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Sacred geometrical basis of the 192 rising & 192 falling intervals between the notes in the 7 diatonic musical scales

First (6+6) enfolded polygons
The first (6+6) enfolded polygons of the inner Tree of Life contain 386 yods when their (35+35) sectors are tetractyses. The topmost corners of the two hexagons coincide with the lowest corners of the two hexagons enfolded in the inner form of the next higher Tree. The 384 yods intrinsic to the first (6+6) enfolded polygons in a given Tree of Life comprise 192 yods in each set of the first six polygons enfolded in it. The 192 intrinsic yods in one set consist of a white yod at one endpoint of the root edge, two white corners of polygons that coincide with Sephiroth (Chesed & Netzach) & 189 yods associated with the six polygons. The other 192 yods consist of a black yod at the other end of the root edge, two black corners of polygons that coincide with Sephiroth (Geburah & Hod) on the other side of the central Pillar of Equilibrium & 189 yods associated with the other six polygons. Therefore,

192 = 3 + 189

and

192′ = 3′ + 189′,

where "3" (the three white yods) denotes the tonic, octave & interval of an octave, "3′" (the three black yods) denotes the unit interval between a note and itself, the next lower octave with tone ratio ½ and the falling octave interval of ½, "189" denotes the 189 rising intervals below the octave between the notes of the seven musical scales and "189′" denotes their 189 falling intervals (reciprocals of the tone ratios of the rising intervals). See also #37 & #38.

Inner Tree of Life
When divided into their (48+48) sectors, the (7+7) separate polygons of the inner Tree of Life have 192 corners, sides & triangular sectors surrounding their centres. Each set comprises three white or black corners of a hexagon that are spaced 120° apart and 189 other geometrical elements.

Sri Yantra
When turned into tetractyses, the 21 triangles in each half of the 3-dimensional Sri Yantra contain 189 yods. Turned into a tetractys, the central triangle has six hexagonal yods (three white, three black) arranged at the corners of two triangles that overlap in a Star of David. The three corners of the central triangle symbolise the triple Godhead outside objective existence. Other than these, each half of the Sri Yantra contains 192 yods when it includes the three hexagonal yods making up half this Star of David. See also #39. The 21 red hexagonal yods and the 21 blue hexagonal yods at the centres of the 42 tetractyses that surround the central one denote the 21 notes between the tonic and octave of the seven musical scales and their 21 inversions (see #36).

64 hexagrams
I Ching (易經), the ancient Chinese system of divination, is a set of commentaries about the 64 hexagrams, which are all possible pairs of trigrams selected from the basic set of eight. The first of the latter is the Heaven trigram composed of three unbroken (Yang) lines. The upper one indicated by three white lines in the diagonal of the 8×8 array of hexagrams symbolises the tonic, octave and octave interval. The seven remaining red trigrams with 21 lines & broken lines denote the 21 notes with tone ratios between 1 and 1024/729. The lower Heaven trigram in the diagonal indicated by three unbroken black lines denotes the unit interval between a note and itself, the octave note ½ & the falling octave interval of ½. The 21 remaining lines & broken lines in the seven blue trigrams denote the inversions of the 21 notes. There are 189 lines & broken lines in the 56 red trigrams above the diagonal that correspond to the 189 rising intervals below the octave between the notes and 189 lines & broken lines in the 56 blue trigrams below the diagonal that correspond to their 189 falling intervals. See also #39.

5 Platonic solids
Suppose that the 50 faces of the five Platonic solids are regarded as Type B polygons constructed from tetractyses, that is, each sector of their regular polygonal faces is a Type A triangle. Each of their 90 edges has four yods, so that (90×2 + 50 = 230) yods line them. Inside each sector/Type A triangle, seven yods line its three tetractyses, so that the 180 sectors of the faces of the five Platonic solids contain (180×7=1260) such yods. 1260 is the number value of Tarshishim, the Order of Angels assigned to Netzach. As well as these, two hexagonal yods lie on the 180 sides of sectors that join vertices to the centres of the 50 faces. The total number of yods lining tetractyses that make up the faces of the five Platonic solids = 230 + 1260 + 50 + 180×2 = 1900. The number of such boundary yods in the 25 faces making up five halves of these regular polyhedra = 1900/2 = 950 = 95×10, where 95 is the number value of Madim, the Mundane Chakra of Geburah. The number of boundary yods in the 50 faces that surround axes passing through pairs of opposite vertices = 1900 − 5×2 = 1890. As four hexagonal yods lie on each axis, the number of boundary yods either in the faces or on these axes and surrounding their centres = 1900 + 5×4 = 1920. On average, the centre of a Platonic solid is surrounded by (1920/5=384) boundary yods that are either in its faces or on its axis. Each half of a Platonic solid on average has 192 such yods, three of them lying on the axis and 189 yods surrounding it. The two sets of three axial yods (white ones in one half and black ones in the other half — see diagram) denote the tonic, octave & octave interval and their falling interval counterparts. The 189 boundary yods on average surrounding the axis that line sides of tetractyses in one half of a Platonic solid symbolise the 189 rising intervals below the octave between notes in the seven diatonic scales, whilst their counterparts in its other half denote the 189 falling intervals between these notes. The inversion symmetry displayed by the Platonic solids, whereby every yod in one half of a Platonic solid has a yod in its other half that is its mirror image, is the counterpart of the musical fact that every rising musical interval of magnitude n between two notes is accompanied by the falling interval of magnitude 1/n.

Seven yods line tetractyses inside each sector. Each set of halves of the five Platonic solids has 90 sectors with (90×7=630) such yods, where 630 is the number value of Seraphim, the Order of Angels assigned to Geburah. There remain (1890−1260=630) yods surrounding their axes that are distributed in both halves of all the Platonic solids. This 630:630:630 division is analogous to what was discovered in #37 for the 189 intervals below the octave between the notes in the seven scales. They were found there to divide into 63 notes & intervals, their 63 inversions and 63 intervals without their accompanying inversions. This triple division is what manifests in the table of 64 hexagrams as the three rows of lines & broken lines in each trigram. The 21:42 division in the set of 63 intervals and in the set of their inversions corresponds to the division of the 90 sectors in each set of half-Platonic solids into the 30 sectors of the faces of the half-tetrahedron, half-octahedron & half-cube and the 60 sectors of the half-icosahedron & half-dodecahedron because the number of yods lining tetractyses in the faces of the former = 30×7 = 210 = 21×10 and the number of yods lining tetractyses in faces of the latter = 60×7 = 420 = 42×10.

Here are five sacred geometries that embody the holistic parameter 384 characterising holistic systems (see also here). Their spatial bifurcation into two halves, one the mirror image of the other, expresses the distinction between the rising and falling intervals between two musical notes. The 192 geometrical elements that compose each half are the counterparts of the 192 rising/falling intervals between the notes in the seven diatonic scales. These geometries demonstrate the sacred geometrical basis of the seven diatonic scales, which constitute a holistic system. See also #60 in Wonders of correspondences. The 192:192 division that is displayed by holistic systems is discussed in The holistic pattern.

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