Hi:
Some of us have decided to move the physics discussion of bores in relation
to the shak elsewhere.
If it isn't too annoying I will just relate here my first posting *there*
so in case there
is anyone on this list that likes the topic and flavor (at least as
exemplified by this
one posting) they can join us or listen in. I also do this with the hopes
that some of the musician
and shak makers might help us out on some things if it look like anything
they might be interested in.
I promise not to post anymore acoustics related (but perhaps culturally
charged) stuff here anymore.
Anyway, first post at ShakuhachiDesign@yahoogroups.com, last post here.
-------------------------------------------------------
A possible place to start a discussion might be the following paper:
A. H. Benade. On woodwind instrument bores, J. Acoust. Soc. Am.
31(2):137-146 (1959).
Unfortunately it is BPDF (before pdf) so the only way to get a copy is to
go to a university library,
or I could scan it if anyone can tell me how to post it here.
I think this might be a good place to start both because the results in the
paper are particularly simple and because it methodology is a simple
example of the one I originally proposed: define the required acoustic
features then infer the bores that fit those features.
His results are for close-ended woodwinds, but I can go home and see if I
can re-do all the proofs for open ended bores. That
is, unless someone can point me to a place where this has already been done.
There are two acoustic features that Benade uses and will be immediately
recognizable to the musician with
no mathematical training whatsoever: (1) The first and second resonance of
a bore (lower and second register) should have
frequencies in the ration of 1:2 (play at exactly one octave apart) and
(2) that (1) should hold if you "chop" the bore down to
any length (that is, for any fingering where all the holes are closed up to
a certain point). So the criteria are that the flute play in tune
in the first and second registers for all the basic notes.
Criteria (1) requires that the bore have the shape where the
cross-sectional area depend exponentially
on the length S(x) ~ x^a (a Bessel horn), where a is called the flare
constant.
If we add (2) this then a = 0 and a = 2 are the *only* shapes that
*exactly* satisfy these two requirements.
These are cylinders and cones. The shak doesn't fit either of these two
categories, but
(generally speaking) is a cylinder connected to a reverse cone, connected
to something that looks along like a Bessel flair
with a>2.
It is possible that there are some major compromises going on here in terms
of playing, but
there are also several ways of relaxing the criteria that could also
explain the shape.
For example, in the shak we don't need the registers to be in tune for
arbitrary lengths
of bore, but only for specified lengths corresponding to the notes of the
scale: that is for
5 lengths of corresponding to the pentatonic scale. Because of this the
shape (based just on these most simple criteria) could be morphed. So
within this "wiggle room"
what kind of shapes would be allowed, and are there any that come close to
the traditional shak shape.
This is something that could be explored mathematically is anyone is clever
enough, but could also
be explore via the computer. Furthermore, more experience shak makers my
already have a "feel" for this
wiggle room, and perhaps they could comment on this, if they are willing.
By the way, hopefully nobody remembers but I said something stupid a while
back. It *IS* the peaks
of the input impedance (not the minima) that correspond to the resonance
because that's
where the wave bound around and build up rather than pass right through.
Sorry about that.
Marko
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