Shaku
Design

Kudado: The Way of Pipe

Updated 4/24/08

With this page we're starting to close in on a suitable flute design made from PVC pipe. It utilizes gray 3/4" PVC (schedules 40 & 80), the Integral Mouthpiece and a special Wedge in the flute's throat. This 'wedge' is the key element so we'll get right to it. The biggest, and perhaps only, problem with straight-pipe flutes is what's called octave flattening. What this means is with a straight pipe the notes of second octave and beyond play progressively flatter. It's just the nature of the physics of straight pipe. Switching to a tapered or conical tube is the standard solution to octave flattening. Flutes and recorders went through this change in Europe from the Renaissance to Baroque period. The natural growth geometry of root end bamboo culms leads naturally to tapered bores in shakuhachi. But a properly tapered bore is a tricky and costly thing to create. The Wedge can be fashioned from wood or other suitable materials or cast in place using polyurethane, epoxy, etc. Here's the basic geometry:

That's it. Stick a properly dimensioned Wedge into the blowing end of your flute and the octave flattening problem is taken care of. It straightens out octave flattening through perturbation. An insertable Wedge opens up a whole new area of tonal experimentation as you can fashion subtly different Wedges to install in a single flute. Slightly convex or concave, grooved, thicker, thinner, longer, shorter--the possibilities abound. For a page on tuning by stock removal follow this link.


Only three design factors remain:

Aspect Ratio (tube length divided by internal diameter), hole size and the blowing edge.

Since most shakuhachi have irregular shaped bores the Aspect Ratio can't be calculated from the throat and length measurements. As most shak owners have little idea of the Aspect Ratio of their flute we'll include a method. It can be determined by measuring the volume of the bore in cubic centimeters. Tape the holes and foot opening of your shak and fill with water (sugar or sand for the less intrepid), then measure that amount in cubic centimeters (cc). The graph below will compute the Aspect Ratio of your flute. High AR is a skinny bore, low AR a fat one. Aspect Ratio determines the playability and character of a flute more than any other single factor.

Since we're using manufactured pipe the internal diameter is fixed. So we can only adjust the Aspect Ratio by changing the length of the tube. Aspect Ratio directly effects how the flute sounds--it's timbre. Higher AR produces brighter tones while lower AR creates darker, 'woodier' tones. Higher AR uses less air to play, very low AR will leave you gasping--which is a not so subtle matter in playability. Higher AR favors the higher notes and the opposite is true with lower AR. Typically, what's identified as 'resonance' correlates closely with high Aspect Ratio. When players have a strong sensation that the flute body is vibrating it's invaribly assocated with an AR over 29. Jack up the AR and the suckers start to pound. Get into the low to mid-thirties and it feels like driving at 55 with the tires badly out of balance.

Another flute characteristic closely tied to Aspect Ratio is attack--the time it takes for the flute to speak. Describing a flute as having a quick attack is synonomous with saying it has a high AR.

Determining the AR of your flutes will give you a very good idea of the correlation between AR and timbre and will reveal the Aspect Ratio to which your ear is partial.

Hole size is a subtler matter:

Generally, hole sizes can range from 8mm to 12mm. Smaller holes favor micro-tones and 'shading' as the holes work more in concert. Smaller holes favor 'softer' sounds somewhat the same way lower AR favors darker tones. Larger holes favor a louder, sharper sound as the holes compliment or interfere (however you want to think of it) with each other less than smaller ones. It's generally felt that larger holes allow for a higher cutoff frequency--meaning more sub-harmonics are produced. Larger holes increase the sensation of resonance as they expose more of the finger-pad area to the air-column vibrations, hence the player has a stronger tactile sensation of vibration. So, for resonance go with high AR first and big holes second. It should be obvious that the air-column vibrates--that's the origin of the sound. That IS the sound. Seen on an oscilloscope the sound wave has a profile--a shape. And shape determines how much 'punch' a wave can deliver. High AR produces a taller, narrower wave--one with more punch and bigger holes deliver a larger amount of impact to the tactile nerve endings in the fingertips.

Think of the whole thing like tsunami. When at sea these waves are often just a few feet tall and sailors are unaware when a wave passes. But as the wave approachs land and the depth of the ocean begins to diminish the wave starts to stand up. A tsunami wave's height is directly related to the depth of the water--the ocean's Aspect Ratio so to speak. When it hits shore the tsunami's power is evident. Shoreline to a tsunami is like a skinny flute is to resonance.

Before leaving the subject of resonance, let's go a step further. Accepting that the receptors for resonance are in the fingertips (and lips); the index fingers are the most sensitive. So if you're going to do anything special to (or near) holes to boost resonance do it to the second and fourth holes. These are the index finger holes and the fourth hole is probably closed a greater percentage of the time than the second--thus that index finger will be in position most often to detect resonance. If there's a single hole to concentrate on in terms of resonance, it's the fourth.

