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The current theme is search: Golden Rectangles which led to finding them in search: St Peter’s Basilica in Rome and in the Umayyid Mosque in Damascus. But the Olmec heads have come to mind after a TV documentary, so that work from 2017 was completed, suggesting “what they mean”.

I have started a theme on Music, part one giving overview of how music derived originally from an ancient world where pure number relationships between tones (ratios) developed much the familiar terminology of modern music, but in the interests of group performance and composition, each interval sacrifices purity for utility.

I have also republished an article on the significence of square roots.

Origins of the Olmec/Maya Number Sciences

ABOVE: Stela C from Tres Zapotes roughly rebuilt by Ludovic Celle and based on a drawing by Miguel Covarrubias.

Introduction

The policy of archaeology regarding the Maya and their root progenitor the Olmec (1500 BCE onwards) is that its cultural innovations were made within Mexico alongside an agrarian revolution of the three sisters, namely squash, maize (“corn”), and climbing beans. This relationship of agriculture to civilizing skills then reads like the Neolithic revolution in Mesopotamia after 4000 BCE, where irrigation made the fertile loam able to absorb agricultural innovations from the northern golden triangle leading to writing, trade, city states, religion, arithmetic and so on. However, the idea that the ancient near east or India could have been an influence through ocean conveyors, of currents and trade winds, has never been accepted when proposed. Yet there are good reasons to think this since the astronomy and monumentalism of the pre-Columbian Mexican civilizations has precedents in the ancient near east and other locations.

The timing of the Olmec and the strangeness of immediately building sacred cities with an almost captive population of around 10,000 people, such as La Venta and San Lorenzo, with strong Jaguar imagery and practices, implies a cultic basis was present from the beginning. And it is now looking likely that the ancient near east was similarly prefigured, not just by agriculture but also by know how involving numbers for the building of sacred buildings with astronomical aspects – a tradition that goes back at least to the megalithic of the Atlantic seaboard of Europe.

Since Columbus, the native populations of North and South America have been largely displaced or marginalized. It may be for this reason that the notion that people from an advanced population had initiated the Olmec civilization requires a high, possibly impossible, level of proof. This Isolationism***, perhaps to avoid “adding insult to injury”, is against the Olmec having derived from the Old World, where the historical records are not that much better. The Olmec origin date is around the time of the quite sudden collapse of the Bronze Age in the Mediterranean around 1200 BCE. And the Olmec, Maya and Aztec appear to have had a definite myth concerning someone called Quetzelcoatl bringing civilizing skills to found their culture, though their culture was also seen as arising from a group of seven underground caves.

***The opposite of Diffusionism: Diffusionism is an anthropological school of thought, was an attempt to understand the distribution of culture in terms of the origin of culture traits and their spread from one society to another. Versions of diffusionist thought included the conviction that all cultures originated from one culture center (heliocentric diffusion); the more reasonable view that cultures originated from a limited number of culture centers (culture circles); and finally the notion that each society is influenced by others but that the process of diffusion is both [subject to chance] and arbitrary . read more

Long Counts and The LUNAR Calendar

Having sketched this background, this article will explore a strange coincidence between the calendrical origins of the Megalithic in Brittany, of a 36 lunar month, 3 lunar year calendar, and the 18 month calendar found in the some of the later Olmec Great Counts, called after the Supplementary Glyphs appended to record the local time in an 18 lunar month calendar. The correlation between long counts and the supplementary data has been invaluable since the long counts can be ambiguous between one or more possible dates but we can predict the sun and moon that far back can compare the glyphs with the alternative dates. Counts have also been found that were eclipses of the sun or moon, resolving a given long count date. It is therefoe interesting to compare the two calendars using the geometrical fact that 36 lunar months is both 2 x 18, 4 x 9 and 3 x 12 since 36 is 4 x 3 x3.

