audible resultant from two supersonic frequencies?

[Roger Jolly]

Hi Robert,
                 Pipe organs have been using resultant pitches for eons.
Typically in the bass where they feel they need a 32ft pedal stop. The 16ft 
C and G are played together. Giving the resultant C at 32ft pitch.
Since the volume is continuous, it's a some what different case than the 
piano decay.
Regards Roger


At 04:39 PM 12/3/02 -0500, you wrote:

>Here's an experiment you can all do, and all you need is a piano.
>
>Play C6 and F#6 together, as loudly as you can.  If the theory is correct, 
>the resultant between F#6 (1484 Hz) and C6 (1050.5 Hz) will be 433.5 Hz, 
>which is just 25 cents below A4.
>
>I tried this, not expecting to hear any resultant.  I was quite surprised 
>to hear a faint A4, or something like it.  But as I listened, I noticed 
>that the A4 resultant behaved strangely.  As soon as the C6 and F#6 began 
>to decay, the A4 resultant suddenly dropped out.  It did not gradually 
>decay.  And if I play C6 and F#6 together softly, I hear no A4 resultant 
>at all.  This leads me to believe that the A4 resultant is not being 
>caused by C6 and F#6 beating together in the usual fashion.  If that were 
>the case, the resultant would decay exactly the same as the two primary 
>notes.  Instead, I think this is a case of non-linear mixing (as alluded 
>to by Sarah Fox several days ago).  When the volume level falls below the 
>threshold for making the distortion or buzzing, then the resultant tone 
>suddenly stops.
>
>There are lots of opportunities for non-linear mixing when playing two 
>notes on a piano.  No piano is perfectly solid.  If there is a loose screw 
>somewhere, or if two parts of the piano are just barely touching, then you 
>will have non-linear mixing.  (In the extreme case, these conditions cause 
>noticeable buzzing.)  At high volume levels, there can even be non-linear 
>mixing in our ears, when the bones that carry the sound from the eardrum 
>reach the limit of their travel.  In fact it is difficult to guarantee a 
>purely linear addition of two tones.  Any distortion of the sound causes 
>the component sounds to interact non-linearly, and thus produce real 
>acoustic energy at the resultant frequency.  For example, Don Mannino's 
>suggestion:
>
>>....Back to your original question, Ric. I made a recording for you, and 
>>you can hear the resultant tone.  I made a wav file of 20K and 20.5K 
>>tones, one in each ear.  Then I combined them into a single mono 
>>file.  If you play it in good headphones and turn up the volume loud, you 
>>can just hear the 500hz tone as a pitch.
>
>The process of making wav files includes time-sampling and quantization, 
>both of which are somewhat non-linear.  If there is a 500 Hz resultant in 
>Don's wav file, it is probably an artifact of the imperfections of the 
>recording process.  If it were possible to record 20 kHz and 20.5 kHz 
>tones and mix them with no distortion at all, then there would be no 500 
>Hz resultant.  Since it is hard to find such a perfect recording system, 
>then the theory can perhaps be tested by finding a poorer recording system 
>- one with more distortion.  If my theory is correct, the resultant tone 
>should be even stronger when played on such a system with more distortion.
>
>-Bob Scott
>  Ypsilanti, MI
>
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