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“The transmission of the meaning of symbols to a man who has not
reached an understanding of them in himself is impossible”

The question arises. Why study the Canon?  The simple answer is that it provides objective evidence that the Work is true.  I know - not because I believe it, or because I have faith - but because my reason proves it to be so.  

Gurdjieff said that in all there are four states of consciousness possible for man, but that ordinary man (that is, man number one, two, and three) lives in the two lowest states of consciousness only.   These are called “sleep” and “ordinary consciousness”.

These two lower states of consciousness correspond to the first two series of the Canon.  

The first series: “Sleep”

.285714/.142857     = 2        C to C    2/1

.428571/.285714     = 1.5        C to G    3/2

.571428/.428571     = 1.3333333    G to C    4/3

.714285/.571428     = 1.25        C to E    5/4

.857142/.714285     = 1.2        E to G    6/5

.999999/.857142     = 1.1666667    G to Bb    7/6

1.142857/.9999999 = 1.1428571    Bb to C    8/7    1

The second series, “Ordinary Waking Consciousness”

1.285714/1.142857    = 1,125        C to D    9/8

1.428571/1.285714    = 1.111111    D to E    10/9

1.571428/1.428571    = 1.1        E to F#    11/10

1.714285/1.571428    = 1.090909    F# to G    12/11

1.857142/1.714285    = 1.0833333    G to A    13/12

1.999999/1.857142    = 1.0769231    A to Bb    14/13

2.142857/1.999999    = 1.0714285    Bb to B    15/14    2

The third state is called “self-remembering”, or “self-consciousness”, and although we believe we already possess it, we cannot create it in ourselves by desire alone.  This third state is man’s natural birthright; but because of the abnormal conditions in which he lives, man does not have or even know about this third state.  If even twenty people decide to make an agreement to awaken, there is a chance to come to the state of self-remembering.  But even that is not sufficient.  What is needed, as reported in Fragments, is to find a man who is already awake and then hire him to keep them awake, not allow them to fall back to sleep.  Without the hire, it is impossible to awaken.  Gurdjieff says unequivocally that “this is what must be understood”.   

What must be understood is that the “hire” corresponds to the third series whose fractions extend to 22/21.  In other words, twenty is not sufficient.  For awakening, we need the hire.  

The third series, “Self-Remembering”

2.285714/2.142857    = 1.066666    B to C    16/15

2.428571/2.285714    = 1.0625        C to C#    17/16

2.571428/2.428571    = 1.058823    C# to D 18/17

2.714285/2.571428    = 1.0555555    D to D#    19/18

2.857142/2.714285    = 1.0526316    D# to E    20/19

2.999999/2.857142    = 1.05        E to F    21/20

3.142857/2.999999    = 1.047619    F to F#    22/21    = 3

In the analogy of the hackney carriage, the “hire” represents the “higher” human driver (self-consciousness, or ego consciousness) who holds the reins of the horses (ordinary waking consciousness) that direct the cart (unconsciousness).  The hire works for itself, for pay; the motives are chiefly mercenary.  In traditional teachings, it is the “self”, lower case.  

There is a fourth state of consciousness, the objective state, in which man can realize, can see things as they really are.  In regards to the hackney carriage, the representative of this fourth state is the “passenger” and owner of the carriage.  This is the “Self”, upper case, the real “I”.  Gurdjieff says that it is the result of long and difficult work on oneself.  The development of this objective consciousness is through self-consciousness - through the lower self, or ego.  In other words, no one comes to the Father-I (the Self) but by me (the self).  

The fourth series, “Objective Consciousness”

3.285714/3.142857 = 1.0454545    F# to Fx    23/22

3.428571/3.285714 = 1.0434782    Fx to G    24/23

3.571428/3.428571 = 1.0416666    G to G# 25/24

3.714285/3.571428 = 1.04        G# to A    26/25

3.857142/3.714285 = 1.0384615    A to A+    27/26

3.999999/3.857142 = 1.037037    A+ to Bb    28/27

4.142857/3.999999 = 1.0357145    Bb to B-     29/28    4

The completion of the fourth series has brought us to the alphabetical letters, Bb and B.  At this point, the thought is aroused that the word, “Being”, so elusive and difficult to pin down, actually references the completion of the fourth body at the ratio 29/28 of the harmonic series.  


In “From the Author”, found at the end of Beelzebub’s Tales, Gurdjieff makes the analogy of the four personalities existing within the common presence of man.  

According to the already indicated seriously instituted experimental investigations carried on over many years, or even according merely to the sane and impartial reflection of even every contemporary man, the common presence of every man - particularly of one in whom for some reason or another there arises, so to say, the pretension to be not just an ordinary average man, but what is called ‘one of the intelligentsia’ in the genuine sense of the word - must inevitably consist not only of all the said four fully determined distinct personalities, but each of them must of necessity be exactly correspondingly developed, to ensure that in his general manifestations during the period of his responsible existence all the separate parts should harmonize with each other.   

