Why numbers manifest living planets

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.

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Phenomenology as a Native Skill

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

  1. overcome the mere functionality of complexity,
  2. enable Will to come into Being,
  3. equip the venue of Life with musical harmony and
  4. 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?

Phenomenology as a Native Skill

Gurdjieff, Octave Worlds & Tuning Theory (2019)

This is a paper I suggested for the All and Everything conference in Cumbria, but it was not selected. It developed a number of strands, which I offer here as a snapshot of my thinking and research around 2019. This text was modified to become appendix 3 of my Sacred Number and the Language of the Angels (2021).

Abstract

The first part presents what has only recently become known about ancient musical theory, to better understand the All and Everything of Gurdjieff’s intellectual output. This must include In Search of the Miraculous (Search, 1916-18) Beelzebub’s Tales (Tales, 1949) and Meetings with Remarkable Men (Meetings, 1963). In part 2 ancient monuments are shown to record a ‘lateral octave’ connecting humanity to the planetary world, 24. Part 3 explores the significance of the Moon in Gurdjieff’s lectures and writing. An appendix reviews the conventional virtues of the Moon as accepted by modern science, stabilising earth, enabling life and beings such as we, to evolve into appropriate habitats.

Contents

Abstract 1
Introduction. 3
Part 1: Ideas relating to a ‘science of vibrations’
The Role of Octaves
LikelySources of Gurdjieff’s Harmonic ‘Ideas’
Did Gurdjieff understand Numerical Tuning Theory?
Was Gurdjieff a Pythagorean?
Pythagorean Tuning
Greek versus Chinese Tuning?

Part 2: Where are the Lateral Octaves?
Part 3: The Significance of the Moon
Appendix 1: A Moon that created Life?
Appendix 2: Reference Charts from Search
Appendix 3: Index of the Moon in Search & Tales
Moon in Search
Moon in Tales
Bibliography

Introduction

Publications about Gurdjieff’s ideas appeared after he and Ouspensky had died. The main works of Gurdjieff’s words are Search, Tales and Meetings. Beyond these lie autobiographical books and compendia of Gurdjieff’s ideas, by his students. Some of his students also looked into traditional sources such as Sufism and Vedanta, and followed up on new scholarship relating to cosmological ideas such as world mythology (Hamlet’s Mill, The Greek Myths); the number sciences of Pythagorean schools and Plato (Source books); ancient buildings (Megalithic Sites in Britain, Ancient Metrology, The Temple of Man); and musical tuning theory (The Myth of Invariance, Music and the Power of Sound).

These subjects appear to have been investigated by the group Gurdjieff belonged to, The Seekers after Truth he describes in Meetings. Like archaeologists and linguists, they sought out ancient records but also contacts with living initiates of esoteric learning. This search was similar in form to that conducted by Pythagoras in 600 BC, after which he contributed much to western knowledge of the numerical and musical mysteries. The Seekers after Truth could today find out much lost knowledge has been recovered for them. For example, the ancient near-eastern musicology discovered within the last century on cuneiform tablets. Similar deciphering of Plato in the second half of the 20th century (by Ernest McClain and others) revealed the technicalities of harmonic numbers and their use within ancient literature, mythology and symbolism.

Part 1:
Ideas relating to a ‘science of vibrations’

The Role of Octaves

Gurdjieff’s early lectures in Search gave (in 1950) a complex but unified theory of everything, based in large part upon the behaviour of musical octaves and the role of a single musical scale in particular: the ascending major diatonic we call C-Major. Search introduced three major diagrams: The Diagram of Everything Living (Diagram hereafter); The Ray of Creation (Ray); and The Enneagram. These charts are reproduced as Appendix 2.

In Ray, the Universe according to Gurdjieff unfolds according to the major diatonic scale, which happens to be foundational in equal-tempered key signatures. This major scale is also the basis for the European Solfege notation (do-re-mi-fa-sol-la-si-do) that was employed when presenting the harmonic meaning of the Enneagram. The use of solfege in Gurdjieff’s musical exposition of the Worlds in Ray, essence classes[1] in Diagram and the six points of the Enneagram connected by its “inner lines” (see Appendix 2), avoided the allocation of use of note classes A, B, C, D, E, F, G; which would have been confusing since do can be any note class

This article aims to interpret Gurdjieff’s harmonic cosmology, using a technique only recently recovered from traditional texts.  The numbers of Gurdjieff’s World Levels (3-6-12-24-48-96) share the factor 3, then simply doubled and, he says, all these worlds have octaves between them[2]. Forming octaves according to number was a lost art of the ancient world from which much of the ancient world’s number symbolism has been shown to derive. Lateral octaves connecting worlds would require higher powers of 3.

