This extract from The Harmonic Origins of the World (p58-62) shows how what are taken to be arbitrary numbers, in the narrative of the Patriarchs, expressed knowledge of planetary resonances.
above: Diagram of the Bible code of “holy mountains”: Products of the powers of five upwards and of the powers of three across, each mountain within a limiting number if note D is {60, 120, 45, 180, 720} .
Female archetype Eve, “mother of all living,” becomes Abram’s wife Sarai. Childless, Abram was encouraged (by Sarai) to have first son Ishmael by concubine Hagar, but the Lord God (the mountain god whose number sums to 345) renames Sarai as Sarah and Abram as Abraham. At a miraculous 90 years of age she gives birth to Isaac (“he laughs”), then reportedly dies at 180 years old. It is harmonically relevant that (a) the giving of heh = 5 to both Sarah and Abraham elevates them from their former selves onto the second row, “stepping up” like the god Ea in Sumeria, and that (b) Adam’s number 45 has now been doubled to 90 and can form an octaval womb in which Isaac can be born, his life to then end at the doubling of Sarah’s 90 years to 180 years.
above: The human essence class related to four other classes in J.G. Bennett’s Gurdjieff: Making a New World. Appendix II. page 290. This systematics presents the human essence class which eats the germinal essence of Life, but is “eaten” by cosmic individuality, the purpose of the universe. The range of human potential is from living like an animal to living like an angel or demiurge, then helping the cosmic process.
The human essence class is a new type of participation within the universe where the creation can form its own creative Will, in harmony with the will that creates the universe. The higher intelligences have a different relationship to the creation than human intelligence. It is based upon this Universal Will (to create the universe) which has manifested a world we can only experience from outside of it. And the creative tip of creation* is the universal life principle that led to the human world where it is possible to participate in the intelligence behind the world, through a transformation into an Individuality, creative according their own pattern while harmonious with the universal will.
*creative tip: The evolving part of organic life is humanity. Humanity also has its evolving part but we will speak of this later; in the meantime we will take humanity as a whole. If humanity does not evolve it means that the evolution of organic life will stop and this in its turn will cause the growth of the ray of creation to stop. At the same time if humanity ceases to evolve it becomes useless from the point of view of the aims for which it was created and as such it may be destroyed. In this way the cessation of evolution may mean the destruction of humanity.
In Search of the Miraculous. P.D. Ouspensky. 306.
Will is not something one does. Rather, it is a participation of one’s being with Will. This creates a transformational action of Will within a human that is receptive to it (rather than merely assertive on their own account). We are born able, through our unique pattern, to participate in our own understanding of the meaning that is this world. In this, numbers are more than data: they form structures of will which do not rely on complexity and are therefore directly intelligible for an intelligent lifeform, enabling what to do, by seeing more deeply what is in the present moment. For example, number is the foundation of that universal invariance: the Present Moment of selfhood*.
The myth of a philosopher’s stone presents a challenge, to find the “stone” itself, which we shall see is probably the numerically favourable environment upon the earth. The stone has been rendered invisible to modern humans by our functional science of infinite complexity, also called instrumental determinism. This has downgraded human expectations to being a walk-on part, an unintentional result of evolution, by natural selection, of intelligent life. To think otherwise it is necessary to see what is not complex about the sky, which is a designed phenomenon related to Life on Earth. Once-upon-a-time, the stone age understood the sky in this right way, the way it had been designed to be read by us, corresponding with the way intelligent life was intended to be, on a habitable planet with a large moon.
1.1 Geocentric Numbers in the Sky
Our pre-digested meanings are those of modern science. Whilst accurate they cannot be trusted in the spiritual sense, if one is to continue looking at phenomena rather than at their preformed conceptual wrapper. Numbers in themselves are these days largely ignored except by mathematicians who, loving puzzles, have yet largely failed to query the megaliths** but, if or when anyone might say the megaliths had a technical purpose, this has annoyed most archaeologists, who live by the spade and not by the ancient number sciences or astronomy.
**Fred Hoyle, Hawkins, Alexander Thom, Merritt and others all found something new in Stonehenge but still failed to explore stone age numeracy as well as the numeracy of metrology. Rather, they assumed measures unlike our own were used, yet the megaliths would continue to have no meaning “above ground”, except as vaguely ritualistic venues in loose synchronization with a primitive calendar.
