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To get a grasp of what lies ahead you'll need to realize you were lied to in freshman physics. Yes Bucky, it happened. Remember all that stuff about organ pipes and lengths and frequencies? And how it all came out nice and even and tidy? Well, that's not quite right. They lied to you and did so with a straight face. It turns out that when a tube resonates at higher octaves the waves lengths aren't exactly even numbered divisions of the fundamental wavelength. They're a tad longer than they told you in physics class and it's this tad that we're dealing with in tuning.
To get started let's agree on some terms: Sharp (in a musical sense) means the wavelength is a little shorter than it should be--thus the pitch is higher. So sharp means shorter, higher. Flat is just the opposite. Sharp, short, higher. Flat, long, lower.
In a tube the higher octaves are flatter than they should be for proper Western even-tempered, chromatic tuning--they're a tad flat, longer, lower than they should be. And the higher octaves are the shorter wavelengths. I shouldn't really say higher octaves as we're only dealing with maybe 1.2 octaves above the fundamental. In a Shakuhachi we have the first octave (the fundamental), the second octave and the beginning part of the third octave.
So what's the game? To make the short wavelengths shorter and the long wavelengths longer. The second octave is flat which means it's wavelengths are long so we need to shorten them. The first octave is sharp which means the wavelengths are short, we need to lengthen them.
Say it again: Make the short wavelengths shorter and/or the long wavelengths longer. Muttering this to yourself as you work makes the time go faster. Keeps you clearheaded and focussed as you know what you're supposed to be accomplishing. It doesn't really matter whether you lengthen or shorten, what we're talking about is the relationship between long and short--between low and high notes. And in practice you'll end up doing some of both.
Now that you understand the game let's cover the rules. How do you change the length of a wavelength in a tube? It's called perturbation and how it works is that if you change (squeeze or expand) the diameter (thus the cross-sectional area) in some portion of the tube the wavelength(s) will get a little longer or shorter. Got It? That's what all the messing with Shakuhachi bores is mostly about. Changing for timbre is another topic for another time, we're just changing for pitch and tuning here. So you're going to add or subtract material from the tube to shorten or stretch the wavelengths to get the notes in chromatic order. Notice that I didn't say add or subtract material from the inner wall of the tube. That's the way it's usually done. But you could insert a stick into the tube and if its shape was correct you'd get the same tuning effect. All we're really after is any method to increase or decrease the cross-sectional area of the tube at specific places along it length. That's the first part of perturbational theory, the second part is a little tricker.
Which part of the bore (hence wave) you squeeze or expand determines what will happen. Where along the wave you mess with it determines the effect you'll produce. Every wave has two types of nodes, let's call them pressure nodes and flow nodes. At a pressure node the air doesn't move much, the pressure changes. At flow nodes the pressure doesn't change much, the air moves one way and then the other--that's what sound is. For the fundamental wavelength (the base note of your flute, figure 1) the flow nodes are at the ends and a pressure node exists at the center of the flute.
An example-- Tom Deaver in Japan has a deep intuitive understanding of perturbation theory as exemplified by his remarks:
One of those exceptional flutes showed up at my shop with a big blob in the bore near the 5th finger hole. The excess unhardened urushi in the hole wasn't noticed and removed and the piece just happened to be placed for drying in a position that let the urushi run out of the hole into the bore. I was asked to remove this blob and refinish the damage from doing so. I refused at once. It was probably this blob that made the flute exceptional or at least the blob was part of it.
Time for the graphics.
Note! From here on we're only talking about CONSTRICTING the tube, adding material to it to affect wavelength shifts, thus the tuning. When removing material from the tube just reverse everything said from here on. How are we going to constrict the tube, squeeze it? By adding/inserting stuff. Putty, paper, 'gi' paste, lima beans--whatever. What the additions do is take up space that's otherwise used by the air column. By controlling the geometry of the space in the tube we control the pitch of the notes.
Figure 1 illustrates the base note of your flute. Flow nodes at the open ends and pressure node in the center. Now for the tricky part. If you add material anywhere in the flow node region (below the line) the wave will lengthen. Add in the pressure region (above the line) and the note is sharpened. So it matters both how much stuff you put in the bore of your Shakuhachi AND where you put it. Put stuff right where the wave crosses the line (getting a bit of flow and pressure) and the addition cancels out--no effect. This crossover point is the Flat Spot and holds particular interest as a candidate for hole placement.
OK, you now know the game and the rules. Time to see the playing field.
The follow is a schematic of the wavelengths (with nodes) which will exist in your flute.
The left side is the top end and right the foot. The measurements are for a standard 1.8 Shaku.
To the right is a graph of the pitch of the notes indicating how far from being in tune each is.
