Pythagoras: Part Four: The Music of the “Spheres” and the Significant Elements.

Armed with the work of Sir James Jeans, and Helmholtz, the study of the music which emerges from the Significant elements of Geometry, leads to the underpinnings of the Electromagnetic Spectrum, where the Octave, is to be considered from the 12 half tones of the cycle.

If we consider the “octaves” of light and sound, it is possible to overlay the color wheel three primary and three secondary colors, with their correlatives of vibrations, measured in Angstroms. It is also possible to see that the rise in “octaves” equates with the entire spectrum of vibrations, which are notably correspondents of real physical phenomena. If we observe the pattern of the “octave”, as “whole notes”, signified by the piano’s keyboard, we find that the three notes c, d, and e, fall on the white keys, while the three notes f#, g# and a# fall on the black keys. As such, the “scale” of whole notes, constitutes not 7 nor 8 notes, but rather 6 notes, evenly spaced according to the structure of the 12 notes that represent the “half tones” of the “chromatic” scale, a term which betrays the corresponds between visible color (not light) and sound. A scale with three “sharp” notes, f#, g# and a#, is one that does not sound like a natural progression of notes common to the generalized structure of music.

The chromatic scale of half tones, and the correspondence with 12 segments of the color wheel, permits the recognition of the relationship between the twelve pointed “star”, which is ruled as a continuous cycle of “fifths”, (clockwise) revealing a progression which travels around the wheel six times, eventually marking each spoke of a twelve spoked wheel, and reaching the point where the rulings first departed. Six “octaves” are covered in the process. With each octave raised, the vibrational length of sine wave which informs each note, is shortened proportionately equating with the tension placed on the string, or the length of the vibrating string, which may be shortened on a finger board of a stringed instrument, revealing octave to be a 1/2 measure of the original string. Each “arrow”: “>”  the single point and tip of the arrow, and the field between the two points of the radials extended back from the tip,  marks the relationship between complementary colors.

The process of constructing “stars” which indicate analogue models of continuous process, graphically portrayed as a two dimensional design, naturally begin with the construction of the isometric matrix, and the raising of its density, according to one of the two primary isometric grids which are ruled from top down and bottom up, or from west to east, and east to west (left to right and right to left). Naturally in a system such as the 12 half tones, of a chromatic scale, the 12 pointed star, is an obvious foundation, from which the generation of a numerical progression- in the case of the twelve pointed star, a progression clockwise, of “fifths”.

The variety of divisions of the 360 degrees of the circle, informed by the rulings of the matrix, either conform to the isometric rule, or not. As such the correspondence between the 2 dimensional image and the three dimensional construction is either true, or not.

With the raising of the density of the matrix to the point that the outermost hexagon shell, has each side divided into 12 segments, the run of natural numbers, from the center, raised by one integral point, with each shell, provides the necessary nodes for each numerical progression that is underpinned by the base 6 numeracy system, which has the greatest number of products, thereby supporting the decimal system of numeracy, from the relation between the base 5 and base 6, the former underpinned by the 6 degrees of separation inherent in each spoke, while the latter is underpinned by the 5 degrees of separations inherent in each spoke. As such the outermost shell divided according to the 72 spokes of 5 degrees of separation, gives the underpinnings of the 60 spokes of 6 degrees of separations, which is marked according the 5th hexagon shell, given that the counting of the hexagon shells, proceeds from the central integral node, treated as a point “zero”, hence the first hexagon shell, defined by each node which informs the 6 radials from the center: “>|<” expands accordingly, in a manner that informs the division of the horizontal diameter line, into 24 segments each degree of separation, having a ratio according to the greater circle which circumscribes the twin circles whose radii may be considered to measure 1 unit, informing the diameter line of the  “ocular” circle as 1 unit, and the greater circle as 3 units in diameter. As such the division of the horizontal diameter line, from the hexagon shells, has the measure of the ratio 30/24 or 10/8 or 5/4=1.25 this latter ratio, is interestingly not equal to the conception of Phi, as rt5/rt4 or rt5/2  = 1. 118033988749895. What is done equally to the numerator and denominator, with regards to the finding of the square root of a number, does not follow unless one returns to the analysis of the relations between the circles themselves, as a geometrically dynamic progression. This subject will be dealt with in the entry regarding the Harmonics of the Circles, as indeed will the study of the variety of spoked wheels, which inform the continuous star analogues, which correspond to the factors which are not apparently underpinned by the products of the sexigesimal system of numeracy.

