Acoustical aspects of the singing bicycle project

When we first designed the Singing Bicycle Project  in 1976, calling it our 'Second Symphony' with some irony, we calculated a series of 24 pipes such that they would all be tuned to ideal harmonics of one single fundamental pitch. The piece was conceived as using 'just intonation'. Although the piece has been performed over a hundredth times since 1980, we never actually verified the acoustic and harmonic result against what we theoretically conceived. In 2014 we performed a series of precize measurements and -not really amazing- found out that the initial concept belongs in the land of fairy tales. We were not really amazed because in the last twenty years we have been involved quite deeply into acoustic research on musical instruments and robotics. This research proved clearly that there is not such a thing as 'harmony' or 'just intonation' anywere in the world of acoustics, be it strings, pipes ,let alone more 3-dimensional vibrating objects.

Loudspeakers driving cilindrical tubes in fact form a pretty complex acoustic system. The traditional acoustic theory with regard to open and closed pipes, does not seem to apply here. The one end, closed by the loudspeaker does neither behave as a closed nor as an open end. The loudspeaker behaves as a membrane driven in its center and shows off a highly non-linear response curve. Thus we have to do with two mutually dependent vibrating systems. The table below gives the results of a long series of measurements, revealing this clearly.

Set-up for the experiment:
Visaton loudspeaker, Type K50WP, 50 Ohms, 3W. Frequency response 180-17000Hz. Resonant frequency 300 Hz. For details and response curves, see spec. sheet.
PVC Pipe 50 mm diameter (internal 46 mm), fault 0.1 mm. Measurement fault on pipe lengths: 0.5 mm. The speakers were glued firmly and airtight to the tubes using PVC cement glue. (Tangit).

The first column in the table gives the pipe length. The second column gives the resonant frequency for such a tube resonating at 1/4 wavelength and with end correction applied according to the textbook formula found in most books on acoustics. The thirth column gives the lowest resonant frequency as measured by sweeping the oscillator from 0Hz upwards. The next colums give the measured frequencies of the next clearly discernable overtones. The excitation was always a pure sinewave.

Measuments performed using an analogue sine wave generator with a digital frequency counter. Resolution 1 Hz. The output impedance is 50 Ohms and thus matches the nominal impedance of the drivers. The precision is better than 1%. Output voltage was kept constant at 3 V rms for all measurements. For most tubes, some resonance was also observed around the octave and other even overtones, but at a level too low to be easily measured. 
The larger the l/d ratio becomes, the closer the first overtone comes to 3.0 times the fundamental. The second overtone seems to move from 3 times to 5 times the fundamental.  The factor for the thirth overtone is from our data not easily discerned. It also seems to increase with increasing tube length.

We used the measurement data to obtain a third degree equation, using a Gauss fit procedure. The equation becomes:

freq = 309.048 - 0.4617146 *  x  + 3.013809E-04 * x^2 - 7.109726E-08 * x^3

The tube length (x) is expressed in mm.

A 6th degree equation, obtained with a Gaussfit program calculated over 20 data pairs taken in the length traject between 395 mm and 1582 mm leads to errors less than 1%. It looks like this:

freq =  317.6497 -.4827873 x  + 2.720424E-04 x^2 + 2.693052E-08 x^3  -5.110973E-11 x^4 -1.656992E-14 x^5 + 1.248304E-17 x^6

(*) The audible result we observed here is two separate tones, one octave different. The fundamental f0 sounds together with f1.