High-tension or Low-tension?

[John Delacour]

At 22:10 20/09/01 -0700, Delwin D Fandrich wrote:

>From: "Stephen Airy" <stephen_airy@yahoo.com>
>> Do you know of a way to measure the individual (and
>> the overall) string tension in my 56" 1913 Ricca (sn
>> 37123) with the strings in the piano and pulled up to
>> pitch?  And, what tools do I need to do it with?
>
>The easiest and least expensive way is with a metric tape measure and a
>micrometer. Neither the tape measure nor the micrometer need be all that
>costly.

But neither will tell you the tension!  You will also need a very cheap calculator.

Measure 'l', the speaking length, in centimetres
Measure 'd', the string diameter, also in centimetres
Look up 'f', the frequency, in a table (or calculate it)

The tension is given by the formula: t = (l*d*f)^2 / K

Every cheap calculator I have used allows me to do this calculation in the following nine key-presses:

   l [times] d [times] f [times] [equals] [divided by] K

What is K?, you ask...

This is a constant which varies according to the density of the wire or string considered as a uniform cylinder.  'm' in the following table is the mass in grams of a centimetre length of the string or wire.  I have seen anything from 7.6 to 7.85 taken as the specific gravity of plain steel wire and I will leave it to those on the list with accurate electronic balances to give you a definitive value.  If you take 18400 (for kg. 40500) as your value for K, you probably won't be miles out.  When it comes to covered strings, K needs to be greater because of the air they contain.  Roughly speaking the fatter the string the less its density, which will vary from 6.9 to 7.3 and again you will be committing no crime if you take K as 20000 for covered strings. Since Wolfenden has been mentioned in a few messages recently, it's worth mentioning that he uses K=18600 for plain wire and K=20000 for covered strings, which means he was working on the basis of about 7.6 g/cc for steel wire.  The value will vary in any case depending on the composition and manufacture of the wire, the finer gauges being denser.

[g45 = 980.616]

 m         K(lb)       K(kg)
6.90       20519       45238
7.00       20226       44591
7.10       19941       43963
7.20       19664       43353
7.30       19395       42759
7.40       19133       42181
7.50       18878       41619
7.60       18630       41071
7.70       18388       40538
7.80       18152       40018
7.90       17922       39511

Examples:

C88     f = 4186.01 c/s     l=5.3 cm.     d = 0.0775 cm.
(f x l x d)^2 / 18400 = 161 lb.

A13     f = 55 c/s     l = 120.0 cm.     d = 0.370 cm.
(f x l x d)^2 / 20000 = 298 lb.

You will discover that using different values for K, the percentage difference in the resulting tension is pretty small, which is why too much heart-searching over the exact specific gravity is not generally called for, though a designer will obviously wish to obtain accurate figures.

JD