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Recipe for Wooden Organ Flue Pipes
with Designers Formulæ, Spreadsheet and Measurements
by  johan.liljencrants@telia.com

viol4.jpg (17 kb)Here is a recipe for pipes that is reasonably simple to follow. The completed pipe has two parts, the resonator body with cut-up and foot, and a screwed-on cover forming the outside flue edge, ears and 'intonation roll', see the attached drawing. Highlights in the design are the ease to assemble it and to control the air band thickness. Most of the voicing work is concentrated to the readily unscrewed cover piece.

In classical pipe making you determine the various measures of a pipe by proportional scaling in a graphical diagram. These diagrams are based on experience from historic pipes and are often jealously guarded trade secrets of the pipe makers.

The approach here is instead to give simple formulas. You have to make a small number of basic selections for pipe type, width scale, note number, and for practical things like foot design.

The rational use of the formulas is to enter them in a computer spreadsheet program and then you can easily develop lists for all parts dimensions for a complete pipe rank. Click here to view or download the Microsoft Excel spreadsheet file,  violpip.xls.
 

Key dimensions

Some key dimensions are found from elementary physical relations where the value for a particular note is selected in terms of its semitone number n on the equally tempered scale. The assignment here is MIDI oriented such that note number n=69 is for the standard a'=440 Hz. Then the 'middle c', or 'keyhole c', is number 60.

In classical organ terminology you talk about pipe ranks being 16 ft, 8 ft, etc. Basically this is oriented toward the organist rather than the pipe maker. That measure refers to the size of an open pipe (that possibly is) activated by the C key on the keyboard (or pedal). On a piano the standard a' is always the one next right of the 'middle c'. But in an organ this is the case only for an 8 ft stop. Other size ranks produce notes transposed one or several octaves, or even fractions of octaves. The 8 ft C=65.41 Hz (or 16 ft c, 4 ft 'C) corresponds to our note number n=36.

Other key dimensions are defined from the designer's selections of scale for a rank, and from some practical considerations.

L = nominal Length = half wavelength

Depends only on which note to render. The frequency of note number n will be f(n)=440*2^((n-69)/12) and the half wavelength will be L(n)=c/2f(n) where c is the speed of sound 343.32 m/s @ 20 deg. C (increasing 0.59 m/s per deg. C - tuning changes 3 ct per deg. C). If we combine the formulas and insert c we have
 
L(n) = 0.3901 * 2^((69-n)/12) [meters]
L(n) = 15.36  * 2^((69-n)/12) [inches
This tells how long the resonator should be in theory for an open pipe. To arrive at the practical length you have to account for several things; this is developed in the following section.

W = inside Width/diameter, and
M = halving number

A major determinant of tonal timbre, designers choice. Suggested 'normal' value for standard 440 Hz a  is W(69) = L(69)/12 = 0.0325 m = 1.28 in. Larger values gives pipes with a loud fundamental and weak higher harmonics in relative terms. Smaller values give weaker pipes rich in harmonics, for instance a violin type pipe may have W(69) = L(69)/18 = 0.022 m = 0.87 in. For other notes you have to select a 'halving number' M, which means the number of semitones you have to go up to have a pipe with W(n+M) = W(n)/2. For any pipe then W(n) = W(69) * 2^((69-n)/M). Conventional practice is M = 16 to 24, but also M = infinity (meaning all pipes in a rank have the same diameter as W(69) ) has been used.
T = tube wall Thickness
Designers choice, essentially to fulfill esthetical and practical requirements like stability and ease of building. Rule of thumb is around T = W/6, perhaps more with small pipes (W < 12 mm, 0.5 in), less with large ones (W > 50 mm, 2 in).
F = Foot mount height
Designers choice, to suit appearance and method of connection to the wind chest. May be the same for all pipes in a rank. The drawing shows a somewhat unconventional example with a conically reamed hole in the pipe foot and a matching tapered metal tube which is integral with the wind chest.
H = cut-up Height
D = flue air band thickness
These parameters are essential for the proper function and timbral characteristics of the pipe. They are related to frequency, power, and blowing pressure. Refer to the Ising intonation number and associated dimensioning chart at http://foxtail.com/Tech/isint.html. Here it is difficult to give any specific recommendations to follow, there are many alternative ways to specify. One simple rule that I have used is to put D=W/100 and then proceed with the Ising formula to find H given frequency and pressure. Another possible rule is to specify that (frequency)*(input power) = constant, having in mind that the input power is proportional to the flue area D*W. Incidentally the first rule gives this result when M=24.
E = tuning cut down depth
Applies to the open end of open pipes only. See below.
Length of a pipe

There are some factors you must account for when calculating the physical length. First thing is to decide whether to make an open pipe or a closed pipe. Generally open pipes are somewhat louder and have stronger harmonics, richer timbre, than closed pipes.

