partial answers

[Frank Emerson]

I like that:  It does because it can.

A jump rope demonstration doesn't answer the question: Why?  But, it does
illustrate how.  The normal rotation of a jump rope is analogous to the
fundamental frequency.  Doubling the speed of rotation, one can make the
rope divide into two equal length segments, as in the second partial, and
likewise with the third.  What you cannot do is force the divisions into
unequal segments, or other than whole-number divisions.  Well, I suppose if
you distributed the mass unevenly along it's length you could, but this is
not done in musical instruments, or at least, it is not intentionally done
by design.  

The harmonic series is so common to "musical" sound that it defines what we
perceive as harmonic or dissonant.  The relative strength of these partials
produces the tonal difference of various musical instruments.  

For purposes of this discussion, you could say that a string of a musical
instrument is one dimensional, with freedom of movement in the other two
dimensions of physical space.  This is true of most musical instruments,
with respect to the sound generating body (disregarding soundboards and
other components whose function is to amplify the sound), including
strings, columns of air, vocal chords, etc.  

There are exceptions, such as bells, marimba bars, xylophone bars, etc. 
While marimba bars and the like are not of any great interest to piano
technicians, an understanding of how they work helps us to understand what
is physically going on with a piano string, ie. understanding the exception
to the rule helps us understand the rule, itself.  These sounding bodies,
which are exceptions to the rule, are very much three-dimensional bodies,
and do not have the same freedom of movement in two of the three dimensions
of physical space.  As a result, they do not have the same familiar
harmonic series.   In the fabrication of marimba and xylophone bars, the
shape of the arch on the bottom not only tunes the bar to the correct
fundamental pitch, but it also forces the second and third partials to
return frequencies that we would consider "harmonic."  Without judicious
removal of material at precise locations, the upper partials of these bars
would be dissonant, as we define musical sound.  I built a marimba for my
daughter, and I can tell you that it is quite a trick to simultaneously
tune the fundamental, 2nd and 3rd partials, and get them to all come out
right at the same time.  By the way, the 3rd partial of a marimba bars is
tuned to a different frequency than is the case with a xylophone bar, which
is part of what gives them distinctively different tonal characteristics.

Frank Emerson


> [Original Message]
> From: Mike Spalding <mike.spalding1 at verizon.net>
> To: <allan at allangilreath.com>; Pianotech List <pianotech at ptg.org>
> Date: 6/29/2007 11:01:34 AM
> Subject: Re: partial answers
>
> Allan,
>
> You're lucky to have such an inquisitive apprentice. Maybe the answer is 
> "Because it CAN", rather than "Because it DOES". It's been way too long 
> since I struggled with the underlying math, but I have a feeling this 
> could be explained by the second law of thermodynamics: Increase of 
> entropy. Mechanical systems always move towards a state of decreased 
> energy, which is often explained as an increase of disorder/chaos. The 
> more modes of vibration in the string, the greater the "disorder". It 
> certainly makes sense on an intuitive level that the vibrational energy 
> of the string will dissipate most quickly when the maximum number of 
> overtones are sounding. When we voice, we reduce or eliminate high 
> overtones, and increase sustain. Maybe. Anybody out there want to offer 
> a more rigorous analysis or rebuttal?
>
> Mike
>
> Allan Gilreath, RPT wrote:
> >
> > Good morning folks,
> >
> > I had a question from my apprentice that someone on the list may be 
> > able to help me with. We all know that vibrating strings divide up 
> > into segments with lengths approximately equal to fractional portions, 
> > i.e. ½, 1/3, ¼, 1/5, 1/6, etc. (we’re not even taking inharmonicity 
> > into account at this level.) His question is, “Why does the string 
> > divide into all of the different available fractional segments and not 
> > just even multiples of two?” I was hoping for a much better answer 
> > than just, “Because it does” but Benade, Helmholtz and Rayleigh, the 
> > best I can tell, all assume this to be a fact and I don’t really find 
> > the “why.”
> >
> > Any thoughts?
> >
> > Allan
> >
> > Allan L. Gilreath, RPT
> >
> > Registered Piano Technician
> >
> > *Allan Gilreath & Associates, Inc.*
> >
> > /The Piano Experts/
> >
> > PO Box 1133 - Calhoun, GA 30703
> >
> > 2612 Hwy 41 S - Calhoun, GA 30701
> >
> > allan at allangilreath.com - www.allangilreath.com
> >
> > phone 706 602-7667 - fax 706 602-0979
> >
>