The implication is that the megalithic calendar over three years, which was based upon noticing that three solar years was the diagonal of a four square triangle whose side length is three lunar years, appears to have resulted in an Olmec/Maya calendar in which each square is 9 lunar months. As was noted in previous books (2004, 2016, 2018), the range 9 to 18 years contains a single lunar month {12}, the Jupiter synod {13.5}, the Saturn synod {12.8} and the Uranus synod {12.5}. This octave range between 9 and 2 x 9 = 18 was therefore possible to manifest as a Mexican city design (Teotihuacan) and as the Parthenon of Athens. A number of other examples can be found as one of the proposed major models used from the megalithic onwards, as discussed in Sacred Number: Language of the Angels (2021).

Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

Music, part 1: Ancient and Modern

We would know nothing of music were it not that somewhere, between the ear and our perceptions, what we actually hear (the differences between different frequencies of sound, that is, different tones) is heard as equivalent musical intervals (such as fifths, thirds, tones, semitones, etc), of the same size, even when the pitch range of the tones are different. This is not how musical strings work, where intervals of the same size get smaller as the pitch at which tones occur, grows larger. On the frets of a guitar for instance, if one plays the same intervals in a different key, the same musical structure, melodic and harmonic, is perfectly transposed, but the frets are spaced differently.

The key is that human hearing is logarithmic and is based upon the number two {2}, the “first” interval of all, of doubling. This can only mean that the whole of the possibilities for music are integral to human nature. But this miraculous gift of music, in our very being, is rarely seen to be that but, rather, because of the ubiquity of music, especially in the modern world, the perception of music is not appreciated as, effectively, a spiritual gift.

Music is often received as a product like cheese, in that it is to be eaten but, to see how this cheese is made from milk requires us to see, from its appearance as a phenomenon, what music perception is made up of . Where does music come from?

Normally a part of musicology, that subject is full of logical ambiguities, confusing terminology, unresolved opinions, and so on. Those who don’t fully understand the role of number in making music work, concentrate on musical structures without seeing that numbers must be the only origin of music.

The ancient explanation of music was that everything comes out of the number one {1}, so that octaves appear with the number two {2/1}, fifths from three {3/2}, fourths from four {4/3}, thirds from five {5/4} and minor thirds from six {6/5}. Note that, (a) the interval names refer to the order of resulting note within an octave, (b) that intervals are whole number ratios differing by one and that, (c) the musical phenomenon comes out of one {1}, and not out of zero {0}, which is a non-number invented for base ten arithmetic where ten {10} is one ten and no units.

Another miracle appears, in that the ordinal numbers {1 2 3 4 5 6 7 8 9 etc.} naturally create, through their successiveness, all the larger intervals before the seventh number {1 2 3 4 5 6 7} leaving the next three {8 9 10} to create two types of tone {9/8 10/9} and a semitone {16/15} thereafter {11 12 13 14 15 16}: by avoiding all those numbers whose factors are not the first three primes {2 3 5}. Almost the whole potential of western music is therefore built out of the smallest numbers!

This simplicity in numbers has now been obscured, though the structure of music remains in the Equal Temperament form of tuning evolved in the last millennium. By having twelve equal semitones that sum to the number two, we can now transpose melodies between keys (of the keyboard) but we have pretty much lost the idea of scales. Instead, each key is the major diatonic {T T S T T T S} (where T = tone and S = semitone intervals) starting from a different key. The fifth is called dominant and fourth subdominant and the black notes (someway fiendish to learn) required to achieve the major key in all keys but C which is all white keys.

The old church scales are achievable by over ruling the clef with accidental notes, and the reason for different keys sounding different is that they contain aspects of what were the scales. So a pop song, for example, is usually in a scale. “Bus Stop” by the Hollies was in the Locrian scale.

Equal Temperament enabled the Western tradition to create its Classical repertoire but it has made ancient musical theory very distant and has abandoned the exact ratios it used to use since every semitone is identical and irrational. Plato described this kind of solution as the best compromise, where every social class of musical numbers has sacrificed some thing of their former self in order to achieve the riches versatility bestows upon modern musical composition.

