The only places this topic is touched upon is in 'Beelzebub's Tales to his Grandson' Chapter 17, 39, and 40 (where the seven 'gravity centers'/notes are called 'Stopinders' by Gurdjieff) and in Ouspensky's 'In Search of the Miraculous' Chapter 7 (where Gurdjieff calls them note accelerations.) This sparse information has caused confusion of how to apply the Gurdjieff method, referred to as 'Work'. Some would say that 'Work' is following the circle from the beginning to the end as in the 'Enneagram of Personalities', others will study the inscribed order of the unique pattern within the circle by dancing the Gurdjieff 'Movement' entitled 'Enneagram', others will practice the 'Work' at 'mi'-'fa' (Gurdjieff's External Activation) with breathing exercises, others will practice Work at 'sol'-'la' (Gurdjieff's 'Harnel Aoot') 'observing themselves', and others will practice at 'ti'-'do' (Gurdjieff's 'Intentional Activation') storing 'bodily energies' and others will ignore the subject as too difficult and rely on words taken from Chapter 39 such as 'being-Partkdolg-duty', 'impartiality', 'conscious labors' and 'intentional suffering', and 'intentional contemplativeness'. In my essay The Secret Enneagram of Gurdjieff and its relationship to Universal Vortex Math' I describe the construction of the Enneagram, but here I will concentrate on the 'Stopinders' of the law of 'Heptaparaparshinokh' in relation to the law of 'Triamazikamno' which are primarily described in 'Beelzebub's Tales to his Grandson' Chapter 39 'Holy Planet Purgatory'.

The two primary laws of 'Triamazikamno' and 'Heptaparaparshinokh' can be easily substituted the 'Law of Three' (Ch 17 pg 138 'Arch Absurd': "'Holy-Affirming', 'Holy-Denying', and 'Holy-Reconciling'.") and the 'Law of Sevenfoldness' (Ch 30 pg 493: "sacred law of Heptaparaparshinokh, or, as they called it the Law of Sevenfoldness").

However, the problem arises in the inconsistency of Gurdjieff's definition where he describes a 'Stopinder' of which there are seven in the 'Law of Heptaparaparshinokh' as:

In Ch 7 pg 125 'In Search of the Miraculous' by P.D. Ouspensky these notes are 'not lengthened, shortened, or disharmonized', but are analyzed according to acceleration of frequency as shown:

"These black strings however he named 'Demisakhsakhsa,' which according to the terminology of the beings of the Earth corresponded to what they call 'half notes,' and these 'half-note strings' on the Lav-Merz-Nokh were not strung between those whole notes between which, according to the indications of the saints Choon-Kil-Tez and Choon-Tro-Pel, there is according to the sacred Heptaparaparshinokh no possibility of the independence of the evolution and involution of the vibrations of sound, and these places they were the first to call 'gaps.' And in the given places of the octave where these gaps ought to be, this learned being King-Too-Toz strung special strings between the whole notes made of the tail-hairs of beings called there 'horse'. These hair strings gave vibrations which were not always the same and King-Too-Toz named these vibrations 'chaotic'."

This experiment did not end there as this 'piano' was improved by adding twice as many strings! The reason for adding ones between the chaotic horse hair strings seems to me to be absurd and perhaps a mistake as I believe only twelve more notes should have been added, not fourteen.:

"On this Lav-Merz-Nokh between these white, black, and hair strings there were also strung in each octave fourteen strings also from 'twisted intestines,' which were painted red and called 'Keesookesschoor,' and if contemporary beings of the Earth were to use these strings they would call them 'quarter notes'."

Contrary to the premise that the 'Law of Seven' was forgotten, here is 24 note scale developed in the Middle East in the eighteenth century:

Although Gurdjieff made fun of the Greeks using only five notes (the pentatonic scale), they used seven notes also. In fact Pythagoras was the founder of music theory by discovering a method to create all the notes in the 'Circle of Fifths'. He did this by choosing the most pleasant tone the fifth (2:3) and added 6 perfect fifths successively in a positive (outward direction in the spiral) to create the intervals Root(do), P5(sol), M2(re), M6(la), M3(mi), M7(ti), (Aug4) and then adding 6 perfect fifths in negative (inner direction) then the intervals P4(fa), m7(li), m3(ri), m6(sil), m2(di), tritone(fi), (dim5th) were obtained. All these could be placed in one octave because of the similarity of all octaves. Although this did have a problem with Aug4/Dim5th, the 'wolf tone' not completing a perfect circle, it set the foundation for the Circle of Fifths which is the foundation of all modern music.

