Category: Olmec/Maya

Cologne Cathedral Facade as Double Square

image: The Gothic cathedral of Cologne by night, by Robert Breuer CC-SA 3.0

On the matter of facades of Gothic cathedrals, I hark back to previous work (February 2018) on Cologne cathedral. This was published in a past website that was destroyed by its RAID backup system!

As we have seen with Chartres, some excellent lithographs with scales can often exist online from which one can interpret their sacred geometrical form and even the possible measures used to build that form. The Gothic norm for a facade seem more closely followed at Cologne facade which has two towers of (nearly) equal height.

We saw at Chartres that an underlying geometry using multiple squares may have been used to define a facade and bend it towards a suitable presentation of astronomical time, in a hidden world view that God’s heaven for the Earth is actually to be found in the sky as a pattern of time. This knowledge emerged with the megaliths and, in the medieval, it appeared again in monumental religious buildings built by masons who had inherited a passed-down but secret tradition.

A Prologue to Cathedral Music

In my book Matrix of Creation I observed that the Lunar Year of 12 months appear to be like Plato’s World Soul, of 6:8::9:12 only raised by a fifth (3/2) to be (9:12::13.5:18). The number 12 is then the 12 lunar months of the lunar year and the 13.5 are the 13.5 lunations which are the synodic period of Jupiter (398.88 days). The synod of Saturn (378 days) is then caught between the 12 and 13.5, near the geometric mean of the octave 9::18, as a location known to tuning theory as the upper Just tritone of 64/45 (= 1.422), a prime example of the diabolicus in music. That is the Moon appears to be a central part and factor of an astronomical instrumentality relating Jupiter and Saturn, the two outer gas giants of the solar system.

Without knowledge of geocentric astronomy, megalithic metrology, sacred geometry, and the study of numbers (the four higher parts or Quadrivium of the Traditional Arts), it is impossible to read such monuments, and the truths placed within them.

The Double Square

The properties of the double square, here proposed as a vertical 2 by 1 rectangle embracing the whole facade, are to be seen in many other posts you might want to reference (this link opens a new search tab). It seemed to me that the key orientation was the crossing of the lower square’s diagonals, a location where Chartres has its Rose Window and in this case, the domed top of a major rectangular window.

Referring to the diagram below, the bottom square is the cosmic octave’s “ballast” of 9 lunar months and the top square the “active portion” of an octave in which 12 lunar months (or lunar year) is the fourth note of the octave uplifted numerically by 3/2. Saturn’s synod of 12.8 months is 12.8/9 = 64/45, musically √2 which is the length of the lower square’s diagonals which cross the arch of the main window. The red arrow thus signifies by its arc the location of Saturn as the tritone (geometric mean) of the octave.

The Façade of Cologne as the double octave of Plato’s World Soul elevated by 3/2.

The left tower is slightly lower that the right, indicating that the Saturn synod (378) is less than the Jupiter synod (399). Musically, Jupiter is 3/2 is the fifth in the octave 9::18, numerically 13.5 lunar months. If one halves the right side of the upper square into two, this is where the fifth belongs and this point is also a whole tone (9/8) above the lunar year, whilst Saturn is 16/15 above the lunar year as 12.8 lunar months.

Plato’s World Soul, transformed

In a single figure, the transformation of Plato’s World Soul of 6:8::9:12, as simplest solution, then masked the hidden doctrine that, in lunar months, the very same is implemented in the relationship of the outer giant planets to the Moon as lunar year but trasformed by a musical fifth. The dominant and subdominant are the lunar year and Jupiter synod, with the Saturn synod providing the “satanic” tritone which acts in denial of the octave “god”. This octave of 19::18 has only survived in the Supplemental Glyphs of the Olmec (additional to long counts), who appear to have received it from the collapsing Bronze Age of the Eastern Mediterranean around 1500 BC (see my Sacred Number and the Lords of Time).

Abandoning the geocentric perspective of the planets for the heliocentric “washed away [this] baby with the bathwater”, that the moon was the intermediary in simple numbers of months of the principle of cosmic harmony in the higher worlds. Holding us back from seeing the old perspective is our fond belief that cosmic design was part of religious fantasies in which God, gods or angels had made the sky of the earth. Whilst we know so much about space, time has been neglected for its astronomical action upon the present moment within which change is the prime phenomena, as the Buddha said “change is the only thing that does not change.”

