Primacy of low whole numbers

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets

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. 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 32 36 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.

Coming soon: Why numbers manifest living planets