See Skye Løfvander's youtube:https://www.youtube.com/watch?v=1DUZsQ2by2s

Green is most pleasing

So beginning with the Root, a simple division of the octave is to multiply the frequency by 1.5 (3:2) which is the Perfect Fifth (being the fifth note 'sol') that Pythagoras used. Its sound is as pure as the octave and can be confused by those who have little musical training. Another simple interval is obtained by multiplying the Root by one and one third (4:3) which is called the Perfect Fourth (being the fourth note 'fa'). In both of these, the frequency blends well and sounds aesthetically pleasing and consonant.
Next in consonance is the Major Third (x 1.25 5:4) and minor Third (x 1.2 6:5) (Third being the third note 'mi' and the half tone down); Major Sixth (x 1.666 5/3) and minor Sixth (x 1.6 8:5) (Sixth being the sixth note 'la' and a half tone down).

In fact a study was done on human speech sounds and it was discovered that there are eight peaks contained these vocalizations that match those that the six we have assigned previously by consonance including two others: the Tritone (x 1.406 45;32) (which is in the middle of the octave) and the minor Seventh: (x 1.8 9:5) (being a half tone down from ti). (Note that the lower the denominator the greater its consonance. Also the more overtones contained, the more its consonance.)

Looking at the gaps three more tones can be added, although they are not as pleasing but dissonant, yet often short dissonance adds flavor. These are Major Seventh (x 1.875 15:8) (being ti), Major Second (x 1.125 9:8) and Minor Second (x 1.06o6 16/15) (Second being the first note 're' and a half tone down). The Tritone is also in this category of dissonant.

The twelve intervals are shown on a piano keyboard octave and the accelerations of the notes are calculated as was done in the previous Ouspensky diagram. In the key of C, it is true that the two places where the black notes are missing do show 'increase retarded' for the white notes, but once the acceleration of of both the white and black notes are calculated, this is no longer true. The reason for this is because the accelerations have been balanced out over the octave. Since the Law of Seven is concerned with seven notes, namely the white keys in the key of C, one might say that the places where the 'increase retarded' (missing black keys) at 'mi'-'fa' and 'ti'-'do' are still valid. Yet, I would suggest that the premise of these being 'discontinuities' is not true and thus would be better called: 'points of action.'

Certainly, a black key could be placed in each of these places but it would serve no purpose as they would not sound perceptively different. The use of "horse hair" would make these intervals worse. There is certainly significance to the fact that an octave contains three important numbers: seven notes, five sharps notes, and thus twelve notes all together. Given this, let us map the white and black notes on the enneagram to see how they match up with the inner triangle. After all, the closing statement from the Ouspensky diagram shown earlier seems to point to the inner triangle having more significance:

This was done to allow the ability of playing in any key without having to retune and thus we have had to accept music that has waves that are out of sync beating against each other. If the objective is to have music that matches Pythagoras' imagined "Music of the Spheres" then one needs to tune to "Just Intonation" and avoid the keys that do not work or retune when desired. Today's technology can do this, but it brings up the question of whether music from nature is superior to mathematics.

The surprising result in this diagram is that all three vertices of the triangle exactly match three of the most consonant intervals: the Root note of the octave (unison), the Perfect Fourth ('fa'), and the Major Sixth ('la'). The sequence of 'do' (Unison), 'fa' (Perfect Fourth), 'la' (Major Sixth) clearly leads to the next octave 'do' and this follows the inner triangle.The most consonant Perfect Fifth 'sol' is in the bottom center directly opposite the apex of the triangle as if echoing the Unison. The righthand side is very crowded from 'do' to 'mi' as if it needs the uplift of 'fa', the Perfect Fourth. The note 'ti' wants to resolve in 'do'.

