A good deal of very interesting information has been offered to me in
response to my question about comb tuning stretch [140129 MMDigest]
I offer my thanks to those who wrote to MMD and those who responded
to me directly.  There's no need for me to recap the replies
published here by Nicholas Simons and Paul Bellamy, but I can add
some information from other sources.

Robbie added a quotation from the Wikipedia article on stretch
tuning, a resource I should have checked myself.  The article
contradicts what I said about music box stretches being far more
than the stretches I have observed in antique combs.  It gives a
reference to http://www.postpiano.com/support/updates/tech/Tuning.htm
which includes a very interesting graph showing a typical stretch for
a "small piano".

The graph shows about three octaves of unstretched tuning in the middle
of the scale and about 30 cents of stretch in the treble and 40 cents
in the bass, for a total stretch of 70 cents.  This is, in fact, the
same order of magnitude as I have observed on the antique combs I have
taken the tuning from.

The size of the piano is relevant because the shorter stringing lengths
result in a greater divergence from the mathematically "ideal" string
having zero stiffness.

My father, Eric Herz, made harpsichords.  He started out as a piano
tuner and got a job with Baldwin tuning new "Acrosonic" console
pianos fresh off the factory floor.  Tuning new pianos is a somewhat
Sisyphean task -- they have to be tuned over and over as they
gradually come into equilibrium.  The Baldwin Acrosonic had such short
strings that he found himself constantly chasing inharmonicities up
and down the octave on each piano, in turn, over and over again.  He
finally left Baldwin in disgust and spent the next 40 years making
harpsichords.

The Wikipedia article also refers to stretch in vibrating teeth, not
for musical boxes but for analog electronic pianos.  Some early ones
used magnetic pickups to get an electronic signal from tuned vibrating
teeth.  In this case, the stretch results from the higher harmonics
traveling shorter distances into the root of the teeth, and thus
vibrating at higher than theoretical frequencies.  (A standing wave
in a string or tooth can be modeled either as a stationary wave where
the height of the peak changes over time, or as a moving wave traveling
up and down the length of the vibrating element where the location of
the peak changes over time).

Nicholas Simons kindly sent me two clippings from the MBSGB journals:
(1) an article from Keith Harding about a proposed tuning temperament
for the middle octaves of a Nicole comb, and (2) an article from J. M.
Powell, showing the stretch of six antique combs.  I should mention
that membership in MBSGB entitles one to a DVD of several years of
past issues as well as a subscription to the current issues of their
journal.  I have applied for membership; although the local information
is not useful to me, the technical articles promise to be very
interesting.

Mr. Harding's article suggested that the single comb he analyzed was
tuned to a Pythagorean scale, where all the notes were determined
through the use only of perfect octaves and perfect fifths.  Several
sources have also proposed a mean tone temperament for some of the
boxes they have heard.

This is a separate issue from the total stretch, however.  It is
difficult to draw clear conclusions from the Powell data.  Although all
combs had increasing stretch in the treble, the bass stretch varied
from as much stretch as in the treble, all the way to having compressed
octaves, rather than stretched.  As for magnitude of stretch, he graphed
all six of his combs without going more than 12 cents above an equal
temperament nor 8 cents below.  I have found combs which stretch by
more than 100 cents.  Paul Bellamy reports a comb stretched by 1.5 to
2 tones!

The Powell article was written in 1980, before the personal computer
era.  The graphs are hand drawn and the curves were drawn by eye.
I have been fitting tuning curves to data with least squares regression
methods.  I have spreadsheets set up for fitting tuning data to linear,
quadratic, and logarithmic curves, which improves accuracy and avoids
the influence of wishful thinking on the curve fitting.  This makes it
easy for me to import comb data and find a best-fit tuning curve.

I would love to have more data to look at and would happily share it
with interested parties. If you have as-found tunings of antique combs
which are accurate to a few cents and would like to have a tuning curve
fitted to the data, I would be happy to do so for both our educations.
My caveats are

1) that the data needs to be in a form easily imported to a
spreadsheet.  A text file with the note in a, a#, b, c, etc., in one
column and cents +/- in the second column is ideal.

2) that you have reasonable certainty that the notes are spelled as
originally intended.  If two adjacent notes are f#, 20 cents and g, 73
cents, then you need to know from the musical program on the cylinder
whether the notes were intended to be a half tone apart or a full tone
apart.  If you do not know this then there can be no useful analysis of
variation from the true note.  In the above example, we don't know
whether the second note is a very sharp g or a very flat g#.

I am, of course, most interested in combs that have retained close to
their original tuning.  In general, combs whose multiple teeth tuned to
the same pitch are in good agreement with each other are likely to be
close to their original tuning.  But I would be happy to plot the
curves for any comb.

Jonathan Herz - Herz Music Box Co.
Montpelier, Vermont
http://www.herzmusicbox.com/