Late Reply to Graham about Harmonic Entropy (and Carl, we need you!)

Hi Graham,

I was trying to trace the beginning of another thread and came across this very interesting set of answers to my questions (below) which, for some odd reason, I completely missed. This is all the more odd because I was very much looking forward to your reply and was quite disappointed, so sorry about that. Since its been a while, I'll put my new remarks in [ ].

In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> rick_ballan wrote:
>
> (Most of this message is one paragraph, so I'll split it.)
> > Thanks Graham, I can now state my case a little clearer.
> > Yes two sine waves an irrational interval apart would
> > still be easily Fourier analysable and predictable, no
> > less so than two rational waves. As you say both would
> > carry very little information.
>
> Being predictable means it isn't chaotic. Carrying little
> information means it has low entropy. Those are the terms
> we were talking about.
>
> > However, the first problem
> > I have with this interpretation is that in music at least
> > waves don't 'carry' (other) information but 'are' the
> > information. (I suppose a bit like McLuhan's "The medium
> > is the message").
>
> I don't think entropy's a problem there. It's a property of
> the signal, not the channel. And it doesn't distinguish a
> meaningful message from a random one.

[Ok, here is where I might be getting confused. Because wave theory led me to consider all waves as having certain properties in common, I probably tend to think of the 'signal' as having the same properties as the 'channel'. But I'm not sure what the distinction is in music. In most technical books, the signal usually means the radio waves that carry information as distinct from the information being transmitted, such as a song on the radio? Below you speak about how predictable the signal is, implying a musical signal, so I'm not sure what the channel means]
>
> > For eg, the perfect fifth 3/2 is a
> > periodic wave and tonal while the flat-fifth sqrt2 is
> > aperiodic and atonal. Hence my concerns about a model
> > which gives continuity around each rational, effectively
> > brushing over this distinction.
>
> Somehow harmonic entropy does distinguish these. I don't
> know what the information-theoretic model is for that. But
> it must come from an unequal probability for different
> intervals occurring. Maybe assuming all rationals up to a
> given limit are equally likely.
>
> That's never made a great deal of sense to me, I have to
> say. But I can see that applying information to harmony can
> be useful. You'd have to talk about probabilities in the
> light of listeners' expectations. It's all about how
> predictable a signal is. If you can always predict the next
> note/chord the entropy's going to be low. And the music
> will be boring. If you can never predict what's going to
> happen next, it's going to sound random, and that's not good
> either. You can see a lot of conventional harmony as rules
> to avoid these two extremes.
>
> > Second, ALL waves are
> > Fourier analysable irrespective of the 'amount of
> > information', so this doesn't seem to be an argument
> > either way. Since waves ARE the information and are not
> > employed into the service of some other theory (both
> > light and matter are composed of waves, and even
> > chemistry is subject to wave theory), then I propose (as
> > a hypothesis) that much of what we have previously
> > defined as chaos, randomness etc...is nothing more than
> > these irrational waves.
>
> Irrationality isn't chaos. There are very simple
> mathematical models for chaos. One of the criteria is that
> the result shouldn't be periodic. But another is that the
> evolution should be sensitive to initial conditions. That
> is, a small difference in the parameters explodes so that
> the state at a future time is completely unpredictable
> unless you have infinite memory.

> [I'll come back to this]

> That doesn't apply to a signal made up of two sine waves at
> irrational intervals (which isn't itself a wave). It isn't
> even truly aperiodic. I think it'd be described as
> "quasi-periodic". It almost repeats at various periods,
> corresponding to rational approximations to the interval.

[Incidentally, I'm not sure what you mean by 'which isn't itself a wave'. Did you mean 'periodic wave' because my understanding is that wave-iness is independent from questions of periodicity, or is there something I don't know?

But concerning "quasi-periodic", this is exactly what I have suspected for a long time but have been unable to formulate, hence the uncertainty principle and my interest in HE. Another possibility I've been exploring is that 'irrationals' are in fact large numbered harmonics with a GCD return period large compared with the initial two eg 5/4, 81/64, 645/512. However after reading what you said it just occurred to me that instead of them replacing the smaller ratios, the number of cycles per same time might be negligible??]
>
> > For example, the situation
> > becomes infinitely more complex when we begin to define
> > frequency as the GCD and then add these in two or more
> > irrational intervals. What if we take two irrational
> > intervals, say sqrt2 and pi, then three, and so on? Each
> > one of these will not repeat all the way to infinity, and
> > each one different to the next! And what's worse, there
> > are an uncountably infinite number of them. This number
> > is so vast that according to Cantor, if we were to throw
> > a dart on a number line, the chances of hitting a
> > rational is so small as to be zero. So if we were to take
> > a small sample of one of these waves, it would certainly
> > appear 'chaotic' or random. This is just an idea and I'd
> > be interested to know what you think.
>
> You're certainly going to have problems defining the GCD of
> irrational numbers. You could say this is what the ear (or
> ear/brain system) does when it looks for the virtual pitch.
> And you could solve the problem by substituting rational
> approximations. The evidence suggests that what the ear
> does isn't this simple, and the threshold for recognizable
> rationality is very low, so theoretical irrational-ness
> doesn't enter in to it.

[Ah great, thanks, that's a load off. And as I just said myself, we can for eg substitute 2^1/3 by 645/512 or any other rational to as much precision as we like. (Btw, I would never try to define GCD for irrationals because not having one is the definition of an irrational)]
>
> Yes, uncountability can be scary. But there are different
> ways of looking at infinities. You can also say that there
> are an infinite number of rational numbers between any pair
> of irrational numbers (and vice-versa). So in that sense
> you can't say there are more of one than the other. You can
> do any calculation to tolerable precision using rationals
> (and "real" numbers as stored in computers are normally
> rational approximations). I practical terms you can go a
> long way by ignoring irrationals.

[You remember Kronecker, Cantor's teacher and eventual nemesis, stating that "God created the natural numbers, mankind created the rest". In fact, after studying and thinking about waves and their place in the scheme of things for many years its now my considered opinion that much of 'pure' mathematics is no more real than the rules of chess. Perhaps we harmonists are Neo-Kroneckerites!]
>
> A small sample of a signal of two sine waves would look very
> simple. If you could sample it with infinite precision you
> could deduce its nature. If you quantize it, it will look
> *identical* to a pair of sine waves at a rational interval,
> and therefore periodic given the GCD.

>
> If you've worked with discrete Fourier transforms, you'll
> have noticed that all intervals between the partials they
> show are rational. That's because it's really a Fourier
> series that assumes you have one period of a waveform. It's
> that uncertainty principle biting you again.

> [And perhaps this is all there is? For if energy is quantized by h then continuous Fourier transforms would be the idealism, not vice-versa]
>
> Graham

[As an aside, you said:] Irrationality isn't chaos. There are very simple mathematical models for chaos. One of the criteria is that
> the result shouldn't be periodic. But another is that the
> evolution should be sensitive to initial conditions. That
> is, a small difference in the parameters explodes so that
> the state at a future time is completely unpredictable
> unless you have infinite memory.

[Just as a matter of interest, I've written a few scientific papers on the relationship between periodicity, time and predictability. Can't go into it here but very basically my hypothesis is that 1. matter is not solid but is composed of standing waves, therefore 2. our sense of 'permanent substance' and our very ability to 'predict' outcomes (repeat the 'same' experiment at different 'times') is only made possible because nature is inherently periodic. Hence, the numbers on a 'time-line' actually represent number of cycles, just as Pythagoras might have intended (if that makes sense). I'll send you a copy later if you like].
>
Anyway, thanks again Graham

Rick