There is no good source of practical information available regarding the construction of pan pipes, so I did a couple of tests to establish some general principles for pan-pipe scaling. The question I was trying to answer was:
Given a pitch, what is the ideal internal length and diameter of the pipe.
Since cane is a natural material, it will not conform to the precise dimensions defined here, however these numbers should provide a reasonable starting point for experimentation.
Acousticians say that the sounding length of a stopped tube (such as a panpipe pipe) is 1/4 the wavelength of the sound produced. In actual practice, the open end of the pipe "loads" the resonating chamber, and the formula I have come up with, (based on simple tests using Lucite tubes of various lengths) is this:
Sounding length = 2.4123*Wavelength
The wavelength of a pitch can be determined as follows:
Wavelength = speed of sound / frequency
Given a sea level speed of sound of 345 meters per second, this produces the following graph. The red line is an idealized curve of all notes between F#3 (261.63) and A#6 (1864.66). The dark blue indicates values physically determined from cut Lucite tubes, and the green line are the intervals from Henry Thomas's Bull Doze Blues, which is a early blues recording featuring the rare American panpipe, the Quills.
This is the table used to calculate the values for the idealized curve shown in the graph above.
| Note | Frequency (Hz) | Calculated wave length (mm) | Pipe length (mm) | Pipe width (mm) |
| F#3/Gb3 | 185 | 1864.9 | 449.87 | 40.90 |
| G3 | 196 | 1760.2 | 424.62 | 38.60 |
| G#3/Ab3 | 207.65 | 1661.4 | 400.80 | 36.44 |
| A3 | 220 | 1568.2 | 378.30 | 34.39 |
| A#3/Bb3 | 233.08 | 1480.2 | 357.07 | 32.46 |
| B3 | 246.94 | 1397.1 | 337.03 | 30.64 |
| C4 | 261.63 | 1318.7 | 318.11 | 28.92 |
| C#4/Db4 | 277.18 | 1244.7 | 300.26 | 27.30 |
| D4 | 293.66 | 1174.8 | 283.41 | 25.76 |
| D#4/Eb4 | 311.13 | 1108.9 | 267.50 | 24.32 |
| E4 | 329.63 | 1046.6 | 252.48 | 22.95 |
| F4 | 349.23 | 987.9 | 238.31 | 21.66 |
| F#4/Gb4 | 369.99 | 932.5 | 224.94 | 20.45 |
| G4 | 392 | 880.1 | 212.31 | 19.30 |
| G#4/Ab4 | 415.3 | 830.7 | 200.40 | 18.22 |
| A4 | 440 | 784.1 | 189.15 | 17.20 |
| A#4/Bb4 | 466.16 | 740.1 | 178.54 | 16.23 |
| B4 | 493.88 | 698.6 | 168.51 | 15.32 |
| C5 | 523.25 | 659.3 | 159.06 | 14.46 |
| C#5/Db5 | 554.37 | 622.3 | 150.13 | 13.65 |
| D5 | 587.33 | 587.4 | 141.70 | 12.88 |
| D#5/Eb5 | 622.25 | 554.4 | 133.75 | 12.16 |
| E5 | 659.26 | 523.3 | 126.24 | 11.48 |
| F5 | 698.46 | 493.9 | 119.16 | 10.83 |
| F#5/Gb5 | 739.99 | 466.2 | 112.47 | 10.22 |
| G5 | 783.99 | 440.1 | 106.16 | 9.65 |
| G#5/Ab5 | 830.61 | 415.4 | 100.20 | 9.11 |
| A5 | 880 | 392.0 | 94.58 | 8.60 |
| A#5/Bb5 | 932.33 | 370.0 | 89.27 | 8.12 |
| B5 | 987.77 | 349.3 | 84.26 | 7.66 |
| C6 | 1046.5 | 329.7 | 79.53 | 7.23 |
| C#6/Db6 | 1108.73 | 311.2 | 75.06 | 6.82 |
| D6 | 1174.66 | 293.7 | 70.85 | 6.44 |
| D#6/Eb6 | 1244.51 | 277.2 | 66.87 | 6.08 |
| E6 | 1318.51 | 261.7 | 63.12 | 5.74 |
| F6 | 1396.91 | 247.0 | 59.58 | 5.42 |
| F#6/Gb6 | 1479.98 | 233.1 | 56.23 | 5.11 |
| G6 | 1567.98 | 220.0 | 53.08 | 4.83 |
| G#6/Ab6 | 1661.22 | 207.7 | 50.10 | 4.55 |
| A6 | 1760 | 196.0 | 47.29 | 4.30 |
| A#6/Bb6 | 1864.66 | 185.0 | 44.63 | 4.06 |
I've found that a ratio of between 10 to 1 and 15 to 1 for length to bore works well. I have been using 11 or 12 to 1 in my experiments. 11 to 1 is used in the example above. Good sounding pipes can diverge from this value and still sound well, which is a good thing because cane is rarely exactly the size you need.
For a practical example of how I've applied this information to the practical construction of a set of pipes, see this page: The Quills