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Hi Mike,

Yeah sorry about that. Out of habit I've been replying via yahoo instead of reply to group at bottom so it comes up as a new topic. I think I did it right this time (except that I can't see your message at the bottom and so find it hard to remember. Also there's no topic in subject. Is that normal?). So 12-tet is 12 tone, got that now, and I'm presuming that 5-limit has to do with 5 cents or something, the vague area where we can't distinguish?

Of course you are right about that. I suspect that the 'creation verses discovery' debate is one of those issues that is never going to be decided, though we might make small steps in either direction, and I can't imagine any definite answer soon. And you are right about upper harmonics becoming so close as to be indisinguishable. Still its heartening that you think some of this can be tested.

Rightly or wrongly, I've come to think of the approach to intervals slightly differently than quantised tuning systems, or at least approaching the problem from the other side. More like the Hammond organ article you sent which has different 'systems' for different intervals (thanks again for that). As we've discussed already, the sharpened 12-tet major 3, while it does not correspond to 5/4, might still agree with other upper harmonics e.g. 645/512 = 1.259765625, where the difference b/w the tempered is only 0.000155424. As we reach the upper harmonics, I suppose more and more overtones can be clustered around each interval.

On the other hand, if we are starting with the 'known' intervals such as the major 3 (fully aware that this is itself up for debate), then the number of overtones separating adjacent pairs will also increase. So you see that I've been trying to find the basis of musical harmony in any tuning system by looking at the individual intervals (Why is the 12-tet and harmonic systems the most common? As you said, tradition or something more? Is there a 'better' or 'other' tuning system, and if so, why?).

Just remembered, in answer to the problem whether maths was created or discovered, it was Karl Popper who said that it was created to have the property that things can be discovered in it. For example, who could have plucked the number pi out of thin air? It had to be deduced. So to me the infinite number of harmonics is more like a field which we discover things in and use at our convenience rather than anything "real".

Thanks Mike

Rick

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