With regards to the “Music of the Spheres” we are forced to consider the significant “image” of light, as it falls inside a hemisphere bowl, whose motion is caused by human hands, which is informs a cycle that may be captured on film, and may be reduced to the graphic representation of two primary “postures”. The correspondence of these postures informs the study of conics, and the ancient Greek study of the  “neusis”, the concoid or conchoid, and its application to the Epicycles of the Planets, as witnessed from the geocentric point of view. The mystification of this pattern is known as the “eternal flame”, of which its concealing/revealing has origins in the Indian tradition of Eastern Imperialism, common to the Hindu culture.

Its “return” to the West, as a mystical influence which enters the public domain, rather than a recognition of the work carried out by Ancient Greek Geometers, occurs by way of illustrations given in Leonardo da Vinci’s notebooks. The influence of Leonardo Da Vinci’s notebooks, is profound, and while he believed that his work supported the Catholic “bull”, he was in his own lifetime, subject to recriminations of heresy, which is not entirely understood by those who presume that there is no significant bridge between the “eastern model” of so called “sacred” geometry, and the primary study of significant graphic patterns which constitute the study of light and motion, and demand a recognition of the three sources of light, motion and mind. The Bowl is actively moved by the holder of the bowl, light is sourced naturally from the Sun, and the mind who is interested in intelligibly realizing the significance of such a pattern, is a human mind. All elements remain accessible on the surface of the Earth, as Kepler states of the Pythagorean insistence, to which he imposes a “correction”, and turns once more to the “quest” of applications employed for the purpose of rationalizing the details of the orbital paths of the planets, and hence leaving the Copernican model, which is topologically considered as a generalization of the planetary cycles, given as concentric circles.

In preservation of the Pythagorean or Platonic realism which infers that all elements are intelligibly accessed on the surface of the Earth, and as such without sophisticated technologies, other than the model building of the isometric matrix, the reconstruction of this modelling of cycles, is inspired by the inner surface of the hemisphere or bowl. The following graphic representations of continuous cycles according a specific numerical progressions that proceed from the center, and out to the perimeter, are given with regard to the modelling of the chromatic scale, yet can be seen to a inverse correspondence, to a different cycle, if the progression is considered figuratively speaking only, in reverse, and hence approaching the center, as it rises through the scale, in an ever diminishing “wave length”.

The unity between between the cycle of fourths, moving towards the center, and as such in an anticlockwise manner, is an attribute of the spiral which in a clockwise direction, counts the degree of separation as a cycle of fifths, though “rising” in the anitclockwise direction appears as a cycle of fourths. A cycle of fourths, spiraling towards the perimeter, increases proportionately to a cycle of fifths, as it reaches the center, and rises in frequency with a diminishing “wave length”.

The cycle of sixths and sevenths, reveals a remarkable correspondence to the structure of the pentagram, while the pentatonic scale, which exploits the structure all “black” notes on a keyboard: c#, d#, f#, g#, a# and these alone, was imposed by the Imperial Regimes of the Far East, tuned according to the “Yellow Bell”, for the purpose of maintaining an “automatic” harmony, constitutes a constraint placed over the experience of music, which permits all players a degree of freedom to engage harmoniously, without the need for a degree of care taken in the choice of note struck. The constraints of such a “scale”, bares no relation to the structure of the pentagon, rather the illusion of the “pentadic” image, is represented as the selection of five notes, and five notes alone. This form of constraint, constitutes a “conditional constraint”, which is culturally determined, and nationally defined. Another indication of a “conditional constraint” which is culturally determined is that which constitutes an imposition placed over the social sphere, common to Indian conditional constraints.

The imposition of the “categorical” hierarchy that is naturally self revealing and witnessed in the increased organizational complexity of the numerous variety of species and types from the four distinctions that are common to Earth, and considered as four basic sovereign “kingdoms”, is realised in India, as a conditional Social Caste, determined by one’s birth right, and hence imposing the conditional constraints of a Social Hierarchy,  that constitutes the “lot” that a “soul” draws, in the process of conception, given that there is a belief which informs the ideology that the  “Soul” either consciously chooses its parents (Buddhism), or is “judged” by the hidden god, according to its Karmic condition, according to the deeds and acts of the mortal’s former life on this planet (Hindu). As such this conditional model, supports the justification of a judgement which transcends the Authority of present judicial systems on earth, and justifies the judicial power maintained as arising from the top most class of Brahmin, or uppermost classes, as a predetermined inheritance of power, and birth right of the individual. This conditional constraint, and its expression which underpins the Hindu and Buddhist characterizations of the elements that inform both the traditional belief systems and the covenant of oaths sworn in allegiance to the “crowning” authority, be it a hidden elite, as in the narratives surrounding Indra, or a rule who assumes the right to wear the imperial crown of the nation, as indeed did Queen Victoria, in honor of the extension of the British Empire, which took authoritative control over the land and its people, and is graphically portrayed as the manifestation of the “lord” of categories,  recognized from the legendary narratives and their historically pertinent commentaries as the figure Ganesha.