Making the flute
Before we examine any more physics, let's cut to the chase – or the willow tree – as the case may be. To make a traditional flute, wait till the sap begins to rise in the spring. Then find a suitable willow twig and twist the bark (as a whole tube) from the core wood of the twig. Part of the core is used to make the mouthpiece. We won't go deeper into that here – instead we'll use PVC pipe. Get some PVC pipe between 16 and 24 inches long. Its inner diameter should be about 35 times smaller than the length, according to one of my sources. (about 0.45'' for a 16" pipe, and 0.67" for a 24" pipe). Measure and mark about 1.5 inches from one end of the pipe. With a fine saw or sharp knife – I've used a moto-tool – cut straight into the pipe until the depth of the cut is VERY close to 3/5 of the total diameter of the tube. Then, mark a 45° angle from the bottom of the cut, toward the far end of the tube. Using this mark as a guide, make a 45° cut down to the base of the first cut, removing a notch from the tube. This slanted cut is the fipple. (See Fig. 1.) Be careful to make these cuts accurately. The 45° bit seems to be important. If you've been using a saw or a moto-tool, you might wish to finish with a good sharp knife. Make the edge of the 45° cut as smooth as possible, and sharp but not knife sharp. One can use sand paper if needed. The flutes I have from Norwegian PVC pipe are very thin-skinned, while the PVC pipe I’ve found here is pretty thick. This makes cutting a good slanted edge critical, and the thickeness may be part of why I’ve not had good luck at getting a fine flute.
Now it's time to make the mouthpiece plug. Get a wooden rod from a hardware or hobby store that will fit tightly into the PVC tubing. Or, for a picturesque touch, find a small branch of the right diameter. You can leave the bark on or not. If you have to, get something with a slightly larger diameter, and whittle it down. Cut the rod to between 2 and 3 inches long. If you're going for the natural look, choose a longer length to show off the natural wood you've found. If the diameter is too big, whittle enough of one end of it down to fit, so you can insert it at least up to the vertical cut into the PVC pipe. Now you need to make an airway in this plug. To make this next bit easier to understand, look again at Figure 1. When the plug is inserted into the flute, you'll blow into the space created between the near end of the flute and the airway cut into the plug. The airway will direct the air at the fipple. The airstream will be split by the edge of this cut, and somehow this sets up the standing wave vibration pattern in the flute's body.
So here goes. To measure the length of the airway, stick the plug into the flute just up the edge of the vertical cut. Mark the plug where it emerges from the end of the flute. Since the airway must extend a bit past this mark to give a place for you to blow into, you might want to draw the mark almost half way around the plug. (If you just mark the top, the mark will be removed as you whittle out the airway.) At the end which will fit into the flute, whittle one side of the plug flat, so that when the plug is inserted, there's an air space of about 2 mm between the round curve of the flute and the flattened upper side of the plug. (See Fig. 2) Now whittle out and shape the channel for the air, curving it smoothly down and back up again. Leave a VERY small flat space between the end of the plug and the end of the upward curve of the airway. I suspect that the stronger your wood, the less flat space you'll need to leave. You may end up doing away with this flat space altogether. At any rate, the upward curve of the wood should aim the air at the fipple. You'll need to experiment quite a bit to get this right. You may also have to exper-iment with exactly how far to insert each mouthpiece attempt – it matters a lot.
Since I've managed to make flutes that work, but not any that I like as much as even the worst ones I've bought, I'd guess you'll need to experiment a while. I've been told that somewhat hard woods work best, but after floundering a bit, I bought some balsa wood to experiment with. When you've got a mouthpiece you like, add a few drops of glue to keep it securely inserted. Decoration of the flute and mouthpiece are up to you. One of the flutes I have, made for the tourist trade, is wrapped with very thin strips of birch bark. The other two have "found" wooden plugs. One is some kind of conifer with a very tight bark. The bark has been partially stripped/whittled down and smoothed so that a little of the bark texture is left, but doesn't shed into one's hand. The other plug has been whittled to show off the wood grain, and the end that sticks out is a kind of "knobbly" shape. Very folky looking. Incidentally – one of these flutes is made from brass tubing. The other is of black PVC pipe which is of a fairly large diameter (3/4"). The far end, which must be covered securely with a finger to play, has been heated to softness and squeezed with pliers so that the opening is oblong rather than round. The PVC pipe I've found in the US is a bit thick to do that with.
