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Calculating the Mouth Height

In reading about pipe voicing, scaling and construction, the height of the mouth is usually given as a ratio of its width. The ratios that are most commonly given are 1/4 for principals and 1/3 for flutes. However, whenever I've had the chance to measure a real rank of pipes or to see the data for a rank of pipes, the ratio of a mouth's height to it's width changes throughout the rank. For stopped ranks, it seems that the ratio decreases as you go up the scale and for principal pipes, the ratio seems to increase as you go up the scale. Fritz Noack mentioned to me and I've read in "The Art of Organ Voicing" that the height of a mouth is more a function of the length of a pipe (or frequency) than a function of it's width. "The Art of Organ Voicing" goes on to say "It (the cut-up) is independent of the width of the mouth of a pipe." p.26.

So I decided I wanted a way to calculate the mouth height as a function of frequency. I found detailed pipe data on the Richards Fowkes website for their Opus 10 and on the Pasi website for his Opus 13 and Opus 14. I took all this data and came up with an average mouth height for each frequency for open principal and stopped flute pipes. Then I generated equations to match the data. Obviously there is a range of mouth heights that will work for any given frequency but the following equations will give the beginner a starting point.

 For an open pipe: MH = 550 2^(ln f)

 For a stopped pipe: MH = [3.018 - 0.233 ln f]^5
• f is the frequency
• MH is the mouth height in millimeters

For example for an A=440 Hz pipe, the mouth height should be about 8.1mm for an open pipe and about 10.5mm for a stopped pipe.

The following table shows mouth height values as calculated by the equations and the averages from the websites.

I'll try to describe how I came up with the equations. I needed to relate the mouth height values to the frequencies. To do this, I needed to linearize each set of data. The frequencies were easy to linearize because they are exponential. Taking the natural log of the frequencies gives you a linear set of data. Next I needed to linearize the mouth height values. I found that taking the natural log of the mouth height values for open pipes gave me very close to linear data. I then took the two linear scales and related them with the following expression:

ln MH = 7.5956 - ln f + 0.30626 * (ln f - 4.1806)

Solving for MH gives you the equation above.

For the stopped pipes, the natural log of the mouth height values didn't give me linear data. I played around and found that taking the fifth root of the values gave me very close to linear data. The following expression relates these two sets of data:

MH^0.2 = ln f - 2.1376 - 1.2334 * (ln f - 4.1806)

Solving for MH gives you the equation above.