For the following flute prototypes we've settled on the hole sizes in the chart below, if for no other reason than they are somewhat average. Should you vary the hole sizes you'll need to adjust the hole locations from the values listed. For a wealth of other flute measurements and possibilities see the Scales and Intervals page. For a prototype computer program to find the correct acoustical location of holes.

3/4" PVC Kuda Flute Measurements
Multiply hole fraction times length for hole placement
For lengths 1.8 to 2.0 Shaku
Pipe
Schedule
Hole
in.
Hole
mm
Thumb
Hole
Fourth
Hole
Third
Hole
Second
Hole
First
Hole
40
3/8"
9.5
0.4149
0.4792
0.5955
0.6806
0.7760
80
27/64"
10.7
0.4296
0.4939
0.6070
0.6922
0.7913

1" PVC Hochiku Flute Measurements
Schedule 80--27/64" holes
Fourth and fifth holes will need opened up a little
Length
Shaku
Aspect
Ratio
Thumb
Hole
Fourth
Hole
Third
Hole
Second
Hole
First
Hole
2.0
25.6
249
289
349
409
469
2.2
28.1
284
324
384
444
504

Let's Talk Edges:

While playing a shakuhachi (other than with all the holes closed) all the 'action' is taking place across two edges: the blowing edge and the edge of the last uncovered hole. Air molecules crosses those edges at the speed of sound and reverse hundreds of times a second. The blowing edge is maybe 15mm long and the hole edge some 31mm for a total of 46mm. If we throw in a few mm of other holes (as one hole doesn't act entirely on its own) we end up with maybe 50mm. Call it two inches. The sound a flute produces is broadcasting from those two inches. Does the profile of those two inches matter? Let's find out.

Cut a tube 8" long and drill a hole all the way through the tube about 3" from one end. So you have two holes--on either side of the tube. Leave one hole rough (with the edges sharp and square) and smooth and round the edges of the other hole (both inside and out). Plug the end of the pipe nearest the holes and blow first one hole and then the other. Just blowing air throught the pipe and out one hole and then the other. Hear any difference? Put your lips around the pipe to ensure the same intonation with each hole. Rotate the tube to ensure that hearing differences between your ears doesn't effect your perception. The sound of the smooth hole is ....well, smoother. The raw hole produces greater hiss and white noise. You might be able to get it to whistle some, while the smooth hole stays steady. Listen carefully and you'll notice that the pitch of the smooth hole is slightly higher.

Now, make a flute and drill just the first hole and drill it all the way through the tube--again we have two holes on opposite sides. We're essentially doing the same experiment as above but doing it with playing conditions. One hole raw, one smooth--now play each hole. Plug the flute and play each hole again. With the flute plugged you'll notice a dramatic difference in the flute's ability to play the base note. With the smooth hole the tone is much steadier and eaiser to sound. The raw hole is more fickle and breaks up more easily. With these two experiments any number of hole edge conditions and profiles can be tested. The point of drilling through the tube, making double holes, is to isolate the differences in edge profiles and make them stand out clearly by direct comparison.

Now for the Blowing Edge:

For most people, instinct tells them that the blowing edge should be sharp. It cleaves the air-steam, right? Well, not exactly, in fact it's a little difficult to tell exactly what's going on at the blowing edge. But we can experiment with the sharpness issue. Make a flute but don't cut the top angle for the blowing edge. Give the blowing edge a nice curve, but no sharpening--leave it square. Will it play? You bet. Now round and smooth that edge like you did with the smooth hole. We've built flutes with blowing edges of up to 3/16" (4.76mm) thick--almost half a hole and they'll play. Go figure.

Attention to edge profile give flutes distinct playing characteristics. Squarer profiles favor the lower notes and lend a darker timbre while rounded profiles do just the opposite. Sharpening the edge and using a thin slant sharpens the sound, making it clear and penetrating. A sharp, thin edge brings out Ro and adds to resonance. The opposite softens the sound and adds mystery.

Cut the top angle at 45 degrees because: 1) you can do it on a table saw and 2) that angle makes the edge a little beefier, thus stronger. Cut the edge about 3mm (about the width of the saw blade) deep on the table saw and then file it down maybe half millimeter more to round and smooth it. Increase the depth of the curve to deepen the register of the flute--decrease the depth to raise the register. If your design intention is to favor the low notes make the curve deeper, high notes just the opposite. Here is the perfect example and design application of kari/meri. Differences in curvature depth is one of the reasons flutes playing the same base note can be different lengths.

Should you want an edge that isn't as fickle as the standard shakuhachi edge, file the slant on the inside of the tube rather than the outside. Just that one difference. Same curve, same curve depth--just reverse the location of the slant. Acoustically, it's a much more stable design besides producing a strong Ro.

Contemplating Mouthpiece Cuts:

The mouthpiece of the shakuhachi is achieved with two cuts: edge and chin. The angle of the edge cut depends on the throat diameter. With a smaller bore the angle of the cut is more oblique to achieve a nice curve in the blowing edge and conversely for a larger bore the cut is sharper. Once the edge-cut has been completed (producing a satisfactory blowing edge) the chin-cut is made. But what should the angle be? Here is where we need to understand the structure and geometry of the mouthpiece.