Whilst the Diagram and Ray schematics are linear, the Enneagram is circular and cyclic, and this means it loosely resembles the Tone Circle for an octave.  However, the special virtue of the Enneagram is its capacity to integrate the law of seven with the law of three within a circular view of ninefoldness as a singular and whole process on any level, in Tales called Heptaparaparshinokh.

Figure 1 (left) The Enneagram in C Major diatonic (Ionian scale ) and (right) Tone Circle in D and the symmetrical Dorian scale native to Invariance of numerical harmony. Note that point 6 is opposite the cosmic D, making the Harnal-Aoot[3] disharmonious in Heptaparaparshinokh.

Likely Sources of Gurdjieff’s Harmonic ‘Ideas’

The official history of European musicology cites Guido of Arezzo (991/992 – after 1033) as the father of the medieval musical theory that led to modern music. One innovation was the solfege used by Gurdjieff to notate do as point 9 and other points inside the octave, connected to the Enneagram’s inner lines as re-me-fa-sol-la. However, Guido did not use solfage within the context of the octave so familiar today. The medieval solfege tradition was hexachordal (do-re-mi-fa-sol-la) rather than octaval: originally there was no si or higher do. The hexachordal system of medieval times enabled novice singers to make rapid progress learning the plainchant of church music where sub-dominant fourths (do to fa) ignored the fixed tonic of octaves.

“In medieval theory the compass of tones was obtained not by joined octaves but by overlapping hexachords. This method, although generally considered inferior to the modern practice, is actually superior in that it produces the scale without at the same time establishing a preference regarding tonality. Indeed, in the modern system the initial tone C automatically becomes the tonal center (in other words our diatonic scale is necessarily a “C-major scale”), whereas in the system of the hexachord such a fixation is avoided.” **HARVARD Dictionary of Music 2nd ed. Willi Apel. Cambridge, Mass: H.U.P. 1969. see Hexachord

The Enneagram is also not just an octave; since point 3 and point 6 can be viewed as new starting points (see figure 2 below, Enneagram Studies); point 3 at fa is then similar to the hexachordal system in the following way. The fa of a lower hexachord was a point of transition to do for the next higher hexachord, triggered to run in parallel to one already started, and this leads to a triple octave in which some octaves are “sacrificial” as with the breath which gives food a needed shock at the first semitone in Search.

Figure 2 The Enneagram viewed as an interaction between three parallel but staggered major diatonics

Search defines the third interval (between mi and fa) as a semitone within all cosmic octaves, then described variously as deflections, retardations or new directions. The purpose of an octave can only proceed through its intersection with the suitable vibrations of another octave, a simple example being: an already developed way of surmounting an obstacle. This idea is functionally like the hexachordal system where at fa (point 4) the singers can adopt the succeeding hexachord, started as a new do (then point 3). The choristers would still remember that this was fa of the preceding hexachord and may yet drop back into that notation. However, by the end section it will be clear that musical theory is inadequate to describe the cosmology of human transformation being presented in Gurdjieff’s vision.

Whilst Benedictine monk Guido of Arrezo has been credited with innovating the hexachordal solfege of the Middle Ages, it was the work of the Islamic philosopher Al-Kindi[4] (left) in the 9th century, an early Islamic world that was very creative, reusing many ideas from the Classical Greeks such as Plato and adding to them. As a Benedictine monk, Guido had access to Jewish translations of Arabic works whilst ironically, Islam would soon dispense with philosophers and musical theory.

The labelling of the Enneagram with the solfege system and the curious starting of new octaves at points 3 and 6 were probably integral to the Enneagram when Gurdjieff first encountered this symbol in Bokhara, Uzbekistan. Bokhara is the centre of the Naqshbandi Order, whose name alludes to seal, pattern, symbolism and the tomb of their founder, Bahauddin (1318-1389) who J.G. Bennett equates with the Bokharian Dervish Bogga-Eddin of Tales [Enigma, 38]

After Enigma (1963) Bennett found another complementary source of Gurdjieff’s “science of vibrations” (The Masters of Wisdom, 1977) in the school of Ahmad Yasavi. (bottom right, d. 1169).