I've also tried to make a flute from a hard clear plastic. It is not only hard, but a bit brittle, so cutting it was some-thing of an adventure in my ill-equipped kitchen. But it's turned out to be one of my best attempts. One thing that led me to try it was its narrow bore (inner diameter) which takes less air. I'd thought of painting it with some properly delicate design as befits its transparency. But the moisture from one's breath steams it up, eventually creating most unattractive water droplets. I may paint the whole thing just to hide this! The interesting thing about the water condensation in this flute is that one can see what I assume is evidence of the standing wave pattern of the fundamental tone.
This brings us back to the physics of this type of flute, and the scale associated with it. I'll try to keep the physics to a minimum, but since it explains the gaps in this flute's unusual scale, I feel I need to provide a bit of explanation. I hope information about the scale will help you play (and listen to) such a flute.
The scale and some physics
The scale produced by the open-ended flute (i.e., not covering the end with a finger) is the overtone scale, be-ginning with the fundamental and consisting of the over-tones of that note. Let's pick a pretty number out of the air – say 100 hertz (vibrations per second) – for the fundamental frequency. This is a pretty low note, the "A" of a violin A string is 440 Hz, an octave below that is 220 Hz, another octave down is 110 Hz – in the bass range of the human voice. Never mind which exact piano note 100 Hz. is, the example makes the math involved easier to get one's mind around. This is the lowest note of the open-ended scale of our imaginary flute. This pitch arises from the shape and length of the standing wave created in an open-ended pipe. The next highest harmonic (or 1st over-tone) is twice that, 200 Hz. This is an octave above the fundamental tone. The 2nd overtone is three times the fundamental, 300 Hz. This is an octave plus a fifth over the fundamental. The 3rd overtone is four times 100 Hz = 400 Hz, two octaves above the fundamental. This sequence of notes creates a "scale" of sorts. A second set of harmonics (or "scale") is created by closing off the far end of the flute with one's finger. These harmonics arise from the set of standing waves created in a closed-ended pipe.
Table 1 shows vibrational frequencies for flutes in three keys, C, G, and A. Closed note frequencies are in parentheses. Note how different some of these frequencies are from the values for the modern even tempered scale, given the left column – even though the fundamental note of each flute is equal to the modern value. This is just how the physics of vibrating things comes out. Drawings of some of the two sets of standing waves are shown in Figures 3 and 4. Figure 5 shows the scale resulting from combining these two scales. Note that the same air pres-sure (blowing strength) will produce two notes: an open end note, and a closed end note. For a given air pressure, the closed end note will always be one step down the "combined" scale from the open end note.
Table 1 - Frequencies (Hz) of C, G, and A willow flute scales
compared with frequencies of the even-tempered scale
|Note Name||Equal Tempered||W. Flute, Key of C||W. Flute, Key of G||W. Flute, Key of A|
|C (octave "3")||130.8||(130.8)||
|C (octave "4")||261.6||261.6||
|G (^ that's middle C)||391.995||(392.4 )||391.99||
|A (std pitch A)||440||
|C (octave "5")||523.3||523.2||
|E (vln's top str)||659.3||(654 )||
|C (octave "6")||1046.5||1046.4||
|C (piano's top note)||4186.0||
This scale not only has skips in its lower
regions which we don't expect a scale to have, but its pitches are not
what we're used to from hearing modern instruments. In modern times we
use what's called an even tempered scale. How this came to be is well
beyond the scope of this
article, but suffice it to say that it's what modern western ears
hear. A nice, and mercifully short, explanation of the even tempered
and how it differs from the willow flute's scale can be found on the
at <www.sju.edu/~rhall/newton> and <www.sju.edu/
~rhall/newton/mathandmusic.pdf>. (These two articles discuss some
aspects of willow flute physics at length.) Other, more theoretical
explanations can be found in any number of books addressing the physics
of musical sound. In the region of the scale which doesn't have the
wide skips of the lowest notes, the natural willow flute scale is
similar to the major scale our ears are used to, but with a raised 4th
and a lowered 6th and 7th notes.
