Wind Instrument Mathematics
Measuring Wind Instrument Lengths (It only looks like rocket science)

Carlos Fernandez - 11:57am May 22, 1998

Hi everybody,

This is my first message in the forum. I apollogize in advance for my poor English. I have been trying to build a fife in G following Rick Miller's plans (PVC flute with cilindrical bore). The results weren't too bad, but I have a few questions about flute design (this message would be better included in the maths/design section, but now is over). Could anybody give me some advice about how to determine ideal bore length for different frequencies? I got a document in which ideal bore length is calculated using the following formula. L=(S/F*2)being L the length of the bore, S the speed of the air flow?????, and F the frequency of the sound.If this is true how can I calculate the speed of the air flow? Any advice about determining the placement of finger holes and their size will be wellcome as well.

WELCOME to the Forum Jennifer Wieringa and Carlos Fernandez!

To answer your questions Carlos,"S" is the "Speed of Sound" at your altitude and temperature (both these factors affect this speed slightly). "F" is the frequency of the note played in "Herts"(hz = cycles per second).

Divide the speed of sound by the frequency (in hz) and divide again by 2. This will give you the "Half wavelength" of the tone you desire.

Example: The speed of sound at sea level temperature of 70Deg F is 13526.5 inches per second (this is "S")

The frequency for a G above Mid C is 392.04hz ("F")

13526.5 ÷ 392.04 = 34.503 ÷ 2 = 17.251 inches. This is the approximate length of a flute fundamental in G.(other factors will shorten this slightly).

The best ratio of a straitbore flute is approximately 1:30 or bore diameter of 1/30th the length, so the approximate bore size for a flute with a half wavelength of 17.251 inches is .575 inches.

I will go into depth on this subject and tonehole placement formulae in the Math/Designs area shortly. Hope this helps Carlos.

P.S. Multiply any measurement (in inches) by 25.4 to get a metric conversion (in millimeters).

Hi Thomas, Thank you for your explanations about the bore length formula. According to your advice I'm going to use a different PVC tube to build my flute in G. I'll use PVC tube with bore diameter of 1,7 cm.(0,55 inches approximately). I think that this tube will fit better the 1:30 bore length to diameter ratio than 3/4 PVC tube. Since I don't know where to drill the finger holes I'll try to follow Rick Miller's design, tunning them from the lowest one to the highest and see what happends.

The tube length is directly related to frequency.

L = Length

Y = (normally lambda) Wavelength

F = Frequency Hz

d = diameter of pipe at first end.

D = diameter of mouthpiece end (flute)

V = Velocity of speed of sound 343.5 meters per second (temperature dependent, 70 deg F is standard) for cylindrical pipe open on both ends.

L = Y / 2

or

F = V / (2 * L)

The recalculate L by removing 0.3 * d for each end of the pipe.

L = L - ((0.3 * d) + (0.3 * D))

When the air column hits the outside world, it slightly blows beyond the pipes physical length, the calculation above works for most pipes. It doesn't work with extremes.

In order to sound properly, you should keep the length to pipe vs bore diameter between 24:1 through 30:1. I like to used between 24 & 26.

If your pipe is closed on one end: Pan pipes, Clarinets, etc, use:

L = Y / 4

or

F = V / (4 * L)

For Conical Pipe

L = Y / 2

or

F = V / (2 * L)

Hope this is a good start!

Mirjan, a tube is a tube, whether it is bent or not, so all the normal wind instrument maths apply. I assume the length you use in the calculations is the length of the center of the bore (for curved instruments, like crumhorn or saxophone, the inside of the bore curve is obviously shorter than the outside). The bending can be taken to exremes. Many brass instruments put all sorts of curves in the tube, some of the curves switchable by keys. Midieval racketts put a couple feet of tube in a 6" package by bending it back and forth inside an otherwise solid cylinder.

