Steorn forum - Anne Elk
- If you keep up with developments in the cognitive science community, every few years someone anounces that octave equivalence has been confirmed in cats, or dogs, dolphins, wombats, monkeys, sheep or what have you. There is surprise and consternation, the results always hotly contested.
To me it seems a daft enquiry in the first place, but for a simple geometric axiom. I think all creatures use octave equivalence in a very fundamental way. Perhaps even plants too. Little green men, almost definitely. Here's why.
Imagine you're humming, whistling or singing a tune. Yodelling would also count (even moreso, you'll see why). You reach a point in the tune where the melody requires you to hit a note that is higher or lower than you can actually reach.
No problemo, we just switch up or down an octave, and hit the 'same' note but in a different register, and continue the melody 'unbroken'. We can do this instinctively, no calcs needed.
Ask any musician what octave equivalence is, and they'll likely say "a doubling or halving of a given freq". But wait a second, we could just as easily have switched up or down more than one octave! If we're just imagining the tune in our 'mind's ear', then the notes we can reach are no longer restricted by our actual physiological ranges. We could switch up or down many octaves and still complete the cadence without interuption. So octave equivalence isn't merely a doubling or halving a given freq... rather, the equivalence occurs at all factors of two of a given freq!
Now stop a second a think about this. What if we had a situation where zero and one were sometimes the same thing? What if we had a form of binary code where, at certain frequencies both bits were equivalent? How the hell would we encode anything with such a limitation? Is it really a limitation...?
I make the analogy to binary because it's a good easy place to start looking at the fundamental nature of information itself. The point i want to make here is extremely simple: in order to encode anything at all, we need just two fundamental elements - sameness and difference.
Sameness and difference is the very stuff of all information. In binary one is the same as one, but different to zero. Likewise zero is identical to zero, but diametrically different to one. This is the bare minimum of variation that can exist - a digital, absolute difference and sameness. In an anologue system we can have many levels of variation, but for now let's stick to digital to keep things clear.
So, as suggested above, how would we manage in a situation where, at certain speeds, zero and one became equivalent? Essentially, they have merged into a superposition where they are now functionally or qualitatively equal. If this happened in computer science would things be easier or more limited?
Coming back to cognition, we have a situation where in the auditory spectral domain all freqs synchronised by factors of two are informationally equivalent. Think about this - G#2 and G#9 are the 'same' note, just higher or lower. Of course the 'G' denomination is superfluous, what matters is this informational superposition.
WTF is the "auditory spectral domain" i hear you ask? Well say the average human hearing range is able to pick up all sufficiently loud frequencies between 20Hz and 20kHz. In reality the extrema become increasingly pitch ambiguous, so we only have accurate discriminable bandwidth of around 7-8 octaves and not the 10 implied by our total hearing range.
In cognitive science one particular are of study is the senses, how they work and the differences - and similarities - between them. One of the most fundamental things our senses do is differentiate between information in time, and information in space. This is done via "temporal integration windows".
- A TIW is the shortest discriminable gap between two discrete events. In audiology it is resolved using click train studies. A subject is played a sample and then indicates whether they heard one event, or two very close together events. In short, everything inside the TIW is coextant information - it is spatial information. Conversely everything outside the TIW is temporal information.
Inside the window, all information has undergone fourier-type reduction to become effectively synchronous, and this is why we talk of "spectral" and "temporal" domains in cognitive science. All it really means is "serial" and "parallel", or "simultaneous" and "sequential".
A point of note here is that in reality, all information is inherently temporal. We only have a sense of space because of our processing limitations. Time is real, but space is an abstraction, only relevant to our processing frame of reference. We're used to discussing the 'observer' frame of reference, but in actual fact this is a processing reference frame, in which we ouselves are 'causing' the illusion of a spatial domain through simple cognitive projection.
It seems that all our senses use this principle. So just as in audition, our visual process also has a TIW (and perhaps more than one - ie. retinal response freqs. vs cognitive response freqs). One way to observe this is the stroboscopic or 'wagonwheel effect' wherein a spinning wheel appears to undergo spontaneous direction changes. However the same geometric principle applies to all modalities, even somatosensory and gustation/olfaction.
