Friday, August 5, 2011

New Perspective on Harmonic Series

The very important and well-known harmonic series i.e. 1 + 1/2 + 1/3 + 1/4 + 1/5 +... is especially associated with Pythagoras who reputedly found in these simple fractions definite links with the manner in which musical notes sound. This pattern of the simple natural fractions, in particular, seemed to correspond perfectly with - what we would recognise as - a harmonious sequence of notes and thereafter it has become known as the harmonic series.

So we have here a very clear link as between simple mathematics and musical harmonics. Subsequently it was shown that this series (and its associated series where the dimensional power of each fraction itself can vary from 1 and ultimately alter over the entire range of complex numbers) has intimate connections with the prime numbers!

So in a certain valid sense there is music in the primes. So just as we are accustomed to give a wave form to musical sounds likewise there is a wave pattern associated with each prime number.

Indeed we could go further and suggest that there are intimate links as between the prime numbers and quantum mechanics with each possessing particle (discrete) and wave (continuous) aspects. The real implication is that just as music itself has quantitative and qualitative aspects which interact to produce the experience that we recognise, likewise - when appropriately understood - prime numbers and quantum mechanics likewise arise from the interaction of quantitative and qualitative elements.

As we have seen the harmonic series is made up of the reciprocals of the natural numbers.

Now each of these numbers has a dimensional aspect as the corresponding fractional powers, which results in a circular - rather than linear - quantitative structure. The corresponding qualitative numbers are then simply the natural numbers, 1, 2, 3, 4, 5.. etc now interpreted in a holistic manner.

Recently I have come to the conclusion that quality in all its aspects ultimately relates to this dimensional appreciation of number (interpreted in a holistic manner). In this context each dimensional number represents a unique way of configuring the fundamental polarities of experience (internal/external and whole/part).

So quality in the most basic sense always arises from a certain manner of configuring these basic polarities (which is given by the qualitative interpretation of dimensional numbers).

Seen in this light the qualitative correspondent of the harmonic series is simply the natural number dimensions. So it is not surprising that our fundamental appreciation of harmony (relating to the manner of configuring polarities) corresponds directly with the simplest dimensional numbers. These are the very ones with which we are implicitly programmed in out attempts to discern qualitative meaning!

Monday, August 1, 2011

A Double Code

In a post on my blog on the Riemann Hypothesis, I referred to a new Mathematical series on the BBC hosted by Marcus du Sautoy called "The Code". I have already seen several such programmes with du Sautoy who I enjoy watching for his enthusiasm and obvious love of the subject. I also enjoyed greatly reading his books "The Music of the Primes" which played a large role in pushing me on to develop my own own insights on the primes. More recently I read his later book on Symmetry - an area in which he specialises - "Finding Moonshine".

Now the code that du Sautoy is referring to to relates to the quantitative use of numbers that is so wonderfully successful in helping to clarify so many of nature's secrets.

However Mathematics equally contains another marvellous code in the qualitative interpretation of these same numbers. However, seemingly there is as yet little or no recognition of the potential significance of this latter code.

I have frequently explained how the qualitative meaning of numbers is intimately tied up with mathematical dimensions. Unfortunately from a conventional perspective, only a reduced linear rational interpretation - that is literally 1-dimensional in qualitative terms - can be given.

In the most fundamental sense this true qualitative nature of dimensions relates to the dynamic manner in which the polarities of experience interact. In the most basic terms these polarities come in two pairings. The first relates to external and internal whereas the second relates to whole and part distinctions.

Now the conventional scientific (and mathematical) perspective is thereby linear as it seeks to define one unambiguous direction with respect to these polarities. In other words reality is here identified in a non-interactive fashion with what is understood as external to the observer. It then deals with wholes and parts in a reduced quantitative manner (whereby parts are simply viewed as fragmented constituents of the whole).

However though explicitly such scientific interpretation is indeed linear, implicitly in experience dynamic interaction in varying ways still takes place allowing for a much more varied dimensional appreciation (where many dimensions can interact to a degree with each other).

What I have recently come to realise is that our qualitative experience of reality in fact relates to the manner in which these various dimensions (representing the interaction of the fundamental polarities) takes place. And as each unique dimensional configuration is represented by a number, this thereby entails that all experience can be represented as the dynamic interplay of numbers (with respect to both their quantitative and qualitative aspects).

In a direct sense this qualitative dimensional aspect relates to the affective function in experience (which can then indirectly be translated mathematically in a cognitive rational manner).

So understood from this perspective our emotional experience with respect to sense and feelings represents the manner in which we configure various number dimensions (from a qualitative perspective).

One especially fruitful area of investigation would relate to musical appreciation. It has been demonstrated - at least since the time of Pythagoras - that there is a marked quantitative aspect to the manner in which we experience sound. So sounds that appear to us especially harmonious relate to the simplest ratios involving the natural numbers.

However an even deeper explanation of why this appears to be the case would relate to appreciation of the qualitative - as opposed to the quantitative - interpretation of number.

What I am suggesting therefore is that because our qualitative experience of dimensions remains somewhat undeveloped in present human evolution, we find it difficult to appreciate all but the relationship as between the simplest natural number dimensions. So the harmony in sound that we thus experience relates to the greater familiarity we have with the easiest dimensional configurations.

Put another way, for someone who in this affective context has implicitly achieved a more developed dimensional understanding, then a much more refined appreciation of sound would then be possible. Then what might appear dissonant at a customary level of experience, might now appear quite harmonious!

This also provides an interesting explanation as to the difference as between artistic and scientific type truth.

The very basis of current scientific truth is the universal acceptance (explicitly in formal terms) of its qualitative 1-dimensional manner of interpretation. However the basis of artistic appreciation is that people implicitly, as we have seen, use varying dimensional number configurations in attaining their qualitative understanding. And as such configurations can vary widely from person to person this implies that universal agreement as to what constitutes artistic merit is not therefore possible.

Of course in some respects a conventional consensus (at least in certain sections of society) may exist as to what constitutes artistic value but this simply reflects a shared qualitative perspective among those with influence, who thereby configure the qualitative dimensions in a largely similar manner.

Also it must be said that even where the same qualitative dimensions are involved, marked variations in subsequent experience (relating in turn to considerable differences as to the degree of spiritual refinement with which they are experienced) can exist.