Immediately following this paragraph, Gurdjieff writes that these four formed personalities “are almost exactly comparable to that organization for conveying a passenger, which consists of a carriage, a horse, and a coachman”.  As Gurdjieff plainly states, these comparisons and parallels, while they are found in most systems, have forgotten the most important thing: that man is not born with the finer bodies, which can only be artificially cultivated under favorable conditions. 

The word personality, when taken literally, is “through (per) sound (sona)”.  The sounding structure able to convey the passenger is the harmonic series.  In series, the harmonic ratios are what harmonize the four parts - cart-horse-driver-passenger - into one unified system, “unity consciousness”.  

Although the harmonic series is considered common knowledge in most present systems of thought, the fact is that what has been forgotten are the harmonics above the usual sixteen, the ones that are most important, and without which man remains in the two lowest states of consciousness only: “sleep” and “ordinary consciousness”.  These ‘higher” harmonics must be “artificially cultivated”, that is, they need to be reasoned out by the human mind.  

I believe I have shown, in objective terms, exactly how this “artificial cultivation” occurs.  I am not asking that others accept what I say merely because I believe it to be so.  I am not asking that others take what I say on faith.  The indubitable mathematical facts of the matter speak for themselves.  One can calculate it for oneself.  Reason proves it to be true.  

This picture is someone’s representation of what they think Plato might have meant, in The Republic, by the Divided Line.  The truth is that the division of the line is notoriously obscure.  Although many erudite theses have been penned regarding this line, no one actually knows the length of the line, or the ratio(s) into which it should be subdivided, or its orientation: whether it should it be drawn horizontally or vertically.  There are those who cleverly argue the proposition that the line should be understood as the Extreme and Mean Ratios (phi), while others refute, just as cleverly, that same argument.  The lack of textual evidence causes some reputable academicians to go so far as to say the length and proportions and orientation do not matter.  There are even those who suggest that the quadripartite idea should be abandoned altogether in favor of a two-dimensional diagram representative of the superior (Intellect) and inferior (Sensory) worlds.  

What is the big deal about the Divided Line anyway?  How does comprehending this enigmatic proportion affect my life?  Who, save for fools and charlatans, cares about such a theoretical construct?  If “sectioning the canon” is what is required for a “liberal education”, then surely it is a waste of time.  After all, the only important thing is to receive a practical education which inputs specific information into the brain in order to enable the development of practical life skills.  An educator is one who desires to instill not what is “idealistic”, but rather what is “practical”.  What does it matter if there is a four-part, or three-part, or even a two-part division of the line?  What academic pedant is inclined to pursue an inquiry into what is deemed, by any sane thinking man, a subject of no practical interest whatever, and in any case, is something impossible to pin down? 

The questions are put to Socrates by his irritated pupils, those whose eyes prefer to avoid the light of Truth.

Yes, Socrates comments, there are two classes of persons: those who pursue truth consciously, with illumined eyes, seeing what is precious; and those of the unconscious class who demean the truth with eyes that see nothing.  Despite protestations to the contrary, the bulk of humanity does not want to know the truth.  Those who profess to be “seekers of the truth” want it, but only on the condition that first their belly is full.  The truth is, most are like the wolf in the story who says it wishes to reform from its dark bloody state of existence, enter a higher state of “goodness” - until the sheep passes by the window; at which point the wolf forgets all about its “idealistic” intentions and contemplates only the “practical” way to get at its next meal.  

The truth is, a liberal education isn’t cheap.  One has to pay for it, pay a lot, and for a long time.  Liberal education does not happen all at once but in stages.  Even the preparatory elementary education takes a long time.

The preparatory training relies on music and gymnastics, the two compulsory subjects.  While music is the most important component, the balance between gymnastics and music must be maintained, for overemphasis on music produces one that is too soft; too much emphasis on gymnastics, however, produces a noble savage who is too hard.  “The man who makes the finest mixture of gymnastic with music and brings them to his soul in the most proper measure is the one of whom we would most correctly say that he is the most perfectly musical and well harmonized” [412].  The only allowable music in the preparatory school is melodic (modal).  It is relatively austere, lacking in the “relishes” and “luxuriousness” (presumably lush tonal harmonies) that Socrates’ pupil, Glaucon, so admires.   (Note that, in pathology, glaucoma refers to opacity of the eye that impairs vision and, if not given early therapeutical treatment, can lead to blindness).  In the early training, only modal melodies are permitted; other strains are forbidden.  The modal melody and rhythm must follow the speech; not the other way around.  Only simple instruments are allowed: strings and pipes.

Speech, or what Aristoxenus referred to as “one of the forms of melody”, has to do with hearing.  On the one hand, speech is a human characteristic, learned in childhood.  To be able to speak in words is a trait not applicable to the lower animals.  

On the other hand, the human attribute of speech, according to the ancient hermetic tradition, is descriptive of the lower part of the soul, called Logos.  Speech does not tell the whole story.  There is another, a higher part of the soul called Nous, or “mind”, that is beyond words, beyond speech, beyond ordinary associative thought.  