Coming back to octave equivalence, we can do away with the maths completely and examine the situation purely in terms of magnitudes of complexity. Let's try this now:
Imagine we have a pure-tone freq F. If we monitor this on some kind of visual display we see a perfect sinusoid. If we paste another copy of F over the first, the peaks and troughs superimpose perfectly and what we now have is exactly the same frequency F, just slightly (3db) louder. We've increased the amplitude of the freqency, but it's otherwise unchanged.
From this experiment we can conclude that a 1:1 freq ratio is not a true harmonic interval, since it is an 'interval' of zero, ie. it doesn't exist. Therefore the unison 1:1 is not an information carrier in the spectral domain, since it only has sameness and no difference. Granted there is a difference in amplitude, and if we wanted to code information using a syntax of temporally-discrete amplitude fluctuations it'd be fine. But it can't encode information as frequency modulation. For this we need slightly more complexity.
Now consider a 2:1 freq ratio. The wavefunction resolves evey other peak and trough! In terms of how often the wavepeaks coincide, it is geometrically the simplest possible frequency relationship!
Now we're getting somewhere. We can say that from our processing perspective, maximum simplicity presents a functional or qualitative equivalencey. Maximum consonance = a functional or qualitative informational equivalence... an informational superposition!
Now get this - at any given moment we may be processing auditory signals across our entire hearing range. A car passing outside, the TV chattering away in the corner, the hum of your computer - whatever, the point is that all the frequencies we're processing are related by this same informational equivalency at all factors of two at all times!
Worse, we can also say that in complete silence, or in the extreme sense even during meditation or sleep, when we are not actually processing any external information, this equivalency domain persists - it is still there, such that if we were suddenly exposed to a tone signal, in isolation, these equivalencies at all factors of two of that frequency would still be there, albeit subliminally.
This raises some surprising conclusions - firstly, that we all posess this equivalency dimension, and that we are using it at all times to process all information, even in the absence of an actual octave stimulus or indeed any external stimulus at all.
- Also, because it is in essence merely a geometric convergence of optimum simplicity, it also applies equally to all information outside our sensory and cognitive TIWs. So for example if we tap out a four-beat with one hand, and an eight-beat with the other, this can be regarded as a 'temporal octave'. The same applies for a 3/6 beat or any time signature based on factors of two of a given fundamental. In fact if we take this argumet to it's conclusion we can make a rather bold statement:
This principle of the convergence of simplicity states that all information we will ever process in any and all domains can be regarded as temporal and spectral modulation of factor-two symmetries. While the maximum complexity of information in a signal may appear divergent or psuedo-random, conversely the maximum simplicity is absolute and finite. Therefore a factor of two frequency relationship is both the simplest type of harmonic interval, and also the simplest type of rhythm. All other harmonic and rhythmic relationships can be regarded in terms of finite complexity relative to this optimum simplicity.
Therefore all music, all language, all sensory perception and eveything we will ever know, think or feel can be described as modulation of factor-two symmetries. As it turns out, the Standard Model of physics is no exception...
I've made some simple geometric analysis re. cellular assemblies and the likey links between this symmetry and the underlying neurology - the main premise being that temporal and spectral factors of two pertain to connective and impulse rate equilibria, and utlimately cortical lateralisation and why we 'chunk' everything into groups of seven, though i won't go into that here.
Of course there's other ways of descibing this equivalency domain - powers of two, log base of two, it's also the most prominet overtone/undertone of the harmonic series, and doubtless other goemetric progressions. What has stunned me though is that it apears to describe all information. There's several salient examples here:
The first periodic table, proposed by John Alexander Newlands, was discovered on the back of this apparent "law of octaves". Newlands found there are seven groups of chemical characteristics, and that these repeat every eighth grouping - IOW there is a functional and/or qualitative octave equivalency intrinsic to all matter!
The quark electric charges in QCD are also defined by modulation of factor of two symmetries - 2/3 and 1/3. Likewise if we take halfspin as the minimum (or fundamental) unit of angular momentum then there's one octave of AM between each fermion and boson.
Therefore it seems all information everywhere, at a most fundamental level, can be described in terms of informational complexity in relation to the same convergent thermogeometric equilibria.
And so ever since i've discovered this i've been wondering; to what extent are the laws of physics - and everything else we know - a projection of how we ourselves process information?
Does the universe work the way we think, or do we work the way the universe thinks? I've written this up in a denser format here if anyone's interested.