Irrationals really are irrational.

Yes Hans all of that is mathematically correct, but I'm afraid I didn't make myself clear. Follow this line of thought:

1) periodic function = tonality: given two frequencies a/b where a and b are whole and a/b is in lowest form, then the wave will be periodic, the frequency being the highest common factor of a and b. Contrary to the conventional scientific explanation, this is the most general definition of frequency. Only b = 1 gives a sine wave corresponding to the fundamental. Further this proof can be extended to infinite trig series. I have found no better explanation of musical tonality where the tonic is always octave equivalent to this H.C.F. frequency.

2) aperiodicity = no tonic: But now look at the interval of the flat-fifth which is 1: sq root 2. Recall Pythagoras' reductio proof that no rational number a/b corresponds to this irrational (I'm sure its on the internet. I can't remember it precisely now). This means that waves in such a relation never repeat in exactly the same way, so that this erratic behaviour extend to infinity. But it gets worse. Georg Cantor proves that there are an uncountably infinite number of such waves. (He said that if we were to throw a dart on a number line, the odds of hitting a rational is so small as to actually equal zero). Given a small section of one of them, it might be near impossible to find where it came from. I imagine that much of what we assign to probability and chance might come down to the existence of these waves, but it's just a thought. Mathematicians are wrong about that: irrational numbers really are irrational in every sense of the word.

3) So when you've spent some time in this world, in comparison harmony (periodicity) ends up seeming like a rare, precious jewel, something to be harnessed and defended, which is what the art of music usually tries to achieve (Aristotle says that we know there are poisons in the world, but this doesn't mean we have to drink it).

4) However, it might be that these irrationals end up being ration after all, which is the problem I've already proposed to the group. If we cannot distinguish beyond a certain number of decimal places, then there will always be rational numbers that are not approximations but are the actual intervals. I imagine a number system where for eg all numbers between 1.9998 and 2.0002 are equal to 2, or something to that effect.

The problem I think is that the western education system over the last few centuries has 'taught' us that maths and science are more fundamental disciplines than music and philosophy. Scientists will study the chirps of some bird, but the entire musical tradition of the human animal is passed by without hardly a second thought. It seems absurd to me that the musical nomenclature would not encode the basics of wave behaviour, or that mathematicians who deal with abstract spatial concepts of there own invention would qualify in these matters.

Thanks Hans

Rick

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