“Ahmad Yasawi’s central school in Tashkent … is of special interest to followers of Gurdjieff’s ideas because it was the main repository of the science of vibrations expressed partly through dance and music and partly through the sacred ritual that came from the Magi. This science distinguished the Yasawis from the main tradition of the Masters …” The Masters of Wisdom. 131.

Before considering that science of vibrations here, from the standpoint of ancient tuning theory, it is important to share Bennett’s strong themic synthesis (in Enigma) between (a) the Pythagorean number sciences west of the Caucassus and (b) Central Asian knowledge about how energies are transformed within cosmic octaves.

In this way, we might agree with Bennett’s conclusion, that the knowledge Gurdjieff taught as his ‘Ideas’ came from putting together two halves of a single truth. One half is found in the Western – chiefly Platonic – tradition and the other half is in the Eastern – chiefly Naqshbandi [and Yasavi] tradition. This fusion of two halves was perhaps hinted at, by Gurdjieff, in the story of the Boolmarshano in Chapter 44 of Beelzebub. [Enigma, 41]

Did Gurdjieff understand Numerical Tuning Theory?

Yes, but his early groups in St Petersburg and Moscow did not have much clue as to numerical tuning theory, despite having been educated in musical forms and the playing of instruments within the equal tempered world of Major diatonic scales using 12 keys. On page 126 of Search, Gurdjieff uses the inappropriate numbers, 1000 to 2000 to explain how octaves and scales worked[5]. These numbers, lacking the prime number 3, are alien to numerical harmony, where an octave’s limiting number (high do) is crucial to the formation of tones within its octave.

All octaves manifest the prime number 2, the first true interval in which doubling creates a boundary only entered by tonal numbers having larger factors, of prime numbers 3 and 5: to “get into” an octave by dividing it. Since the limiting number of 2000 has no prime factors of 3 to “give” to any new integer tones within its octave range 1000-2000, whole number tones of the Pythagorean kind were not possible. One concludes from this; the poor number of 2000, presented as the limit, was either (a) not correctly remembered by the students or (b) was deliberately inadequate to scale formation, so that only the diligent would calculate the correct octave range. Gurdjieff says,

“the differences in the notes or the differences in the pitch of the notes are called intervals. We see that there are three kinds of intervals in the octave: 9/8, 10/9, and 16/15, which in whole numbers correspond to 405, 400, and 384.” Search, 126.

There is only one number that can form these three intervals to these three numbers: 360, and 360 is low do (in tuning theory) for the lowest possible limit forming five different scales, namely the high do of 720. This limit (see later) used in the Bible’s earliest chapter Genesis (written c. 600 BC in Babylon), to define Adam (whose letters, equalling 45[6], double four times to 720)

Figure 3 Harmonic Mountain and Tone Circle of 720.

The mountain for 720 (see figure 3 above) shows the initation of three scales from D (=360) to E as 405 (Mixolydian scale), to e as 400 (Ionian) and eb as 384 (Phrygian). It is as if Gurdjieff was referring to Ernest G. McClain’s “holy mountain” for 720, and if any student followed up on this clue it would show the limits intended for the scale (in the octave in the 1000-2000 example) as 360:720. At which point, they would have stepped into the world of ancient tuning theory and found the octave numerically fecund, with its five, now largely antique, scales within.

Was Gurdjieff a Pythagorean?

The ethos of ancient tuning was exactly like that of the Diagram, in that everything emanates from the number one to form the first new World numbered 3. This is exactly as Socrates and Lao Tsu[7] stated and the Pythagoreans (600 BC onwards) have given us two diagrams through which they thought the world was created: The Lambda and Tetraktys. The Lambda mixes the powers of prime numbers 2 and 3, where the numbers grow in a triangular fashion. The Tetraktys is a similar triangular shape, having, like the Lambda, 1 at the top and three rows below of 2 and 3 then 4, 5, 6 and 7, 8, 9, 10; the first ten numbers.

Figure 4 Key Cosmological Diagrams of the Pythagoreans. The Lambda form of the Tetraktys generates rows of musical fifths propagating downwards in a repeat of 2 and 3 as 2 x 3 = 6, and 6 is 3/2 of 4 and 2/3 of 9.