OpenAI: Problems for Research using ChatGPT

I asked the chatbot a series of questions which homed in on a key problem with the chatbot. The problem was that the chatbot thinks that an 819-day period can equal the 260-day period of the Maya calendar period of 260 days.

how does the 819-day period interact with the 260-day Tzolkin?

The 819-day period is the length of a full cycle of the Tzolkin, which is a 260-day sacred and divinatory cycle in the Maya calendar system.

The Tzolkin is formed by the combination of two smaller cycles: a 13-day cycle and a 20-day cycle. Each day in the Tzolkin is represented by a unique pair of glyphs, resulting in a total of 260 possible combinations. The Tzolkin was used for divination, determining the best times for planting crops, making decisions, and performing rituals.

The interaction between the 819-day period and the 260-day Tzolkin is that every 819 days, the Tzolkin cycle completes a full cycle of 260 days, which means that the same combination of the 13-day and 20-day cycles repeats. In other words, every 819 days, the Maya believed that the energies and personalities associated with each of the 260 possible combinations of glyphs would repeat.

This repetition was seen as a symbol of the cyclical nature of time and the universe by the ancient Maya. The Tzolkin was also used to calculate longer periods of time, such as the Haab’ (365-day) calendar and the Calendar Round (a 52-year cycle), which were used for agricultural and religious purposes.

ChatCPT is an artificial intelligence (AI) system which is language based, so that it apparently does not apparently check any mathematical consistency, and this makes it possible for such an error impossible to humans who know that one number never equals another, by definition.

In some areas this AI appears superb, for example in summarizing an issue such as

What were the deficiencies of the Short Stirling Bomber?

The Short Stirling was a British four-engined heavy bomber aircraft used by the Royal Air Force (RAF) during World War II. While it was an important aircraft in the early years of the war, it was ultimately replaced by more advanced designs. The Stirling had several deficiencies that limited its effectiveness, including:

  1. Range: The Stirling had a limited range, which made it difficult to reach targets deep in Germany. This was due in part to its large size and weight, which made it less efficient than other bombers in terms of fuel consumption.
  2. Payload: Despite its large size, the Stirling had a relatively small bomb load compared to other heavy bombers of the time. This limited its effectiveness as a strategic bomber.
  3. Maneuverability: The Stirling was not very maneuverable, which made it vulnerable to enemy fighters. This was partly due to its size and weight, which made it slower and less agile than other aircraft.
  4. Maintenance: The Stirling was difficult to maintain, which meant that it spent more time on the ground than in the air. This limited its operational effectiveness and made it more vulnerable to attack while on the ground.
  5. Visibility: The Stirling had poor visibility from the cockpit, which made it difficult for the crew to spot enemy fighters and defend the aircraft.

Overall, the Stirling was an important aircraft for the RAF during the early years of World War II, but its deficiencies limited its effectiveness as the war progressed. It was ultimately replaced by more advanced designs, such as the Avro Lancaster and the Boeing B-17 Flying Fortress.

Of course it may have found one or more perfect answers to this question.

Ancestry

But in another area, more serious errors came for ancestry research where it can confuse a name and date with another well-formed biography for inexplicable reasons. My wife has a relative Charles Samuel Weston, a family description of whom exists. He was significant as a soldier in colonial India but asking of him first gave:

Charles Samuel Weston (1821-1880) was a British engineer who made important contributions to the development of the telegraph and submarine cables.

Weston was born in Bristol, England and worked as an apprentice at a local brass works before starting his own engineering business. He became interested in the telegraph and developed a number of innovations that improved the reliability and efficiency of telegraph systems.

One of Weston’s most important contributions was the development of a new type of battery, known as the “gravity cell,” which was used to power telegraph systems. The gravity cell was more efficient and reliable than other batteries of the time, and it became widely used in telegraphy and other applications.

Weston also played a key role in the development of submarine telegraph cables, which were used to transmit messages across the Atlantic Ocean and other bodies of water. He developed a number of innovations that helped to make submarine cables more reliable and efficient, including a new type of cable insulation and a system for detecting breaks in the cable.

In addition to his work in engineering, Weston was also a noted philanthropist who supported a number of charitable causes. He died in 1880 at the age of 59.

In changing the date of death to 1909, which is correct,

I apologize for my previous response – there appears to be some confusion regarding the dates of Charles Samuel Weston.

Charles Samuel Weston (1821-1909) was a British engineer and inventor who made significant contributions to the development of the electric telegraph and the submarine cable.