More important is to observe and hear how the three sets of three within the octave demonstrate how the Law of Three acts within the Seven Notes:

1. (Active) 'mi'-'fa', 'sol'
2. (Neutralizing) 'sol'-'la', 'ti'
3. (Passive) 'ti'-'do', 're'

"Our COMMON FATHER OMNI-BEING ENDLESSNESS, having decided to change the principle of the maintenance of the existence of this then still unique cosmic concentration and sole place of HIS most glorious Being, first of all altered the process itself of the functioning of these two primordial fundamental sacred laws, and HE actualized the greater change in the law of the sacred Heptaparaparshinokh."(do, re, mi, fa, sol, la, ti) (As to the Law of Three, the alteration is never addressed.)

"These changes in the functioning of the sacred Heptaparaparshinokh consisted in this, that in three of its Stopinders HE altered the what are called 'subjective actions' which had been until then in the Stopinders, in this respect, that in one (between the third and fourth deflections) HE lengthened the law-conformable successiveness; shortened it in another (between its last deflection and the beginning of a new cycle); and in a third, disharmonized it."

"In an ascending octave the first 'interval' comes between 'mi' and 'fa'. If corresponding additional energy enters at this point the octave will develop without hindrance to 'si', but between 'si' and 'do' it needs a much stronger 'additional shock' for its right development than between 'mi' and 'fa', because the vibrations of the octave at this point are of a considerably higher pitch and to overcome a check in the development of the octave a greater intensity is needed."

"In a descending octave, on the other hand, the greatest 'interval' occurs at the very beginning of the octave, immediately after the first do and the material for filling it is very often found either in 'do' itself or in the lateral vibrations evoked by 'do'. For this reason a descending octave develops much more easily than an ascending octave and in passing beyond 'si' it reaches 'fa' without hindrance; here an 'additional shock' is necessary, though considerably less strong than the first 'shock' between 'do' and 'si'."

The statement above that "the material for filling it ('do'-'ti') is very often found either in 'do' itself or in the lateral vibrations evoked by 'do'" supports my creation of 'ti'-'do' as the AO Alpha-Omega.

"And, namely, with the purpose of providing the 'requisite inherency' for receiving, for its functioning, the automatic affluence of all forces which were near, HE lengthened the Stopinder between its third and fourth deflections. This same Stopinder of the sacred Heptaparaparshinokh is just that one, which is still called the 'mechano-coinciding- Mdnel-In.'"

"And the Stopinder which HE shortened is between its last deflection and the beginning of a new cycle of its completing process; by this same shortening, for the purpose of facilitating the commencement of a new cycle of its completing process, HE predetermined the functioning of the given Stopinder to be dependent only upon the affluence of forces, obtained from outside inside* through that Stopinder from the results of the action of that cosmic concentration itself in which the completing process of this primordial fundamental sacred law flows. And this Stopinder of the sacred Heptaparaparshinokh is just that one, which is still called the 'intentionally-actualized-Mdnel-In'."

* I have changed 'outside' to 'inside' because it seems to violate the whole point of results within the octave being manifested in the completed 'Do'. It may have been confused in the sense that in the Creation of the Universe, the shortening allowed our Endlessness to project his 'Theomertmalogos' or 'Word-God' down through the octaves of Creation, thus from the outside.

"As regards the third Stopinder, then changed in its 'subjective action' and which is fifth in the general successiveness and is called 'Harnel-Aoot', its disharmony flowed by itself from the change of the two aforementioned Stopinders. This disharmony in its subjective functioning, flowing from its asymmetry so to say in relation to the whole entire completing process of the sacred
Heptaparaparshinokh, consists in the following:

#1. (chaos) If the completing process of this sacred law flows in conditions, where during its process there are many 'extraneously- caused-vibrations,' then all its functioning gives only external results.

#2. (quiet) But if this same process proceeds in absolute quiet without any external 'extraneously-caused-vibrations' whatsoever, then all the results of the action of its functioning remain within that concentration in which it completes its process, and for the outside, these results only become evident on direct and immediate contact with it.

