[ Ref. MMD 981129 "Peterson Autostrobe Stretch Profiles" ]

The question the Mike Haydon asks is one that has no one answer as
pianos are not all the same and the amount of stretch in the tuning on
one piano probably won't be the same on the next piano.

The 'stretch' required will depend on the condition and the scaling of
the piano strings and to some smaller degree on the conditions under
which the piano is tuned.

The reason that pianos are 'stretched' when tuning is the variance of
the harmonics from the fundamental frequency of a given piano string.
Due to the variations mentioned above the variations will seldom be
exactly the same.

The harmonics (the octaves, for instance) will not be a mathematical
integral multiple of the fundamental because basically, although the
vibrating length of the string is halved to produce the 2nd harmonic,
or octave, the diameter and the resiliency of the string is not reduced
proportionately.  Therefore, the harmonics will be slightly sharper
than they would be if they were true mathematical integers of the
fundamental.  This is what makes piano tuning an art rather than a
science.  Fortunately, this problem does not exist in most other
instruments.

An organ, especially an electronic organ, does not require stretch
tuning, and indeed would not sound correct if its tuning were
stretched.  This is one of the reasons that tuning a piano to an organ
as is sometimes done for churches, among other, can be a problem for
the conscientious tuner.

Hal Davis

 [ The harmonic modes of wind instruments couple to the fundamental
 [ mode such that all possible overtones are related by integers;
 [ For example, the stopped flute pipe supports all odd harmonics,
 [ so a fundamental tone of 100 Hz will have overtones at 300, 500,
 [ 700 ... Hz.  The overtones of the stringed instruments increase
 [ slowly beyond the integral harmonic series, so one might observe
 [ 100 Hz (fundamental), 201 Hz, 302 Hz, etc.  Percussion instruments
 [ like chime tubes and carillon bells support multiple resonance
 [ modes independently; when the chime is struck a whole "chord"
 [ of tones resound.  -- Robbie