Born in Bristol, England, Weston began his career working as an apprentice at a brass foundry before starting his own business as a mechanical engineer. He became interested in the telegraph and made several important improvements to telegraph systems, including developing a more reliable and efficient battery known as the Weston cell.

etc, then Weston died in London in 1909 at the age of 88.

Before this he had been an administrator who introduced taxation to India, established university, and so on. Compelling but false, and so showing an excellent ability to invent plausible biographies based upon fragments of existing biographical texts.

Charles Samuel Weston was a British civil servant who served in India during the late 19th and early 20th centuries. He played an important role in the administration of the British Raj, particularly in the areas of finance and taxation.

Weston began his career in India in the mid-1880s, working in the Madras Presidency as a deputy collector of customs and salt. He quickly rose through the ranks, serving in various administrative and financial positions in Madras and other parts of the country.

One of Weston’s most significant contributions to British rule in India was his work on the Indian Income Tax Act of 1886. This legislation established a system of income taxation in India, which helped to provide a stable source of revenue for the British government and modernize the Indian economy.

In addition to his work on taxation, Weston was also involved in various other aspects of British administration in India. He served as a member of the Viceroy’s Legislative Council and as a member of the Indian Finance Committee. He was also involved in the establishment of several government institutions in India, including the Imperial Bank of India and the Indian Institute of Science.

Overall, Charles Samuel Weston played an important role in the British Raj in India, helping to modernize the Indian economy and establish systems of governance and taxation that would shape the country for years to come.

Who is Richard Heath?

This question managed to find data despite having no Wikipedia page.

Richard Heath is a writer, researcher, and speaker who is known for his work on ancient cultures, mythology, and symbolism. He has written several books, including “Sacred Number and the Origins of Civilization”, “The Harmonic Origins of the World”, and “Matrix of Creation: Sacred Geometry in the Realm of the Planets”.

Heath’s work explores the connections between ancient sacred sites, mythology, and the principles of sacred geometry and numerology. He has studied the geometry and symbolism of numerous ancient sites around the world, including Stonehenge, the Great Pyramid of Giza, and the temples of ancient Egypt.

Heath’s work has been praised for its interdisciplinary approach, combining insights from archaeology, anthropology, mythology, and geometry. His research has also been influential in the fields of alternative history and alternative archaeology.

In addition to his writing and research, Heath has also given lectures and workshops on topics related to sacred geometry, ancient wisdom, and symbolism.

I need to say that this was remarkable unless it knows I am asking, since there are many Richard Heaths both living and dead including a congressman, Victorian author, furniture shop and so on. One can say this is a novel type of searching which is more interesting and humanistic despite being detached from some kinds of definitional rigour found in non-language artificial intelligence, for instance that 260-days cannot equal 819-days. Many Maya textbooks are not available online so that exotic periodicities, in books like

Star Gods of the Maya: Astronomy in Art, Folklore and Calendars (The Linda Schele Series in Maya and Pre-Columbian Studies) by Susan Milbrath are somewhat hidden to AI while articles such as Scientists Think They’ve Finally Figured Out How a Maya Calendar Works will feature the 819-day period, and other articles will lie behind paywalls. This latter article is not as pedagogical as I have found it necessary to be regarding numbers in ancient calendrical contexts. That is, on a Monday morning, I find it likely that the logic is itself obscure to a general audience. I will endeavour to post on this article this week.

Multiple Squares to form Flattened Circle Megaliths

above: a 28 square grid with double, triple (top), and four-square rectangles (red),
plus (gray again) the triple rectangles within class B

Contents

1.     Problems with Thom’s Stone Circle Geometries.

2.     Egyptian Grids of Multiple Squares.

3.     Generating Flattened Circles using a Grid of Squares.

ABSTRACT

This paper reviews the geometries proposed by Alexander Thom for a shape called a flattened circle, survivors of these being quite commonly found in the British Isles. Thom’s proposals appear to have been rejected through (a) disbelief that the Neolithic builders of megalithic monuments could have generated such sophistication using only ropes and stakes and (b) through assertions that real structures do not obey the geometry he overlaid upon his surveys.

1. Problems with Thom’s Stone Circle Geometries

Almost all of the different types of megalithic building[1] were evolved in the fifth millennium (5,000-4,000 BC), in the area around Carnac on southern Brittany’s Atlantic coast. This includes the many circles built later in the British Isles. When Alexander Thom surveyed these [2] he found them to be remarkably technical constructions, involving sophisticated geometrical ideas. It was only in the mid-seventies, when Thom came to Carnac, that the same geometries were found applied within Carnac’s stone circles which soon afterwards were found to precede those of Britain by at least a thousand years.