When it comes to books about musical scales, their tone numbers are often calculated from the “bottom up”, from starting numbers like 24, 27, 30, and 36 as low do and then applying successive intervals to achieve the scale you already have in mind. This is a lesser method since one does not then see the true behaviour of the number field in generating the scales between numerical octave limits of any size. Ancient number science had come to the more holistic approach, in which limiting numbers could be investigated to explore the evolution of scales, in the octave beneath these limits, enabling systematic discovery of those key limiting numbers associated with musical phenomena. It is this work that led to a great deal of the number symbolism found within ancient stories, buildings and art, that are part therefore of Gurdjieff’s notion that they become legominisms from which one can understand lost knowledge.

The Lambda diagram, named after the Greek letter Lambda (Λ), can locate the number factors found in Gurdjieff’s “Pythagorean” numbered worlds (2 & 3) growing from One at the vertex. The top triangle is 1 then 2 (left) and 3 (right). We are told that, between the Absolute and the Eternal-Unchanging of the Diagram, a “conscious manifestation of the neutralising force (3) … fills up the ‘interval’ between the active (1) and the passive forces (2).” [Search, 137: brackets added]

From then on, (see figure 5 below) the passive force (2) travels downwards separately from the reconciling force (3) as two ‘legs’ formed by the increasing powers of 2 and 3. The mixed powers of 2 and 3 then combine throughout the middle region, to form numbers made of all the possible combinations of 2 & 3. For example, the number 3 tracks along the left-hand side of the lambda, being doubled to create new lower World-numbers next to them, starting with world 6 as 2 x 3, world 12 as 4 x 3, and so on until the lowest world in the Ray is 96 which is 32 times 3 whilst the lowest essence class in the Diagram is 1536 which is 512 x 3. It is therefore true that Gurdjieff’s worlds, numbered according to the number of their laws, emanates from world 6 or 2 x 3, as the left-most blending of 3 with powers of two[8].

Figure 5 Creation of Worlds, in yellow, through reconciliation (3) of the denying force (2), in red, by the Will of God.

Since it is the material nature of existence which forms the passive (denying) force (Etherokrilno) of the creation, then the worlds are the blending of the reconciling force (3) and denying force, the powers of 2. This leads to numbers for Gurdjieff’s worlds in Search as 2n × 3. The Lambda diagram marks the field of possible numbers of the form 2q × 3p and the reconciling force of 32 = 9, 33 = 27, etc. appear unused. However, tuning theory has many symbolic correlations for these higher powers of 3, the most obvious having to do with the formation of the tones within scales. This will show that Diagram and Ray were indicating one portion of a greater whole relevant to the formation of octaves within these worlds. If so, Gurdjieff’s sources included a different fragment of the secret teachings of the Pythagoreans, or shared their sources.

Pythagorean Tuning

One must differentiate between the practical tuning order of a musical scale, found in the Lambda, and the more familiar ascending order of its tone-numbers found within a piano keyboard. The tuning order for a Pythagorean scale requires the successive application of powers of three, which the Lambda diagram naturally generates as its rows get wider. The Pythagorean scale of the heptachord was probably a 1st Millennium BC development, at least exoterically.[9]

Between worlds 6, 12, 24, 48, 96 only two tones can form, the fourth fa and the fifth sol, a situation called by Plato the World Soul, created by a creator god or Demiurge. Numbers with 3 squared in their makeup can generate two further tone-numbers, namely re and si-flat. The word octave means “eight notes” and therefore, if there are to be octaves between Gurdjieff’s world numbers, they cannot be found between the numbers of those worlds but must be found in the rows the Lambda provides, these inheriting ever greater higher powers of three, as the rows descend.

This reveals why the cosmic octaves were based upon C-Major since, in the tuning order for Pythagorean heptatonic octaves creates descending and ascending tones around the primordial tonic of our note D (sometimes called Deity but here perhaps Demiurge). After two descending fifths (=2/3), D (as 864) becomes depleted of two threes and is then C (as 768), the world of the essence class METAL in Diagram.

Figure 6 (above) The row belonging to world 96 seen as in C (which is yellow) and (below) normalized to a single octave

The white region to the right of the yellow Worlds 96 to 48 is the region were lateral octaves should be, following the major diatonic pattern of tone-tone-semitone-tone-tone-tone-semitone, where do is naturally C as in figure 6.

One must accept that the Lambda diagram of the worlds probably formed an introduction to a much more complex subject which later introduced the role of prime number 5 within octave ranges. This was presented three centuries after Pythagoras by Plato, yet cryptically hidden by him from our scholars until the later 20th century, when American musicologists (such as Ernest McClain) were then able to see a world-wide tradition of harmonic numbers that included factors of 5 in heroic stories and great time periods. The number two was considered female because the octave was a womb impregnated by male numbers. Plato called the number 3 a divine male and the number 5 he called the human male number, these differently creative within the octave. Through this, new diagrams emerge around the number 360-720, alluded to by Gurdjieff alongside the off-putting 1000-2000 diagram in Search, referred to above.

