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The reasons for a tapered bore are at least threefold:
1) The natural growth geometry of the root end of bamboo tends toward a decreasing internal diameter the closer to the root one gets. Thus while forming the bore there is a natural (and therefore traditional) tendency to follow the growth pattern.
2) The timbre (tone color) of the flute is changed (improved?) by choking it.
3) Octave tuning (the problem of notes being flattened the higher they are) may be helped by employing a taper.
European flutes and recorders went through this same evolution toward a tapered bore, moving from Renaissance to Baroque. Central to the question of bore design is the question of what a Shakuhachi should sound like? To some a taper sounds mellow, to others it's stuffy. To some a straight tube sounds bright to others it's shrill. Part of the problem in shak bore design is that timbre and tuning are inner-mingled in the same geometry. It's hard if not impossible to discern whether some bump or undulation affects timbre or tuning the most. Fortunately, octave tuning and changes in timbre can both be achieved in ways other than employing a taper and in ways which tend to separate the problems. Timbre is directly affected by Aspect Ratio (tube length divided by diameter) and tuning can be done by way of perturbations.
But first, let's look at some bore profiles. For the uninitiated, be aware that these graphs amplify and thus distort the look of the taper--that's their purpose. The vertical scale is 1/10 that of the horizontal scale. Actual bores are much slimmer and svelte looking. All the following graphs represent a D4/1.8 shak.
Figure 1
It's hard to figure out much from these examples. What part of a given curve is dedicated to tuning and what part to timbre? 20mm is a good general throat size for a 1.8 shak, so the range is maybe 19 to 21mm, with a choke point diameter of 13 to 16mm. In a way, these two measurements are inversely related to each other. A big throat can be compensated by a smaller choke and conversely with a smaller throat the choke can be opened up some. Think of it this way: if the throat continued to get smaller and the choke larger, at some point you'd have a straight-walled tube.
Another way to think of the throat/choke relationship is to make the cross-sectional area of the tube at the choke point 1/2 the area of the throat. Square the throat diameter, divide by 2, then take the square root. For a 20mm throat this amount to 14.14mm
Part of the difficulty with tapers is that a taper can be devised which makes any single note sound great but begins to fail on the other notes. Compromise, compromise, compromise. And after all that compromising is done you have to compromise still further to get the tuning right. Buried under all that compromising and tuning is some idea of what a taper should be but its nearly impossible to dig it out. With tapers people tend to end up using what's worked before so bore evolution is a slow process. Kind of like the Red Barn phenomena in the Midwest. When farmers are asked why they paint their barns red they explain that red paint is the cheapest. At the hardware store if you ask why red paint is the cheapest they'll tell you it's because they sell so much.
Now let's approach the bore from a more theoretical standpoint. It's possible to create a waveform through the addition of the Sine waves of notes (with any number of harmonics)--kind of the reverse of doing a Fourier Transform. We'll build our waveforms by specifying what we want the flute to sound like. And then it's possible to generalize the waveform into a smoother result.
Figure 2
We can easily stretch/shrink the waveform to fit any throat and choke dimensions--let's call these the vertical dimensions. But the interesting part is that an 'Offset Point' naturally emerges from the algorithm and it's value is based on the number of notes and harmonics we specify. Notice in figure 1 there's an Offset Point in each bore profile. Never noticed it? Well it's there.
Let's learn a little more about the location of this point. First, the existence of an Offset Point seems to arise from specifying the number of harmonics we want--as can be seen in the next figure. From the third harmonic on, the waveforms are similar beyond thirteen centimeters. By the sixth harmonic the waveform has leveled off and is pretty flat across the first 13 centimeters of the flute's throat. Beyond the 12th harmonic the first 13 centimeters just flattens out until it becomes virtually a line.
Figure 3
Where this line breaks into a taper is the location of the Offset Point (horizontal dimension) and is governed by the number of notes you want your flute to play well. And by notes, I mean the pentamic notes the Shakuhachi is designed to play. So designing for 11 notes, for example, we're designing to extend to the first note of the third octave (F6).
Figure4
If the algorithm works (and that's still an if) we can design a generalized bore optimally configured for different note ranges. Tuning would be completed after shaping a particular profile for a particular range. The closer to the mouth the Offset Point is placed the greater range the flute should have. Notice also that there's a gradual shrinking of the Choke Point that goes along with this. Part of the process of generalizing the waveform is computing a very high number of harmonics--thousands, and so it can safely be said that the generalized form contains all harmonics--at least all you could ever hear.
Figure 5
So much for theory, now let's investigate doing it another way--messing with a straight tube. As mentioned before, timbre and range are both greatly influenced by Aspect Ratio. The following graphic will give you an idea of Aspect Ratios for shaks in the D4-C3 range. The bottom of the 'Third Octave Zone' (an Aspect Ratio of about 30) is the dividing line between second and third octave (the range) flutes. As the Aspect Ratio rises (numerically) the timbre gets brighter--lower and the timbre is darker. A D4 shak with an internal diameter of 18mm is right on that line. So generally, if you want a straight-walled shak to favor the low notes design it with an Aspect Ratio of 30 or less--the high notes, 30 to 32. Simply said, whether tapered or straight if you're having trouble getting the high notes shrink the diameter and conversely, expand to boost the low notes.