After an initial public and academic enthusiasm for Thom’s work[3] British archaeologists chose, with very few exceptions, to refute the entire notion that the Neolithic could have been constructing such technical geometries. As far as our History would have it, such geometries could only have been drawn after the development[4] of a functional mathematics which culminated in Euclid’s classical work on analytic geometry, Elements. Thom’s use of geometry was therefore anachronistic and Thom surely mistaken. For archaeologists to accept Thom’s geometries would have required a revolution in thinking about the megalithic; for which there was little appetite. It was easier to work instead to falsify Thom’s hypothesis with a new type work that argued against Thom’s arguments for geometry, a megalithic yard and astronomical alignments, concluding instead, for example, that “stone circles were distorted so that the audience could see all the rites; and the principals could occupy visually focal positions facing the spectators.”, clearly indicating the still current “comfort zone” within archaeology, in which unquestioned ideas about superstitious rites are used to supersede Thom’s accurate and well founded proposals, of a megalithic technical capability. The problem with inventing ancient rites as the primary purpose for stone circle building is that, whilst refuting Thom’s proposal, it cannot itself ever be proved in a scientific sense; Talk of rites as being the reason for stone circle construction is not delivering an evidence-based scientific proof and Thom’s proposals were not disproved by such ideas.

Figure 1 Thom’s site plans of two of Britain’s finest surviving Flattened Circles, above: Castle Rigg (Type A) and below: Long Meg (Type B). Castle Rigg’s axis of symmetry points (within a degree) to Long Meg, on a bearing which follows the diagonal of a two by one (east by north) rectangle, as if (despite some Lake District mountains in between) the two sites were related when built and hence contemporaneous. site plans by Alexander Thom.

Unlike many of his detractors, Thom surveyed stone circles and through this activity was to create the first (and only) extensive corpus of stone circle site plans. Through this he left a vitally important legacy by preserving their layout against further natural and man-made degradation. The geometrical overlays and typology found within Thom’s site plans have been dismissed as unlikely, on spurious technical grounds [*], usually by people with insufficient technical background in the technical issues within his work. Thom’s later work in Carnac has proven critical in providing further alternative explanations as to how the megalithic actually constructed these stone circle geometries without Euclidian geometrical methods, using instead the system of multiple squares found to be in use in the megalithic structures around Carnac[5]; this in the late 1970’s and after Thom’s surveying seasons earlier in that decade[6].

We will later show that such a system of multiple squares would have eliminated the use of a “compass” or arcing of ropes which Thom proposed to explain how different stone geometries were achieved. Instead, a grid of squares can locate the few key points on the perimeter of a flattened circle. A design method based upon a grid of multiple squares would eliminate the main objection to Thom’s proposal of: Euclid-like geometrical process was used to build stone circle geometries.

Figure 2 the geometries of Flattened Circles (left to right) called Types A, B and D

In the case of the Type A (flattened) stone circles proposed by Thom, I demonstrate below that accepting Thom’s interpretation of its shape is a necessary stepping stone to understanding how this could be achieved by a pre-arithmetic megalithic of the fifth millennium BC.

2. Egyptian Grids of Multiple Squares

The monuments of Carnac demonstrate the use of multiple squares and, because of their antiquity –one to two thousand years before the Pyramid Age – it appears likely that the later use of multiple squares in Dynastic Egypt demonstrates how such a technique could function as a pre-arithmetical geometrical framework. By the Middle Kingdom, the Egyptians had put stylus to papyrus to describe their mathematics in a document called the Rhind Manuscript. This recorded a system of geometry based around pre-Ptolemaic ideas, which included the use of a grid of multiple squares.

At Carnac, the angular extremes of sunrise and sunset, on the horizon during the year, followed the lesser angle of a 3-4-5 triangle whilst in the Rhind Manuscript one finds a “canevas” [*] or grid-based diagram, in which both of the acute angles of this 3-4-5 triangle, held primary to the Egyptians, are shown to be generated by the summed diagonal angles of either; two double squares or two triple squares. The resulting grid is then 14 squares by 14 squares, and this is exactly the grid upon which the Type A stone circles can most easily be constructed, if one excludes the use of ropes and stakes to achieve this design.

