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Design |
for the Design of Didgeridus |
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This page is part of an ongoing attempt to create a general set of guidelines Didgeridus belong to the family of musical instruments called areophones in which the sound is produced by the vibration of air. A subgroup of lip-buzzed horns contain simple or primitive trumpets, alphorns and Didgeridus among others. Generally these instruments are conical (with the larger end away from the player) and often curved. We'll first consider cylindrical tubes to gain a general understanding of a portion of sound physics. The rate of vibration of air in an open tube (like a Didge) is determined by the length of the tube along with other smaller factors like the Aspect Ratio and how much friction the surface of the inner tube wall produces. The cross-sectional shape of an air column has relatively little acoustic consequence. Instead, cross-sectional area is the important consideration. An air column which is square in cross section, and of uniform dimensions over its entire length, behaves very nearly the same as a cylindrical pipe of the same cross-sectional area. A straight-sided square pipe which increases in cross section at a uniform rate behaves like a conical pipe. It is also generally assumed that as long as the cross-sectional area retains the intended value, it doesn't matter how the tube may curve and snake around--a cylindrical pipe will still behave acoustically like a cylindrical pipe even after bending, as long as there are no sharp angles or kinks.
One of the first considerations in design is setting the pitch (frequency as measure in Hertz or cycles/second). Fortunately, it's one of the most straight forward parts of Didge design. Ignoring a handful of niggling small factors, pitch is determined by tube length. The pitch of most Didges is between 50 and 100 Hz so that's the range we'll concentrate on. How long should a tube be to get a D note? The answer, and it should also be applied to all such questions like it, is that it's the length of the tube when sounding a D note. This isn't a trick answer, it is meant to convey that much about Didge building is experiential instead of theoretical. Theory will get us close but actual construction and tuning is the final word. Instead of attempting to precisely account for a dozen or so variables (large and small) it's usually easier just to build the thing.
An open tube will support a standing wave (sound) who's frequency (pitch) is proportional to the length. The frequency is equal to the speed of sound (1087.1 ft/sec) times 3, times the length in inches. F = 3261.3 / L Or to say it another way: 3216.3 / Length = Frequency. And yet another way, the tube length divided by three, times the fundamental frequency equals the speed of sound. A 50" tube will produce a sound of 65.2 Hz. Slightly different formulas factor in bore measurements: L = (3216.3 / F) - 1.35 B for a pipe open at one end M. Cavaillé-Coll, the French organ builder, used: L = (3216.3 / F) - (5/3) B for cylindrical pipes open at one end We get various answers:
If having a 'tuned' Didge is a goal, remember, this table is for a straight-walled tubes. Since it derives from ideal, theoretical circumstances, when cutting add a couple inches to compensate for any general weirdness which might creep in. Altitude makes a difference. I'm at 7000' so my Didges run long. Toward the end of fabrication tune by trimming to length.
Another important consideration in design is the Aspect Ratio (AR)--the ratio of tube length divided by its diameter. This ratio largely determines the overall tonal quality of sound--the timbre. 'Fat' air columns (low AR) are generally poorer in overtones than skinny ones. At the extremes, excessively fat pipes will not speak at all, and excessively slender ones tend to break up into harmonics rather than produce the fundamental. You can see this effect at work in the ranks of a large pipe organ: the thickest pipes are the ones called "flute pipes" which are characterized by a strong fundamental and relatively weak higher overtones. The ones called "string pipes" are much more slender, and they show prominent harmonic overtones. Is there an ideal Aspect Ration one can strive for? For a 24" flute a diameter near 1" is considered ideal by some--a good balance of fundamental and harmonics. But as the example of the organ above indicates, people will select different ratios as they seek different tone qualities. It seems logical that if we adopt the AR (24) from the flute above that this ratio should produce the same quality of sound in any length pipe. However, this isn't the case. It turns out that the ear hears lower pitched pipes as relatively dark, and the higher ones as relatively bright. To create greater unity of timbre in a pipe organ, the lower pitched pipes should be a bit thinner relative to length, creating a slightly richer harmonic spectrum. A rule of thumb developed by European organ builders over the years has been to make tube diameter one and two thirds (5/3) as large for each doubling in length. So in pipe organs the AR isn't constant; it increases as a pipe's length grows. For the sake of this example let's adopt an Aspect Ratio of 24 as optimal for a two foot tube. The graph below plots both a constant AR of 24 and the 5/3 doubling used in large pipe organs.