All men have the advantage of speech, but few are granted mind, as the following dialogue illustrates:  

Teacher: Speech God imparted to all men, but mind He did not; not that He grudged it to any for the grudging temper comes from souls of men who are devoid of mind.


Tell me, why did not God impart mind to all?


It was His will that mind should be placed in the midst as a prize that human souls may win.


Where did he place it?


He filled a great basin with mind, and sent it down to earth; and he appointed a herald, and bade him make proclamation to the hearts of men: “Hearken, each human heart; dip yourself in this basin, if you can, recognizing for what purpose you have been made, and believing that you shall ascend to Him who sent the basin down”.  Now those who gave heed to the proclamation, and dipped themselves in the bath of mind, these men got a share of knowledge; they received mind, and so became complete men.

To become complete men with mind was the aim of the higher mystery schools.  The training began with the arousing of Thought, spelled with a capital “T”.  In other words, not the ordinary subjective thoughts of the mechanical type produced by the data processor which Gurdjieff termed the “formatory apparatus”.  Real Thought could not be produced automatically; only consciously.  The “arousing of Thought” required preparation and study and took a fairly long time.  To be able to produce “a Thought of one’s own” was the purpose of the human being. 

In the Pythagorean tradition, conscious thought was called “Reason”.  From the ability to reason, one could see and comprehend the higher things which were unperceived and unperceivable by those whose eyes were blind.  

The training began in early childhood.  

In Plato’s Republic, Socrates explains the training of these “noble puppies”.  The young pups are in preparation to become guardians, guard dogs.  At first, the pups are guarded from the truth.  Instead, they are treated like children, are shown picture books and taught by means of allegories and fairy tales.  The education relies on carefully-crafted stories that mold and shape the moral nature, and only a young and pliant puppy can be shaped completely to correspond to the values of the state.  Any who rebel against the ideology will, one way or another, be rejected.  These noble puppies are watched and tested, to see which of them, if any, might exhibit a just and proper demeanor.  Very few stand out above the crowd.  

The training encourages bodily skills that make the puppies spirited, swift and strong [376].  Those who can best withstand hardship (extremes in temperature, lack of sleep, grinding toil) are held up as heroic models of stamina and endurance.  Youth, Socrates says, is the time for extraordinary work [536].  As Socrates reports, “The trial of who is first in gymnastic exercise is one of the most important tests to which our youth are subjected”.  That is the only kind of knowledge which takes lasting root.  (Those pupils fortunate to have acquired Gurdjieff’s little booklet titled “Herald of Coming Good” will recall that the name of the heart of all activities at the Chateau du Prieure was “Gymnasium”).

Later on in the dialogue, Socrates acknowledges that this type of education is, at best, incomplete.  The education of puppies remains in the preparatory stage.  In fact, the word pupil, means “a minor under the care of a teacher or guardian”.  

Anatomically, the pupil is the contractile opening in the iris of the eye through which light reaches the retina - the retina itself (? L. rete, “net”) being the inner membrane at the back of the eyeball.  Obviously the pupa is not all there is.  There is the retinue that direct and instruct from behind the scenes.  These guardians running the show are the “pupa-teers”, those who manipulate the movements of the puppets. 

Socrates tells Glaucon that if a pupil wishes to go on to a higher level, it must be diligent, must toil at learning music as well as the more popular gymnastics.  Do not be satisfied with “popular exposition” (that is, the gymnastics that appeal to the coarser sensory effects rather than the intelligent causes) and believe there is no need to search further.  Yes, Socrates says, there cannot be any worse fault in a guardian.  

The stated aim of the preparatory educational system is that only a certain number will become “noble puppies”.  Not every puppy will succeed and matriculate.  Not every puppy will become what might be termed a Big Dog (canis Major) or even a Little Dog (canis Minor).  Some candidates are found to be “lame” and, although they may be allowed to remain and serve, do not go on from puppy to dog school. 

Lameness, Socrates says, is of two types.  One is “lame” who loves gymnastics and all bodily exercises, but is a hater of learning or listening or enquiring.  Conversely, one who loves the labor of learning but hates gymnastics is also “lame”. Those attending lower school are carefully considered and must demonstrate soundness in both body and mind, and must, above all, be lovers of work.  However, as regards the work of these noble puppies there is mental work and work of the body, and “the mind more often faints from the severity of study than from the severity of gymnastics”. [535]  

There can be little doubt that the Gurdjieff Work is the preparatory training ground for would-be “noble puppies”.  The training is no small thing.  Only a rare breed even seeks it out.  Of those selected as candidates, the teaching received is different from what we call “education” today.  Even so, there are not many “noble puppies” selected from the rank-and-file.  

Many puppies are called, but few are chosen.  The rest stay on, working in the den and paying dues, and are mostly unaware that there is another higher level beyond the ordinary “puppy” education to which they aspire.  The Work, in its “outer” aspect, traditionally serves as a crafts guild.  The apprentices who are taken in learn by imitation, and skills come from repetition.  If intellectual truths are revealed, their appearance is largely accidental, for the formal knowledge is intentionally lacking. 