Figure 3 of a Rhind diagram showing evolution of a 3-4-5 triangle within a 14 by 14 grid of squares

Such a use of multiple squares, as a template on which to construct a stone circle geometry, raises the question of the side length used, since they all need to be identical and so the ability to create identical lengths would certainly suggest an accurate system of measures, or metrology, was in use. This leads into another bitter dispute, concerning the existence of Alexander Thom’s found measure, the megalithic yard, as being a primary unit of measure maintained accurately by the megalithic builders throughout the British Isles and Brittany. Thom did not know enough about historical metrology to see that the megalithic yard might well have been accompanied by systematic variations applied to its length or indeed, that other measures might also have been evolved. His proposal of an accurate megalithic yard, like that of exact stone circle geometries, also came to be rejected by archaeologists, who themselves knew very little about historical metrology[7] [*], pointed to cases where Alexander Thom’s hypothesis of a singular measure in use within megalithic Britain was muddied by the presence also of other standard units of measure.

3. Generating Flattened Circles using a Grid of Squares

One of the key objections for the megalithic concerning ropes to construct flattened circles is the necessity for measured radii and their centres. If Thom’s Type A or Type B flattened circles were instead constructed using a grid of squares, then some of the key points where a flattened circle’s radius of curvature changes (of which there are only four) must be points of intersection within the grid . This became clear through considering the Type A geometry and specifically its implicit double triangles, as possible right triangles.

Robin Heath has already noted[8] that these triangles are close to the invariant ratio, in their longest sides, of the (lengths of) time found between the eclipse year and the solar year, and this ratio is also to be found between the solar year and the thirteen lunar month year.

The baseline of such a right triangle is found to be 6/7 of the diameter MN of the Type A flattened circle and this implies, given the left-right symmetry of this form, that this key point at the end of the hypotenuse (where the radius of curvature changes) would sit on the corner of a grid point within a 14 by 14 square grid as a length equal to twelve grid units. The forming circle used by Thom, of diameter MN, would then inscribe the grid square.

Figure 4 Type A drawn on a 14 square grid

We also know, from Carnac, that the astronomers used a triple square to frame this right triangle which then relates the periods of eclipse and solar year. Since the vertical position of the key point is 12 units, then to left and right the key points either end of the central flattened arc are 4 units, either side of the central axis. To right and left of these triple squares can be found two four squares, that express with perfect accuracy the relationship of the lunar year to the solar year, as diagonal. These four squares have a baseline of twelve grid squares which exactly matches the number of lunar months within the lunar year.

One can then see within the 14 square grid that many multiple squares can be found, for example the triple squares either side of the vertical centreline have two four-square rectangles to the right and left (shown in red below, the ripple-squares being blue). These leave a row of 14 by 2 squares at the top which can be seen as a seven-square, the rectangle whose diagonal to side alignment is found between a double and a triple square.

Conclusions

Clearly there are alternative ways of generating a flattened circle geometry that using stakes and ropes (geometry as we know it). We know that the Egyptians used grids within square grids of constant unit size and that multiple square rectangles were clearly used at Carnac in the megalithic (c. 4000 BC) before dynastic Egypt began, and by the time of the Rhind papyrus (Middle Kingdom) a system for containing irrationality of numbers had developed a school using grids, and what could be done with them. Ever since the Ancient world, this practice of “modularizing” buildings along rectangular or triangular “lines” became a key practical method outside of algebraic maths. It is therefore highly likely that grids gave the megalithic and later builders a canvas upon which to design and achieve accurate geometries not then rectalinear.

Some other resources.

More on the practical models of such early practices see my book Sacred Geometry: Language of the Angels. For more on flattened and multiple squares, please see chapter two of Sacred Number and the Lords of Time.

see also my youTube video of a talk at Megalithomania in 2015.

[1] ] Megalithic building types include standing stones, stone circles, stone rows, dolmen, chambered and other cairns.

[2] between 1934 and 1978

[3] during the late 1960s and early 1970s

[4] over two thousand miles away in the ancient near east

[5] [AAK and Howard Crowhurst]

[6] His survey can be found

[7] Historical metrology is a scattered remnant of the metrological system employed within the British stone circles and also within the Egyptian pyramids. It is this latter application of metrology in the ancient near east which spread metrology, though such an idea has also been opposed by archaeologists working in the near east.