The fundamental is hard to get on most flutes – one must blow VERY gently and the resulting note is extreme-ly soft. There are also notes above the ones shown here. These get progressively closer and closer together. They're hard to get because one must blow so hard, and it's hard to control which note you'll end up playing. Incidentally, this is the same scale that the munnharpe (Nor) (munngiga - Sw., mouth/jews harp - Eng.) produces.
Try making a willow flute this winter, and have fun with it.
Note: The munnharpa’s open and closed notes are made by opening and closing off the throat while playing. Changing one's internal mouth shape, much as one does while whistling, does the rest.
While researching this article, I found a web article on making native American flutes, which are also fipple flutes, but with finger-holes. One blows directly into the end of these flutes. Instead of a plug which directs air at the fipple, this flute has an internal barrier near the blowing end, creating a small chamber with a hole in the top. The hole is away from the blowing opening, and beside the barrier. The air comes out this hole and strikes a plate attached to the outside of the flute. The plate has a channel scored into its underside to direct the air back down toward the fipple, and so back into the flute. This webpage has a list of flute lengths and internal bores for several keys. Since the lengths are valid for any type of flute, I'm listing that information in Table 2. These bores are a bit big, though. My willow-flute sources suggest an inner flute length (between the far end and the fipple) to inner diameter ratio of 35/1. The Indian flute author suggests a length/bore ratio of 23/1. The values given in this table are 20/1. The bore size affects the tonal quality of the flute, as well as ease of playing. The differing bore sizes are most of what make the sharp-voiced willow flutes and the mellower-voiced Indian flutes sound so dif-ferent. This web page is at: <www.wycliffe.org/events/music/NativeAmerican.pdf>
Books on the physics of musical sound: I have several, none of which I'm fond of. There are lots of them which cover the subject in varying detail and depth. The bare basics are also usually covered in introductory physics texts. They all say approximately the same thing, and all that I've seen are aimed at the person vitally interested in physics. Somewhere out there, there must be one aimed at the minimally mathematically inclined musician! While I did consult mine to check details for this article, they're pretty dry, and one is about as good as another if you just want basic infor-mation.
On the Internet: all in English unless otherwise noted.
Just for fun:
<www.soundwell.com/multi flute-e.htm> and associated pages.
This group of pages wants to sell you "willow" flutes made with flexible tubing of some type. This allows really low pitched scales and flutes in a variety of keys. They even have one with interchangeable tubes to allow for more flute keys. It has a nice explanation of how they work; you could probably make one yourself from the info given here. There are also some sound examples so you can hear what these things sound like. The author has a definite sense of humor, and the instruments are inherently pretty funny.
If you want to make a real willow flute:
And in Norwegian (with diagrams):
Guide to building other types of flutes:
The physics of woodwinds:
<www.phys.unsw.edu.au/~jw/woodwind.html> The physics of strings are on some associated pages.
The physics of willow flutes:
<www.sju.edu/~rhall/newton> & <www.sju.edu/~rhall/newton/mathand music.pdf>
Many university pages on the physics of music come into and out of existence with courses being taught. These pages tend to have clear and accurate explanations, although some of them refer the reader to the class text, which doesn't do the non-class member much good. They're worth investigating if you are interested in this stuff. I've also seen good instrument build-er’s pages, but some of them also seem to have rather fleeting lifespans. The page given above under woodwind physics, from the University of New South Wales in Australia, has been around for at least a couple of years, so it seems fairly stable. §