Numbers are not abstract once incarnated within Existence. In their manifestations as measurements, they have today become abstracted due to our notation and how we transform them using arithmetic, using a positional notation based upon powers of two and five {10}, called the decimal system* (https://en.wikipedia.org/wiki/Decimal). The so-called ordinal numbers {1 2 3 4 5 6 7 8 9 10 …etc.} are then no longer visually ordinal due to the form in which they are written, number-by-number, from right-to-left {ones, tens, hundreds, thousands …} * (the reversal of the left-to-right of western languages). Positional notation awaited the invention of zero, standing for no powers of ten, as in 10 (one ten plus no units). But zero is not a number or, for that matter, a starting point in the development of number and, with the declaring of zero, to occupy the inevitable spaces in base-10 notation, there came a loss of ordinality as being the distance from one.
Before the advance of decimal notation, groups within the ancient world had seen that everything came from one. By 3000BC, the Sumerian then Old Babylonian civilization, saw the number 60 perfect as a positional base since 60 has so many harmonious numbers as its factors {3 4 5}, the numbers of the first Pythagorean triangle’s side lengths. Sixty was the god Anu, of the “middle path”, who formed a trinity with Fifty {50}, Enlil* (who would flood humanity to destroy it) and Forty {40} who was Ea-Enki, the god of the waters. Anu presided over the Equatorial stars, Enlil over those of the North and Ea-Enki over those of the South. In their positional notation, the Sumerians might leave a space instead of a zero, calling Sixty, “the Big One”, a sort of reciprocal meaning of 60 parts as with 360 degrees in a circle from its center. So the Sumerians were resisting the concept of zero as a number and instead left a space. And because 60 was seen as also being ONE, 60 was seen as the most harmonious division of ONE using only the first three prime numbers {3 4 5}.
These days we are encouraged to think that everything comes from zero in the form of a big bang, and the zeros in our decimal notation have the unfortunate implication that nothing is a number, “raining on the parade” of ordinal numbers, Nothing usurping One {1} as the start of the world of number. The Big Bang, vacuum energy, background temperature, and so on, see the physical world springing from a quantum mechanical nothingness or from inconceivable prior situations where, perhaps two strings (within string theory) briefly touched each other. However, it is observations that distinguish meaning.
In what follows we will nevertheless need to use decimal numbers in their position notation, to express ordinal numbers while remembering they have no positional order apart from their algorithmic order as an infinite series in which each number is an increment, by one, from the previous number; a process starting with one and leading to the birth of two, the first number.
Whole (or integer) numbers are only seen clearly when defined by
(a)their distance from One (their numerical value) and
(b) their distance from one another (their difference).
In the Will that manifested the Universe, zero did not exist and numerical meaning was to be a function of distances between numbers!
Zero is part of one and the first true number is 2, of doubling; Two’s distance from one is one and in the definition of doubling and the octave, the distance from a smaller number doubled to a number double it, is the distance of the smaller number from One. This “strange type of arithmetic” *(Ernest G McClain email) is seen in the behavior of a musical string as, in that kind of resonator, half of the string merely provides the basis for the subsequent numerical division of its second half, to make musical notes – as in a guitar where the whole string provides low do and the frets when pressed then define higher notes up to high do (half way) and beyond, through shortening the string.
This suggests that a tonal framework was given to the creation by Gurdjieff’s Universal Will, within which many inner and outer connections can then most easily arise within octaves, to
overcome the mere functionality of complexity,
enable Will to come into Being,
equip the venue of Life with musical harmony and
make the transformation of Life more likely.
Harmony is most explicit as musical harmony, in which vibrations arise through the ratios between wavelengths which are the very same distance functions of ordinal numbers, separated by a common unit 1.
Take the number three, which is 3/2 larger than two. Like all ordinal numbers, succeeding and preceding numbers differ by plus or minus one respectively, and the most basic musical tuning emerges from the very earliest six numbers to form Just intonation, whose scales within melodic music result as a sequence of three small intervals {9/8 10/9 16/15}, two tones and a semitone. Between one and those numbers {8 9 10 15 16} are the first six numbers {1 2 3 4 5 6}* (note absence of seven between these sets), whose five ratios {1/2 2/3 3/4 4/5 5/6} provide any octave doubling with a superstructure for the melodic tone-semitone sequences; their combined interdivision, directly realizes (in their wake) the tones and semitones of modal music.
We will see that the medium for such a music of the spheres was both the relationship of the sun and planets to the Moon and Earth, and this manifested quite literally in the lunar months and years, when counted. But Gurdjieff’s octaves cannot be understood without disengaging modern numerical thinking, procedures and assumptions. It is always the whole being divided and not a line of numbers being extended, though it is easier to look within wholes by expressing their boundary as a large number: Hence the large numbers of gods, cities, time and so on. For example, creating life on earth requires a lot of stuff: perhaps the whole solar nebula has been necessary for that alone and billions of our years. Were you worth it?