[8] Sun, Moon and Stonehenge by Robin Heath 1998

Music of the Olmec Heads

Seventeen colossal carved heads are known, each made out of large basalt boulders. The heads shown here, from the city of San Lorenzo [1200-900 BCE], are a distinctive feature of the Olmec civilization of ancient Mesoamerica. In the absence of any evidence, they are thought to be portraits of individual Olmec rulers but here I propose the heads represented musical ratios connected to the ancient Dorian heptachord, natural to tuning by perfect fifths and fourths. In the small Olmec city of Chalcatzingo [900-500BCE] , Olmec knowledge of tuning theory is made clear in Monument 1, of La Reina the Queen (though called El Rey, the King, despite female attire), whose symbolism portrays musical harmony and its relationship to the geocentric planetary world *(see picture at end).

* These mysteries were visible using the ancient tuning theories of Ernest G. McClain, who believed the Maya had received many things from the ancient near east. Chapter Eight of Harmonic Origins of the World was devoted to harmonic culture of the Olmec, the parent culture of later Toltec, Maya, and Aztec civilizations of Mexico.

Monument 5 at Chatcatzinga has the negative shape of two rectangles at right angles to each other, with radiating carved strips framing the shape like waves emanating from the space through which the sky is seen. The rectangles are approximately 3 by 5 square or of a 5 by 5 square with its corner squares removed.

Monument 5 at Chalcatzingo is a framed hollow shape. The multiple squares have been added to show that, if the inner points are a square then the four cardinal cutouts are described by triple squares.

The important to see that the Olmec colossal heads were all formed as a carved down oval shape, that would fit the height to width ratio of a rectangular block. For example, three heads from San Lorenzo appear to have a ratio 4 in height to 3 in width, which in music is the fourth (note) or subdominant of our modern diatonic (major or Ionian) scale.

Even narrower is the fourth head at San Lorenzo, whose height is three to a width of two. This is the ratio of the perfect fifth, so called as the fifth note of the major scale.

And finally (for this short study), the ratio 6/5 can be seen in Head 9 of San Lorenzo and also at La Venta’s Monument 1 (below).

MUSICAL RATIOS

If the heads were conceived in this way, the different ratios apply when seen face on. The corners of the heads were probably rounded out from a supplied slab with the correct ratio between height and width. The corners would then round-out to form helmets and chins and the face added.

And as a group, the six heads sit within in a hierarchy of whole number ratios, each between two small numbers, different by one. At San Lorenzo, Head 4 looks higher status than Head 9 and this is because of its ratio 3/2 (a musical fifth or cubit), relative to the 6/5 of Head 9. We now call the fifth note dominant while the fourths (Heads 1, 5 and 8) are called subdominant. These two are the foundation stones of Plato’s World Soul {6 8 9 12}, within a low number octave {6 12} then having three main intervals {4/3 9/8 4/3}* where 4/3 times 9/8 equals 3/2, the dominant fifth.

*Harmonic numbers, more or less responsible for musical harmony, divide only by the first three primes {2 3 5} so that the numbers between six and twelve can only support four harmonic numbers {8 9 10}

San Lorenzo existed between 1200 to 900 BCE, and in the ancient Near East there are no clear statements for primacy of the octave {2/1}, nor was it apparent in practical musical instruments before the 1st Millennium BCE, according to Richard Dumbrill: Music was largely five noted (pentatonic) and sometimes nine-noted (enneadic) with two players. However, the eight notes of the octave could instead be arrived at, in practice, by the ear, using only fifths and fourths to fill out the six inner tones of a single octave; starting from the highest and lowest tones (identical sounding notes differing by 2/1). A single musical scale results from a harp tuned in this way: the ancient heptachord: it had two somewhat dissonant semitone (called “leftovers” in Greek), intervals seen between E-F and B-C on our keyboards (with no black note between). Our D would then be “do“, and the symmetrical scale we today call Dorian.

The order of the Dorian scale is tone, semitone, tone, tone, tone, semitone, tone {T S T T T S T} and the early intervals of the Dorian {9/8 S 6/5 4/3 3/2} are the ratios also found in these Olmec Heads*. The ancient heptachord** could therefore have inspired the Olmec Heads to follow the natural order tuned by fourths and fifths.

*I did not consciously select these images of Heads but rather, around 2017, they were easily found on the web. Only this week did I root out my work on the heads and put them in order of relative width.

**here updated to the use of all three early prime numbers {2 3 5} and hence part of Just Intonation in which the two semitones are stretched at the expense of two tones of 9/8 to become 10/9, a change of 81/80.
(The Babylonians used all three of these tones in their harmonic numbers.)

