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.

Double squares: Venus and the Golden Mean

The humble square, with side length equal to one unit, is like the number one. It’s area is one square unit and, when we add another identical square to one side, the double square appears. Above right the Egyptian Djed column is shown within a double square. The Djed is the rotating earth which the gods and demons have a tug of war over. This is also a key story in the Indian tradition, called The Churning of the Oceans, where the churning creates both the food of the gods (soma) and every wonderful thing that emerges upon the Earth. In this, the double square symbolized the northern and southern hemispheres of the Earth. The anthropomorphic form Djed shown above has elbows indicative of the Double square.

Figure 1 The churning of the ocean (Samudra Manthan in Sanskrit)

The Djed appears to be the general principle of rotation of, and apparent motion around, the earth.

The god Isis is (as a planet) Venus and is shown (fig.2) offering up the sun disk: another Djed is below, with her Ankh symbol of Life atop the Djed, now having female arms . This sun most probably points to the practical year as 365 days which is 5/8 of the Venus synod of 584 days. (This ratio of 1.6 is the sixth note of the octave 1 to 2.)

In figure 2, two female attendants provide the duality which one might take to be her two famous manifestations of (firstly) the brightest Evening Star, as the sun goes down, and then (after that) the brightest Morning Star before the sun rises. Above there is duality again with three baboons either side of the sun, perhaps representing the six visible planets: Moon, Venus and Mercury: Jupiter, Saturn, Mars and their “tug of war”.

Figure 2 The creation of Horus-Ra from out of an ankh with female arms atop a djed. from Budge 1899, also fig. 7.8 of Richard Heath, The Harmonic Origins of the World.

The numbers 5 and 8 are Fibonacci approximations {1 2 3 5 8 13 21 34 …} to the golden mean, a transcendent number {1.618034…} which rational numbers can only approximate. Venus and the Earth have clearly settled into orbits around the sun resonant with Fibonacci ratios since the Venus orbital period (224.701 days) is 8/13 of the solar year. And it is this fact that eventuates in what we see on Earth, namely the manifestations of Venus every 8/5 of a practical year. of 365 days.

Figure 3 The double square, its in-circle and out-circle manifesting golden rectangles around itself.

In this post, I developed a result sent to me, that a square drawn within the upper hemisphere of a circle must define a golden mean rectangle either side from its height of 1 and the remaining radius of 0.618034… and so it can be seen that the divine principle of the Golden Mean emanates from the double square, either side of each square, when the double square is embraced by a circle drawn from its center. Obviously, on Earth and between orbits (of Venus and Earth), the Golden Mean (also called Phi) has to be approximated by whole number ratios but the principle is present within the geometry and its out-circle. Schwaller de Lubicz thought the dynastic Egyptians held the Golden Mean to be “the fundamental scisson” (literally “scissor cut”) in the range one to two and, its reciprocal can be seen to share the portion over 1 (figure 3).

One can see that geometry and the early numbers would have been seen as two aspects of what we call space and time, in which “things” are separate from each other in Existence but somehow conjoined within Eternity. What we call order is in fact an achievement of harmony made possible by the arranging and fitting of parts to form a coherent whole. It is this insight which gave meaning to their study of geometry and numbers from the prehistoric onwards, into the recorded history of early civilizations. The meaning for Life on Earth became encoded within ancient and prehistoric symbols, whose geometrical and numerical language of expression went to the heart of phenomena.

Story of Three Similar Triangles

first published on 24 May 2012,

Figure 1 Robin Heath’s original set of three right angled triangles that exploited the 3:2 points to make intermediate hypotenuses so as to achieve numerically accurate time lengths in units of lunar or solar months and lunar orbits.

Interpreting Lochmariaquer in 2012, an early discovery was of a near-Pythagorean triangle with sides 18, 19 and 6. This year (2018) I found that triangle as between the start of the Erdevan Alignments near Carnac. But how did our work on cosmic N:N+1 triangles get started?

Robin Heath’s earliest work, A Key to Stonehenge (1993) placed his Lunation Triangle within a sequence of three right-angled triangles which could easily be constructed using one megalithic yard per lunar month. These would then have been useful in generating some key lengths proportional to the lunar year:  

  • the number of lunar months in the solar year,
  • the number of lunar orbits in the solar year and 
  • the length of the eclipse year in 30-day months. 

all in lunar months. These triangles are to be constructed using the number series 11, 12, 13, 14 so as to form N:N+1 triangles (see figure 1).

n.b. In the 1990s the primary geometry used to explore megalithic astronomy was N:N+1 triangles, where N could be non-integer, since the lunation triangle was just such whilst easily set out using the 12:13:5 Pythagorean triangle and forming the intermediate hypotenuse to the 3 point of the 5 side. In the 11:12 and 13:14 triangles, the short side is not equal to 5.

Continue reading “Story of Three Similar Triangles”

The Approximation of π on Earth

π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and south poles have a shrunken radius and, in pre-history, the shape of the earth’s Meridian between the poles was quantified using approximations of π as was seen in the post before last. In some respects, the Earth is a designed type of planet which has to have a large moon, 3/11 of the earth’s size and a Meridian of such a size that the diverse biosphere can be created within the goldilocks region of the Sun’s radiance.