In the Bible, Abraham and Sarah were given the hey (=5) in their names by the Lord God, so that they could have Isaac when Sarah was 90 years old. Isaac would die at 180 years old, whilst their primal ancestor was Adam whose gematria 1.4.40 equals 45 (9 x 5) when summed and 1440 in position notation. Doubling 45 gives 90, doubled again 180 and doubled again the 360 alluded to by Gurdjieff (above) then 720 and 1440. This indicates that in the Lambda, 9 creates a set of different numbers that, times 5, created the numbers of the Patriarchs. And in the decimal world of the Semites, dividing by 5 happens when you divide by 2 and add a zero, so that 144 x 5 = 720. The row with 9 at its head sits in the Lambda beside the Worlds until 9 × 16 generates 144 which, times 5 is 720. The octave 360 to 720 sits next to and connects worlds 24 and 48. There are different ways to add 5 as a factor, see Part 2.

In the Diagram, the human essence class is centered in world 24 though humanity has generally fallen to 48, the mechanical. The Lambda is probably the best way to present the creation of the Gurdjieff’s Worlds, but it was only the first rung of a possible transmission to his students

Greek versus Chinese Tuning?

In Chapter 40 of Tales we meet the Chinese twin brothers, Choon-Kil-Tez and Choon-Tro-Pel who were the first people (after Atlantis sank) to lay anew “a science of vibration adding two of the three Mdnel-Ins to the ‘seven-aspectness-of-every-whole-phenomenon’ and form the law of ‘ninefoldness’. This appears to be the Enneagram, which is then also the Heptaparaparshinock of Tales[10], in which the law of three has been combined with the law of seven, to enable the independent and ever-renewing Trogoautoegocratic function on which the megalocosmos was “newly” based.

By page 860, Greek music is stated to have influenced the formation of a deficient modern theory, in which the seven intervals of an octave are considered to be of two types, tones and semitones. According to Beelzebub, this causes a wrong idea of there being five main notes called ‘restorials’ (‘gravity center sounds’) instead of seven in the Chinese system. One must say that here notes (aka tones) are being confused with the intervals between them and the succeeding note, and vice versa – so be warned. This was the case in ancient Indian music[11] and in Tales where the notes of the octave found in Search were superseded by seven Stopinders, the seven intervals.

In Tales we are told that one Gaidoropoolo (i.e. Guido of Arezzo[12]) saw no difference between the seven Chinese ‘whole note’ intervals and the five Greek whole tones: “in the Chinese ‘seven-toned octave’ those whole notes [sic] called ‘mi’ and ‘si’ are not whole notes at all, since the number of vibrations which they have almost coincides with the number of vibrations of those Greek half tones”, found between ‘re’ and ‘fa’ and between ‘si’ and ‘do’. This seems aberrant with respect to modern theory since the Chinese and Greek semitones differ by just 81/80, the syntonic comma found between Pythagorean semitones (256/243) and the Just semitones of (16/15) and also between Pythagorean whole tones (9/8) and Just whole tones (10/9).

In the Pythagorean tuning of the Chinese, the octave is achieved by successive applications of ascending and descending musical fifths from D. The two semitones of 256/243 arrive last, in the third “turn” and this causes the gap left in the semitone position to be equal to the eighth power of 2 over the fifth power of 3, a mere “leftover” or leimma. Since the worlds of Search were couched in Pythagorean numbers involving factors of 2 and 3, one has to add octave limits and intervals involving the number 5. And Search does talk about the Just tone and semitone 10/9 and 16/15.

In the Greek Just tuning (and before them, the Old Babylonian, Akkadian and Sumerian tunings) the extra prime 5 was identified as making simplest and more harmonious scales for human music. The new semitones of 16/15 and smaller tones of 10/9 use smaller numbers and are better sounding than the purely Pythagorean tone-set. Gurdjieff appears to suggest that, in the creation of the Megalocosmos, 5 was not employed and all the seven intervals whole in their ability to become divided into seven secondary units and then divided again (Tales p827) exactly as one finds in Search on page 135-136, and its figure 17. One should correlate the remark of Pythagoras that God preferred the tone set produced without using 5. It may be that the 1st Millennium found, in the octave and the heptatonic scale, a plausible theory at a time when the forming of theories was emerging, rather than the following of formulaic methods by rote.

It is likely Gurdjieff’s octaves between the Worlds require the human number five, enabling the transformations of alchemical traditions to act within the cosmic octaves between Worlds. The ‘science of vibrations’ seen in Search was explicitly alchemical and since the early teacher of Ahmad Yasavi was a local and idiosyncratic “alchemist and magician Baba Arslan” (Masters 128)), this may be the realistic source for some of Gurdjieff’s ideas that lie behind the monastery of Meetings With Remarkable Men.

The apparent conflict over semitones must be in connection with the cosmic octaves themselves. It may be the human purpose to create their own further evolution. As Bennett said (above) in Enigma, despite the western Pythagorean tradition having had a developed tuning theory, Gurdjieff’s notion, of transformation for the whole man through a musical cosmology, was not present, a notion which Bennett says was found in Persia, Babylon, and then the “stans” of the Masters of Wisdom: the Kazakstan of Ahmad Yasavi and Uzbekistan of Baha-ud-Din Naqshband.


NOTES

[1] Essence class is Bennett’s considered word for what were presented as “classes of creatures”. Bennett took over the development of the diagram by realising that these classes had five terms, the Creature e.g. Man, the range of its evolutionary possibilities (higher and lower terms) e.g. Angel and Animal and what feeds them and what they feed (the Trogoautoegocratic reciprocal maintenance of Tales). The two classes above and the two below a given class were obviously part of a five-fold scheme of each class if you follow what the Hydrogen numbers of Diagram tell you.

[2] “In the study of the law of octaves it must be remembered that octaves in their relation to each other are divided into fundamental and subordinate. The fundamental octave can be likened to the trunk of a tree giving off branches of lateral octaves.” Search. 134.

[3] Tales. 754. Beelzebub blames the “asymmetry so to say in relation to the whole entire completing process”, an asymmetry caused by do not being on the vertical axis of symmetry opposite D when starting the ascending major diatonic starts in C.

[4] Al-Kindi was the first great theoretician of music in the Arab-Islamic world. He is known to have written fifteen treatises on music theory, but only five have survived. He added a fifth string to the oud. His works included discussions on the therapeutic value of music and what he regarded as “cosmological connections” of music. Wikipedia

[5] Gurdjieff may well have had the numbers changed by the compilers of ISM (upon whose excellent work we depend).

[6] A.D.M = 1.4.40 = 1 + 4 + 40 = 45 or, in position notation 1440 which is 32 x 45.

[7] “The Tao begot one. One begot two. Two begot three. And three begot the ten thousand things.” Tao Te Ching 42, “The ten thousand things carry yin and embrace yang. They achieve harmony by combining these forces. Men hate to be “orphaned,” “widowed,” or “worthless,” But this is how kings and lords describe themselves. For one gains by losing And loses by gaining.”

[8] This led to the study decades later of Triads, the six different permutations of the three forces, Affirming, Denying and Reconciling. Bennett saw the increasing of a world’s number as the substitution of an essential force with an existential force, thus doubling the number of triads in lower worlds (perhaps instead of the inner octave explanation of doubling).

[9] It is now thought the ancient near east, like hexachordal music, did not have an octave fixation. Richard Dumbrill notes that Just intonation, without consideration of numerical generation using primes 2, 3 and 5, used the Fifth and Thirds to fit structures like the tetrachord (Fourth) or an enneatonic, 9 toned range beyond the octave, popular for example in traditional instruments like the bagpipe today. The heptachord and its octave only enters the present historical record in the 1st Millennium BC. ICONEA 2010 “This paper will have sufficiently shown that heptatonism did not appear spontaneously on the musical scene of the Ancient Near East. The textual evidence is unambiguous as there are no traces of any heptatonic construction before the first millennium B.C.”

[10] Tales. 841. 2nd Para “… concerning the fundamental cosmic law of the sacred Heptaparaparshinokh then called the law of ‘ninefoldness’ …”

[11] Sachs. 1943. 165 para 2 and 3

[12] Arezzo is south east of Florence. Between are the farms of the delicious and sought after Valdarno chicken, or poolo for poulet. I suggest Gurdjieff had studied the solfege of Guido and Valdarno chicken came to mind.

Bibliography

Bennett, J.G.
1963, Gurdjieff: A Very Great Enigma, Coombe Springs Press.
1973, Gurdjieff, Making a New World, Turnstone Press.
1977, The Masters of Wisdom, Turnstone Press.
1983, Enneagram Studies, rev.ed.. NY: Samuel Weiser.

Blake, A.G.E.
1982, An Index to in Search of the Miraculous, DuVersity Publication.

Bremer, John
2005, Plato’s Ion: Philosophy as Performance, Texas: Bibal.

Dumbrill, Richard
2010, ICONEA Proceedings, Music Theorism in the Ancient World, 107-132, esp 126.

Gurdjieff, G.I.
1950, Beelzebub’s Tales, Routledge & Kegan Paul, London.
1963, Meetings with Remarkable Men, Routledge & Kegan Paul, London.

Heath, Richard
2002, Matrix of Creation, Inner Traditions.
2007, Sacred Number and the Origins of Civilization, Inner Traditions.
2011, Precessional Time and the Evolution of Consciousness, Inner Traditions.
2014, Sacred Number and the Lords of Time, Inner Traditions.
2018, Harmonic Origins of the World, Inner Traditions.

Heath, Richard and Heath, Robin
The Origins of Megalithic Astronomy as found at Le Manio, https://independent.academia.edu/HeathRichard

Ernest McClain (see also www.ernestmcclain.net for pdf)
1976, The Myth of Invariance, Shambhala .
1978, The Pythagorean Plato, Shambhala.

P.D. Ouspensky
1950, In Search of the Miraculous, Routledge & Kegan Paul, London.

Sachs, Curt
1943, The Rise of Music in the Ancient World, East and West, New York: Norton.

Traditional Studies Press
1971, Guide and Index to G. I. Gurdjieff’s All and Everything, Toronto.

The Knowing of Time by the Megalithic

The human viewpoint is from the day being lived through and, as weeks and months pass, the larger phenomenon of the year moves the sun in the sky causing seasons. Time to us is stored as a calendar or year diary, and the human present moment conceives of a whole week, a whole month or a whole year. Initially, the stone age had a very rudimentary calendar, the early megalith builders counting the moon over two months as taking around 59 days, giving them the beginning of an astronomy based upon time events on the horizon, at the rising or setting of the moon or sun. Having counted time, only then could formerly unnoticed facts start to emerge, for example the variation of (a) sun rise and setting in the year on the horizon (b) the similar variations in moon rise and set over many years, (c) the geocentric periods of the planets between oppositions to the sun, and (d) the regularity between the periods when eclipses take place. These were the major types of time measured by megalithic astronomy.

The categories of astronomical time most visible to the megalithic were also four-fold as: 1. the day, 2. the month, 3. the year, and 4. cycles longer than the year (long counts).

The day therefore became the first megalithic counter, and there is evidence that the inch was the first unit of length ever used to count days.

In the stone age the month was counted using a tally of uneven strokes or signs, sometimes representing the lunar phase as a symbol, on a bone or stone, and without using a constant unit of measure to represent the day.

Once the tally ran on, into one or more lunar or solar years, then the problem of what numbers were would become central as was, how to read numbers within a length. The innovation of a standard inch (or digit) large numbers, such as the solar year of 365 days, became storable on a non-elastic rope that could then be further studied.

The 365 days in he solar year was daunting, but counting months in pairs, as 59 day-inch lengths of rope, allowed the astronomers to more easily visualize six of these ropes end-to-end, leaving a bit left over, on the solar year rope, of 10 to 11 days. Another way to look at the year would then be as 12 full months and a fraction of a month. This new way of seeing months was crucial in seeing the year of 365 days as also, a smaller number of about 12 and one third months.

Twelve “moons” lie within the solar year, plus some days.

And this is where it would have become obvious that, one third of a month in one year adds up, visually, to a full month after three years. This was the beginning of their numerical thinking, or rationality, based upon counting lengths of time; and this involved all the four types of time:

  1. the day to count,
  2. the month length to reduce the number of days in the day count,
  3. the solar year as something which leaves a fraction of a month over and finally,
  4. the visual insight that three of those fractions will become a whole month after three full solar years, that is, within a long count greater than the year.

To help one understand this form of astronomy, these four types of time can be organized using the systematic structure called a tetrad, to show how the activity of megalithic astronomy was an organization of will around these four types of time.

J.G. Bennett’s version of Aristotle’s tetrad.

The vertical pair of terms gives the context for astronomical time on a rotating planet, the GROUND of night and a day, for which there is a sky with visible planetary cycles which only the tetrad can reveal as the GOAL. The horizontal pair of terms make it possible to comprehend the cosmic patterns of time through the mediation of the lunar month (the INSTRUMENT), created by a combination of the lunar orbit illuminated by the Sun during the year, which gave DIRECTION. Arguably, a stone age culture could never have studied astronomical time without Moon and Sun offering this early aggregate unit of the month, then enabling insights of long periods, longer than the solar year.

The author (in 2010) at Le Manio Quadrilateral
where megalithic day-inch counting is clearly indicated after a theodolite survey,
over three years of its southern curb (to the left) of 36-37 stones.

The Manio Quadrilateral near Carnac demonstrates day-inch counting so well that it may itself have been a teaching object or “stone textbook” for the megalithic culture there, since it must have been an oral culture with no writing or numeracy like our own. After more than a decade, the case for this and many further megalithic innovations, in how they could calculate using rational fractions of a foot, allowed my latest book to attempt a first historical account of megalithic influences upon later history including sacred building design and the use of numbers as sacred within ancient literature.

The “output” of the solar count over three years is seen at the Manio Quadrilateral as a new aggregate measure called the Megalithic Yard (MY) of 32.625 (“32 and five eighths”), the solar excess over three lunar years (of 36 months). Repeating the count using the new MY unit, to count in months-per-megalithic yard, gave a longer excess of three feet (36 inches), so that the excess of the solar year over the lunar could then be known as a new unit in the history of the world, exactly one English foot. It was probably the creation of the English foot, that became the root of metrology throughout the ancient and historical world, up until the present.

The southern curb (bottom) used stones to loosely represent months from point P while, in inches, the distance to point Q’ was three solar years.

This theme will be continued in this way to explore how the long counts of Sun, Moon, and Planets, were resolved by the megalithic once this activity of counting was applied, the story told in my latest book.

Legominism and the Three Worlds

Above: Altaic shaman’s drum depicting the cosmos

The general ordering of the cosmos throughout history was phenomenological, following the very apparent division between the sky and the earth, with the living principle between called a “middle earth”. A summation of its symbolism was placed within Dante’s trilogy The Divine Comedy; of an inferno, purgatory and paradise which were the three worlds of the geocentric experience. But how does it come about that the phenomenological was translated into ancient literature, buildings or, as Gurdjieff names these, legominisms in the literal sense of being made of meaning-making and the naming of things – a power given to Adam but not the angels.

Continue reading “Legominism and the Three Worlds”

EARLY INDO-EUROPEAN (c. 5000 BC) mystical numbers

The mystic status of numbers which led to this intense concern with their properties and relationships seems to have existed also, even before Sumerian times, in the beliefs of the neighboring Indo-Europeans. Modern scholars interpret similarities between word roots as signs of deep and original connections, just as ancient sages had long done with similarities between the sounds of words, or between the ways to write them. Based on this principle, some of the moderns have shown that the religious view of numbers among speakers of Indo-European languages goes back to the prehistoric period when the words for their relationships formed.

Here is what David R. Fideler says about these early word roots:

“Cameron, in his important study of Pythagorean thought, observes that harmonia in Pythagorean thought inevitably possesses a religious dimension. He goes on to note that both harmonia — there is no “h” in the Greek spelling — and arithmos appear to be descended from the single root “ar”. ¤This seems to ‘indicate that somewhere in the unrecorded past, the Number religion, which dealt in concepts of harmony or attunement, made itself felt in Greek lands. And it is probable that the religious element belonged to the arithmos – harmonia combination in prehistoric times, for we find that ritus in Latin comes from the same Indo-European root’.”

Guthrie’s “The Pythagorean Sourcebook and Library”

Such traces of early reverence for invisible but knowable Numbers suggest that if some ancient mathematicians were aware of the major constants, they might have ranked these mysterious “super-numbers” even higher than the natural numbers. They would have assigned them important religious and symbolic roles, and they would have explored their properties and permutations as a means to understand the relationships among the gods they represented or were. ¤Reference: http://www.crcsite.org/numbers.htm


The mystic status of numbers which led to this intense concern with their properties and relationships seems to have existed also, even before Sumerian times, in the beliefs of the neighboring Indo-Europeans. Modern scholars interpret similarities between word roots as signs of deep and original connections, just as ancient sages had long done with similarities between the sounds of words, or between the ways to write them. Based on this principle, some of the moderns have shown that the religious view of numbers among speakers of Indo-European languages goes back to the prehistoric period when the words for their relationships formed.