"The invisible tool"

[BobDavis88@aol.com]

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Michael Gamble writes:

> This is a late reply to your most interesting comments on the 
> partial/harmonic frequencies on my model "A" S&S for which I thank you.
>  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 supplied by S&S? I should be very interested to see a full 
> listing, in data-base format, of such accuracy so that, when tuning, the 
> beats we hear can positively be identified as issuing from the result of 
> which-ever partials beating together.
> 
Michael,

S&S would not be able to supply such figures, because they vary due to 
variance 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 
sophisticated 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 A440, I 
just carefully tune the note to the AccuTuner, then change the AT to read the 
2nd partial. Uncorrected and in "tune" mode, it will display the cents 
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 
pick up "x" (the number of cents) from a cell.

Printing out all the partials in database format might be interesting and 
instructive, but would also be time-consuming and unnecessary. Two notes 
comprising an octave, tuned to different coincidences, will suffice. The easiest place 
to demonstrate the beating of the different partials is in the bass, because 
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, Accutuner, 
Verituner, TuneLab Pro) that you can borrow, I'll bet you find it illuminating.

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 clean, 
because the other coincidences will beat, but there will be no beat at the 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 note down at 
all, a beat will develop pitched at G4, but the E5 beat will slow 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 fundamental 
matching the 2nd partial of the lower), and actually calculate how many beats 
you will hear at each "matching" partial coincidence, using the methods 
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





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