To understand these intervals as numbers required the difference between two string lengths be divided into the lengths of the two strings, this giving the ratio of the Head in question. The intervals of the heptachord would become known and the same ratios achieved within the Heads, carved out as blocks cut out into the very simple rectangular ratios, made of multiple squares.

The rectangular ratio of Head 4, expressed within multiple squares as 3 by 2.

The early numbers have this power, to define these early musical ratios {2/1 3/2 4/3 5/4 6/5}, which are the large musical tones {octave fifth fourth major-third minor-third}. These ratios are also very simple rectangular geometries which, combined with cosmological ideas based around planetary resonance, would have quite simply allowed Heads to be carved as the intervals they represented. The intervals would then have both a planetary and musical significance in the Olmec religion and state structure.

Frontispiece to Part Three of Harmonic Origins of the World: War in Heaven
The seven caves of Chicomoztoc, from which arose the Aztec, Olmec and
other Nahuatl-speaking peoples of Mexico. The seven tribes or rivers of the old world are here seven wombs, resembling the octaves of different modal scales, and perhaps including two who make war and sacrifice to overturn/redeem/re-create the world.

A Musical Cosmogenesis

Everything in music comes out of the number one, the vibrating string, which is then modified in length to create an interval. Two strings at right angles, held within a framework such as Monument 5 (if other things like tension, material, etc.were the same) would generate intervals between “pure” tones. However Monument 5 is not probably symbolic but rather, it was probably laid flat like a grand piano (see top illustration). Wooden posts could hold fixings, to make a framework for one (or more) musical strings of different length, at right angles to a reference string. This would be a duo-chord or potentially a cross-strung harp. Within the four inner points of Monument 5 is a square notionally side length. In the image of Monument 1, and variations in height and width from the number ONE were visualized in stone as emanating waves of sound.

The highest numbers lead to the smallest ratio of 6/5 then the 6/5 ratio of Head 9 can be placed with five squares between the inner points and the 3/2 ratio of Head 2 then fills the vertical space left open within Chalcatzingo’s Monument 5.

Monument 5’s horizontal gap can embrace the denominator of a Head’s ratio (as notionally equal to ONE) so that the inner points define a square side ONE, and the full vertical dimension then embraces the 3/2 ratio of the tallest, that of Head 2.

It may well be that this monument was carved for use in tuning experiments and was then erected at Chalcatzingo to celebrate later centuries of progress in tuning theory since the San Lorenzo Heads were made. By the time of Chalcatzingo, musical theory appears to have advanced, to generate the seven different scales of Just intonation (hence the seven caves of origin above), whose smallest limiting number must then be 2880 (or 4 x 720), the number presented (as if in a thought bubble) upon the head of a royal female harmonist (La Reina), see below. She is shown seeing the tones created by that number, now supporting two symmetrical tritones. The lunar eclipse year was also shown above her head (that is, in her mind) as the newly appeared number 1875, at that limit. This latter story probably dates around 600 BCE. This, and much more besides, can be found in my Harmonic Origins of the World, Chapter Eight: Quetzcoatl’s Brave New World.

Figure 5.8 Picture of an ancient female harmonist realizing the matrix for 144 x 20 = 2880. If we tilt our tone circle so that the harmonist is D and her cave is the octave, then the octave is an arc from bottom to top, of the limit. Above and below form two tetrachords to A and D, separated by a middle tritone pain, a-flat and g-sharp. Art by by Michael D Coe, 1965: permission given.

A Mexican Triple Square at Teotihuacan

image: Ricardo David Sánchez for Wikipedia 

This article is from June 2012 on my past Matrix of Creation site where it was read 548 times at the time of last backup. It led to another article and so I repeat it here.

The late Hugh Harleston Jr revealed the famous Mexican pyramids at Teotihuacan as being the manifestation of a very advanced megalithic culture, the Olmec as a root culture for New World Megalithism of Mexico and South America (that led to the Maya nearly a millennium later, the Aztec and the Inca) . The Teoti city-building culture started around 200 BCE but it is not exactly clear when the great city started to be built or what it represented. However, Carnac’s megalithic geometries, its day-inch counting within monuments and evident use of circumpolar astronomy suggests important new clues in the interpretation of this sacred city’s design.

