gavriel wrote:In the case of combining
the frequency of F3+F5
the main picks appear at the multiple of 174.61 Hz - hence at the frequency of an F.
BUT the secondary picks appear at 440 Hz !
generating a "phantom 'A'-ness"
I still don't think I understand. What are picks? Do you mean overtones? Or do you mean peaks? Or do you mean the points at which the wave crosses zero level? Like in this picture that Chris showed before:
If you mean the zero points, then I think I understand your F3+F5 example. If you distort a signal, like overdriving a guitar amplifier, the distortion creates new overtones that are sums and differences of the original signals, and the sum of the frequencies F3 + F5 = 880Hz, or A5.
There is potentially a problem here, because E6 on its own has the same zero crossings as A 440, and so does C#7. And if you take A 110Hz as your starting point, then E4 has the same crossings, as does C#5 and B5. And you really need pure tuning here to get the crossings really correct.
But if I'm understanding you correctly, then you are suggesting that we should be essentially listening to overtones generated by distorting sinewaves of the same chroma, because these overtones have some other chroma. And the sinewaves don't really need to be deliberately distorted, because there is always some distortion in the ear anyway.
Did I get your point?
gavriel wrote:another example originates in the art of counterpoint of the 15th century.
paralel fifth and octaves were not allowed.
One fo the reasons is that in a church with a huge reverb, when a two voices sing an octave that is sliding upwards in paralel. The signal is phasing with the echo of the octave, generating a different hidden pitch, that was perceived as a disturbing dissonance.
I've heard that consecutive fifths and octaves were forbidden because if two voices sing an octave or a fifth apart, they sound like one voice, and for a four part harmony to sound good you can't have two voices sounding like one, because then it sounds like a three part harmony.