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Updated
This from Tom Deaver:
I've noticed over the years that, in some cases, when the sound of ro (kan) changes when opening the 4th finger hole, the change can be eliminated with adjustment between the second and third finger holes. It seems that most people prefer flutes such that opening or closing the 4th finger hole causes no change in the sound of ro. I have no idea what's going on but know that the area between the second and third finger holes is about a third of the length of the flute. It's also easy to mess up Chi and chi working in that area. It's probably near the end of the half-length of the Chi sound wave.
Tom has developed a self-referential test for well-balanced shakuhachi. You can now test your own and any shakuhachi you might play. Simple, quick, definitive--the best of analytical feedback.
To achieve this balanced condition, the fourth hole needs to be placed and sized to also serve as a register hole. To make a register hole work on a traditional shak it has to lie at the flute's acoustical center. This may not be at the linear center as the bore of a traditional shak varies along its length, but with PVC we can step around this problem and investigate the whole idea of a register hole and general sonic balance.
So what does the Deaver test achieve? What it's really about is determining the proper hole size in relation to bore. In Tom's description, he can work the other way round from what we will. As do all traditional makers, Tom has the option of adjusting the bore to match the holes he's already located and drilled. In our example, we'll select hole size based on a preexisting wall thickness and bore diameter. Since the wall thickness and bore diameter of PVC is uniform we can place the fourth hole exactly at the acoustical center of flute length and be done with it.
So the problem we need to solve is what sized holes to use in order to have the fourth hole at the acoustical center of the flute length? Smaller holes migrate toward the head of the flute and bigger holes toward the foot. "Hole size", in this context, is not just the size drill you use. What we're talking about is the effective hole size--how easily the hole vents. And that is also dependent on wall thickness. In this example we'll use 3/4" Schedule 40 PVC, so the question is what hole size will make the fourth hole in Sch40 PVC fall at the acoustical center of the flute? And the answer is 9.5mm or 3/8"
If you'll notice on a previous page, the dimensions for the Sch40 9.5mm C flute places the fourth hole right at the acoustical center. So, for any length Sch40, 9.5mm sized holes will create the balanced condition we're trying to achieve.
Since lower pitched flutes are often preferred while playing honkyoku, let's make this one a 2.1, pitched to B
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Note
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Hole
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Distance (mm) & Percentage
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Hi
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5
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275 -- 0.428
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Ri
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4
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314 -- 0.489
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Chi
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3
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388 -- 0.604
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Re
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2
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443 -- 0.690
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Tsu
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1
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504 -- 0.785
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Ro
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642 -- 1.000
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The fourth hole can serve as a register hole for ro. By moving the fifth hole up slightly it could also serve a register hole for tsu.
The wavelengths signify the effective length of the flute while playing second octave notes.
Deaver continues:
There is another little sort of test. Do you know the fingering technique called "kara-kara" in Tozan notation? Don't see it so much but believe it is repeatedly striking the 1st finger hole while playing ha meri in kan. (That's a ha without the capital letter so its kan, not Otsu. Did I get my personal notation switched around? Hope not...Otsu has capitals and kan does not.) This repeated striking might also be noted as ru again and again quickly.
What happens is that the pitch goes to a higher frequency while the first hole is momentarily closed. Now, play chi and chi meri and apply the same technique, ru, in rapid succession. Aren't these fingerings called cross-or forked-fingerings, closing a hole below other open holes in the lattice? Benade touches on this in his textbook, saying that the pitch of the forked fingering may go up or down depending upon the situation. Well, on a preferred shakuhachi, the pitch always goes up. You can imagine that not all the flutes in the house now work this way. Some flutes send the pitch up, others send it down. I'd be happier if the shift was always up.
So, what's going on when one plays chi and then ru and the pitch goes down rather than up? Hole placement, as one might suspect, seems to have an effect.
If the first hole is a little lower than usual it can be close enough to the nodal point (see the blue ring above) to create the effect described.
Along these lines there's another trick.
Suppose you want a single-piece flute with the utaguchi cut at the node AND want it tuned to a specific note. If you are very lucky and find a correctly dimensioned culm it'll all work out. Otherwise, you can raise the pitch with a tuning hole(s). Native American flutes have used this technique for a long time. Drill a very small hole (usually on the back side) about halfway between the first hole and the foot. Place it in the bamboo node to hide it if you want or proudly put it on the front of your flute between nodes. Drill it just like you would a finger hole--straight into the bore. Start the hole small (1/16") and slowly enlarge until the flute raises to the pitch you want. Then place finger holes and finish in a regular manner. What a tuning hole does is allow the flute to vent a little more than it would ordinarily--thus raising the pitch. In different flute cultures these holes are often called Devil holes and/or Ghost holes.
See The Synthesis for a final flute design.