Figure 6
For those with an strong interest in subject of Aspect Ratio, this is an experiment really worth doing. Get a length of 3/4" CPVC (surprise! ID 18mm) and hook it up to a mouthpiece from a recorder or whistle. I just taped the mouthpiece from an old 16" recorder onto the CPVC with electrical tape. You'll need a fipple mouthpiece like that of a recorder because blowing pressure is ultra-sensitive in this experiment. Don't have a mouthpiece lying around? Make one or buy a cheap plastic recorder, band-saw off the mouthpiece and proceed. Cut the CPVC to 5 feet (or so)--we're talkin' an Aspect Ratio of somewhere around 85! Now play and try to get the tube's fundamental. Usually, if you blow softer and create a more quiestent airstream you'll find another lower note clear to the limit of hearing. Anyway, start at the (a) low note (about 300Hz) and you should be able to play every hundred Hz up to 1200 or 1300. About there my dog starts going nuts and interrupts the playing. And this is all just with the breath and a very high Aspect Ratio. You'll be jumping octaves, thirds?, fifths? and I'm not sure what else.
Playing this instrument is a rather peculiar and meditative experience. You can gently 'push' notes and listen as they gradually evolve and break into the next (other?) note(s). 'Other worldly' doesn't quite describe the sound, but it's probably unlike anything you've ever heard. Playing is pleasant, easy, and informative. Call this instrument a Hiasra (HIgh ASpect RAtio) and enjoy and learn about Aspect Ratios.
Another nice thing is that the Hiasra takes very little air, basically you can play as long as you can hold your breath. Mouth, tongue and throat effects easily translate into music. Didge techniques apply. You can even indulge in 'Tuvan throat singing' by humming a drone sound. Because the Hiasra is so sensitive to airstream characteristics water droplets in the windway can foul the works. Just blow them out and proceed when the instrument goes mute. Slap on an end cap (pick one up when you buy the pipe) for less interesting 'minor notes'. After you've completed this experiment you're on your own with other pipes and Aspect Ratios.
The Sound Color Analyzer and Tuner for Shakuhachi is a big help during this experiment. It's a free download for Mac (PPC only) or Windows 95, 98, 2000, NT.
There is a lot of traditional lore about which materials are best for wind instruments. In practice, however, any reasonably heavy, hard, smooth and rigid material will sound much like any other and have the same potential for producing a good-sounding instrument. However, extremely hard reflective surfaces are not necessarily the ideal: people sometimes prefer the mellower tone of somewhat damped resonances.
Wind instrument tubing materials that are light and/or yielding will dampen air resonance within the tube to some degree, and lower the resonant frequencies slightly. Walls made of rough or porous materials also have noticeable damping effects. Increased damping leads to poorly defined resonance peaks, especially in the high frequencies, creating a sound that is less than bright. Although not generally realized, any deviation from a circular bore leads to the possibility of wall flexing. In other words, the walls of a very thin-walled, strongly elliptical bore will flex. Whether you want this or not is another question. Some people believe this is the sign of a superior shak. Vibrations in the walls while playing is the result of wall flexing. If you want it, thin the walls and increase the elliptical nature of the bore--the less circular the stronger the effect. On the downside, wall vibrations are inefficient as far as turning the energy of the airstream into sound--which only runs about 1% anyway.
Timbre is greatly influenced by air friction in the bore.
We don't ordinarily think of friction within a shak, the air flow doesn't seem to be traveling very fast. But we're thinking at the wrong scale. What's going on in your shak is happening at Mach 1--the speed of sound. The air volume in the bore of a shak may turn over at a leisurely rate but the molecules making up the sound waves are moving at Chuck Yeager speeds. Think about the surface of something you would put in a wind-tunnel to be tested a Mach 1. The smoother, more polished, mirror-like the surface the less drag it has. Such a surface is 'slippery'. And the same circumstances are true in the bore of a shak.
So what kind of inner surface is best for a shak? Well, it depends. A smooth surface brightens the timbre, rough darkens it. And we're talking more about the micro-surface rather than obvious irregularities. If the surface is rough, air drag increases and thus a greater force is needed to oscillate the air. The instrument isn't as 'quick' and supple as it could be. So in answer to which surface is preferable we need to realize that the texture of the inner bore surface needs to be weighted against the Aspect Ratio.
We can brighten a shak's tone by: 1) raising the AR, and/or 2) polishing the bore.
The tone can be darkened by: 1) lowering the AR and/or 2) increasing friction in the bore.
A thin pipe with a slippery (fast) bore is doubly bright.
