Chartres, in north-west France, is a very special version of the Gothic transcept cathedral design. Having burnt down more than once, due to wooden ceilings, its reconstruction over many building seasons and different masonic teams, as funds permitted, would have needed strong organizing ideas to inform the work (as per Master Masons of Chartres by John James).
As shown below, Chartres main towers are unequal in height and the “western” facade itself does not align to east-west, as normal Christian churches do. The left tower is also higher than the right tower and, it has been said, the left represents the Sun and the right the Moon. The height of the left tower, to its globe below its cross, is indeed the solar year of 365 days in feet. But the height of the shorter right tower, to its own globe, is not the 354.367 days of the lunar year (of 12 months); rather, it is the top of its cross, sporting a crescent moon suggesting it is a moon tower, that is 354 and a third feet high.
The cosmic time coding of the two towers as solar year->lunar year between the globe’s height (on left in red) and the top of the cross (on right in blue). But the left tower also indicates the Saturn synod of 378 days to the top of its cross. The for-square rectangle, geometrically relating the solar (diagonal) and lunar years, is shown.
That is, the height of the lunar year in feet, from the same starting point as the solar tower’s height as the solar year, the lunar year would be to the top of the lunar cross, where the crescent is attached, and not to its globe. There is then a reasonable connection between the solar and lunar years and the two towers. However, it is also interesting to see the number of days, as feet, of the left tower to its own cross. It is exactly 378 feet, the synodic period of Saturn in days. Readers of my books and this site will remember that the ratio between the lunar year and Saturn synod is exactly 16/15: a musical semitone within the ancient tuning system called Just intonation.
This arrangement suggests Chartres was built to be a time-factored monument, which may be why the cathedral was aligned to midsummer sunrise (which was a megalithic norm) rather than being aligned east-west. Built on top of a solitary promontory, horizon events would have been clear across the flat fertile plains.
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, TheGreek 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 possiblecombinations 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’ …”
[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.
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 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 the human intelligence, this based upon this Universal Will 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 only possible to participate in the intelligence behind the world, through a transformation into an Individuality, creative according their own pattern harmonious with the universal.
*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 favorable 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.
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.
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 a distance from 1.
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”. 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}.
How do numbers “create”?
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.
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 definedby
(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 unnecessary due to the first true number {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” *(EGM) is seen in the behaviour of a musical string as, in that kind of resonator, half of the string merely provides the basis for the subsequent numerical division of its second half, to make musical notes – as in a guitar where the whole string provides low do and the frets when pressed then define higher notes up to high do (half way) and beyond, through shortening the string.
This suggests that a tonal framework was given to the creation by Gurdjieff’s Universal Will, within which many inner and outer connections can then most easily arise within octaves, to
overcome the mere functionality of complexity,
enable Will to come into being,
equip the venue of life with musical harmony and
make the transformation of Life more likely.
Harmony is most explicit as musical harmony, in which vibrations arise through the ratios between wavelengths which are the very same distance functions of ordinal numbers, separated by a common unit 1.
Take the number three, which is 3/2 larger than two. Like all ordinal numbers, succeeding and preceding numbers differ by plus or minus one respectively, and the most basic musical tuning emerges from the very earliest six numbers to form Just intonation, whose scales within melodic music result as a sequence of three small intervals {9/8 10/9 16/15}, two tones and a semitone. Between one and those numbers {8 9 10 15 16} are the first six numbers {1 2 3 4 5 6}* (note absence of seven between these sets), whose five ratios {1/2 2/3 3/4 4/5 5/6} provide any octave doubling with a superstructure for the melodic tone-semitone sequences; their combined interdivision, directly realizes (in their wake) the tones and semitones of modal music.
We will see that the medium for such a music of the spheres was both the relationship of the sun and planets to the Moon and Earth, but this manifested quite literally in the lunar months and years, when counted.
Coming soon: The Musical Octaves of Gurdjieff’s Worlds
What we call numbers start from one, and from this beginning all that is to follow in larger numbers is prefigured in each larger number. And yet, this prefigurement, in the extensive sense {1 2 3 4 5 6 7 etc.}, is completely invisible to our customary modern usage for numbers, as functional representations of quantity. That is, as the numbers are created one after another, from one {1}, a qualitative side of number is revealed that is structural in the sense of how one, or any later number, can be divided by another number to form a ratio. The early Egyptian approach was to add a series of unitary ratios to make up a vulgar* but rational fraction. This was, for them, already a religious observance of all numbers emerging from unity {1}. The number zero {0} in current use represents the absence of a number which is a circle boundary with nothing inside. The circle manifesting {2} from a center {1} becomes the many {3 4 5 6 7 …}.
The number one manifests geometrically as the point (Skt “bindu”) but in potential it is the cosmological centre of later geometries, the unit from which all is measured and, in particular, the circle at infinity.
