<HTML>Michael Gamble writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT= : 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">This is a late reply to your most interesting comments on t= he partial/harmonic frequencies on my model "A" S&S for which I thank yo= u. The figures you gave in = that post make me ask how you derived them! Do you have a "Sierra" frequency= meter to give such accurate read-outs? Or are these figures which are suppl= ied by S&S? I should be very interested to see a full listing, in data-b= ase format, of such accuracy so that, when tuning, the beats we hear can pos= itively be identified as issuing from the result of which-ever partials beat= ing together. </BLOCKQUOTE> Michael, S&S would not be able to supply such figures, because they vary due to v= ariance in string diameter, bridge placement, and soundboard stiffness, even= on pianos of the same model. We have two S&S A's at home and two at the= shop, and each measures out ve-e-e-ry slightly differently. I don't have a Verituner (alas, no, Isaac!). I used something less sophistic= ated but equally accurate - an AccuTuner II and a bit of math. The AccuTuner= is readable to 0.1 cent, and accurate beyond that; so for instance with A44= 0, I just carefully tune the note to the AccuTuner, then change the AT to re= ad the 2nd partial. Uncorrected and in "tune" mode, it will display the cent= s deviation from 880. In this case I think it was 1.8 cents. The frequency represented by 1.8 cents above 880 is found by multiplying 880= by 2 raised to the x/1200 power where x is the number of cents: =880*(2^(1.8/1200))   <--outside parentheses for clarity only= =880.92 Hz A spreadsheet can be set up to calculate this for you, having the formula pi= ck up "x" (the number of cents) from a cell. Printing out all the partials in database format might be interesting and in= structive, but would also be time-consuming and unnecessary. Two notes compr= ising an octave, tuned to different coincidences, will suffice. The easiest = place to demonstrate the beating of the different partials is in the bass, b= ecause of the richness of partial content as well as the longer sustain. If = you have a colleague with one of the better tuning aids (Cybertuner, Accutun= er, Verituner, TuneLab Pro) that you can borrow, I'll bet you find it illumi= nating. Demo 1: Take a bass note such as C3 (the one below middle C). Set the tuner = to listen to G4, which is the third partial of C3, and zero the tuner. G4 is= also the sixth partial of C2. Leaving the tuner alone, tune C2 so that the = lights stop. This will produce a 6:3 type octave, because you have tuned the= 6th partial of C2 to the 3rd partial of C3. This will NOT be perfectly clea= n, because the other coincidences will beat, but there will be no beat at th= e pitch of G4. Listen carefully and you will hear other beats, especially at= E5, which is the 10:5 coincidence. If you now move the pitch of the lower n= ote down at all, a beat will develop pitched at G4, but the E5 beat will slo= w down. Next: If you measure the first however many partials of these two notes, you= can tune them any way you want (for instance with the upper note's fundamen= tal matching the 2nd partial of the lower), and actually calculate how many = beats you will hear at each "matching" partial coincidence, using the method= s above. You can verify it by ear, and they will "positively be identified as issuing from the result of= which-ever partials beating together. I've got to go to bed. More later, maybe this's something to chew on. Let me= know if this is unclear. Bob Davis </HTML>