The velocity of sound in air (sea level) is 344 + 0.6 (t-20) meters per second, where t is temperature in degrees C. Daniel, 70 degrees F (about 20 C) sounds a little too low to use, especially for smaller wind instruments (ie fifes and flutes) since the air driving it is at 37 C (body temp). I'll go home tonight and stick a thermometer in one of the holes in my flute to see how hot the air actually is. It is well known that flutes get noticably sharper when they are first played as they heat up. Every article I have read on building flutes cautions not to adjust the pitch of a cold flute, because it will be too far off when played for any length of time.

Most beginning flute makers seem to follow the same method of drilling starter holes in convenient positions, and iteratively tune them from the end back towards the embouchure. The actual positions of the holes do not need to be precise, the hole sizes are adjusted to set the pitches. While this certainly works for the 'normal' notes, I can't see how it would get the cross-fingered notes right. Does anyone know how you can design the placement and sizes of the holes to minimize the errors for all the normal notes and a selection of cross-fingered holes (ie, F# or C natural on a normal D flute)?

Hello all you amateur designers!

Daniel has graciously given us some formulae on designing flute/whistle bores. (Thanks Daniel)

Here are some "Basics" from me. The speed of sound at sea level and 70°F is 13526.5 inches per second and as you increase the temp or the altitude the pitch gets sharper.This pitch sharpening, in the case of altitude, is caused by lower air pressure (less resistance/higher frequency). In the case of temperature the pitch increases because the bore "expands" and constricts (gets narrower).

A flute/whistle/crumhorn etc. length is determined by solving for the "Half-wavelength" of the frequency you desire, that formulae was stated by Daniel earlier but here is another one S ÷ F = W. The speed of sound (S). The frequency (F) desired (example A=440hz concert). and (W) is the wavelength.Divide the wavelength by 2 and you get the approximate length (certain other factors affect/shorten this and I will go into it further next post)

Bore size in flutes/whistles is approximately 26 or 30 to 1, so a flute with a bore of .5inches could be between 13in (26 times) and 15in (30 times) on average. A clarinet on average is 45 times the bore diameter in length or more (Basset horn etc.) Personal preference plays a role here, a 30/1 bore (long and narrow) will sound mellow and accent the lower notes and a shorter wider bore will accent the higher ones.There are acceptions to this too, such as in the case of the Ney and some Kena's.

Most people have trouble with the notehole placement and these calculations can be tedious to aquire. Once made/calculated for flutes and whistles in various Keys you need never "start from scratch" again, only manipulate what you have to suit large or small hands.

I am going to post exact formulae and how to calculate these tone holes and modify them in my next post (tommorrow the 11th). I want to "refine" my methodology to simplify these formula's for our begining readers/makers.All you need to do this is a simple calculator with memory and scatch paper. Please be patient till tommorrow,Thomas.

Hi , to all experts. I am interested in builting a fife. Is there anyone knows the formula since it's pitch is higher than flute . Thank you !

What key is your fife in?

I really happy when i read the reply, I thought no one would bother me , I really appreciate ! I would like to built a B-flat fife . But I saw a fife, make in steel with six hole, and an extra hole at the back, the extra hole is covered by the thumb, a man told me it is a D fife,is it really a fife or a flute ? Besides B-flat, is there any other key ?

thank you !

Hi Brian, Some keys are more popular than others but it is possible to make fifes in all 12 tones and varying pitches of A , no problem , when they cost more than 50$ they are called flutes <g>.

Seriously , the B-flat fife is the most popular with the " Fife and Drums " crowd . Made of wood or metal , they have 6 holes ( some time an extra hole with a key on it ). The fingering is similar to the tin whistle ( wich is generally in D : shorter ).

I have a fife/flute instrument made of white plastic , made by Yamaha , 8 holes ( one thumbhole ) and the fingering is like the Recorder .It is not a traditional instrument but is handy for people who already play the recorder .

Generally you can guess the tonality by the lenght: there are small G fifes/whistles/flutes(8") ,A,B flat,C ... and tall G (like the one on the PVC flutes plans (25")

The B-flat fife is in Trevor Robinsons book "The Amatuer Wind Instrument Maker". The book is in the MIMF Bookstore . The plans are located on page 30. The instrument is 347 mm long with an 11 mm bore. The instrument list appear to be a straight bore with 6 toneholes, it appear to be a very early design.

I hope this is a good start.

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