#3. (chaos & quiet) And if however during its functioning there are neither of these two sharply opposite conditions, then the results of the action of its process usually divide themselves into the external and the internal."

"Try very hard to understand everything that will relate to both these fundamental cosmic sacred laws, since knowledge of these sacred laws, particularly knowledge relating to the particularities of the sacred Heptaparaparshinokh, ... data are crystallized for engendering that Divine property which it is indispensable for every normal three-brained being to have and your favorites call it 'impartiality'."

"The knowledge about this will give you additional data for elucidating by a concrete example certain particularities, which I have already explained to you, of the fundamental cosmic sacred law of Heptaparaparshinokh. After the loss of the continent Atlantis, certain knowledge concerning the origin and significance of this same 'being-Exioehary' (Gurdjieff's Second Stopinder 'ti'-'do') also survived, and this knowledge also began to pass from generation to generation. ...

In these though fragmentary yet nevertheless authentic informations, it was very convincingly indicated that by means of the substances 'Exioehary' or sperm formed in them, it was possible to perfect oneself, but unfortunately for them there were no indications, in this information which had survived and reached them, what and how precisely this had to be done...

However hard these same beings of your planet who were first interested in this question strove to get clear about this, they arrived at nothing, and it was only the second generation of them who ultimately, after long conscious observations and intensive active mentations, categorically understood that this was indeed possible, exclusively only on condition of a ceaseless fulfillment of being-Partkdolg-duty (Fifth Stopinder ('sol'-'la) item #3 above: combination of chaos and quiet), ... and those of them, beings of the said generation as well as certain of the subsequent two generations, who began seriously to actualize this, did indeed attain the expected results.

But already the fourth generation of those beings who were first interested in this question and who were followers not from essence-conviction but from a property called 'to imitate,' which had by that time also become inherent in these terrestrial three-brained beings, also began to exist together and to do as it were the same thing.

So from that time it began and until now automatically continues, that such followers organize themselves in separate groups and sometimes form solid sects of various denominations and, putting this same 'abstinence' as the basis of their aim, exist together segregated. Just these same places of theirs for segregated common existence together, are called there 'monasteries,' and the separate beings belonging to these sects, 'monks.' At the present time, very many of these 'monasteries' exist there, and these innumerable 'monks' who enter them do indeed strictly abstain from the ejection from themselves in the customary way of the being-Exioehary or sperm formed in them; but of course, no sensible result at all is ever obtained from this abstinence of theirs, ...

It is however unjust to say that no sensible result at all is obtained among these monks there. There are even obtained among them 'sensible results' of two independent kinds:

The Monks
monks?	Fifth Stopinder Harnel-Ahoot "#1" (chaos) 100% external	"The first kind of action of this definite substance consists in this, that it promotes the depositing of superfluous what is called 'Karatsiag,' or, as they call it there, 'fat'."
Jesuits?* (Sorry I went to high school and college under them)	Fifth Stopinder Harnel-Ahoot "#2" (quiet) 100% internal	"And its second kind of action promotes the arising and the dispersing over the whole planetary body of what are called 'Poisonioonoskirian-vibrations.' And in the second case, on the contrary, these abstaining monks become, as it is also usually said there, 'meager-thin'; and the action of the 'Poisonioonoskirian-vibrations' which penetrate through them is chiefly evident in their general psyche which becomes sharply dual and the manifestations of which are divided into two diametrically opposite kinds—the outer, visible and for show, sensed by everyone around them, and the inner and hidden, which the ordinary beings there, especially the contemporary, are entirely incapable of ascertaining or perceiving - namely, in their outer visible manifestations, these 'Poisonioonoskirian-monks' appear to be what your favorites would express as 'bigots' of a high degree; and in their hidden inner manifestations, not shown to others, what your favorites would call 'expert cynics,' also of a high degree."
being- Partkdolg-duty?	Fifth Stopinder Harnel-Ahoot #3 (chaos & quiet) 50%external/ 50%internal	"neither of these two sharply opposite conditions"

