A machine that could experience the equality of octaves in the same innate way as an animal, would, i believe, be a step closer towards a machine that could embody a true mind; whether synthetic or natural.
Synopsis:
Exploring the evidence for a hypothesis that the perception of tonal equivalence between octaves may be borne of a more fundamental and widespread property of general information processing; specifically, that it may be a key indicator to the manner in which higher-level information is bound to that of lower-level neuronal function.
It is often theorised that our affinity and propensity for music must be some kind of extraneous bonus feature arising from the interplays between other higher-level functions - that it simply 'piggybacks' upon all our other faculties. While doubtless largely correct, this view nonetheless seems wanting - music is such a visceral and innate experience, that it seems to speak the very language of the soul. Yet when it comes to trying to explain this intimacy, things tend to get very wishy-washy indeed..
For instance it is commonly suggested harmony may be reminiscent of soothing maternal cooing in infancy, or likewise with regards to rhythm, of her heartbeat during our most formative stages of development. These explanations too contain threads of reason, yet surely we can find firmer ground than this? Music is intricately mathematical, yet how might its structures relate to the goings-on of our multitudinous neurones? Where to start on such an abstract and complex problem?
Perhaps a slight shift in perspectives could help. What if, rather than viewing music as a 'happy accident' in the coming-together of all our other higher faculties, we were to consider instead that all of these other faculties are themselves, individually and in tandem, dependent upon the same underlying phenomenon that lead to the emergence of music? Inverting the previous appraisal: what if all of our other faculties were piggybacking on the same fundamental aptitudes that give rise to music?
With regards to the organic basis of harmony and and rhythm, I believe much of what has traditionally been all but dismissed as mere cultural fluke can, in fact, be neatly correlated to basic considerations in the challenge of processing natural information. In a nutshell, perhaps we like solving music because we do so in precisely the same manner that we solve everything else. In music, our minds are free to do en-masse what they like doing best... it's spoon-fed brain candy, processor pr0n.
A common code?
We experience a continuity and consistency of existence that, Occam would suggest, must be formed of a common processing system - as if there's a unified way information is represented, exchanged, integrated and processed internally across our full range of faculties, from sensation and motor control to higher functioning. Somehow, we effortlessly pull together information from disparate sources, filtering and mapping these diverse input streams into unified and coherent representations and models of reality. It stands to reason, then, that in some important but as yet little-understood way, all of these systems are sharing a related format, if not the self-same, unified format.
There are of course various 'levels' of information formatting, from the lowest-level of synaptic impulses and the pre-processing functions of their respective ganglions, up to the highest levels of conscious awareness, language, maths and arts. Clearly, impulse rate codings are the most obvious system-wide format, but this is too low-level to be able to address our problem - like trying to understand a chess program from its binary code. At the higher level format of sentience, we're oblivious to the workings of our neurons and so the question persists as to how information is bound to the things we actually experience, and how these data are then bound together appropriately.
If synapses were loosely akin to binary bits, and perhaps spike trains to bytes (both gross over-simplifications, but for broad analogy) then what might the computational principles of the processor be? Could it employ Boolean-like functions? How do we get from there to qualia such as the redness of red, the scent of a rose, or the memories and emotions it evokes?
The following is an attempt to clarify some hitherto under-appreciated points with regards to how we may process the 'stuff' of information at this higher level.
Everyday value
Faced with a page of mysterious binary code, we'd first look for common hexadecimal strings. The neural code of course is far more complex, however it too has apparently binary components - with the difference that they can vary in magnitude and duration, and also relate in parallel, as well as in series. Yet this is only one aspect of how we represent information. Clearly, how the network self-organizes and cooperates is another core substrate of the processing scheme. But to reduce all this complexity down into its most general possible terms; at root, any such system is going to have temporal and spatial dimensions.
That's a simple, free head-start from first principles. Next we can fill in some more blanks using similar points:
- all the information we process has temporal and spatial components.
- the coding scheme we're using can represent / encode these components, and store and reproduce them
- it can integrate and process them between multiple fields and modalities 'on the fly'
By necessity, then, the system is based upon processing of temporal and spatial signal components - it's concerned with relationships between frequencies in time and space, or simply, sequential and parallel relationships, and therein, meaningful patterns or 'forms'. It ascribes meaning to them as appropriate associations.. And most fundamentally, it has to represent these forms with some inherent value. That is, some malleable property or currency intrinsic to the system that can be used as a proxy substance for internal representations.
This inherent value is the stuff of information. It's the analog of hexadecimal to digital code. It's how information is evaluated. It can represent anything, and could be represented by any number of systemic factors, but one particularly critical consideration to mind is clearly efficiency; processing information costs energy, and natural information is usually highly complex; yet accuracy is absolutely crucial. Given that our central nervous systems consume around 20% of all the fuel we metabolise, whatever system of processing we've arrived at must be supremely efficient. It would probably be no risk of overstatement to suppose that the system has largely been designed by efficiency..