(An expanded view)
Timbre is greatly influenced by the Aspect Ratio. •Increasing the tube's diameter increases the mass of the air in the tube making it harder to oscillate. If you have any doubt about this put a coin in your hand and see how rapidly you can shake your fist. Now try a brick. Greater tube diameter makes a Didge sound woodier, darker, breathier; this will increase its fundamentals. Increased diameter requires a greater force to play. •Decrease the diameter to create a brighter, airier, clearer tone--more string-like with harmonic overtones. Decreased diameter requires less force to play.
Since a Didge sounds only one 'note' it's important that this note be pleasing to the player. Conveniently, the two lines in the graphs work out, at four feet, to be perfect for ABS plumbing pipe. For ABS pipe the ID's are actually 2.03" and 1.58"--an excellent match to the graph. Experimenting with these two diameters of unmodified straight pipe will give you a sense of the effect of Aspect Ratio. And will provide a good idea of the general AR you want your Didge to have. With the dimensions mentioned here we're talking about ARs of 24 and 30. So, as a general rule the Aspect Ratio of Didges should probably be between 20 and 28--plenty of room for individual preference. A low AR favors the fundamental--a high AR favors harmonics. Again, remember the Aspect Ratio doesn't remain constant when scaled. That 3' Didge you love the sound of will not have the same timbre when both length and diameter dimensions are doubled. 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.
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 such, and have the same potential for producing a good-sounding instrument. A prize shakuhatchi, for example, has a bore like a mirror--it's like looking into the navel of infinity. However, extremely hard reflective surfaces are not necessarily the ideal: people sometimes prefer the mellower tone of somewhat damped resonances--in a Didge, for example. Timbre is greatly influenced by air friction in the bore. Wind instrument tubing materials that are light and/or yielding will dampen air resonance within the tube to some degree, and lower the resonate 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. We don't ordinarily think of friction within a Didge, 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 Didge is happening at Mach 1--the speed of sound. The air volume in the bore of a Didge may turn over at a leisurely rate but the molecules making up the sound waves are moving at Chuck Yeager speeds. So 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 Didge. So what kind of inner surface is best for a Didge? 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 Didge's tone by: 1) raising the AR, and/or 2) polishing the bore. A thin pipe with a slippery (fast) bore is doubly bright. So if a Didge'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 Didge 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 another possibility. 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 an ACE bandage down the 4' test pipes (above) you were using to explore ARs. Try it stretched and then loose. 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 pipe. You can 'place' strips of cloth using string tied to the ends. 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 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.
The end of a wind instrument tube, where the enclosed air meets the outer airs, is a critical point. A large opening is good for sound projection, since it creates a lot of 'surface area' for radiating the sound. The sound that radiates into space from a small opening will be restricted, even when the internal wave is quite strong. You can increase the size of the opening, and thus increase radiation efficiency, by flaring the bell at the end of the tube. The presence of a bell affects the tuning and relative prominence of the overtones Brass instruments makers have learned to compensate for these effects, but the factors involved are rather subtle. Estimating the effective tube length for a tube with a belled end is difficult, but it is safe to act as if the tube effectively ends at some point mid-bell. Didges are usually (often?) tapered, ending with a flair or bell toward the end. Do we want a straight-walled taper, a curved taper? Here is where ideal musical physics and didges part ways. The shape of a Didge's taper doesn't derive from any ideal mathematical standard as do many modern brass instruments. It comes from what the eucalyptus tree thinks is the best taper for growth. That being said, the graph below uses power (X to the n) curves to sketch out possible taper variations. Creating a Didge out of any but naturally occurring materials requires that we make decisions about taper. For the purpose of these exercise stick with ABS pipe in the four foot length as above. In both 1.5" and 2" ABS pipe wall thickness runs about 0.160". With a modicum of care it can be heated and stretched until it's half as thick. Therefore, we can create a diameter twice as great. A bell on a 2" pipe can reach 4" without much trouble. Using ABS pipe we can double the internal diameter--let's use that as a design setting. Next we need to decide how much of the tube we want to taper. A quarter? Third? Half? To get started let's make it a third or 16" for a 4' tube. Cut a mandrel out rigid material (1/8" masonite?) to check and measure the taper.