It would be impossible to quantify the earth as a physical object without the use of approximations to π, a technique seen as emerging in Crucuno between its dolmen and famous {3 4 5} Rectangle where the 32 lunar months in 945 days was used, through manipulation of proximate numbers to rationalize the lunar month to 27 feet (10 Drusian steps) within which days could be counted using one Iberian foot (of 32/35 feet) as described here and in my Sacred Geometry book.

John Michell (1983) saw that different types of foot had longer and shorter versions, different by one 175th part and corresponding to the north-south width of two parallels of latitude: 51-52 degrees, which is the mean earth degree, and 10-11 degrees. The ratio 176/175 is interesting as for its primes.

  1. The harmonic primes {2 3 5} are 16/25 times 11/7.
  2. The 11/7 is half of the pi of 22/7 and the harmonic ratio is the inverse of 25/8.

From this it is clear that these two latitudes are related by the approximation to 1 of a π (22/7) and a reciprocal 1/π (8/25).

But John Neal (2000) saw that some feet also expressed 441/440 which is the ratio between the mean radius of the earth and its polar radius, visually clear in the Great Pyramid. This ratio is also the cancellation of two different πs, namely 63/20 and 7/22 since 7 x 63 = 441 and 20 x 22 = 440. From this emerged an ancient model of the earth that was embodied within the ancient metrology itself. I call this the metrological model rather than the (earlier) geometrical model based upon equal perimeters and the singular π of 22/7.

The metrological model gave a set of regular reference latitudes that accurately defined the geoid of the planet’s meridian by 2,500 BC. One can ask how those developing the model came across the idea of using proximate ratios of π like 176/175 and 441/440, since the system works so well that one may say that the meridian appears to have been designed that way.

The geometric model already defined the mean radius as 3960 miles and so that gives a mean earth meridian of 22 x twelve to the power six. One 180th of this gives a degree length of 364953.6 feet and this is only found at the parallel 51-52 degrees. It is this that defines the megalithic in England, Wales, Scotland and Ireland, an obvious candidate for the metrological survey whose complementary latitude was probably 175/176 of this (362880 feet) in Ethiopia, south of the Great Pyramid. The parallel of the Great Pyramid is 441/440 longer (362704.72) than that of Ethiopia while Athens and Delphi are 440/441 of the mean earth and Stonehenge parallel, that is 364126 feet.

This system was first set by Neal in All Done With Mirrors 2000 as I was writing my first book Matrix of Creation. Are we to think Neal made it up or are we dealing with an exact science that had developed through the megalithic enterprise. And if the Egyptians had an exact science of the earth’s geiod, what are we to make of the fact that the earth appears to follow such a numerically inspired pattern of relationships still true today, in the age of global positioning satellites.

One clue lies in the mind, and how ancient number sciences focus holistically upon the balancing mean. A mean earth that did not spin never existed, since it was only the collision with another planet which created the Moon 3/11 smaller than the Earth. The mean earth radius is these days established as the cube root of the equatorial radius squared times the polar radius. This is less, by 3024/3025, than the geometric model’s mean earth radius of 3960 miles, again maintaining rationality.

It would appear that, in entering the physical and spatial, any planetary design might have been based upon precise rational approximations, about the mean size, of π. To this mystery must be added the musical harmony of the outer planets to the Moon, the Fibonacci harmony of Venus to the Earth itself and the extraordinary numerical relationships of planetary time created by the Sun, Moon and Earth documented by my heavily-diagrammed books and website. From this, more and more can be understood about our prehistory and about its monuments.

Books on Ancient Metrology

  1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
  2. Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
  3. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  4. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  5. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
  6. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  7. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.

Working with Prime Numbers

Wikipedia diagram by David Eppstein :
This is an updated text from 2002, called “Finding the Perfect Ruler”

Any number with limited “significant digits” can be and should be expressed as a product of positive and negative powers of the prime numbers that make it up. For example, 23.413 and 234130 can both be expressed as an integer, 23413, multiplied or divided by powers of ten.

What Primes are

Primes are unique and any number must be prime itself or be the product of more than one prime. Having no factors, prime numbers are odd and cannot be even since the number 2 creates all the even numbers, meaning half of the ordinals are not prime once two, the first “number” as such, emerges.

Each number can divide one (or any other number) into that number of parts. In the case of three (fraction 1/3) only one in three higher ordinal numbers (every third after three) will have three in it and hence yield an integer when three divides it.

Four is the first repetition of two (fraction ½) but also the first square number, which introduces the first compound number, the geometry of squares and the notion of area.

Ancient World Maths and Written Language

The products of 2 and 3 give 6, 12, etc., and the perfect sexagesimal like 60, 360 were combined with 2 and 5, i.e. 10, to create the base 60, with 59 symbols and early ancient arithmetic, in the bronze age that followed the megalithic and Neolithic periods.

Continue reading “Working with Prime Numbers”