Two: Potential spaces
From one {1} springs two {2}, to which we owe all forms of doubling as reproduction without sexual division, called “barren” by Plato, yet giving the possibilities of complex worlds of different scale, in terms of their limiting number. This is the first true number of Creation which gives the quality of polarity between the two halves of (as yet) nothing, halves of a world that will create the beginnings of an everything. Super dense, as an initial Form of forms, all things will come to rotate around this Axis of axes*. (Axes, when pronounced with a long e, is the plural form of the word axis, meaning imaginary lines that run through the middles of things. The word axe is derived from the Old English word æces, the axe which divides into two. ) This is the birth of duality, as with the centre and circumference of a circle or positive and negative (opposite) charge, and the medium of the wave or vibration, which gave birth to dynamic systems, such as planetary rotation of an axis or an orbit.
When number was incarnated in our own planetary creation, it was Saturn who visibly delimited the outer limits of the visible planets. His name is close to Seth and Satan (as the necessary adversary of the heroic Horus) and he was seen as limiting unbounded growth within existence. Saturn expresses 5 synods of the planet Saturn in 64 {26} lunar months (but this is to jump in numbers, though not too far, to the planetary double octave {24 48 96} lunar months. Sixty-four governs the “eye of Horus”: a government deriving from the mythical conflict between the god Horus with his rival Set; in the context of two eyes, here the right eye was torn out but then restored, to then see the role of two within the creation, in the “afterlife”.
Figure 1 The Egyptian icon of the Eye of Horus as the components used to represent vulgar fraction as a series of powers of two.
https://en.wikipedia.org/wiki/Eye_of_Horus: It derives from the mythical conflict between the god Horus with his rival Set, in which Set tore out or destroyed one or both of Horus’s eyes and the eye was subsequently healed or returned to Horus with the assistance of another deity, such as Thoth.
The Moon finds an exact reunion with the earth day after exactly 945 days, which equal 32 lunar months of 945/32 (29. 53125) days, very close to the actual lunar month of 29.53059, being effectively exact as 57 seconds different.
While the number two can, in being divided, create new areas of interaction (including cosmic octaves), its mere extension {2 4 8 16 32 64 128 …) forms only a backbone of potentialities, these then (see later page) borrowed by higher limiting numbers such as 720, a number containing favourable numerical factors for the creation an ideal “family” of limits, metaphorically presented by Adam and the Patriarchs of the Bible.
Figure 2 The vision of the Godhead asleep on a pre-creational ocean (of his sweat) attended by the Goddess until he awakes for a new creation.
The number {2} as dyad manifests as the geometry of the line. The line {2} from a center {1} as a rotational vector becomes the geometry of the circle.
Three and four: Actualization
The actual creation has a different planetary symbol, the equilateral triangle with three sides, seen also as the capital Greek delta, whose value is 4 because the planetary system is an Activity involving forces. These numbers are reconciled as 3 x 4 which equals 12, the number of Autocracy * and balanced action. We are told in myth that Jupiter’s twelvefold nature was “the receiving of the measures” from Saturn, as 4. Jupiter is the planetary demiurge which Plato describes carving out the World Soul “octave” {6 8 9 12}, using ratios involving Three, the cubit and its reciprocal {3/2 4/3} * (Timaeus). Only when we reach the lunar octave {24 27 30 3236 40 45 48} will the diatonic scale of eight notes emerge, the first and last being the same note, doubled.
The first true doubling {2 3 4}, between 2 and its square, holds the first type of penetration of the octave, by Three {3}. With three, the Demiurge forms his World Soul using intervals involving only Two and Three {3/2 4/3}, which can create the fifth and fourth notes (“dominant and subdominant”) of an octave.
The number {3} gives form to the first geometry of area, manifest in the triangle which, given a right angle, becomes trigonometrical, as the functional mediation between the line and the circle.
The number four connects the relatedness of the Triad (3) with the existentially actual to provide an engine in which Form can become Substance through an intermediate pair of terms that fulfill the gap {2} between form and the actual situation. One could say this is the first instance of filling the octave with tones {8}, intermediate between 2 and 8. Four is the first square number which in geometry is called square as an area equal to 4 has sides equal to 2
Five: Vitality and Life
Coming next, Five {5} will also be able to divide the coming “octave” {3 4 5 6} in a superior way than three and two can alone, by redefining a new tone {10/9} for Just intonation and a corresponding semitone {16/15}, these cancelling the excessive powers of three produced by tuning only with three {3}, called the cycle of fifths, which used successive fifths and its inverse{3/2 4/3} because the ear can best define the larger musical intervals. The octave between three and six defines the framework of Just intonation where three intervals span the octave {4/3 5/4 6/5}, summing to doubling {2}.*
*This was probably referred to as “the three strides of Vishnu”, Trivikrama (‘having three steps’) being one of his 1,000 names.