What we call numbers start from one, and from this beginning all that is to follow in larger numbers is prefigured in each larger number. And yet, this prefigurement, in the extensive sense {1 2 3 4 5 6 7 etc.}, is completely invisible to our customary modern usage for numbers, as functional representations of quantity. That is, as the numbers are created one after another, from one {1}, a qualitative side of number is revealed that is structural in the sense of how one, or any later number, can be divided by another number to form a ratio. The early Egyptian approach was to add a series of unitary ratios to make up a vulgar* but rational fraction. This was, for them, already a religious observance of all numbers emerging from unity {1}. The number zero {0} in current use represents the absence of a number which is a circle boundary with nothing inside. The circle manifesting {2} from a center {1} becomes the many {3 4 5 6 7 …}.
The number one manifests geometrically as the point (Skt “bindu”) but in potential it is the cosmological centre of later geometries, the unit from which all is measured and, in particular, the circle at infinity.
Two: Potential spaces
From one {1} springs two {2}, to which we owe all forms of doubling as reproduction without sexual division, called “barren” by Plato, yet giving the possibilities of complex worlds of different scale, in terms of their limiting number. This is the first true number of Creation which gives the quality of polarity between the two halves of (as yet) nothing, halves of a world that will create the beginnings of an everything. Super dense, as an initial Form of forms, all things will come to rotate around this Axis of axes*. (Axes, when pronounced with a long e, is the plural form of the word axis, meaning imaginary lines that run through the middles of things. The word axe is derived from the Old English word æces, the axe which divides into two. ) This is the birth of duality, as with the centre and circumference of a circle or positive and negative (opposite) charge, and the medium of the wave or vibration, which gave birth to dynamic systems, such as planetary rotation of an axis or an orbit.
When number was incarnated in our own planetary creation, it was Saturn who visibly delimited the outer limits of the visible planets. His name is close to Seth and Satan (as the necessary adversary of the heroic Horus) and he was seen as limiting unbounded growth within existence. Saturn expresses 5 synods of the planet Saturn in 64 {26} lunar months (but this is to jump in numbers, though not too far, to the planetary double octave {24 48 96} lunar months. Sixty-four governs the “eye of Horus”: a government deriving from the mythical conflict between the god Horus with his rival Set; in the context of two eyes, here the right eye was torn out but then restored, to then see the role of two within the creation, in the “afterlife”.
Figure 1 The Egyptian icon of the Eye of Horus as the components used to represent vulgar fraction as a series of powers of two.
https://en.wikipedia.org/wiki/Eye_of_Horus: It derives from the mythical conflict between the god Horus with his rival Set, in which Set tore out or destroyed one or both of Horus’s eyes and the eye was subsequently healed or returned to Horus with the assistance of another deity, such as Thoth.
The Moon finds an exact reunion with the earth day after exactly 945 days, which equal 32 lunar months of 945/32 (29. 53125) days, very close to the actual lunar month of 29.53059, being effectively exact as 57 seconds different.
While the number two can, in being divided, create new areas of interaction (including cosmic octaves), its mere extension {2 4 8 16 32 64 128 …) forms only a backbone of potentialities, these then (see later page) borrowed by higher limiting numbers such as 720, a number containing favourable numerical factors for the creation an ideal “family” of limits, metaphorically presented by Adam and the Patriarchs of the Bible.
Figure 2 The vision of the Godhead asleep on a pre-creational ocean (of his sweat) attended by the Goddess until he awakes for a new creation.
The number {2} as dyad manifests as the geometry of the line. The line {2} from a center {1} as a rotational vector becomes the geometry of the circle.
Three and four: Actualization
The actual creation has a different planetary symbol, the equilateral triangle with three sides, seen also as the capital Greek delta, whose value is 4 because the planetary system is an Activity involving forces. These numbers are reconciled as 3 x 4 which equals 12, the number of Autocracy * and balanced action. We are told in myth that Jupiter’s twelvefold nature was “the receiving of the measures” from Saturn, as 4. Jupiter is the planetary demiurge which Plato describes carving out the World Soul “octave” {6 8 9 12}, using ratios involving Three, the cubit and its reciprocal {3/2 4/3} * (Timaeus). Only when we reach the lunar octave {24 27 30 3236 40 45 48} will the diatonic scale of eight notes emerge, the first and last being the same note, doubled.
The first true doubling {2 3 4}, between 2 and its square, holds the first type of penetration of the octave, by Three {3}. With three, the Demiurge forms his World Soul using intervals involving only Two and Three {3/2 4/3}, which can create the fifth and fourth notes (“dominant and subdominant”) of an octave.
