summer sharpness in low tenor

[Joe DeFazio]

on July 15, 2008, David Love wrote:


> An examination of the formula for frequency of a string as a  
> function of tension (or BP%), diameter, length (BP%) and looking at  
> the differences in the rate of change depending on tension levels  
> should yield more clarity.
>
> For example, take two strings of equal length producing equal  
> frequency (the dependent variable) but with different diameters  
> (gauges—they will have different amounts of tension and they will  
> also have different BP%) and then change the length equal amounts  
> and you should see a difference in the change in frequency between  
> the two.

Although I am far from an expert in rescaling or the physics of piano  
strings, I don't believe that the part of this about BP% is accurate.   
My understanding is that two strings of equal length producing equal  
frequency will both be at exactly the same percentage of their  
respective breaking points.  For instance:

Note 40 (C4), speaking length, 715mm , diameter of 0.040",  tension  
199.957lbs, %breakpoint 49.436%
Note 40 (C4), speaking length, 715mm , diameter of 0.038",  tension  
180.461lbs, %breakpoint 49.436%

(figures from Pscale)

In other words, though the second string is smaller in diameter, and  
will be at a lower tension when producing the note "middle C," since  
its diameter is in fact smaller, its breaking point will be lower, and  
it will be at exactly the same percentage of its breaking point.  As I  
understand it, this is because, as its diameter changes, its breaking  
point changes proportionally.  See pages 33-35 in John Travis' "A  
Guide to Restringing" (in the section by James Hayes) for a simple  
experiment you can do to demonstrate this empirically.

The aspect of the string's behavior that does change in the example  
above is inharmonicity:

Example above with 0.040 diameter, inharmonicity = 0.192
Example above with 0.037 diameter, inharmonicity = 0.173

If I am wrong in my conceptual understanding, someone please correct  
me.  I want to be sure that I understand this correctly.

I will also point out that, from my archive reading and past posts,  
there are at least two formulae out there for calculating breakpoint  
and breakpoint percentage, so some of you might plug my numbers into  
your formula and get a different value for %breakpoint. However, that  
value should be identical for the two wire diameters given above.

Joe DeFazio
Pittsburgh
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