"Thanks to the new particularity of the fifth Stopinder of the sacred Heptaparaparshinokh, these emanations issuing from the Sun Absolute began to act at certain definite points of the space of the Universe upon the primesource cosmic substance Etherokrilno from which, owing to the totality of the former and the new particularities of the sacred primordial laws, certain definite concentrations began to be concentrated. "

"Further, thanks to these factors and also to their own laws of Heptaparaparshinokh and Triamazikamno which had already begun to arise in these definite concentrations with their action upon each other, everything which had to be gradually began to be crystallized in these concentrations, and as a result of all this, those large concentrations were obtained which exist also until now and which we now call 'Second-order-Suns'." (Stars: the Sun)

"When these newly arisen Suns had been completely actualized and their own functionings of both the fundamental laws had been finally established in them, then in them also, similarly to the Most Most Holy Sun Absolute, their own results began to be transformed and to be radiated, which, together with the emanations issuing from the Most Most Holy Sun Absolute into the space of the Universe, became the factors for the actualization of the common-cosmic fundamental process of the sacred law of Triamazikamno, and that is to say:

The Most Most Holy Theomertmalogos began to manifest itself in the quality of the third holy force of the sacred Triamazikamno; the results of any one of the newly arisen Second-order-Suns began to serve as the first holy force; and the results of all the other newly arisen Second-order- Suns in relation to this mentioned one newly arisen Sun, as the second holy force of this sacred law.

Thanks to the process of the common-cosmic sacred Triamazikamno thus established in the space of the Universe, crystallizations of different what is called ‘density’ gradually began to be formed around each of the Second-order- Suns out of that same prime-source Etherokrilno, and grouping themselves around these newly arisen Suns, new concentrations began to take form, as a result of which more new Suns were obtained, but this time 'Third-order-Suns.' These third-order concentrations are just those cosmic concentrations which at the present time are called planets."

"At this very place in the process of the first outer cycle of the fundamental sacred Heptaparaparshinokh, namely, after the formation of the Third-order-Suns or Planets, just here, owing to the changed fifth deflection of the sacred Heptaparaparshinokh, which as I have already said is now called Harnel-Aoot, the initially given momentum for the fundamental completing process, having lost half the force of its vivifyingness, began in its further functioning to have only half of the manifestation of its action outside itself, and the other half for itself, i.e., for its very own functioning, the consequences of which were that on these last big results, i.e., on these said Third-order- Suns or planets, there began to arise what are called, 'similarities-to-the-already-arisen'."

"And as after this, surrounding conditions of actualizations were everywhere established corresponding to the manifestation of the second particularity the fifth Stopinder of the fundamental sacred Heptaparaparshinokh, therefore from then on the actualization of the fundamental outer cycle of the sacred Heptaparaparshinokh ceased, and all the action of its functioning entered forever into the results already manifested by it, and in them there began to proceed its inherent permanent processes of transformation, called 'evolution' and 'involution'."

"And, namely, they named the Most Most Holy Prime- Source Sun Absolute itself: 'Protocosmos.' Each newly arisen 'Second-order-Sun' with all its consequent definite results they called 'Defterocosmos.' 'Third-order-Suns,' i.e., those we now call 'planets,' they called 'Tritocosmos'."

"The smallest 'relatively independent formation' on the planets, which arose thanks to the new inherency of the fifth Stopinder of the sacred Heptaparaparshinokh and which is the very smallest similarity to the Whole, was called 'Microcosmos,' and, finally, those formations of the 'Microcosmos' and which also became concentrated on the planets, this time thanks to the second-order cosmic law called 'mutual attraction of the similar,' were named 'Tetartocosmoses'." (humans)

"And we are such Keschapmartnian beings because the last fundamental Stopinder of the sacred Heptaparaparshinokh, which at the present time almost all the beings of the Megalocosmos call the sacred Ashagiprotoehary,' is not in the centers of those planets upon which we arise (as it occurs in general in the majority of the planets of our great Megalocosmos) but is in the centers of their satellites, which for our planet Karatas is the little planet of our solar system which we call 'Prnokhpaioch', and for the planet Earth, its former fragments now called the Moon and Anulios."

