My book, "Piano Servicing, Tuning and Rebuilding", includes a four-step
procedure for tying a knot in a string, as illustrated by Douglas Strong
in the "Piano Technicians Journal" issue of November 1972 (and used with
permission).  It is shown on p. 106 of the second edition of my book.

When I pull up the pitch of a piano, I start in the middle and work
toward the treble, then finish in the bass.  If the strings are in
such bad condition that they will break when pulled up to pitch, this
condition usually manifests itself in the midrange.  My rule is that if
more than two midrange strings break during a pitch raising, I replace
them and make no further attempt to raise the pitch of that piano until
it is restrung.

Tuning thousands of pianos in the 1970s, I rarely had the occasion to
tie a bass string, because the fragile, rusty treble strings almost
always warned me not to proceed.  However, on one occasion, a perfectly
good looking bass string broke in a little early 1960s Cable grand made
by Aeolian-American.  As the strings still sounded and looked almost
like new, I thought I'd try the knot-tying technique.  The result of
my first attempt is illustrated on page 111.  As the photograph shows,
this was in an area with very little working room.

After forming a coil in the new leader, placing it on the tuning pin,
and pulling the string up to pitch, the knot pulled tight -- just in the
right place -- and the string sounded just like its neighbors.  I later
decided that since I was able to do it on the first try, it was a tech-
nique worthy of inclusion in the book.  As I note in my text, "this
repair can be very challenging due to the stiffness of the wire; but
with strength, persistence, and great care taken not to cut your
fingers, it can be done."

Of course, if the string breaks in the sounding portion (between the
agraffe and the winding) it's difficult to tie a knot that doesn't
end up touching the agraffe as it tightens.

Art Reblitz