At a certain age, Socrates tells us, gymnastics are over and “sleep and exercise are unpropitious to learning”.  After that time, “those who are selected from the class of twenty years olds will be promoted to higher honor, and the sciences which they learned without any order in their early education will now be brought together, and they will be able to see the natural relationship of them to one another and to true Being”.  Socrates suggests to Glaucon that now the study of philosophy takes the place of gymnastics and continues “diligently and earnestly and exclusively for twice the number of years which were passed in bodily exercise”.  After five years, the trainees will be sent back into the world, where they will acquire life experience, and where they will be tested to see whether or not they succumb to temptation.  Here they remain, if they survive, until the age of fifty, when they are obliged to toil for the public good (their “being-duty”).  If, during that time, they manage to train other candidates who can take their place, only then can they depart the drudgery and go to the Isle of the Blessed to rest in peace.  

There are many things to take into consideration here.  The word “twenty”, for instance, recalls Gurdjieff’s statement that “if twenty people awaken, there is a chance to come to “self-remembering.”  Assuming that the training of “early childhood” (the “play time”, music and gymnastics) refers to the first two series of the Canon (the numbers 7 and 14); then the “higher education” (the study of philosophy) would pertain to the third and fourth series (the numbers 21 and 28).  The final number of the upper division (28) is “twice the number of years” of the end of the lower division (14).  

The “succumbing to temptation” reflects upon the idea of the “tempter”, the devil’s interval F#, which, being “two-natured” (both F# and Gb, both “angel” and “devil”) can rise to the higher virtuous spheres of sharps, or alternatively, can succumb and descend to the world of vice in the lower world of flats.  To be “led into temptation” would presumably mean to remain confined “below the line”, in the first two sets of the harmonic series, content to remain an immature child, intent on pleasure and enjoyment, caring nothing for wisdom and knowledge, failing to do one’s “being-duty” as a mature adult.  In many cultures, “responsible age” means age twenty-one.  This number, as we know, appears at the end of the third series, at the ratio 22/21.  The third series is “above the line”.  

The third series, “Self-Remembering”

2.285714/2.142857 = 1.066666    B to C    16/15

2.428571/2.285714 = 1.0625        C to C#    17/16

2.571428/2.428571 = 1.058823    C# to D 18/17

2.714285/2.571428 = 1.0555555    D to D#    19/18

2.857142/2.714285 = 1.0526316    D# to E    20/19

2.999999/2.857142 = 1.05        E to F    21/20

3.142857/2.999999 = 1.047619    F to F#    22/21    = 3

It so happens that the twenty-first letter of the Greek alphabet is “phi” .  It is surely no coincidence that it gives us the first syllable of our word “phi-losophy”, which means “lover of wisdom”.  Nor is it likely coincidental that the master-teacher Gurdjieff named his little puppy “Philos”.  

That other great teacher of the wisdom tradition, Plato, in his Academy, taught his pupils that phi is the most binding of all mathematical relationships.  Geometers know that phi is symbolized by the pentagram, and the number five.  Plato said that only those who comprehended geometry could enter the Academy.  A further hint of the pentagonal number and phi appears in the Republic where we read that “five years” are spent in the diligent study of philosophy before the select trainees are sent into the world to teach others.  Plato insisted that the knowledge of phi was so critical that not only did it afford man the wisdom of the gods; by it, man could become as one the gods.  He made it the key to the physics of the cosmos.  Or perhaps one should say he made it the three keys of the cosmos.  

The word “key”, in the Italian language, is “clef”.  As musicians know, there are three clefs (keys) that open the door to musical literacy: the treble (or G) clef, the key to the scale system of sharps; the bass (or F) clef, the key to the scale of flats; and the one in the middle, the C clef, which serves as the “bridge” between the upper and lower regions of the musical grand staff.  The G and F clefs denote the “female” and “male,” respectively (the higher and lower voice ranges); while the C clef, lying in between, can delineate both, depending upon where it is placed on the lines of the staff.  On the one hand, it sets out the female, the alto clef; on the other hand, it defines the male, or tenor clef.  Putting the three clefs together, there are a total of twenty-one lines and spaces.

Keep in mind that it is not my idea.  The tripartite division of the 21 notes on the staff, their formation into the three systems, is a historical fact.  In musicological terms, the three systems have precedent as the hexachord durum (the hard hexachord which began on G, and used square neumes which eventually became the sign for the sharp); the hexachord molle (the soft hexachord beginning on F which had the round sign, now our sign for flat); and the hexachord naturale (the hexachord from middle C, placed in the middle between the two staves, thus linking them by means of the inner ledger line).  

What is not historical information - and therefore remains occult for contemporary cultures - is “the addition”: that is, the low F#, which I have taken the liberty of adding below the bass staff.  I believe that this F# is crucial for the completion of the third series, “Self Remembering”.  Without the F#, no liberation.  

