Jonathan Herz will find articles on comb tuning in publications of the
MBSGB.  Both Keith Harding and John Powell have written articles about
tuning and stretching and these appear in our magazine, "The Music Box".
The MBSGB publication, "The Nicole Factor", has a brief section devoted
to this subject which includes three graphs of octave stretch taken
from extant instruments.

There appears to be no 'rule' for stretching.  The combs from different
makers exhibit different curves of pitch variation along the comb.

There are many caveats when researching this topic.  Remember that the
original 'perfect' scale will not have been our modern equal tempered
scale and most likely will have been created to sound right for just
the tunes played, as the musical box is never played in concert with
other instruments.  Some curves show a steady rise in pitch relative to
the pure mathematical scale whereas others show a rise and then a fall
as the notes go up the scale.

One aspect of scale stretching that I have never seen analysed is
the degree of stretch compared with the true pitch of the harmonics
[overtones] of the lower notes.  The stretch in a piano, for example,
is automatically produced when tuning by ear, since the upper note
in an octave is tuned in unison with the harmonic of the fundamental,
and the harmonic will be slightly sharper than twice the frequency of
the fundamental due to the mechanics of the string's vibration.

This is called inharmonicity.  Tuning in this way will lead to a
'beatless' tuned piano.  A piano tuned to an electronic standard
with a purely mathematical scale will sound dull as the octaves will
be flat to the harmonics.

It is said that some tuners will stretch beyond 'beatless', but how
far is even correct or desirable?  Has anyone analysed a piano tuned
by a concert standard tuner and compared it with its beatless state?
I've heard it said that barrel pianos should be stretched beyond
beatless, but is that simply because they go out of tune quickly so
naturally return to a better sounding state?

So I return to the music box comb.  Is the stretching of the scale
simply the result of the tuner working to the natural harmonics of
the other teeth (remembering that a tooth will have a different set
of harmonics to a stretched string) or simply using his ear to tune
to produce the best sound, or is he working to a 'house standard' for
stretch?

Nicholas Simons, GB