Here’s the measurements that has been calculated for plastic tubing brass instruments. (Editor’s note: we can’t call ’em “hosaphones,” as that word has been, believe it or not, trademarked.)

The formula for calculating the length of a tube, given that you want it to resonate at a specific frequency is as follows:

The speed of sound

V(sound) = 340 m/s at sea level my measurements are in inches, so we have to convert the speed to in/sec by multiplying 340 m/s * 1/0.0254 in/m)
V(sound) = 13385.826 in/s

Frequency

The frequencies listed in the chart below are calculated by this formula: F(note – 1 semitone) = F(note)/(2^(1/12)) (that’s the 12th root of 2 in the denominator) Where F(A) = 220 Hertz

Effective Length vs Actual Length

The formulas for calculating the resonant frequency of a pipe were in three categories: both ends closed, both ends open, and one end closed. Brass instruments are in the last category. This means that one end is where the pressure disturbance is created, and is sealed to the atmosphere around it. The other end is open to the atmosphere and the pressure must drop to atmospheric pressure very shortly after leaving the end of the tube. How far out does the pressure node go? Well, it extends beyond the end of the tube at a distance roughly equivalent to the interior radius of the tube. With the pipe I was using, this is a little short of 1 inch.

    L(eff) = L(actual) + DeltaL(radius of the tube)


                   Figure 1            Single End Closed Pipe

  |< ----------------Length of Pipe---------------->|
  |< -------------Effective Length of Pipe------------>|

                                    Delta Length -->|  |< --

 Length = (V(sound) / (2*freq)) + interior radius of the tube

 The Chart:
    Note     Freq      Length (in) | Made
  ------+-----------+-------------+------
    AA# |   58.270  |   115.860   |   Y
    BB  |   61.740  |   109.405   |   N
    C   |   65.410  |   103.322   |   Y
    C#  |   69.300  |    97.579   |   N
    D   |   73.420  |    92.259   |   N
    D#  |   77.780  |    87.049   |   N
    E   |   82.410  |    82.215   |   N
    F   |   87.310  |    77.657   |   N
    F#  |   92.500  |    73.956   |   N
    G   |   97.999  |    69.296   |   Y
    G#  |  103.826  |    65.463   |   N
    A   |  110.000  |    61.845   |   Y
    A#  |  116.541  |    58.430   |   N
    B   |  123.471  |    55.206   |   Y
    c   |  130.813  |    52.164   |   Y
    c#  |  138.591  |    49.292   |   N
    d   |  146.832  |    46.582   |   Y
    d#  |  155.563  |    44.024   |   Y
    e   |  164.814  |    41.609   |   Y
    f   |  174.614  |    39.330   |   Y
    f#  |  184.997  |    37.178   |   N
    g   |  195.998  |    35.148   |   Y
    g#  |  207.652  |    33.231   |   N
    a   |  220.000  |    31.422   |   Y