This is set up for a uniform tube so the phenomenon of the sharper low notes and flatter high notes (green line) is made clear.
You've got 45 flow nodes and 30 pressure nodes for 15 notes. Leaving out the flow nodes on the left (at the mouth of your flute) and it's thirty and thirty. Kind of like a Go board on peyote--and it's your move. Your assignment (should you choose to accept) is by way of adding stuff (and thereby constricting the bore) to make the tone line straight and vertical. Do this and you've won the game.
But first, let me show you what you're up against.
The next two figures show the tonal effect of adding 25mm of stuff at different locations along the bore. Figures 3-5 all have the slant of the Tone Line in figure 2 as a base. It's something like ground zero of the basic tube. I've left it in as a reference so you can see the changes the additions make. Compare the Tone Lines of figures 2-5.
Had it not occurred to you, it should be evident by now that addition of stuff usually affects ALL notes--and not in an immediately obvious way. By carefully examining which part of the wave (above or below the line) the colored strip crosses you can predict what pitch difference it will make to each note.
Now let's view a tuning game in progress.
Hmmm, not going well for the Home Team--maybe by the playoffs.
Remember: The goal is to make the Tone Line straight and vertical--ducks in a row.
There are a couple tricks which utilize the length of the additions made to the bore. Look at the bottom of the yellow bar in figure 5. It spans the crest of a pressure node to the crest of a flow node. Because it's that particular length it has no effect on the pitch of the G6 note--the two effects cancel out. So it's possible to make additions to the bore which are transparent to a particular note. For example, you want to make an addition but want to leave D5 untouched? Make your addition (in the case of a 1.8 Shak) 545/4 mm long and D5 will remain untouched. And that's regardless of thickness of the addition or its location anywhere along the bore. To engineer such 'transparencies' just calculate the distance from the crest of a pressure node to the crest of a flow node for the note you want left undisturbed.
As a general rule-of-thumb, long additions affect the low notes more than the high and short additions affect the high notes more than the lower.
So you have two conceptual tools at your disposal: The length of the material you place in the bore and the location of where you put it. The thickness just amplifies the effect.
For a literal demonstration of perturbation effects visit the Throated Flutes page.
For Nodal Tuning of Jinashi flutes.
Notice that we haven't talked about the flute holes. From a perturbational/spatial point-of-view just consider them as negative space that has to be compensated for. In the end we're just talking about space, hence volume, and how it affects note pitch. One good measurement, which is seldom if ever done, is to determine the volume of the bore. Tape the holes and the bottom and fill with water and then measure in cubic centimeters. If water scares you use sand or some other fine medium. Knowing the volume of flute bores will tell you a lot about how they behave.
The way it works is when you're ready to tune, carefully play each note and make a Tone Line graph--which notes are high, which low and by about how much. The holes, the basic tube shift, the effect of the shape of the bore, etc. will all be reflected in this graph. Then by consulting the Node Graphic (figure 2) you can calculate and make judicious additions to the bore--thereby making the Tone Line straight and vertical. Remember, when removing stuff from the bore of your shak just reverse everything said above.
That's it. You're now equipped with the basic tuning theory and a detailed map of the terrain. If you want to get into shak tuning pull up your socks and pack a lunch, it'll be a long day. Good luck!
There's a great Sound Color Analyzer and Tuner for Shakuhachi.
It's a free download for Mac(PPC only) or Windows 95, 98, 2000, NT.
While we're at it let's talk about holes and their location. For those who have read Benade and/or Hopkin talking about Benade, the following is for you. The Benade/Hopkin algorithm strains normal patience and understanding. About the third time through trying to remember what the 'tube cut-off location' is your brain does a cut-off. Anyway, the following graphs will erase any fears that they know something you don't. Figure 6 is calculated for a standard straight walled tube of 1.8 shak length and 4.5mm wall thickness, tuned to D4 (293.66hz). The only two variables (in figure 6) are tube internal diameter and hole size (10&11mm). By-and-large there is small difference between 10 and 11mm holes until you get to larger diameters. Notice the 10mm holes peeking out from behind the 11mm holes in the 25mm row. Given the conditions listed this is where Benade/Hopkin say the holes should go. Now you can sleep at night.
In figure 6 we varied tube diameter, in figure 7 we'll vary wall thickness keeping a constant 20mm ID.
Figures 6&7 give you a comprehensive picture of how changes in tube diameter and wall thickness affect the locations of holes.
For some precise hole measurements and instructions for making some great practice shaks visit The Synthesis.
For more read:
Air Columns and Toneholes by Bart Hopkin, (pp.10-12 for tuning, pp. 14-24 for hole location) Get it from Monty Levenson.
Fundamentals of Musical Acoustics by Arthur H. Benade (pp.473-476 for tuning)