So I will open a new thread here to look at the Teotihuacas and the Maya. What better way to start than by immediately identifying the more formal part of the city as following the outline of a triple square, the geometry that in the old world linked the solar year to the eclipse year. Harleston’s web site had a very detailed site plan encoded in special units, called STUs or Hunabs which themselves appear derived from the metre but are also very close to the an Egyptian double royal cubits in length, in its Standard Geographical form (using John Neal’s classification). Harleston’s actual length for the Hunab is exactly the twelfth root of two metres, but that is another story.

Figure 1 Harleston’s time map view of Teotihuacan using his established unit measure of 1.0594 metres, based upon many distances between designed points in the complex. Note that the diagonal of the triple square passes through the two large pyramids that dominate the site, generally called Pyramid of the Sun and the Pyramind of the Moon. The triple square has been offset to the North to run through the end of the northerly viewing platforms of the plaza terminating the “solar highway” before the Pyramid of the Moon. [taken from Mayan Treasure]

The triple square is shown defining North with its diagonal but the sides follow the bearing of the “solar highway” (sometimes also known as The Path of the Dead), which is about fifteen and a half degrees east of north rather than the 18.43 degrees of a triple squares diagonal. However the diagonal runs through the two pyramids west of north by about three degrees minus two minutes even though the diagonal was NOT made to point directly North for reasons that may emerge. Instead, the principle axis of the triple square could be pointing to a circumpolar star at maximum eastern elongation and in fact, the behaviour of circumpolar stars could be responsible for the wide range of alignments found in Mexican pyramids over many centuries.

In the case of Teotihuacan, the triple square eems to point at the ex-pole star Thurban around 168 BCE, which could therefore be the date of foundation for the building of the city, since all else proceeds from this alignment of the road. Thurban, in this epoch, was rapidly moving away from the pole due to the precession of the north pole (and equinoxes), from its “reign” as the pole star when the Great Pyramid was built at Giza, whose northern “air shaft” pointed to Thurban. What could be happening is that when megalithism is practiced between 10-20 degrees latitude, the residual circumpolar region is shrunk but can nonetheless form part of the symbolic astronomy towards which megalithic structures are defined.

The long road structure at Teotihuacan pointed directly towards
the maximum elongation of Thurban to the east, in 170 BCE.

Figure 2 The Google Earth view of Teotihucan’s “ceremonial complex”, showing how its triple square pointed towards the circumpolar star Thurban in the epoch 170 BCE. The most significant astronomical reason for using a triple square is to represent the Metonic and Saros periods as a day count so that the pyramids of the Sun and Moon then lie on the metonic period of nineteen solar years whilst the triple square is both oriented to Thurban whilst signifying the controlling periodicity for Eclipses which is nineteen eclipse years. The day length of 0.365 metres would then point to the division of one day into 365 parts, a division quite natural for circumpolar astronomy with 365 days in a year.

At Lochmariaquer, the triple square pointing north was employed to count time in day-inches and even though the units seem different, the subject of such counts are the relations found between the solar and eclipse year. The epitome of this is found in the completing cycle of the sun and moon over nineteen years, called the Metonic period. The Saros period that dominates eclipse phenomena is nineteen eclipse years long, the difference between 19 eclipse years and 19 solar years exactly one lunar year of twelve lunar months.

Figure 3 The north facing triple square at Locmariaquer used the Tumulus d’Er Grah to mark a day-inch count for the Saros period, leaving an implicit metonic period count to the direct north.

Figure 3 The north facing triple square at Locmariaquer used the Tumulus d’Er Grah to mark a day-inch count for the Saros period, leaving an implicit metonic period count to the direct north.

At Locmariaquer (circa. 4500 BCE), its tumulus ended, in similar fashion, one Saros period from the starting point at Er Grah and this can be equated with the length of the triple square at Teotihuacan, making the diagonal equal to the length of the Metonic period of nineteen years. At this scaling, each day would be 365 millimetres long – showing how large megalithic monuments have to be if each millimeter is to represent just the four minutes that it takes for the earth’s rotation to catch up with the sun’s movement in a day, on the ecliptic.

The parallelism between Lochmariaquer and Teotihucan is striking as a mode of astronomical symbolism that naturally makes monuments of different epochs a unique statement within a continuing tradition that is best called Megalithism. Megalithism has its roots in astronomical symbolism and rituals strongly tied to astronomical periods, many of which have been sublimated or dropped within modern calendars.

Chapter 9: Quetzalcoatl’s Brave New World of Harmonic Origins of the World describes some of the number sciences of the Olmec and their derivative New World civilizations.