To be continued.

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.

Umayyad Mosque: Golden Rectangles from Squares

photo above of Umayyad Mosque, Damascus by Bernard Gagnon for Wikipedia CC BY-SA 3.0.

In previous articles on double squares and then St Peter’s Basilica, it became clear that squares and double squares have been embodied, within sacred buildings and art, because circles can then spawn golden rectangles from them. A golden rectangle has one dimension related to its other dimension as the golden mean {1.618034…}. Firstly, the original square plus golden rectangle is a larger golden rectangle but, secondly, the new golden rectangle (beside the square) shares its side length as one unit {1} but its other side is then the reciprocal of the golden mean (0.618034).

The golden mean is the only irrational number whose reciprocal, and square share its fractional part {0.618034 1.618034 2.618034}: there can be only one real number for which this is true. But it is in its geometrical expression, living structure and aesthetics (as in classical architecture) that lead its uniqueness to be seen as a divine ratio. Therefore, it seems, ancient human civilizations sought this golden form of harmony within the form of the Temple, especially in Dynastic Egypt and Classical Greece. The planet Venus must have reinforced this significance since its synod {584 days} is 8/5 of the solar year {365 days} and its manifestation such as evening and morning stars, move around the zodiac tracing out a pentacle or five-pointed star, the natural geometry of the golden mean.

The natural geometry of the Golden Mean is the Pentacle, traced out by planet Venus upon the Zodiac as evening and morning star. (from Sacred Number and the Origins of Civilization)

In the renaissance, the Classical tradition of Ancient Greece and Rome was reborn as neoclassicism, a famous proponent being Palladio, and further neo-classicism arose in the 19th Century and continues in the United States. From this, the previous article on St Peter’s saw its original square become rectangular in a golden way. The whole basis for this is due to the nature of squares and circles, that is: golden rectangles are easily formed geometrically through squares and circles.

The extension of St Peter’s from a square, by adding a golden rectangle, can be seen to also apply within the original square. Furthermore, there is a medium-sized square within the golden rectangle plus a small golden rectangle (see below).

The overall golden rectangle of St Paul’s of a square and golden rectangle below. Using the square within the golden rectangle, the original square above can have four such overlapping squares, to create a cruciform pattern, the upper part of which was used to lay out the Umayyad Mosque.

The medium square can be tiled four times within the large square to overlap the other medium squares, as shown above. This creates a small central square while the four regions that overlap are smaller golden rectangles. The lower golden rectangle is also repeated four times with overlapping, twice horizontally and twice vertically. It is seen that squares and golden rectangles can recede within a square, into smaller sizes, or expand around a square. It is as if all levels of scale hold a kind of fractal, based upon the golden mean.

The top six elements of the square can be seen to match the site plan of the Great (Umayyad) Mosque of Damascus, built 900 years before St Peter’s Basilica, on the site of an Orthodox Cathedral and, before that, a Roman temple to Jupiter. In other words, any golden rectangle design can contain resonances of somewhat different golden mean designs, that may express a different meaning or context; in this case the Mosque gives the notion of two squares overlapping to generate an intervening region of blending and the rectangle of overlap will then be phi squared in height (shown yellow below) relative to the width being unity – the central square’s side length.

The geometry of the Umayyad Mosque

My thanks to Dan Palmateer, for his emails and diagramming whilst on this theme of golden rectangles. One of his own pictures (below) shows the central square of the main square, by tiling the main square with the small golden rectangle.

The central square within the greater square is revealed in St Peter’s as a square within a circular area, noting that this plan (held by The Met Museum) was made after the building had been completed.

There was obviously a vernacular of golden rectangular building in Islam which was carried forth in Renaissance Europe. The potential for golden rectangular building can be all-embracing, as it is a property of space itself, due to numbers.