609.36, instead of 3520)<BR><BR>If A3 is tuned to 220 Hz., <BR>its 2nd = partial vibrates @ 440.48 instead of 440<BR>its 4th partial is 882.6, instead of = 880<BR>its 8th partial is 1774.8, instead of 1760<BR><BR>This creates significant beating, even though the fundamental of the upper note is = twice that of the lower. You can see that at the 4:2 level, you've got 880.92 = versus 882.6, (over a beat and a half per second). At the 8:4, it's even = worse.<BR><BR>If A3 is then retuned so that its 2nd partial is also 440,<BR>its fundamental = will be at 219.76, but the other coincident partials will come a little closer = to matching.<BR><BR>Fortunately, we do some of this automatically by ear, = but it's still important to know when to go for smooth and when we are likely to = have to squeeze it a little to get better results elsewhere in the = piano.<BR><BR>Bob Davis<BR>Stockton, California<BR><BR></FONT></FONT></BODY></HTML>