Tim,
I'm an amateur also, but isn't the pitch determined by the density and
breaking strength of the string as well as its length? And, of course the
response of the instrument is determined by a number of factors, including
volume of the bowl, size of the rose, and the makeup and bracing of the
soundboard.
Tim
Quite correct, but there is a bit of an anomaly involved which which makes
the length of the instrument the over-riding factor in the highest pitch for
the top string. The lute, like the guitar and unlike the harp or psaltery,
has the same length for all strings. The achievable pitch for the highest
string will define the overall tuning of the instrument. Please pardon me if
this reads a bit like a primer, I'm trying to sort out the competing
factors - and those with more knowledge please allow that I'm trying to show
the principles, not all the details.
The force required to break a string is a function of the tensile strength
of the material and the diameter of the string. The tensile strength is
measured in force per square unit, i.e. lbs. per square inch or kg. per
square mm. or whatever (no arguments on force vs. weight, etc., please). The
thicker the string the more force required to break it.
The pitch of a string is a function of the length, the tension, and the
mass. Note that it is the mass of the string, not the density of the
material. The mass is a cubic measure - the combination of the square
measure that is guage (or diameter) and the length. There is an obvious
interplay here as the vibrating length affects the pitch, and also affects
the mass.
The anomaly that started me on my book on the topic is something understood
by the old luthiers, but it came as a surprise to me when I first made my
own instruments after over fifty years of playing guitar. (The book is on
hold for the moment, I've been unable to complete my testing of the various
string materials due to a medical matter - I carelessly left my right leg at
the hospital some time ago and the testing involves lifting some heavy
weights). I tuned up my first (and so far only) lute made from a kit and
found I couldn't tune the chanterelle to g'. I took smaller and smaller
guage strings, nylon and Nylgut, and they still broke at the same pitch.
That got me started looking into it.
The apparant anomaly is that, given the material and the length, any string
will break at the same pitch no matter the guage. When you think of it, it
is no longer an anomaly. The greater guage requires more tension, given a
fixed vibrating length, to come up to pitch as the string has more mass. It
requires more force to break the thicker string, but also more tension to
bring it up to pitch. Conversely it takes less tension to achieve the pitch
on a lighter guage, but also less force to break it. It happens that the
trade off is directly proportional and there is a breaking pitch for each
material at a given vibrating length.
One thing that is an anomaly, however, is that the most commonly used string
materials have factors that trade off and give them similar breaking
pitches. Steel, gut, nylon and Nylgut have similar breaking pitches (about a
tone and a half). Bronze and brass are quite different. I first worked this
out mathematically from standard figures on the density and tensile strength
of the materials. As I was unsatisfied with the standard specifications I
decided to do actual tests to make averages for the various manufacturers -
and that is what is on hold for the moment. (Breaking steel strings of some
guages requires 150lbs of weight on my home made jig). But so far my results
have confirmed the thesis, and also shown the standard figures to be a bit
off.
BTW, I was able to tune my lute to G minor - I found a nylon fishing line at
a sporting goods store that was just a bit more tensile strength than
musical nylon. Nylgut would hold g' for an hour or so and f#' for a few
days, musical nylon held the g' for a few days - the fishing line has held
it for five years. I haven't tested gut yet, but from the standard figures
it should break about a tone and a half below the nylon and Nylgut. I also
converted a Bolivian charango to a Scot's mandora (g'' top string). No way,
I tune it to d'', although it will take f''. That gives me a d'', g', d',g,d
which can work with other instruments.
Again I beg the pardon of the list, most of you know far more than I. I have
algebraically recast some of the string calculation formulae, and designed
some graphic forms for the data. I am not comfortable with using the several
computer programs for string selection as I don't know the underlying
assumptions - I do mine on an ancient TI - 35 scientific calculator (and can
do them quicker that booting the computer and using a program <g>). I'd be
pleased to offer the form of calculation to any who want it. I've set them
up so that one can make a single calculation of most of the variable - then
use the two relevant ones (whichever they may be) to get the result, using
the first calculation as a constant.
Best, Jon
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