The number {3} gives form to the first geometry of area, manifest in the triangle which, given a right angle, becomes trigonometrical, as the functional mediation between the line and the circle.
The number four connects the relatedness of the Triad (3) with the existentially actual to provide an engine in which Form can become Substance through an intermediate pair of terms that fulfill the gap {2} between form and the actual situation. One could say this is the first instance of filling the octave with tones {8}, intermediate between 2 and 8. Four is the first square number which in geometry is called square as an area equal to 4 has sides equal to 2
Five: Vitality and Life
Coming next, Five {5} will also be able to divide the coming “octave” {3 4 5 6} in a superior way than three and two can alone, by redefining a new tone {10/9} for Just intonation and a corresponding semitone {16/15}, these cancelling the excessive powers of three produced by tuning only with three {3}, called the cycle of fifths, which used successive fifths and its inverse{3/2 4/3} because the ear can best define the larger musical intervals. The octave between three and six defines the framework of Just intonation where three intervals span the octave {4/3 5/4 6/5}, summing to doubling {2}.*
*This was probably referred to as “the three strides of Vishnu”, Trivikrama (‘having three steps’) being one of his 1,000 names.
The planet Venus brings a new type of harmony, which is also the sixth note {8/5 (= 1.6)} of a diatonic octave (see this page) since her synod of 584 days is 8/5 of the practical year of 365 days. The Fibonacci series allows whole number approximation to the Golden Mean {φ} between adjacent members obtained as being the sum of the two preceding numbers {0 1 1 2 3 5 8 13 21 34 55 …} unlike the ordinal set {1 2 3 4 5 6 7 8 9 …}, the latter instead obtained by more simply adding one more unit {1} to the preceding number. Unlike the musical tone and semitone of Jupiter and Saturn relative to the lunar year, Venus is resonant with the Earth’s orbital period of 365 whole days, and this type of orbital resonance, with each other, is mutually attractive, providing the lowest and most stable energy between the two planets. The inner orbital diameter (semi major axis) divided by the difference in orbital diameters, equals 2.618, or phi squared {φ2}*. (See later page for more on their orbits) Structures of growth, based upon Fibonacci ratios, are commonly found within living bodies, which must achieve this algorithm in which their present size added to their digestion of previously eaten food results in the sum of the two.
Figure The Fibonacci series in two dimensions are common forms of living growth.
The Venus synod will be seen to fit inside the octaves of the Moon because 20 lunar months is 590.6 days which is less than the synodic comma {81/80} of her 583.92 day synod*.
*The synodic comma is the exact ratio connecting Pythagorean and Just versions of the same note. One of the Indian temple designs is a nine-by-nine square grid which makes the number of equal-sized sub-squares {81) divided by the count where the central square is not counted gives the ratio of the synodic comma {81/80}.
Music: Child of the First Six Numbers
The larger intervals of numerically larger octave doublings are in this way foreordained in the first six numbers {1 2 3 4 5 6} and their relative size to each other, are five musical intervals {2/1 3/2 4/3 5/4 6/5}. Doubling has led to the pillars of Plato’s world soul {3/2 4/3} and three when doubled {3 4 5 6} has led to the three strides {4/3 5/4 6/5}, both sets summing to Two {2}.
The first six numbers, creating all the large tones of musical harmony, punctuated by Seven.
Between the five musical intervals, the tones and semitones of Just intonation are to be found {9/8 10/9 16/15} so that, as a tuning system, the Just system leads automatically to the tones and semitone of the seven modal scales, in both melodic and polyphonic harmony.
When the World Soul {6 8 9 12} is twice doubled {24 48} and doubled again {48 96}, the two octaves express the world numbers of Gurdjieff {24 48 96}, but now these numbers correspond to lunar months and, as with music when heard, all of the possible intervals are compresent in the instrument, the Moon illuminated by the Sun, since one can count from any lunar month, over any number of lunar months, to achieve any of the larger and smaller intervals between these octaves. And it is now true that the three principle planets of Jupiter, Saturn and Venus are present at the second, fourth and sixth notes, each of these relevant to Gurdjieff’s theory of octaves as stated by him in Russia, and his cosmic epic Beelzebub’s Tales. And J G Bennett continued to build on what Gurdjieff had expressed, without knowledge of the astronomical references, to populate his own Dramatic Universe, in 4 volumes and many compendia (see Bibliography). Of particular importance is how human beings figure within the cosmic vision, without which a planetary cosmos involving consciousness and creativity would be meaningless. If one resists the modern functional view of cosmogenesis: music, or other forms of harmony, can be seen to redeem the creation of a world like ours, through the short cuts numerical systems naturally provide for us, through a gravitational environment that can provide these.
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.
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.