Liberation concerns man’s “egoistic individuality”.  Great Nature needs a certain number of liberated beings.  “Although the said liberation is possible, nevertheless whether any particular man has the chance to attain it - this is difficult to say”.   There are a mass of reasons which may prevent liberation, the chief reason, Gurdjieff says, being our heredity and the conditions under which the process of our preparatory age flows.  In other words, to be properly prepared would mean having attained the ratio 16/15, the linking factor to the third series, without which one stays “below the line”, wrapped up in one’s subjective opinions, trapped in the “faith of one’s fathers.”  

It goes without saying that these facets of the first two series, Plato’s eikasia and pistis, are the chief factors that lead to bloody wars and revolutions.  Such a devilish world, filled with strife, exists without conscience.  To rise above the line, to come to a peaceful place, requires dianoia, the word that connotes science and mathematics, the tools of higher reason.  The higher world comes about with knowing: con (“with”) sciens ? ppr. of scire (“know”).  There is no more denying it.  Conscience means knowing with [the mathematical tools of] science.  

The idea returns us to the ratio at the completion of the third series, 22/21.  

It doesn’t take a “rocket scientist”, as they say, to know that at the close of the third set, at the 22nd seventh - the F# - we arrive at what mathematicians refer to as the “transcendental number”, pi , for which one of the ancient values in decimals is 3.142857, and when expressed as a fraction, is 22/7.  Transcendent, for a philosopher, means that which is beyond ordinary knowledge.  In Kant’s system it is beyond experiential knowing; in other words, it requires reason for its understanding.  From the reasoned calculations of the harmonic series, we have arrived at the transcendent number, pi, from which it becomes possible to derive that symbolic diagram called the enneagram.  The whole number 3 expresses the three points of the “outer triangle”, while the decimal .142857 gives the “inner movement”.  The enneagram symbol “was given such significance by those who knew that they considered it necessary to keep the knowledge of it secret”.   Let us diverge for a moment and look at the transcendental information afforded by this astonishing symbol. 


Anticipating the question that inevitably arises regarding the Divided Line (“What does all this have to do with me, with my life?”), and knowing that answering that question poses no easy challenge, Socrates offers the famous Allegory of the Cave, in order to better acquaint his pupil, Glaucon, with the arrangement.  

The majority of mankind, Socrates explains, lives like cave-dwellers confined to a dark underground world, and don’t know that other worlds exist.  These cave-dwellers recognize nothing but the gray shadows of statues projected on the wall of the cave.  Because the shadows are all they ever see, the concerted opinion of these cave dwellers is that the shadow world is all there is.  These prisoners are unaware even of the fact that they are prisoners, and thus have no wish to be liberated from prison.  They don’t even know that they don’t know.  What these prisoners don’t know is that the shadows are of real statues, which are manipulated by another group existing up on the wall behind and above the cave dwellers.  Because of a fire burning bright and out of sight, statues cast their moving shadows.  However, the cave dwellers are unable to turn their heads, and so remain facing straight ahead, fascinated by the moving picture show.  In terms of harmonics, they represent the first series, eikasia, the unenlightened state of sensible appearances.  

Socrates tells that a certain prisoner is loosed from bondage and made to turn and look at the first and statures.  The select prisoner begins to comprehend how the things causing the shadows are more real than the reflections he had always presumed to be reality.  He sees how the fire and statues together make the moving shadows on the cave wall.  The contact with the “real things” (the statues) has made him have faith in a greater reality than the shadows on the cave wall.  He knows he doesn’t know this greater reality, or of the manipulators causing it to happen.  Nevertheless, he begins to organize what little he has perceived of this higher reality into a belief system, or “religion”; and he and the others he convinces about “black-and-white truth” design beautiful artworks that reflect these beliefs.  The arts and crafts and architecture may reflect great skill of workmanship; they may finely model and accurately portray the shadowy things perceived; but this art has no truth in it, for the formal knowledge is lacking.  In terms of harmonics, this is the second series, pistis, belief or faith, without understanding the reason for such beliefs.  

Next, this prisoner is dragged outside the cave.  At first he is blinded by the sunlit world, so bedazzled that his eyes see nothing but a glare.  He is certain he is dead.  However, once the “blindfold of light” is removed, he can focus on the objects (house, people, trees, flowers, and so on), and starts to realize that the moving statues were not real, but only copies of these objects.  Instead of two-dimensional reflections, he now sees the real form of things as they are; their three-dimensionality.  He knows that he doesn’t know much about these forms, but he wishes with every fiber of his being to be able to know.  He knows that what is needed now is the formal knowledge, the reasoned knowledge of science and mathematics.  


Since the time of the ancient Greeks, the great philosophical issue of form and how to describe it has not been resolved.  Kant was perhaps the last of the modern philosophers to recognize the problem.  The conceptions of an antecedent ideal form and consequent sensate matter “lie at the foundation of all other reflection, so inseparably are they connected with every mode of exercising the understanding”.  He likened form to pure order divorced from the senses.  “That which in the phenomenon corresponds to the sensation, I term its matter”, he said; “that which effects that the content of the phenomenon can be arranged under certain relations, I all its form”.  

