If the 421 polytope is truly an archetypal object whose E8 symmetry embodies the superstring physics of matter, one should expect to see correspondences in its geometry with not only sacred geometries but also the seven types of diatonic scales or the 8 musical modes, which the author has shown in earlier articles correlate with the 64 hexagrams of the I Ching, the Sri Yantra, the inner form of the Tree of Life and other sacred geometries. And, indeed, one does see this! This page will reveal how the 8 Church modes manifest the 24:168 pattern that is characteristic of holistic systems and which appears in the Petrie polygon of the 421 polytope, as previous pages in this section have shown. The musical analogy is too detailed to be due to chance (see below). Instead, it indicates that a fundamental pattern embodied in sacred geometries pervades different types of systems ranging from E8×E8 heterotic superstrings and DNA to the ancient Mesopotamian heptachords and the ancient Greek musical modes that were part of the historical origins of the 8 musical modes used in the plainsong of the Catholic Church (for background, see Article 14 and pp. 1-16 of Article 16). This holistic pattern is discussed in detail in The holistic pattern.

The 12 types of notes comprise the 6 notes between the tonic and octave of the Pythagorean musical scale and 6 notes that are non-Pythagorean (written above in red). Including the tonic and octave, there are 7 pairs of notes and their inversions (N.B. the inversion of a note N with tone ratio n is the note whose tone ratio is the interval 2/n between N and the octave):

If these two Kabbalistic numbers were the sums of numbers randomly taken from the list shown above, it could not be convincingly argued that this is evidence for the number values of both words being naturally generated. But what are the chances that the first 3 types of Pythagorean notes with tone ratios 9/8, 81/64 & 4/3 (i.e., a set of consecutive types of notes) would, by coincidence, generate the number value of one of these words? Obviously, they are small. This argues against the appearance of Kabbalistically significant numbers being due to chance because, if this were so, they would be the sum of numbers scattered randomly throughout the list. They would not be the sum of three consecutive numbers (all referring to Pythagorean notes) that start from the lowest one — hardly a random sequence!

For a musical scale with 8 notes, there are 8 unit intervals between each note and itself and 28 intervals between pairs of different notes. This makes a total of 36 intervals. They include the unit intervals between the 6 intermediate notes and themselves. There are (36−6=30) other intervals. They comprise the 21 intervals between the 7 notes above the tonic, all 8 note intervals and the unit interval (the tonic, of course, also has a tone ratio of 1, but this is conceptually different from the unit interval, which is not a note but the ratio of the tone ratios of a note and itself). A minimal set of 30 intervals (tonic+28 intervals+unit interval) define each Mode when we view it as a sequence of 8 notes that can start from a given note of any octave, i.e., the tonic need not be exactly the same note for each Mode (it could be a different octave), so that is why it must be counted for every Mode, despite its tone ratio having the same magnitude as the unit interval. The 8 Modes have (8×30=240) such intervals that exclude the redundant unit intervals between the 6 intermediate notes of each Mode and themselves They comprise the (8×21=168) intervals between their last 7 notes and the [8×(1+8)] = 72 intervals that are either notes or unit intervals. The 12 basic notes between the tonic and octave are repeated 36 times as notes, although, of course, not in equal numbers. As intervals that are not notes, they are repeated 168times. According to the table above, the 168 intervals consist of 110 Pythagorean intervals and 58 non-Pythagorean ones. They comprise 118 intervals up to 1024/729 and 50 higher intervals (inversions of the former).

These results may be summarised in the following equations:

The division of the number 10 (Decad) into 3 and 7 expresses the separation of the 10 Sephiroth into the 3 members of the Supernal Triad and the 7 Sephiroth of Construction. Its arithmetic consequence is the dividing of the number 240 into the numbers 72 and 168. It appears in the 8 triacontagons as the division of the 30 corners of each triacontagon into 3 decagons (coloured red, green & blue in the picture shown below):

