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

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 20^th century (by Ernest McClainAmerican Cryptologist and Pythagorean Musicologist who decoded Plato's cryptic numerical ciphers in The Pythagorean Plato. The Myth of Invariance showed limiting numbers had been an ancient way of defining the onset of key musical tuning realities, then coded into many religious texts. Wikipedia. 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 2^nd 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.

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).

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 2^q × 3^p 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 1^st 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 SoulPlato's description of how the Creator designed the world using only the intervals of musical fifth (3/2), whole tone (9/8) and fourth (4/3), within a purely numerical framework (6 8 9 12)., 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 20^th 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 1^st 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.

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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 intonationA musical tuning system improving the Pythagorean system of tuning by fifths (3/2), by introducing thirds (5/4 and 6/5) to obtain multiple scales., 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 1^st 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. 2^nd 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.

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