The length of an open pipe is somewhere near L, at its basic resonance it contains a half wave of the sound. To fine tune it a convenient method is to cut down one wall of the pipe at its open end and to partly cover that opening with a plate that you can slide up or down.

The length of a closed pipe is somewhere near L/2, at its basic resonance it contains a quarter wave of the sound. To fine tune it you plug its end with an adjustable stopper. The pipe must have some extra length (W suggested) to hold the length of this stopper.

Another factor is the so called 'end correction'; a real pipe will resonate at a lower frequency than one might believe from its physical length because some air outside the ends of the pipe takes part in the resonance motion. Already 150 years ago Cavaillé-Coll gave a good rule: with an open, square pipe you should subtract twice the width from the nominal length. With a closed pipe you subtract only once the width. The latter rule is not accurate with an extreme width where we tend to get a Helmholtz cavity resonator (like the ocarina and many whistles) rather than a transmission line tube resonator.

For tuning the pipe a reasonable tuning range is +/-50 ct, a half semitone up or down. Then you have to make the untuned pipe to be 50 ct low, that is, you should increase L(n) by 3%.

Finally, adding W+F for the flue and foot areas we arrive at the total length of open and closed pipes to be respectively
 

Lop(n) = 1.03L(n)  -2W(n)+W(n)+F      = 1.03L(n)-W(n)+F,
Lcl(n) = 1.03L(n)/2- W(n)+W(n)+F+W(n) = 0.52L(n)+W(n)+F.


In practical tuning measurements I found that opening up one side by the length E is approximately equivalent to shortening the pipe by dL=E*E/(E+W). When the opening E is less than the width W this has a comparatively small influence, but when larger than W the influence is about proportional. To tune up to 50 ct high you should decrease L(n) by 3%, a total difference of 6% from the previously incremented L(n). But since cutting only one wall is less efficient, solving the empirical equation above gives the open pipe tuning cut down E = 0.03L*(1+sqrt(1+4W/L)). This ranges from E=0.06L for an extremely narrow pipe to 0.072L for a very wide one having L/W=4. Forget about this small variation and just select E=0.07L.

Be warned that varying the cut-up height H and the possible insertion of an intonation roll (f) will change the tuning of the pipe. Also fabrication tolerances give their share. The final decision of total length should be left to the last moment when you voice the pipe. The resonator length measures given appear to be conservative such that you can expect the final length to be a little shorter than indicated.
 

Parts dimensions

The following table lists the dimensions of all parts, expressed in the key dimensions. See Figure 1 below for the parts referenced by "RefDes" (reference designation). The indication "+x" means you should add a small extra margin (1..3 mm, 1/16..1/8 in) which is to be shaved off later, during assembly and finishing. The lengths for the body pieces (a) and (b) have two alternatives, select the appropriate one for closed pipes or for open pipes respectively.

                   RefDes  Qty   Length       Width      Thickness
Closed: front, back:    a    2   0.52L+F+W+x  W+2T+x     T
              sides:    b    2   0.52L+F+W+x  W+x        T
Open:   front, back:    a    2   1.03L+F-W+x  W+2T+x     T
              sides:    b    2   1.03L+F-W+x  W+x        T
Bottom plug             c    1   F+W+x        W          W+x
Cover                   d    1   F+W+x        W          W/2+x
Ears                    e    2   F+2W+x       W/2+x      T
Roll                    f    1   W            W/4        W/4
Gasket                  g    1   F+2W+x       W+2T+x     D
Closed: Stopper plug    h    1   0.7W         W-         W-
Open: Tuning plate      i    1   0.07L+3T     W+T/2
      Tuning cut down E          0.07L


Spreadsheet

An example framework spreadsheet for MS Excel including the key dimension formulas is  violpip.xls . Here you can enter your specific selections and also decide whether to use meters or inches.

Assembly

To insure alignment it is wise to drill the axial foot hole in the bottom plug (c) before assembly, perhaps even in a lathe.

Clamp cover (d) between upper ends of sides (b) in order to keep distance. Glue plug (c) between the sides (b) and clamp. Be very careful to put all pieces in parallel, align by simultaneous pressing parts against a flat support to avoid any warp. Let dry and then plane the front and back surfaces to be accurately flat and to measure.

Keep cover (d) clamped as a distance holder between the ends of the sides (b). Glue on the front and back pieces (a), clamp and let dry. Remove (d) and shave off the surplus width of (a) with a plane to make the tube sides flat.

