Richard Heath's Introduction:
This is a statement received from Ernest G McClain about his system for calculating holy musicological mountains of rationally related tones entirely generated from an explicit, limiting whole number and an anciently secret, scribal procedure or algorithm - something like that given below. I publish it so as to retain it as an important document on this website.
This subject is introduced in chapter three of my recent book, Precessional Time and the Evolution of Consciousness, but has never before been presented "from the horses mouth". Formerly readers would have to realise the method their way from McClain's presentation which, fortunately, I came to do.
After Anthony Blake and I received this email (with many copied in including the entire bibal yahoo group) I was already frustrated with calculating by hand or even using a spreadsheet. This led to the Harmonic Explorer Application soon after, which not only calculated, but presented both holy mountains and the tone circles (or mandalas) distinctive of Ernest's ancient musicology.
The new application has made possible in a new vision and work in progress embracing the world of ancient limiting numbers, coded into most ancient texts, sacred and mythological, and new harmonic understandings, perhaps unknown to the ancients.
The characterisation of different holy mountains via limiting numbers correlates with both the plot and the characters found in ancient stories. This means that ancient stories had an objective basis in musical harmony that now, with this application, can be easily explored. Correlations never before seen, due to this easy computation and presentation within Harmonic Explorer, has a master user in Ernest McClain, who is currently devising tutorials. Richard
12th October 2011:
Dear Richard and Tony: I cannot guess how comfortable you are in reading them, so I review the principles here. I learned from Plato how to do this analysis. Because I never know the meaning of one until it is completed, several steps are involved. For musicians like Pete Dello, Howard Schatz and me, this is simple as apple pie--but for non-musicians it is a NIGHTMARE!
- It proves convenient with modern habits to do so as upright, male, phallic mountains like Tony's [lattice]. This means the base is a row of triples, and every higher row quintuples, to the top. Thus all elements at equal distances on straight lines anywhere map the SAME ratios. ANY pattern can be moved anywhere within its borders, or rotated 180 degrees to map reciprocals. The base has ONLY triples. All other elements are products of 3x5. The left ascent has ONLY quintuples. The right descent is hazardous, but 5:3 when no "shoulders" intrude." This couldn't be either simpler, or more powerful, BUT ONLY FOR THE MUSICALLY DISCIPLINED!!!
- Then I rotate the FRAME 180 degrees (as the clock does with everything every 12 hours) and the result becomes a "Venn diagram" of "class inclusion" (WITHIN the areas that overlap elements belonging to BOTH the Mt. AND to its reciprocals). Thus Tony's MT. can rotate 180 degrees within mine. The result is always "4-cornered," as Ezekiel saw his wheels, with reciprocals on diagonals, and so all become similar wheels "turning in the same direction" later.
- Now all elements are doubled to the maximum within the LIMIT, which can turn out to lie anywhere, and now is SEEN emergent as "throne" of the current King. It is usually these central symmetries that matter most, but the limits of the Mt. must be studied carefully, for doubling is often repeated to control events to the far right and ABOVE. It is THIS final doubling that determines LINEAR scale order when re-aligned in ascending or descending NUMERICAL scale order that now becomes fully cyclic with linear permutation order. The numerical result is frightful chaos, so it seems until studied carefully, but from left to right within each row we notice wholetones of 9:8 in alternate values (i.e., as if "bull-leaping").
- Now I "musicalize" by assigning "D" as center of symmetry to the throne as largest integer, and the other 12 tones always surround in the same pattern as its "central fire." The central row is "middle earth and holds Spiral 5ths tuning in capitol letters (upper case), sometimes extended even beyond 12 tones to become "wings of the great eagle" (watching over Egypt, or Israel, etc.) But now 5 pairs of "small wings" also appear at right angles, with 4 pairs in perfect vertical alignment (that prove to be always Just semitones of 25:24 in SMALL CASE (as reminders that neighboring rows are displaced by a comma TOWARD THE MIDDLE ROW). This "displacement" means that useful tonal meanings lie ONLY within the 3 middle rows. From them one can always locate the radial positions of ALL values. Why? Because EACH ROW is a small "Spiral of 5ths," and all fifths 2:3 embrace 7 semitone/hours, and/or fourths of 3:4 embrace 5 semitone/hours (while giving us the choice of reading the clock EITHER as one cycle of 24 or as two cycles of 12, reversing at "soltices" and speeding through equinoxes. But "small wings" are limited by tritones of a-flat as first in the row BELOW the throne (and to its left), and by g-sharp as last in the row ABOVE the throne (and to its right). And a-flat BELOW almost coincides with G-SHARP on the far right of the throne row, while g-sharp in the row above correlates instead with A-FLAT in far left of the throne row (i.e., when the Mts. are large enough).
- WITHIN the severely confining triangles of the superimposed MAGEN DAVID, there now lie ONLY 3 kings named by me G-D-A who "frame tetrachords" of 3:4 within fifths of 2:3. The four "little wings" ABOVE or BELOW provide 7-tone modes either for the Greek Dorian "true Hellenic mode" in one direction (as eb bb f c) or its reciprocal (modern Major mode) in the opposite direction (as e b f# c#).
- There can be ONLY 11 perfect symmetries in either SPIRAL 5ths or Just tuning, but there is always an optional "tritone" to function as asymmetric twelfth, with an error of a comma or less.