Later philosophers were less interested in the before and concentrated on the after.  Heidegger, for example, skipped over the Platonic “ideal” and conveniently began his materialistic discourses with the idea of “substance”.  Unfortunately, lacking the foundation necessary, his reflections were not based on reality.  Heidegger, it seems, was basing his insubstantial formulations upon the shadow world.  Granted, he realized more than most: that there is a shadow-world.  Nevertheless, he saw nothing above this unreality, and his perceptions were valid only for that limited shadow-scenario.  In the final analysis, what he espoused was no more than another belief system.  One who follows Heidegger must accept what he says on faith.  

Heidegger’s philosophy is lacking, at least by ancient Greek standards.  The ancient philosophers, unlike their modern counterparts, were not satisfied merely with beliefs (pistis).  Their ideology demanded a higher knowledge than what can be perceived by the deluded senses.  In the Platonic view, the “knowledge of the Forms” constitutes real knowledge.  The form is the actual cause of everything existing in the shadow world.  Form is the reason for all order and intelligibility, and nothing exists at all without form and structure.  Form is first, matter is second.  

Plato’s “theory of forms” was set forth to solve the two problems of (1) ethics and (2) impermanence).  How, Plato asked, regarding the ethical, can man attach value to that which is merely subjective and relative?  To know truth, beauty, justice, and goodness is impossible in man’s ordinary state.  In the world of shadows, everything is opinion and conjecture, and everything changes with the fashion of the times.  What is just in one age is unjust in another.  What is beautiful for one person is ugly for another.  How can a man be ethical in such a relative framework?  What, in such a world, constitutes truth?  

Regarding impermanence, Plato asked, how can a sane and reasoning man expect to find joy or peace in a world where the only constant is change and dissolution?  The solution is to detach oneself from the relative and subjective material nature.  Matter contains the seeds of its own destruction.  Rather than caring about the mutable mortal planetary body and its desires, seek instead to develop the discernment of the immutable realm of form, the understanding of which comes through the higher reasoning mind.  Socrates, Plato’s teacher, held that Forms or Ideas are the immutable objects of our highest knowledge.  Form is unchanging, not admitting of variation at all, or in any way or at any time, unlike the perishable things of the sensible world. 

There is only one problem with that.  No one today seems to know what it means. 

Plato’s Republic affords clues.  In Book VII, Socrates (Plato’s mouthpiece) gives something of “universal application” which, as he puts it, is “the little matter of distinguishing one, two, and three - in a word, number and calculation”.  He asks Glaucon, “Do not all arts and sciences necessarily partake of them?”[523]  Number, Socrates goes on to say, can lead the mind to truth and being in a very remarkable manner [525].  

After discussing the benefits of the quadrivium - the mathematical sciences of arithmetic, geometry, astronomy, and music - he then stresses the idea of pattern and proportion and the right way to investigate exact truth.  The right way goes beyond the purview of the senses.  For truth, Socrates insists, one needs dialectic.  

Because it moves in the realm of Idea, dialectic deals not with relativities and probabilities inherent in matter; but rather is the tool that deals with absolute truth.  It forms the “pure reason” of which Kant speaks, and about whose “misemployment” creates all fallacy and illusion.  

Dialectic has sometimes been termed the ascent from appearance to reality; and over the centuries, philosophers have taken it to mean the pursuit of truth.  Logicians consider it the contemplation of being.  Scientists, however, have frequently disparaged dialectic, saying that it flies in the face of experimental proofs and empirical research which are the cornerstones of scientific methodology.  Yet in the hands of Socrates, dialectic is the very instrument of science: “There is no other method of comprehending by any regular process all true existence or of ascertaining what each thing is in its own nature”.  Dialectic, he says, “is the coping-stone of the sciences, and is set over them; no other science can be placed higher - the nature of knowledge can go no further”.  In other words, for Socrates, dialectic is the highest science, meta (above) science.  Dialectic is the science of metaphysics.  

A “dialectician” is one who appeals to reason rather than to experience.  Experience is part of the sensory world and depends upon the unreliable senses to apprehend reality.  The prisoners in the cave rely upon their “experience” to determine their shadow-reality.  Yet there are many today who insist that we must begin not from metaphysics but from experience.   Perhaps what is meant is that we must first wake up to the fact that we are prisoners in the cave.  Without waking up to that primary experience, we fail to turn around.  Turning around, we experience secondarily how the statues cast their dream shadows on the wall.  Leaving the cave (which constitutes a kind of death), we die to the dark world we only believe we know and come into the bright light of day.  Now able to see the truth, we are reborn into the real world, the third state.  The series is: first to awake, then to die, then to be reborn.  


The “real world” isn’t what we believe it is at all.  

The real world can only be apprehended by real numbers.  For a mathematician, pi is a real number.  (A real number is defined as a rational number joined with an irrational number).  From pi, we can begin to glimpse views of the real world.  

Remember that it is not until the end of the third set of the harmonic canon, at the 22nd seventh - 22/7, the F# - that we arrive at the real number “pi”, one of the ancient values of which, in decimals, is 3.142857.  