"it is necessary for you to bear in mind, for a clear representation of these processes, that in our Megalocosmos (from results which have already flowed from every kind of Trogoautoegocratic process) there are many hundreds of independent 'active elements' with various specific subjective properties which take part in new formations. These many hundreds of 'active elements' with various properties wherever they might be, proceeding from the seven Stopinders of the fundamental common-cosmic Sacred Heptaparaparshinokh, are, depending upon the Stopinder from which they received their primordial arising, divided and localized, according to what is called the 'affinity of vibrations,' into seven what are called 'Okhtapanatsakhnian classes.' And all without exception of both the large and small already definite concentrations in all our Megalocosmos are formed from these active elements belonging to seven independent classes, and, as I have already told you, they have their own subjective properties."

"And these subjective properties of theirs and likewise their what are called 'proportions of vivifyingness' are actualized

1. firstly, according to what form of functioning of the fifth Stopinder of the Sacred Heptaparaparshinokh was flowing during their arising, and
2. secondly, whether the given active elements arise owing to the conscious intention on the part of some independent individual, or
3. whether they arose automatically, merely owing to the second-grade law called 'Attraction-and-Fusion-of-Similarities'.

Unfortunately, 'being-piandjoehari' is misplaced as this corresponds to 'sol' whereas the meaning intended is the 'missing Stopinder' which I have named as 'Hormoneary' (See 'The Key to Soul Is Within The Missing Stopinder in the Food Octave'. Therefore 'sol' needs to transform into 'Hormoneary'.

As to 'mi' or 'si'('ti') this relates to the 'gaps' which is impossible to prove in a vacuum because in a vacuum there is no sound!

"It is necessary at this point in connection with the actualization of the fifth Stopinder of the sacred Heptaparaparshinokh to trace a parallel between two processes which externally have nothing in common with one another, namely: in the same manner as the first being-food cannot acquire its vivifying power until after its transformation into being-piandjoehari, in the same manner on this piano the vibrations of a chord do not acquire a corresponding vivifying power until they have been fused with the preceding vibrations produced, starting from the center of gravity of the totality of the vibrations of the note 'sol'."

"This last particularity of the sacred law of Heptaparaparshinokh is absolutely certain in this given case, that is to say on the piano, but uniquely in consequence of the fact that if the vibrations of 'mi' and 'si' are produced in a hermetically sealed room, these vibrations either cease instantaneously or else the notes 'mi' and 'si' by reason of the momentum obtained from the first shock given for their arising undergo involution and immediately cease, that is to say, as soon as the note 'mi' reaches the note 'do' and the note 'si' the lower 'fa.'"

1. First Point of Action (mi-fa) is the 'Automatic Activator' (AA)
2. The Second Point of Action (sol-la) is the 'Intentional Activator' (IA)
3. The Third Point of Action (ti-do) is the 'Alpha Omega' (AO)

1. The Active side is 'mi'-'fa'-'sol' and its 'point of action' (the 'Automatic Activator' (AA)) is between 'mi' and 'fa'. This is automatic because the energies pass through it by some automatic means. For example, 'mi'-'fa'-'sol' passes from 'mi' to 'fa' by means of Air from the outside taken in by the lungs.
2. The Neutralizing side is 'sol'-'la'-'ti' and the 'point of action' (the 'Intentional Activator' (IA)) is between 'sol' and 'la'. It is intentional because it only passes through 'sol' to 'la' with intention. For example, the ''sol'-'la'-'ti' passes from 'sol to 'la' by means of 'conscious sensing of photons'.
3. The Passive side is 'ti'-'do-'re' and its 'point of action' is (Alpha/Omega (AO). It is the starting 'do' and the completed 'Do, an octave higher. The interval 'ti'-'do', being the completion of the octave 'do','re','mi','fa','sol','la', 'ti', 'do' has a greater function than mere completion. Ti'-'do' is like a 'converging point' as in mathematics and thus should be considered as part of the 'higher Do'. Its action is best described by the Christian symbol from the Book of Revelation: 'AO' ('Alpha/Omega' with its connotation of the first shall be last and the last shall be first. This is also similar to the Ouroboros, the snake that bites its tail.
   
   'do/Do' 'incepting/converging AO'