Taking all of these points together, we're now ready to take the plunge into some intriguing yet kick-yourself obvious insights into what the stuff of our everyday experience is actually formed of...
Common (mis)conceptions
This "continuity" and "consistency" of experience is of course perhaps tenuous at best, if not outright illusory. While a perfectly agreeable impression, look too closely and cracks start to appear - we're subject to the limits and constraints of our wetware. But cognitive illusions aside, the wonderful thing about logical paradoxes in particular is their ability to instantly peel back the wallpaper on a previously seamless reality to reveal an uncertain world beyond our projected screen of common, safe presumptions. Moreso, the resolution of a such a paradox might be a source of great contemplation in itself! Zeno's paradox of motion and infinitesimals, or causality vs free will, for example.
However one of the greatest sources of paradox, and subsequent revelation, for me, has been the percept of intrinsic "sameness" with respect to frequencies in a factor of two relationship. One tone remains distinctly higher, the other just as suredly lower, yet in some other, evidently not purely-tonal but "harmonic" sense, they're perceived as "the same". This paradox of octave equivalence turns out to be a rabbit hole so deep, and winding so far beyond even the most musicologically erudite speculations to date, you will shortly see that the matter simply has not been afforded the full investigation and final resolution it so deserves...
Far from being a mere tonal artifact, octave equivalence points to a general principle of processing per se, that is not unique to audition, and speaks to matters much more fundamental in the course of information processing in multicellular organisms generally. The cause of the equivalence is an innate consequence of network processing, and it reveals to us a chink in the armour of the neural code's complexity - it allows us to crack the value of zero, and differentiate (and thus begin to characterise) the substance of information in relation to that reference!
If at this point you're reminded of Newlands' ultimate humiliation before the Royal Society with his "law of octaves" proposition for organising the nascent periodic table, and wondering if i'm about to suggest what the harmonic forms for various qualia might be, rest assured, i'm only going to offer some simple axioms, and the conclusions are entirely up to you. Any speculation i offer will be clearly framed as such, but we'll be mainly concerned with delineating the apparent contradictions of the paradox, and trying to untangle the knots of familiar but etymologically subjective and unhelpful terminologies.
Consonance and dissonance are but degrees of equivalence!
First up on the chopping block are our cherished notions of consonance and dissonance. Notwithstanding that the subjective qualities of their meanings may prevail (ie. as synonyms for "sonorous" or "harsh"), their objective value at this point is nil, serving only to muddy the waters. We can return to such qualia later. The reason is that the simplest possible frequency relationship, in time or space, is a factor of two of a given fundamental or root. All other frequency relationships are, by definition, more complex. But none can be simpler, insofar as resolving to an integer minimum. The next most simple, or least complex is a factor of three, the harmonic fifth. Next we have a factor of five - since factors of four include those of two (4f = 2 x 2f), they're octaves. Next, 6f, and so on, which can likewise be expressed as ascending inverse-proportion integer ratios, or the product ratios of recursive divisions by two, etc. ie. the harmonic series, by whatever function one prefers.
It is significant, then, that maximum consonance correlates to maximum simplicity - factors of two don't merely sound consonant; they're maximally consonant! Consonance has an upper limit, and it's what we've been referring to as "octave equivalence", underlying our concept of pitch class. Ergo, what we've been calling "consonance" is, actually, simply this equivalence...
There is only equivalence, and degrees of inequivalence, or difference. Maximum consonance, AKA equivalence or "sameness", is an output of "zero difference" with respect to markedly different inputs, albeit in a factor of two relationship. If any assigned value would suffice, then "zero" is a perplexing and fascinating choice. Why would nature plump for such an apparently incongruous means of valuation? What conditions might cause such an outcome?
The mark of Zero
So you can probably see where this is going. It appears we're processing all auditory information, at the very least, in relation to some kind of entropic minima, where the simplest possible input signal elicits a null output.
The resolution of the paradox is that this "zero" value must be the stuff of metadata - because it is information we have generated about the input we're processing. It is a response that applies an internally subjective value to a relationship between stimuli, in a system wherein minimum entropy equals "no information", and anything else = "some information".
This harmonic valuation system is our default metadata response. In other words, what we've been calling "harmony" is something rather more profound - this 'harmonic' value, this 'difference from equivalence' assessment that we robotically apply in reaction to all auditory stimuli, all of the time, can only be the lingua franca of the broader processing system itself. It must be the base stuff oral language is constituted of, just for starters..
In upcoming posts i'll explain why oft-quoted explanations for the phenomenon invoking Deutsch's and Sethares' related works are lacking, and go on to show that this same "2f=0" principle extends universally beyond the human auditory system to other modalities and organisms... but for a closing thought, consider that all the natural sound you're hearing, in any environment, contains some frequencies in this state of equivalency, and all of the other frequencies are being modeled in relation to this system-wide "informational ground state"...!