The third series

                    2.285714/2.142857 = 16/15        B to C

                    2.428571/2.285714 = 17/16        C to C#

                    2.571428/2.428571 = 18/17        C# to D

                    2.714285/2.571428 = 19/18        D to D#

                    2.857142/2.714285 = 20/19        D# to E

                    2.999999/2.857142 = 21/20        E to F

                    3.142857/2.999999 = 22/21        F to F#

Like other of the most important real numbers (which include e and v2, for example), the number pi carries its own special symbol.  

Those in the Work are fond of quoting the statement “The transmission of the meaning of symbols to a man who has not reached an understanding of them in himself is impossible”.   For them, it is an excuse not to have to study symbols.  Perhaps they would do well to look at the corollary found in the Rg Veda: “Knowledge is structured in consciousness, and the level of one’s own consciousness determines how deeply one can comprehend that knowledge”.  

Granted, symbols in and of themselves are virtually useless.  A symbol is merely a tool.  What use is a tool without knowing what the tool is for?  A slide rule in the hands of a monkey is merely a funny-looking stick.  As Gurdjieff put it, a symbol can only be understood by one whose consciousness already knows what is comprised in that symbol.  Let’s look a little closer at that tool, pi, the symbol itself looking strangely like a portal, or perhaps a bridge, and which makes its appearance at the end of the third series.  Available to only to human consciousness, perhaps it will bring “glimpses of truth” to those in the Work whose profession is “Seekers of the Truth”.  


Our ordinary levels of consciousness are structured in what the Amerindians called the “tonal”, the known world within oneself.  In Gurdjieffian terms, the “tonal” represents the two lower states of consciousness, the flowing data which is perceived by that data processor called the “brain”.  The human brain, according to researchers, is divided into three distinct strata: the lower “reptilian”, the middle “old mammalian”, and the higher “new mammalian”, or neo-cortex.  For ordinary man - Man number one, two, and three - the three-tiered brain system provides the data for the three dimensions, the world that begins at birth and ends with death.  For the enlightened shaman who has a “window” open to another and more “real” world, however, the “tonal” is not all there is.  There is the “nagual”, which lies outside one’s time.  For ordinary mortals, the nagual remains hidden, an Unknown quantity.  

The stories of Carlos Castaneda and his sorcerer Don Juan bring a certain familiarity to these ideas.  However, nothing in those tales comes close to explaining the real underlying symbolism of “tonal” and “nagual”.  I believe our current study uncovers hidden data that bears upon the real underlying meaning of those words.  We need look no further than the musical staff to find astonishing correlations.  

The “tonal” is fairly simple to intuit.  There are the three “registers” - lower, middle, and higher - which represent the three tiers of the human brain.  

The three distinct segments marked in brackets divide the grand staff into three octaves of seven notes, a total of twenty-one notes.  As already stated, the twenty-first letter of the Greek alphabet is phi, designated by the mathematical symbol ?.  In the harmonic series, the twenty-first note is “F”.  The tonal data provided by the three brains afford the necessary vibrations by which we register the “phi”-nomena of the ordinary world.  

The Hindus, for one, considered phi to be the binding force, and in their sacred text put it thus:

                I am that which binds,
                I am the golden navel of the universe
                He who knows this knows Upanishad.

There is another mathematical operator, designated by the symbol pi, 3.142857, which lies “outside the system”.  This twenty-second note, F# (the only one sporting the sharp sign, #), is surely the secretive “nagual”.  In biblical terms, it suggests the “serpent”, that strange and seemingly irrational alien presence whose sorcery, as recounted in the creation story in Genesis, entices Eve to eat of the fruit of the tree of knowledge.  

The warning, one recalls, had been put out to the primordial pair: “Lest ye eat of the fruit, ye shall surely die!”  The word “die” itself is suggestive of the dieses, the Greek microtonal intervals whose very name (“diesis”) means “sharp sign”.  To leave the tonal system composed solely of placid colorless white keys was surely to risk the dangerous regions wherein lay in wait the dangerous micro-intervals (“thorns and thistles”), bringing clashing dissonance.  According to the ancient Greek philosophers, twenty-eight dieses occur within the circuit of the complete universe.  The number 28 occurs at the end of the fourth series.  We return to this critically important idea at a later point.  

The F# comes from “outside the system”.  Zooming around in the “three-dimensions” of the solar system, it can, so to say, be two places at once.  Since it is not confined to the “ordinary laws” in operation, it can easily alter its “appearance”, so that, instead of F#, it can assume the appearance of Gb.  In fact, it can be both at the same time.  Inside the system, this masquerade is very confusing.  Is the “nagual” good or evil?  

Those familiar with quantum mechanics might see a startling resemblance between certain subatomic particles and the strange behavior of this “alien invader”.  For example, this “outsider” has the capability of moving in two directions: from top down (involution), or from bottom up (evolution), and the shift in direction causes the strange sensation, for those experiencing it inside the system, of moving forward and backward in time.  Such a thing is deemed possible in quantum physics.  

There is also the question of gravity and the “graviton”.  Although no one really knows what a graviton is, it is deemed necessary for laws of quantum gravity to work as they should.  Einstein said that accelerating is what gives the sensation of gravity.  But is the sensation experienced as rising, or is it actually a sense of the law of falling?  Is one experiencing going “up” or “down?”  Or should it be called “in” and “out?”  No one really knows for sure.  No one really knows what is moving, or even if it is moving.  The whole thing is extremely disconcerting.  

Thankfully, for a musician at least, one thing is relatively clear.  When traveling in the “forward direction”, this “alien” takes on the “sharp” sign (#); and when moving in “reverse”, it dons the sign of the “flat” (b).  A mythologist might recognize it as “Janus”, the god who looks both ahead and behind.  For a pupil of the Work, the idea recalls Ouspensky’s “divided line” in Fragments (p. 119), with its dual arrowheads pointing opposite ways.  

What is the point of this two-faced sorcerer?  The answer lies in its function.  The F#/Gb is the link between the higher and the lower regions.  Before the idea can be grasped, first one needs to see what the higher and lower parts are, what is meant by the included middle, and how the diabolical tritone F#/Gb insures that the three parts blend together to form one whole unified system.  

Look!  There are two “incommensurable” regions.  The “low” part is composed of the sharp keys D, and E, the “high” part of flat keys Ab, Bb, the key of C being “neutral” in both.  In order to connect the E to Ab, a middle region is necessary.  This mid-section, or “bridge”, is composed solely of the three octave ladder of “white keys”.  The key factor linking the whole is the tritone, F#/Gb.  How curious to realize that what we have here, in essence, is a “double crust pi” (the “dough”, or Do); and what comes in between is the “pie filling” of lower-case letters!

C D E         F#    gabcdefgabCdefgabcdef        Gb         Ab Bb C
 Low         link                    Middle                    link           High

The “pie” and “filling” may concern the two mathematical operators, pi and phi.  We may deem the symbolism important as soon as we realize it has to do with the geometry of ellipses.  

The area of an ellipse is calculated as pi times phi.  

Ellipses and their study go way back in time, at least to the ancient Greeks.  In the twentieth century, elliptical equations are studied by mathematicians and it has long been assumed that elliptic equations (and the “E-series”) exist in a totally different framework from so-called “modular equations” (which use the “M-series”).  I bring these seemingly unrelated equations into the discussion here for future reference, and reserve the right to recall these witnesses at a later point.

The two irrationals have a direct connection to Gurdjieffian cosmology.  A long, long time ago, remember, there was the cataclysmic shock with a comet that entailed serious consequences: specifically, two fragments broke off began making elliptical orbits around their fundamental piece, just as Earth makes elliptical orbits around the Sun.  This unfortunate situation made the now-elliptical trajectories of the cosmic bodies difficult to measure.  Due to the errant comet the former, and simple, measure of three no longer worked; instead, the calculations required the two irrational values, pi and phi.   

At this point we may wish to ponder that perhaps Gurdjieff’s so-called mythos not myth at all, but the return of an objective science known in ancient times and then forgotten.  This knowledge - quite literally, about the “fragments of an unknown teaching” (the words in quotes referring to the subtitle of P.D. Ouspensky’s book, In Search of the Miraculous) - concerned the knowledge of a battered Earth, one whose orbit was no longer exactly round and whose measures were no longer perfect and rational.  (In other words, the octaves did not precisely match up at the Cs).  

Consider, now, the Sumerian tale of Enlil.  Long before humans were put upon the earth, the great god Enlil was put in charge of supervising the work of digging out the beds of the Tigris and Euphrates rivers.  According to scholars who have studied the ancient Sumerian tablets, the gods had come to Earth to mine for gold.  The lesser gods who were responsible for actually carrying out the ceaseless labor of gold-digging became exhausted and eventually a rebellion ensued.  Enki (the Babylonian Ea) suggested that a new race called human beings be created to take over the work.  Translations of the ancient writings tell that these humans were artificial creations, genetically manufactured on the Earth by the gods in order to do the hard labor.  The process was accomplished by mixing the blood of the god with an already-existing primitive creature to form the Worker.  

The story, even if ever proven historically, must nevertheless be treated as allegory.  The hybrid artificial creation that “digs for gold” is the three-brained human being, whose third brain is artificially designed to accomplish the required work.  The “gold” mined from the depths of the earth suggests the twenty-first Greek letter, phi, that “golden” number of the philosophers symbolic of the higher consciousness ascribed to humans.  The ratiocinations produce the “words” of human speech, the calculations being derived logarithmically, according to the “real numbers”: that is, by square roots and pi.  The ratios are not natural, but rather are artificial formations, the cuts on the circle producing the twenty-two srutis.  

One thing must be understood.  The word “symbol” means “to throw together”.  Symbols throw together what otherwise cannot be mixed.  

Hermetically speaking, there are two incommensurable extremes: the Great Mother (“earth”), the physical realm represented by the square and square root (); and the Heavenly Father (“fire”), the higher immaterial spiritual realm, represented by the circle and pi (л).  

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