Each Mode has 27 intervals below the octave between its 8 notes, of which 6 intervals are those between the tonic and the next 6 notes and 21 are intervals between the last 7 notes. The total number of intervals for the 8 Modes = 8×27 = 216 = 8×(6+21) = 48 + 168, where "48" is the number of notes in the 8 Modes between the tonic & octave and "168" is the total number of intervals betwwen their last 7 notes. This has a natural interpretation in terms of the 8×8 array of 64hexagrams used in I Ching, the ancient Chinese system of divination, which was compared with the triacontagons on page 9. "48" is the number of lines & broken lines in the 8 diagonal hexagrams and "168" is the number of lines & broken lines in the 28 off-diagonal hexagrams on one side of the diagonal. So there are 216 lines & broken lines in a diagonal half of the array. The number of lines & broken lines in the whole array = 384 = 168 + 216 = 168 + 48 + 168. This has a musical interpretation in terms of the 48 notes between the tonic & octave (24 in each set of Modes, or, alternatively, 24 notes up to that with tone ratio 1024/529 and 24 notes higher than that), the 168 rising intervals & the 168 falling intervals. These compare with the two interior angles of 168° in two adjacent sectors and their diametrically opposed counterparts and with the pair of vertex angles of 24°:

We are not claiming that these symmetry groups actually describe the 8 Modes. For the present purpose, it is sufficient to demonstrate the existence of numerical analogies between the Modes and these symmetries that imply the existence of a universal pattern because these correspondences are too detailed to be dismissed as due to coincidence. The following implications of the existence of this pattern are profound: 1. E8×E8 heterotic superstrings exist; 2. the UPA is such a superstring because its structure, paranormally described over a century ago by Annie Besant & C.W. Leadbeater, conforms to this pattern embodied in both sacred geometries and holistic systems, such as the Church Modes.

The tonic, octave & unit interval in the set of 30 intervals associated with each Mode correspond to the 3 successive corners of two adjacent sectors of each triacontagon whose interior angle is 168°. The (8×27=216) intervals below the octave between the notes in the 8 Modes correspond to the (8×27=216) remaining corners of the 8 triacontagons. The number 216 of Geburah, the 6th Sephirah of Construction, is a structural parameter of the inner Tree of Life, being the number of yods per set of 7 enfolded polygons that line the 88 sides of their 47 tetractyses:

It is also the number of yods that either surround the centres of the first 6 separate polygons with 36 corners, themselves a holistic system (see Article 4), or line the boundaries of the first (6+6) separate polygons. 36 is the number value of ELOHA, the Godname of Geburah, whose gematria number value is 216.

The tetractys is the Pythagorean representation of holistic systems. A tetractys array of the number 24 generates the interval composition of the (4+4) Modes:

As pointed out in #2, each set of 4 triacontagons with 120 corners are the Coxeter plane projections of the 120 vertices of 5 24-cells that make up a 600-cell. This means that the (4+4) triacontagons are the projections of the 240 vertices of 10 24-cells, the Coxeter projection of the 421 polytope being that of a compound of two 600-cells. This in turn means that possibility 2 is the correct one, for the following picture shows that the 24 vertices of each 24-cell actually coincide with 6 corners of a heagon/hexagram in each of the 4 triacontagons that are the Coxeter projection of the 120 vertices:

Now let us see whether the representation shown above of the number 240 as a tetractys of 24s is more than just arithmetic, i.e., do the numbers 72 and 168 into which it splits factorise when they denote intervals as 3×24 and 7×24, respectively? Let us denote classes of intervals by underlined numbers to distinguish them from numbers of classes. Let us use primes, double primes, etc to distinguish different sets of the same number of intervals. By definition:

72 = 2×4×(3 + 6)**,

where 3 = {tonic, octave, unit interval} and 6 = 6 notes between the tonic & octave. Hence,

72 = 8×{tonic, octave, unit interval} + 2×4×6

= 24' + 2×24,

where 24' = 8×3 and 24 = 4×6. The set of 72 intervals comprises 3 classes of 24 intervals. By definition:

168 = 2×4×21.

21 = 6' + 15,

where 6' = 6 intervals between the octave and the 6 notes and 15 = 15 intervals between the latter. The 6 notes may be represented by a dot at the centre of a pentagon whose corners denote the other 5 notes: