Category: Planetary Harmony

Celestial time periods forming musical intervals

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.

Numbers of a Living Planet: Preface

The image above is Kurma avatara of Vishnu, below Mount Mandara, with Vasuki wrapped around it, during Samudra Manthana, the churning of the ocean of milk. ca 1870. Wikipedia.

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Numbers, Constants and Phenomenology
  5. Phenomenology as an Act of Will

Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.

It is impossible to talk of a creation outside of the time and space of Existence, though from it, other dimensions can be inferred such as an “Eternity” visible in the invariances of numbers and structures. It is this higher dimensionality that leads to

  1. The recurrence of celestial time periods,
  2. The mental powers to recognise manifested patterns,
  3. The use of spatial geometries of alignment,
  4. The numerate counting of time,
  5. A phenomenology which is neither factual nor imaginary.

The quantification and qualification of Existence, adequately conducted, reveals harmonious structures within time and space, especially in the spacetime of our planetary system, when this system is as seen from our planet. The harmonious nature of our planetary system helped the late stone age to develop a large numerical and geometrical model of the world through counting astronomical recurrences. This model, which shaped ancient texts, implies that solar systems may have an inherent intelligence which makes them harmonious.

Harmony in a planetary system must therefore employ invariances already present in the number field, by exploiting the recurrent orbital interactions between planets and large Moons, this in a connected set of three-body problems. Before our exact sciences and instruments, prehistoric naked-eye astronomers could understand the planetary world by counting the duration of planetary time cycles: the subject my books explore. Through counted lengths of time, the megalithic age came to understand the invariances of the number field and so evolve an early and distinct type of numeracy. This numeracy lived on as the basis for the ancient Mysteries of the early civilizations, embodied in their Temples and in the Pythagorean approach to ordinal numbers and geometries, expressing the “number field” in two or three dimensions, areas and volumes. (see Sacred Geometry: Language of the Angels for an introduction to this)

That is, this early human numeracy naturally manifests within the maths governing rotational systems, this involving key transcendental* constants such as π, these regulating what is actually possible, mathematically, within dynamic planetary systems that are gravitational attractors of each other: these constants include pi {π}, √-1 {i}, e, and phi {φ}.  The first three { π, √-1, e} are surprisingly well-organized rotational frameworks making the behaviour of vectors relatively simple using geometry. For example, the lunar year of twelve lunar months has become strongly resonant with the two outer gas giants, Jupiter and Saturn. The Golden Mean (or Phi {φ})1 can be approximated by orbital ratios between planets through exploiting the Fibonacci number series2, most visibly in the orbital recurrence of Venus and the Earth, seen in the 8/5 {1.6} relationship of its synod* to the solar year. Phi φ is also expressed in living forms of growth, since growth is often based upon the present size of a living body and what it has previously eaten.  Fibonacci ratios are ideally suited to creating the “strange attractors” which can create stable patterns out of otherwise chaotic orbital interactions.

1 My use of curly braces is borrowed from a stricter world of set notation. It offers an ability to place groups of numbers, symbols and other non-grammatical element next to their grammatical context.

2 The series reinvented by Fibonacci uses addition of two previous number to create the next number. His version of that algorithm is {0 1 1 2 3 5 8 13 21 34 55 and so on}. These numbers are found within natural form of life, where such numbers can be generated from two previous states or when two counter rotating spirals of seeds will fill the surface of an egg shape with maximum packing. More on this later.

Through universal mathematical laws and constants, rotational and recurrent systems will effectively provide numerical shortcuts* (J.G. Bennett’s null-vectors) expressing Musical or Fibonacci ratios, and without those ratios being available, relationships within existence would be more complex, less synchronous, and truly accidental. Harmonic shortcuts have therefore given the planetary world a simplified mathematics when viewed from the surface of the earth, within the geocentric pattern of time. This synchronicity provided the stone age with a path towards a direct numerical understanding of time through phenomena (that is, a direct visual and countable phenomenology).

In this way, the megalithic cultures of prehistory found that the geocentric planetary system expressed numerical invariances (these already within the number field itself) thus making the time world of the sky unusually harmonious and intelligible. This contrasts with the now-popular modern notion that, while the solar system is a large and impressive structure, its origins come only from the mathematical laws of physics, these forever operating in a mechanical way. That is, the modern way-of-seeing planetary time is heliocentric and causal and this has hidden an ancient view, gained through the megalithic study of the phenomena in the sky using megaliths as large instruments with sightlines to the horizon events of sun and moon, to simply count of time-as-length and, evolve a very basic numeracy based upon numerical lengths (a metrology) and triangular geometries to compare lengths.

Megalithic methods employed the properties of circles, ellipses, squares, rectangles, and right triangles before the analytical geometry of Euclid, Greek math, or ancient near-eastern arithmetic. This was only possible because key parts of the mathematics of complex numbers, for example, are directly visible in the form of the right triangle and unit circle; as the natural form of two vectors: a length at a given angle (or direction) and another length at different angle gives access to ratios. A right triangle can therefore express two vectors of different length and differential angle, and this applies to a pair of average angular rates in the sky, without knowing the math or physics behind it all. If the two vectors are day-counts of time, then the right triangle can study their relationship in a very exact way. Such a triangle may also have been seen as the rectangle that encloses it, making the diagonal (vector), the hypotenuse of the triangular view.

The properties of the imaginary constant i (√-1) represents, through its properties, the rotation of a vector through 90 degrees. It is this that gives the right-angled triangle its trigonometric capacity to represent the relativity of two vector lengths. My early schoolroom discoveries concerning vectors in applied math classes, that right triangles can represent vectors of speed for example, was without any knowledge of the mathematical theory of vectors. This geometry enabled prehistorical astronomy to study the average planetary periods as vectors. That is, rotational vectors enabled the sky to be directly “read”, from the surface of the third planet, through simple day-counting, comparing counts with right triangles, and forming circular geometries of alignment to astronomical events found on the horizon; all without any of our later astronomical instrumentation, maths, or knowledge of physics.

Physics has not yet explained how the time constants between the planets came into a harmonious configuration, because it is unaware that this is the case. The mathematization of Nature, since the Renaissance, has hidden the harmonious view of geocentric planets and all preceding myths, cosmologies and beliefs were swept aside by the heliocentric world view (see Tragic Loss of Geocentric Arts and Sciences, also C.S. Lewis’s The Discarded Image).

The modern approach then emerged, of blind forces, physical laws and dynamic calculations. That is, while the simplifying power of universal constants is fully recognized by modern science (these having made maths simpler) the idea that these simplifications came to be directly reflected in the sky implies some kind of design and hence an intelligence associated with planetary formation.

Furthermore, modern way of seeing things cannot imagine that the megalithic could conducted an astronomy of vectors (using geometrical methods while not understanding why they worked) and that this empowered a simple but effective type of astronomy, without our mathematical or technical knowledge. This is an anachronistic procedural heresy for the history of Science and also for the present model of history, where science for us is the only science possible, evolving out of near-eastern civilization after the stone age ended.

Foundational myths of modern civilization are threatened by the notion that the world is somewhat designed by a higher intelligence. Until these subconscious conflicts of interest are overcome, prehistory will remain the prisoner of modernity where mysteries remain mysteries because we don’t wish to understand.

2. Primacy of low Whole Numbers

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.

On the Harmonic Origins of the World

Does the solar system function as a musical instrument giving rise to intelligent life, civilization and culture on our planet? This 2018 article in New Dawn introduced readers to the lost science of the megalithic – how our ancestors discovered the special ratios and musical harmony in the sky which gave birth to religion and cosmology. The musical harmonies were the subject of my book released that year, called The Harmonic Origins of the World.

After the ice receded, late Stone Age people developed the farming crucial to the development of cities in the Ancient Near East (ANE). On the Atlantic coast of Europe, they also developed a now-unfamiliar science involving horizon astronomy. Megalithic monuments were the tools they used for this, some still seen in the coastal regions of present day Spain, France, Britain and Ireland. Megalithic astronomy was an exact science and this conflicts with our main myth about our science: that ours is the only true science, founded through many historical prerequisites such as arithmetic and writing in the ancient near east (3000- 1200 BC) and theory-based reasoning in Classical Greece (circa 400-250 BC), to name but two. Unbeknownst to us, the first “historical period” in the near east was seeded by the exact sciences of the megalithic, such as the accurate measurement of counted lengths of time, developed by the prehistoric astronomers. With the megalithic methods came knowledge and discoveries, and one discovery was of the harmonic ratios between the planets and the Moon.

The idea that the planets were gods had been born before the ancient world, through the data of megalithic astronomy and this megalithic idea was the basis for the religious ideas of the East. Megalithic astronomy and Near Eastern religious and harmonic ideas have both been written out of our history of civilization, leaving us with enigmatic monuments and ill-defined religious mysteries. How this slighting of our real history happened is perhaps less important than our discovering again the purpose of the megalithic monuments and of those religious ideas that sprang from the discovery that the planets were harmonically related to life on Earth.

Le Menec Alignments indicate a profound astronomical work in the new stone age by 5000-4000 BC. Composite mash up by David Blake using Blender, Google Earth elevation and imagery plus Alexander Thom geometry and digitized stone locations.

Is human history lacking something fundamental?

Continue reading “On the Harmonic Origins of the World”

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. Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
  4. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  5. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  6. —-. 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.
  7. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  8. —-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology.  The Squeeze Press: 2024.
  9. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

Book: Matrix of Creation

Sacred numbers arose from ancient man’s observation of the heavens, and represent the secrets of cosmic proportion and alignment. The ancients understood that the ripeness of the natural world is the perfection of ratio and that the planetary system–and time itself–is a creation of number. We have forgotten what our ancestors once knew: that numbers and their properties create the forms of the world.