MUSIC VS NOISE PART FOUR: TETRAHEDRAL LOGIC : SOME BASICS What exactly is tetrahedral logic? The term ‘tetrahedral logic’ is the term I utilize to describe a form of logic based on the tetrahedron as the core unit of one. This is not an original concept by any means. For me, the first and most significant influence regarding this form of analysis is without a doubt R. Buckminster Fuller, author of Synergetics (vol 1,2) and inventor of the geodesic dome. While he authored many other books and invented countless other artifacts, none are of the same magnitude as the aforementioned. Distilled to its underlying principle, Synergetics is a science based on the notion that a LIVING SYSTEM is MORE than the sum of its parts. When any given additive is discovered to result in more than its classically correct sum, the result is said to be synergetic. Case in point – we are certainly more than the sum total of the elements that comprise our bodies. However, in general science finds it difficult to account for or identify with specificity exactly what the excess is comprised of in most synergetic systems. Bucky Fuller proposed some very ingenious ideas in the two aforementioned volumes, and for those interested in this subject I cannot stress how vital the reading of his material is. The second item of note, his geodesic dome, is of particular importance because it is the only structure that increases in stability as it increases in size. This is because it is based on the tetrahedron as its generative unit as opposed to the cube, which is the traditional primary unit utilized for modern computation and analysis. Note that when we raise a number to a given power, it is said to be ‘squared’ or cubed’ as opposed to ‘triangled’ or ‘tetrahedral’. This nomenclature alone suffices as evidence that the vast majority of computation is reckoned in cubic terms. Further evidence is found in almost all architecture; namely, our houses, apartment complexes, office buildings, right on up to our most grandiose high rise skyscrapers are all some form of stacked cubes, and in the final analysis tend to be rectangular boxes situated on their minor face. Incidentally, the pyramids in Egypt and all similar are NOT tetrahedrons, they are pyramids, the difference being that a pyramid has a square base and five faces, whereas a tetrahedron has four identical faces, each being an equilateral triangle. The tetrahedron is the first possible polygon consisting of flat faces and straight edges, and as such the first manifestation of three-dimensionality with said characteristics. It is often referred to as the first of the five Platonic solids, though an argument can be made for any proposed order of consideration. However, the fact remains that it is the first three-dimensional possibility and this is why it is the shape upon which computational logic must rely if it is to be rendered in the most stable and succinct terms. Until recently, I had not seen the term ‘tetrahedral logic’ specifically utilized in any formal sense, and considered it more or less a part of my own personal lexicon of ideas insofar as I have applied the logic in analogous fashion to music, as well as to some other, more experimental (and often incorrect) propositions. It is the application to music that I am primarily discussing in these posts, but herein I will identify and illustrate one or two other manifestations in the spirit of enhancing the reader’s understanding of how I utilize the term. The reason I take the time to do this is because recently (as in just earlier this year) it seems that several websites claiming to familiarize one with tetrahedral logic have appeared, most of which are not even remotely connected to anything I have ever offered. There are one or two exceptions, including one entitled “Principles of Physical Complexity” posted by an entity calling itself ‘Memes Ltd”. I have not thoroughly examined every page of the site, but have read and internalized enough of it to satisfy myself that any similarities to my ideas are the result of the common influence – Buckminster Fuller – and nothing more, as the author of that site seems to be concerned with an entirely different agenda, not to mentioned it is a much more thorough presentation than any of my own. However, I must mention one passage presented early in that site, namely: Logical reasoning was carried out with the aid of a so-called logical tetrahedron consisting of four different, but mutually dependent variable properties and the six possible relationships between these four properties (see figure above). This tetrahedral model allows examination of all possible interactions between four different variable properties. The reason this phrase jumped out at me is because I am almost certain that it may be found in nearly exact terms in one of the Synergetics volumes. However, as the volumes are in excess of a thousand pages, I have yet to scrutinize them again for this purpose . I leave that to a terrifically bored reader Anyway, it is this principle that is the underlying theory of tetrahedral logic. So how does this apply to music? Well, by transference we may posit that if valid, the solfeggio frequencies may be modeled in tetrahedral terms if they are in fact ‘structurally sound.’ In this case I reckon structural solidary as the equivalent of harmonic accuracy (meaning specifically the rate or occurrence of deviation from geometric progression as one progresses through the scale. Again I refer to the 53 tone scale generated by the C-128 tuning utilized in classical music and outlined in the previous installment I posted on this issue.) Lets start with the solfeggio scale generated from C 128, or the A 432 scale. While consisting of twelve tones, I propose that it in fact may be a product of four tones that correlate with the four faces of the tetrahedron – meaning that four tones of the scale will be found to have a common property, analogous to the common triangular faces of the tetrahedron. And? First we must assign a numeric value to one face. If the tetrahedron itself represents a three dimensional unit of one, then one face might be reckoned as .33333 --- in 3D terms. However, if we want to consider a face as a wholeness, in flat terms, we must compute a face in the context of a presupposed existence (as true flatness has a thickness of zero and therefore is non-existent) so we may therefore transfer the analogy one step ‘downward’ and compute it as the sum of three vectors, representing one fourth of the whole, in which case we simply put a ‘1’ in front of the decimal, in the same fashion as we transfer units of ten as wholeness in math. This gives us 1.3333333 as a functionary. So what now? Recognizing that the tetrahedron is in fact the sum of two vectors, as opposed to four, we will look for two ‘musical vectors’ connecting ‘four vertices’ in the solfeggio scale. And lo and behold, we find that Solfeggio [396] x [1 .3333333~] = Solfeggio [528], Solfeggio [639] x [1 .3333333~] = Solfeggio [852] Furthermore that these are the only frequencies connected thusly. May we then conclude that the analogy of the connection between each set of two tones to that of the two vectors of the tetrahedron is valid? Or furthermore that the 1.33333 –ness of the base reference unit of commonality as relating to the whole, is valid? Or that the fact that four and only four of the tones, so reckoned, might be analogous to the four faces of the tetrahedron? This is just the beginning. More to come shortly.
Posted on: Sun, 15 Sep 2013 03:55:28 +0000
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