Primarily, the mouthpiece does one thing: position the edge at the proper height to meet the airstream and it does so by resting on the 'chin bone'. So the chin to edge (Chedge) distance is the critical dimension. Every tube has a minimum Chedge, a distance that can be no shorter--throat diameter plus one wall thickness (see graphic below). The Chedge can be lengthened by increasing the angle of the chin-cut and theoretically Chedge has no upper limit. Usually Chedge is between 25 and 30mm depending on the chin structure of the player--Jay Leno being a special case. Measure the mouthpiece of a flute that fits you perfectly. Measure from the center of the blowing edge to the outside of the chin rest on the opposite side of the tube. Yes Bucky, actually do it! That's your personal Chedge. Remember it, write it down. MEASURE the Chedge of any flute you're contemplating buying lest the attractive finish get you distracted.

Back to the question of the angle of the chin-cut. The angle should be that which is required to produce the proper Chedge. Often the chin-cut angle is selected to set the slope of the flute when played. But that's a secondary consideration. If the Chedge isn't right it doesn't much matter what the slope of the flute is as playability will be adversely effected. Get the Chedge right and the rest will follow. As far as playability goes Chedge is the most important dimension, if it isn't right you'll struggle from now to eternity--every moment you attempt to play. Want to know where that sore chin comes from? The flute you're struggling to play has a Chedge that's too long for you. Either get another or get out the file. Here is where a millimeter makes a big difference.

The two numbers we'd like to see attached to flutes are Chedge and Aspect Ratio. These two will tell you whether you can play a particular flute and whether you'd want to play it.

Minimum Chedges for 3/4" PVC are: Schedule 40--23.5mm and Schedule 80--22.5mm, meaning that the Schedule 80 pipe will need a greater chin-cut angle to achieve the same Chedge.

Casting Wedges in place:

The following schematic pretty much says it all. The flute needs to be set at the proper angle which is adjusted by moving the Incline Block. Drill 9/64" holes through the tube wall to allow the resin to anchor. Tape the anchor holes and place a 'dam' (black electrical tape) across the lower part of the mouth opening and then introduce the resin, letting it seek it's own level. SmoothCast 315 polyurethane resin has a low viscosity and is UV resistant. It's naturally white but can be colored to about any color you desire. After mixing, the resin is injected into the tube with a 10cc syringe from your pharmacy. The syringe allows you to place the resin precisely and more importantly, gives control over the volume of the Wedge. The 'proper' Wedge thickness is 1/5 of bore diameter and varying the injected volume varies thickness--another place for experimentation.

Since the Wedge is proportioned both to the internal diameter and the length by the same factor (one fifth) we're really talking about the ratio of Length/Bore. Thus, the Wedge is directly related to and is a manifestation of the Aspect Ratio. To set the proper angle of the Incline divide length by bore (L/B--the AR) and set the length to the Incline Block and the height of the Incline Block to that ratio.

Example:
545mm length tube with 20.5mm bore. 520 / 20.5 = 26.59
Set one inch Incline Block 26.59" from beginning of Incline.
Or set 3/4" incline Block 19.94" from the beginning of Incline.
Or set 20mm Incline block 532mm from the beginning of Incline.
The point being that the ratio of the height of the Incline Block and the distance from the beginning of the Incline is the same ratio as the Aspect Ratio.

For the flutes specified in the first table (Kuda) above use about 3.5 to 4.5 cc of resin to construct the Wedge. The basic difference between schedules 40 and 80 pipe is wall thickness--80 being thicker. The internal diameter of 40 is 20.5mm and 80 is 18.5mm--outside diameters are the same. So in general, use schedule 80 for high AR flute and schedule 40 for low. With our local suppliers there's also a difference in color and outer texture between the two schedules. Schedule 80 is often easier to find at electrical supply (rather than plumbing supply) as it's used as electrical conduit.

Use Acetone on a rag to clean and smooth PVC--it's semi-miraculous. Rub all the places you filed and sanded and any roughness will vanish. Add a thin coat of mineral oil (baby oil) for a little shine or use fine steelwool for a satin finish. Following these design specifications will produce a decent practice flute. Durable, simple to make, water-proof, little upkeep, inexpensive--what a deal!

All the design examples on this page have been built using the measurements given. They all play. Feel free to adapt any and all of the information on this page to your circumstance and preference.

See The Synthesis for a final flute design.


Flute Performance:

Altitude affects the pressure/density of air which by 14,000 feet (4267 meters) had dropped by over 40%. Each 1000 foot (305 meters) change in altitude results in about a 3% change in pressure. But altitude makes no difference on the pitch of flutes and humitity has only a small effect. It's temperature that has a direct and significant impact. The only variable in atmospheric speed of sound is temperature and pitch is directly related to speed of sound--if it changes so does pitch. There's some evidence that altitude affects the timbre of flutes and especially those with lower Aspect Ratio. Rising in elevation decreases the effective Aspect Ratio.

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