The planet Venus brings a new type of harmony, which is also the sixth note {8/5 (= 1.6)} of a diatonic octave (see this page) since her synod of 584 days is 8/5 of the practical year of 365 days. The Fibonacci series allows whole number approximation to the Golden Mean {φ} between adjacent members obtained as being the sum of the two preceding numbers {0 1 1 2 3 5 8 13 21 34 55 …} unlike the ordinal set {1 2 3 4 5 6 7 8 9 …}, the latter instead obtained by more simply adding one more unit {1} to the preceding number. Unlike the musical tone and semitone of Jupiter and Saturn relative to the lunar year, Venus is resonant with the Earth’s orbital period of 365 whole days, and this type of orbital resonance, with each other, is mutually attractive, providing the lowest and most stable energy between the two planets. The inner orbital diameter (semi major axis) divided by the difference in orbital diameters, equals 2.618, or phi squared {φ2}*. (See later page for more on their orbits) Structures of growth, based upon Fibonacci ratios, are commonly found within living bodies, which must achieve this algorithm in which their present size added to their digestion of previously eaten food results in the sum of the two.
Figure The Fibonacci series in two dimensions are common forms of living growth.
The Venus synod will be seen to fit inside the octaves of the Moon because 20 lunar months is 590.6 days which is less than the synodic comma {81/80} of her 583.92 day synod*.
*The synodic comma is the exact ratio connecting Pythagorean and Just versions of the same note. One of the Indian temple designs is a nine-by-nine square grid which makes the number of equal-sized sub-squares {81) divided by the count where the central square is not counted gives the ratio of the synodic comma {81/80}.
Music: Child of the First Six Numbers
The larger intervals of numerically larger octave doublings are in this way foreordained in the first six numbers {1 2 3 4 5 6} and their relative size to each other, are five musical intervals {2/1 3/2 4/3 5/4 6/5}. Doubling has led to the pillars of Plato’s world soul {3/2 4/3} and three when doubled {3 4 5 6} has led to the three strides {4/3 5/4 6/5}, both sets summing to Two {2}.
The first six numbers, creating all the large tones of musical harmony, punctuated by Seven.
Between the five musical intervals, the tones and semitones of Just intonation are to be found {9/8 10/9 16/15} so that, as a tuning system, the Just system leads automatically to the tones and semitone of the seven modal scales, in both melodic and polyphonic harmony.
When the World Soul {6 8 9 12} is twice doubled {24 48} and doubled again {48 96}, the two octaves express the world numbers of Gurdjieff {24 48 96}, but now these numbers correspond to lunar months and, as with music when heard, all of the possible intervals are compresent in the instrument, the Moon illuminated by the Sun, since one can count from any lunar month, over any number of lunar months, to achieve any of the larger and smaller intervals between these octaves. And it is now true that the three principle planets of Jupiter, Saturn and Venus are present at the second, fourth and sixth notes, each of these relevant to Gurdjieff’s theory of octaves as stated by him in Russia, and his cosmic epic Beelzebub’s Tales. And J G Bennett continued to build on what Gurdjieff had expressed, without knowledge of the astronomical references, to populate his own Dramatic Universe, in 4 volumes and many compendia (see Bibliography). Of particular importance is how human beings figure within the cosmic vision, without which a planetary cosmos involving consciousness and creativity would be meaningless. If one resists the modern functional view of cosmogenesis: music, or other forms of harmony, can be seen to redeem the creation of a world like ours, through the short cuts numerical systems naturally provide for us, through a gravitational environment that can provide these.
How the Moon was seen to be like Saturn in expressing the Sacred Number 28.
illustrated SCRIPT
To measure the length of a lunar orbit, one needs to see any prominent stars that the moon is passing by in her orbit.
This sort of procedure is implied in Indian astronomy, and other traditions, where the sky was first divided into 28 or 27 equal parts, called lunar mansions (nakshatras).
The lunar mansion on a given day was identified by the name of the lunar mansion seen behind it. In B.C. 2350, the system started with the spring equinox which stood on the seven sisters or Krittikas.
This was in the Zodiac constellation Taurus the Bull, but the mansion, one of 28, was called Ashwini – the Horse’s Head. In this way Indians still speak of “the moon in Ashwini”.
The division into 28 probably came from the observation of Saturn’s loops which resembled the twenty seven and a third days for a lunar orbit, though the first is a frequency of synods while the latter is a frequency of earth rotations, both relative to the ecliptic / Signs / Mansions.
This video introduces an article on a Scottish iron-age stone tower or brock which encoded the size of the Earth.
In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court.
Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15 as proposed by Mackie).
The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7, the simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods., generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.
If we take the size of the moon in that model, as being 3/11 of 396 feet this would give a circle radius 108 feet and one can see that, using the moon, the outer perimeter of the brock was probably elliptical too.
Thank you for watching.
You can view the full article on sacred dot number sciences dot org, searching for BROCK, spelt B R O C H.
