Pitch at breaking points

[Richard Moody]

----- Original Message -----
From: Stephen Birkett <birketts@wright.aps.uoguelph.ca>
To: <pianotech@ptg.org>
Sent: Friday, March 31, 2000 10:02 PM
Subject: Re: Pure Tone Strings



Newton writes
> > Given one note location and speaking length every wire size will
> > break at the same pitch when drawn up that high.
> >

Stephen writes
> This is not true, but you can be excused Newton since even Grant O'Brien
> made that mistake in his book about Ruckers harpsichords,
>Thicker wire will break at a lower pitch. How
> much lower depends on various factors. I don't have the data for modern
> steel wire, but you will find that doubling diameter will probably lose
> you about 4 semitones at breaking point.

Newton, as usual is right.  OK, within 0.489 cents for one (specific) size
to the next.  Strange as it may seem wire of the same length but from size
to size does break near the same pitch. If you include all 21 sizes used in
the piano, you get a spread of 131 cents of the breaking frequencies.  You
can get that figure to 88 cents if you favorably select from the range of
breaking points.

    Thicker wire does break at a lower pitch but not nearly as much as 4
semitones when doubled, at least not in the catagory of music wire.   So
Stephen is partially right.  I don't know what O'Brian wrote, but that would
be interesting to see because it would never occur to me that different size
wire will break near the same freq. for the same length.  Amazing!!  Can
this be predicted by looking at the formula and noting the proportions?  See
Footnote.

     I came up with .031 (#13) breaking at  111cps and .052 (#23 )  breaking
at 102.9 cps. (Both at 100 inches)  I used the breaking strength figures in
McFerrin.  These are the low points.  The high ones show the same pattern.
Following that pattern I don't expect to see a 400 cent difference from
double the diameter.  The breaking point will vary at least 10 lbs or more
for the same size wire. At .052 it is a 30 lb difference giving a frequency
difference of 103 to 105.5.    From the smallest breaking point of #13 to
the largest breaking point of #23 the difference is 88 cents, well within a
semitone.

    McFerrin gives 273-285lbs  to break a #13 wire   and 635-667lbs to snap
a #23.
The spread of breaking points against frequency would be interesting to
graph. That way you could see F where it (d) doubles.  If someone has
breaking points for sizes 12 on down I would be happy to plug those in.

    ALSO  please note this is a first run with a "new" formula for computing
frequency while varying tension or Length, or diameter.  If someone wants to
double check  it would be appreciated esp if they agree.  for #13  L = 100
d = .031  T = 273lb  resulting a freq. or F of  111.06.  How conveinent---we
are at A2.

---La ric


FOOTNOTE
      F = Square root of tension times the inverse of Length times the
inverse of diameter times a constant     F= T^(1/2)*1/d *1/L* k.
Technicians now know that if F changes at all, it changes very little when
he/she only  changes d   My question is can a mathematician predict the
nature of change in F by looking at this formula.?   My guess is he should.
d can only change by so much, in this case, the increments of piano wire
sizes or  .001.   That  will cause  F to change by X amount.   And also T.
The changes in d and T result in (=)the changes of F.
The only thing I know how to do is plug in 20 sizes of d and look at the 20
different results of F.  I can also ask Quatro Pro to graph F against D.  I
don't need the formula of change execpt out of curiosity, and the challange
of solving a problem. One would think if it could graph the changes it could
show the formula of the changes...