Do the outside beveling and inside wedge wise cut on the cover (d). Try to make the flue edge precisely aligned to the cover inside surface. Glue on ears (e) and clamp, align against a flat support. When dry, shave off the surplus to make both sides flat and thickness to measure. Here is the time to fine adjust  the inside wedge cut such that the outer flue edge is precisely flat with the inside of the cover.

Drill the hole connecting the outside to the foot hole. The drawing shows the front piece to be a wide one, covering the whole front. Of course you may instead elect to let one of the pieces (b) form the front. If you do so, then the natural thing would be to have made this front piece shorter than the other pieces (a) and (b) by the tuning cut down measure E.

Draw accurately the location of the cut-up on the outside of the tube, use a stick to check against the top of the internal bottom plug. Drill through the wall one or several places and shape the rectangular opening with a knife and a file. Then sharpen the labium edge by oblique cutting away on the inside. Most easily done with a very sharp, pointed knife. Pull the knife while cutting small chips at a time. Finish with a small file. This is the most difficult part of the making and it is wise to protect the flue edge against dents by clamping a thin metal plate over the foot area. There is no merit to make the labium extremely sharp, you can leave an edge width of perhaps 0.5 mm or 1/64 inch.

Now cover the cut-up with adhesive tape and fill the upright pipe with thin hot hide glue (or whatever sealing agent you may prefer), leave for perhaps 30 seconds and then pour it out again. Remove the tape, let the excess sealant drip off and let the pipe dry in upside down position.

The gasket defines the air band thickness D. It can be made from different materials, for instance cardboard, leather, or veneer. Let it have some oversize thickness, cut it out with some extra margin and glue it to the cover. When dry, sand the outsides on a flat sandpaper on the table to be flush with the cover ears. Inside the ears, trim with a sandpaper file or a knife.

Mark the places for screw holes on the cover and clamp it in place on the pipe body. Drill the screw holes through the cover into the body with a drill corresponding to the core size of your screws. Then remove the cover and re-drill it to match the screw outer diameter.

For an open pipe, cut the narrow grooves for the tuning slider plate using a hacksaw blade. The depth of these grooves may be some T/3 and they should extend 2E down the tube. Draw a line across the front where the tuning cut down should end, distance E from top. Drill a sequence of holes just above this line and saw along the inner sides. Break loose the chip and finish the edges of the cut down with knife and file. Cut the slider plate from 0.5..1 mm sheet metal of your choice. If you cut it with shears you may have to flatten the plate afterwards, use a wooden mallet to avoid hammer marks. File notches at its upper end corners and fold the upper edge into a handle. Finish the sides of the plate with a file and match it to its grooves in the pipe. Most probably you will have to go a second round with the hacksaw in the lower part of the grooves. Bend the plate slightly concave towards the front such that it slides with appropriate friction.

For a closed pipe, make a stopper plug to match the interior of the pipe when clad with a soft leather strip around its circumference. Make a handle to taste. You should bevel all sides of the stopper, both at top and bottom, such that its side and front views about fit in a circle. This in order to avoid breaking the pipe in case you some later time have to rock the stopper loose because it is hard stuck. Glue on the leather strip around the stopper and be careful to apply glue only as a narrow string along the extreme low end of the stopper. To fine adjust how stiff the stopper runs inside the pipe you slide paper strips into the unglued area between the plug and the leather.

Voicing

For voicing the primary parameter is the thickness of the gasket (g). This is easily controlled sliding the cover over a sandpaper, flat on the table. If you install a 'harmonic brake' in form of a roll (f) you can use a relatively high intonation number (big D, small H, high pressure) without over blowing the pipe. Then the precise position of the roll is critical and has to be found by trial and error. When satisfied you can lock the roll in place with drops of cyanoacrylate glue and/or needles drilled in through the ears (e).

The result

Three pipes are shown in the photograph (top of page); from left a'=440, a''=880, and a'''=1760 Hz. All are of fairly narrow scale, they have W(69)=L(69)/18, M=24, T=4 mm, F=20 mm. The cover of the middle one is removed to show the inside. (The fixing screw placement on the left pipe is not recommended).

The left one has H=12 mm, D=0.4 mm and when open at its far end it sounds like  o440.wav  when blowing pressure is 2 kPa (8 inches WC). Operating it as a closed pipe for the same note (special stopper with a long handle pushed about half way down the resonator) changes its sound into c440.wav . A bigger one in the series sounds like  o220.wav , and the two smaller ones in the picture like  o880.wav  and  o1760.wav .

Johan Liljencrants
Tue, 21 Dec 1999 11:55:11 +0100
 

Appendix

violpip1.gif (9 kb)

Fig 1.  Wooden Flue Pipe [violpip1.gif]


30 December 1999

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