A fat pipe with a rough (slow) bore is doubly dark.
So if a shak's AR is a little low we could polish the bore and come out about right. How to polish a bore? A good coat of varnish, lacquer, polyurethane, etc. is a quick, easy way. Part of the reason for the tradition of pouring water down a Didge at corroboree is to "polish" the bore a bit. A wet surface is smoother to air molecules. Slop some water down the bore of an untreated rough wooden shak and it's tone will brighten.
How to roughen a bore? There are the obvious ways--steel wool, etc. Affix a big wad of coarse steel wool to the end of a rod and then use an electric drill. Gluing sand on the bore surface is an extreme possibility and probably should only be used for instructional purposes. Mix up a diluted mixture of wood glue and water, coat the inside, fill with dry sand, leave overnight, then drain the sand out. But there are other more interesting and subtle possibilities.
To get and/or increase that nice mellow, breathy, woody, dark timbre we need to increase air friction in the bore. But this doesn't necessarily mean on the bore wall. To get a feel for the effects of air friction, snake a strip of cloth down the bore using string tied to the ends. It's a simple matter to experiment with different materials inserted in the bore to alter the air friction. Make sure they're fairly thin or their cross-sectional area will alter the AR of your tube. Try adding friction in the top half of the tube, the bottom, the middle. You get the picture.
What surface do we want? The one which coupled with the Aspect Ratio produces the tone we like. These two: AR and surface friction need to be complimentary and balanced. So it's quite possible to have a very rough bore and a fairly bright tone or a glassy-smooth bore and a dark haunting tone. One of the woodiest, darkest tones I ever got was from a polished Plexiglas tube--but it's AR was quite low.
For some precise hole measurements and instructions leading to some great practice shaks visit PVC Shakuhachi. The Aspect Ratio's role in the sound of the shakuhachi becomes clear in these examples.
At a meta-level, the fundamental thing a shak (as well as other woodwinds) does is constrain air movement in subtle, interesting and useful ways. There's a lot of talk about impedence--just think of it as resistance. Everything we've talked about in this page (and the others) can be boiled down to messing around with the ways and means of creating and lessening air resistance. That's what the tube does, what the holes do (both in size and location), what the bore shape does, what the Choke Point does, what the inner wall surface does and so on. Air is tricky stuff--light, elastic, very quick and the most frustrating part is that you can't see the stuff.
It helps to begin thinking of the air as moving at Mach 1 speeds instead of the speed of the airstream. There's an F-16 aircraft in the bore when you play and the instrument is a wind-tunnel. This wind-tunnel can be shaped differently, doors can be opened and shut along its length. We can put in baffles, wedges, carpet--whatever we want. We can paint the walls of our wind-tunnel or leave them raw--whatever. And all of these changes have an effect on the resultant sound. Some more pronounced than others. Why taper a bore? To increase its air resistance. But now you know other ways to accomplish that. Once you move your mind to the concept of air resistance; many, many possibilities suddenly open up. Repeat this Mantra often: Friction and Constriction create Resistance. We're just fiddling with air--there's nothing holy about it. A shak is just an example of what you can do with one breath of air and some resourceful uses of friction and constriction. Lengthening the tube or thickening the wall both lead to an increase in constriction thus air resistance. Begin thinking of the air-load on a butterfly's wing--that subtle.
So we're left with the debate of tapered vs. straight-walled. Which way to go? Since we've separated out tuning the debate can now be seen a little clearer. With a straight-walled tube the timbre is easily selected and is more uniform across notes. With a taper, between the low and high notes in each octave, timber shifts--the Aspect Ratio (thus the timbre) undergoes a bigger shift with a taper than in a straight-walled tube. Remember, when the holes are opened you are in effect cutting off the tube, so with a taper you're leaving the choke point behind. The problem with tapers can be boiled down to trying to compute the Aspect Ratio for each note. It's not an easy proposition. Figuring out the true Aspect Ratio for the fundamental note is hard enough. Do you use the Choke Point? That's not quite right. Average? Maybe, but still not right. If you were to really try and figure it out you'll appreciate that the whole thing is a non-linear problem.
Since a straight-walled tube doesn't have a Choke Point the shift in Aspect Ratio is more linear and gradual. So at the end of the day it's what you want the timbre of your shak to do, how you want it to vary while playing. More constant or more change? Kind of a Three Bears deal. Too much, too little and just right. I hope the foregoing will be of some aid in achieving your 'just right'.
Ultimately, the perfect flute would change internal diameter as you played it. Figure out how to achieve that and you'll have something.
There's a strange paradox to the Shakuhachi. If you sell someone a wonderful sounding shak, what are you selling? Some cleverly shaped emptyness--that's about all. Designed nothingness--that moves at the speed of sound.
For a book that's something like this discourse read The Structure of Delight.
For your visual enjoyment following is a graphic generated by the algorithm--a 'dimensional' view of the waveform created by plotting the values as they're computed and added up.