Template of a 16" Didge Taper Mandrel for a 2" ABS Tube (Note that the length and width scales are different so the schematic will fit on this page) Probably in the end, applying air pressure to a heated tube is the most reliable way to expand it. What you'll attempt to do is create a bubble who's shape you have control over. Since both ends of the tube need to be 'stoppered' when applying air pressure you'll end up losing a foot or so of tubing when cutting the end off the bell. A way around this is to create a 'double' mold which will produce two Didges at a time--bell to bell. Then just cut them apart. Construct a tapered mold and place the heated third of the tube within it--then apply air pressure. Such a mold doesn't have to be elaborate, it ends up being a frame more than a mold. Wire mesh seems a good choice from which to create experimental 'molds' as you can apply heat through the wire. Tapering at least a third of the tube seems a good general guideline. Any of the tapers shown in the graph will produce a respectable sound.
The three main musical qualities are pitch, timbre, intensity. The first two have already been discussed. Intensity is determined by the force of the air the player uses, the 'efficiency' of the tube and the 'broadcast' portion of the bell. Why can the tuba in the marching band be heard blocks away? One factor is that a tuba is low pitched (like a Didge) and low frequency sound travels better. But the factor we're interested in here is the 'broadcast ' portion of the bell. Think tuba. Think that big round, nearly flat portion of the bell. If you want your Didge to BROADCAST then a surface is required. That means flattening out the edge of the taper/bell. The sound waves need something to 'push' against. 'Loud' horns have large effective broadcast plates. The first two examples in the graph below are extreme cases: No broadcast surface and one specifically designed to do so.
The upshot is this: if you want your Didge to have greater intensity, start flaring the lip of the bell. The greater it's turned and flared the greater the sound amplification and projection. Try different mandrel shapes and curves until you get the balance of timbre and sound projection you like. Everything effects everything. Change the bell shape to increase the intensity and the timbre begins to go south. Cut the tube to raise the pitch and the Aspect Ration starts to tank. Each change favors one quality over another. It's kind of like herding turkeys--they all want to go different directions. There is no perfect Didge--there's only a balanced Didge. Designing a Didge (or most any musical instrument for that matter) is an exercise in compromise. Keeping that in mind will making designing go much easier.
Back to the tuba analogy. If the Didge were to have a mouthpiece other than a wax rim it would be something like the mouthpiece of a tuba. The mouthpiece in the schematic below is intended to fit into the end of a 1.5" ABS tube. Such a mouthpiece makes the Didge much easier to play.
The mouthpiece has three elements: Cup, Rim and Throat. As always, each element must be balanced with the others. •Cup Cup width is determined by mouth size--1.25" is a good general adult size. A wide cup favors endurance, while a narrow width favors flexibility and range. The cup can be deep or shallow and the drawing above (using a half-circle for cup shape) is fairly deep. A deep cup favors sonorous dark tones (lower registers), increases volume and lowers pitch. Shallow is the opposite The qualities of cups are deep/shallow and large/small. •Rim Rim width had to do with lip comfort and playing dexterity. A round rim favors comfort while a sharp rim favors brilliance and precision of attack. The inner rim edge should be sharper than the outer. Rims are wide/narrow and round/sharp. •Throat Throat diameter should run between 3/8" and 1/2". Start small so you can drill them out. A small throat favors the higher registers and endurance. The throat size should create a little 'resistance' or back pressure. If it gets too large the sound becomes 'breathy'. Throats are small/large and short/long. The following table is a very general set of parameters --someplace to start.
Another advantage of using mouthpieces is that you can get different qualities of sound from a single Didge by switching mouthpieces. Possible methods of manufacture: standard lathe techniques, ceramics (building from clay and firing), molding using a thermosetting polymer such as Sculpy (available at most craft/art stores). Put some grooves around the outer side of the rim so it's easier to pull the mouthpiece out of the tube. If needed use wax for tight fit.
Domestic cats all purr at the same frequency! Big cats, scrawny cats, old tom cats, kittens--same frequency. Which indicates that purring isn't some physiological phenomenon but must be driven centrally from the brain. Purring is associated with contentment, well-being, good health and a trancey kind of behavior. Do cats purr to 'massage' their neurons? Do the neurons operate to 'massage' the cat with sound? In any event, the frequency for cats is 25.9 hertz. A Didge needs to be about 10.5 feet long to resonate at that frequency. But if we employ frequency doubling (octave jumping--we're talkin' G# here) we can end up with a 'relative' to cat purring at 51.8 Hz (63") and 103.6 Hz (31.5"). Something special might be happening with sound in these ranges emanating from Didges about those lengths. Do Bandicoots purr? If so, we have a good idea of the possible frequencies. This whole vibratory deal may be specific to neuron health and as possessors of a few billion of the suckers humans might be well disposed to 'purr' occasionally. What better way to do so than with a Didge. It's a thought. |
