Readers will note that the first quality of the first multiplication in each triad corresponds to the man number in Gurdjieff's system; thus, man number one corresponds to material (body), number two, emotion (desire), number three, intellect (power), number four, real "I" (agency), number five, higher emotional man (purification) and six, man with a developed higher intellectual connection (wisdom.) Each level defines an inner order which (in the case of the first triad) exists and can be worked with, and (in the case of the second) must be completed before the next stage can be reached.
Man number seven represents completion of the path, and lies outside the realm of the multiplications.
The chart indicates the character, properties and challenges facing man at each stage on the path of inner development. As Gurdjieff explained, the diagram is constantly dynamic, with the same elements (notes) constantly changing their relationships to one another in a harmonic interaction of inner principles. The scale of inner development reflects this.
One can furthermore predict the weaknesses and strengths of man according to his level of development. Readers familiar with the tarot may notice that this system is related to some of the conventions associated with card order, indicating a common origin at some time in the distant past.
Each triad hence represents a specific step on the path of inner development, with consequent iterations of vibration and defined obstacles. A great deal more could be said about each multiplication, both about the indiviual triads and their relationship, along with the obstacles and intervals they incorporate. In general, on each level of development, the first multiplication represents what is given, or, the initial environment under which the work on the path takes place. The second defines the aims that must be undertaken.
The point is that treating the diagram as a completely impenetrable mystery does it little justice. With careful study, the elements and principles it incorporates can absolutely be known. The ennegram defines an ancient science of Being, whose principles are today almost completely lost; and this explains Gurdjieff's succinct remarks to Ouspensky about the diagram.
To take one intriguing example, the story of Faust represents 428 / 571. Faust fully embodies the first triad in his embrace of wisdom, and his desire for power. Unfortunately, his failure to understand his own personal responsibility for himself (agency), his lack of purification, and his misunderstanding of the material nature of his quest, cause him to lose his soul, because he doesn't know what he's doing. He has arrogantly reached into and even past the level of development represented by the second half of the octave he is in, in a deal with the devil, presuming that the power he gains will be enough to offset what it costs him.
This particular multiplication also has other interesting esoteric meanings.
The iteration of each first triad in a multiplication represents a danger, because within its own context, it represents what one would call the mechanical side of that level of development, in other words, one with an extended interval (the first stopinder) that requires a special kind of conscious labor to pass it. Failure to do so will always cause the initiate to overestimate their ability; and each, in a different way.
The second iteration of each triad represents qualities that must become completely acquired and incorporated into Being, purified, understood, and then surrendered. So each left side triad of the enneagram represents a heroic labor made without any expectation of recompense; in other words, service. The results of that service, furthermore, must be fully surrendered as an offering to God.
Readers may be interested in considering the possibility that Beelzebub's error lay in the multiplication 857 / 142, which is the rough equivalent of the development level of man number six. (Within each octave, at each level, there is an equivalent progression of development from 1-6. Entities such as angels are subject to this law, just like everything else.) Like Faust, he thought he would be able to master the second triad because he overestimated his progress in the first.
Beelzebub had real "I" (agency); He was wise, and he had purified himself. Unfortunately, he had not conquered his egoistic desire to exercise power over material reality (rebellion against the natural order, where God is the only entity with this ultimate power.)
This carefully crafted character intentionally represents the last temptation of Christ by the devil, which gives us some intimations of what level Gurdjieff felt Beelzebub was on.
This final iteration of development represents the ultimate submission to God: material, power, and desire, all practice Islam, that is, they must be surrendered after they are attained.
It's important to stress this consistency of the development of vibrations within octaves at every level. It doesn't matter whether you are a molecule, a man, or an angel: within your own octave, the given material conditions, the necessary shocks, and the obstacles are consistent for each level of development. They don't change over and over. Although different language would be used to describe the conditions within each triad, or multiplication, depending on the level, the overall principles underlying the situation remain absolutely consistent.
Gurdjieff stressed this consistency of principle across multiple levels of the cosmos over and over again in In Search of the Miraculous. Even if one argues that the enneagram somehow means something quite different than what has been proposed in these studies, one cannot escape this particular conclusion.