James,
> Very pretty. Is there a simple explanation of the derivation, or was
> it a lot of trial and error?
It's long been know that length and bore scale at different rates. Length
doubles every octave or 12 notes, bore doubles every 14-15 notes. So we have
a fraction ( 12/14.4) which, surprisingly, equals 5/6. After that, things
fall into place pretty readily. Using 5/6ths the energy equations come to
unity and it's no great trick to fashion direct length/bore relationships.
> Do you think there might be a similar expression for a tapered bore?
Tapered isn't all that different, the trick is to arrive at a measurement of
the tapered bore volume as if it weren't tapered. There are a couple ways:
the first has to do with tuning. If your D flute is 545mm long the bore
average is about 17.5mm in diameter. It doesn't matter a great deal how the
bore is arranged. See the second graph at:
http://www.navaching.com/shaku/long.html
The second way is to measure bore volume and the first graph at
http://www.navaching.com/shaku/kuda.html will return Aspect Ratio. Just
divide length by AR to get bore.
By and large, the mass of the air in the bore is what makes the shak work
correctly. And that, of course is dependent on bore volume. When the air
mass is about right for the length the flute can reach it's potential.
There's great emphasis on tapered bores, particularly in the shakuhachi. But
the physics of acoustics finds no compelling reason for tapering. Musicians
swear by it, yet scientists find little measurable difference. The mystery
remains, go figure.
In traditional construction of shaks, length is a no-brainer--just get out
the saw. But adjusting the bore volume to the length is wrapped up in a lot
of arcane schemes. What's often missed by way of all the adding and
subtracting is that all changes directly effect overall bore volume. The
local view of where to put the putty or grind away some bamboo often fails
to recognize the global fact that all subtractions or additions effect the
air mass. Changing the air mass a few percent makes all the difference.
Nelson
This archive was generated by hypermail 2b29 : Tue Jan 06 2004 - 14:09:33 PST