This post is part of a series about the octave as a way of understanding the world.
The previous post was about The Octave and the Tree of Life
Saros Philosophy presents a different approach to the octave or Law of Seven, showing how its structure can arise from very simple laws. The full process is described in The Book of Jubilee  and is outlined below:
The root of the octave is in the law of three. In any interaction, three forces act: an impulse, a reaction, and a result. Here the three are labelled Affirming, Denying and Unifying (A, D and U).
The three forces can act in different orders, giving six combinations called constructors: ADU, AUD, DUA, DAU, UAD, UDA. The constructors can be represented as shown below.
The six constructors can link together, one following on from another in a tail-eating sequence. For example UAD ends with the denying force D, and so it could lead on to a constructor which begins with D, such as DUA. This would make the pair UAD-DUA, or written more simply, UADUA, the D in the middle being both the last part of UAD and the first part of DUA.
The six can combine in different patterns, but if we require that each of the six should appear once and only once, there are only three possibilities. One of these is the following:
This sequence of six constructors repeats again and again, because the last one leads back to the first one. One representation of this sequence is shown below:
Rewriting the sequence without duplicating forces gives a repeating sequence of 12 forces:
U A D A U D A U D U A D –
It is in this sequence of 12 forces that the octave is found. The diagram below shows the sequence U A D A U D A U D U A D drawn out in a circle, starting at 10 o’clock and moving clockwise.
The full tones do-re, re-mi, fa-so, and la-si are mediated by the Affirming force. The sequence of forces in each transition is UAD or DAU.
At the first semi-tone interval or shock point mi-fa, the transition is from U to D with no mediating third force. At the other shock point, si-do, again the transition is unmediated, this time from D to U.
The disharmonised point (Gurdieff’s Harnel Aoot ) so-la, is a full tone, made of three forces, but this time two of the forces are the same – UDU, so the transition seems like UD followed by DU.
The glyph in the middle of the diagram shows the common forces of the notes within the octave. The three Denying notes (si, re, fa) and the four Unifying ones (la, do, mi, sol).
The following diagram shows the same structure related to an octave on a piano keyboard. The seven white notes are the notes of the octave, all U and D forces. The 5 black notes are the semitones – four Affirming forces, and one Denying.
I think that the key contribution of Saros Philosophy to the theory of the octave is showing how a very simple method of linking three elements can lead to the structure of the octave with its patterns of tones and semitones. The method of producing the octave structure is entirely independent of the mathematical ratios and musical interpretation used by Pythagoras, and in addition to the two semi-tone intervals, it also highlights the so-la disharmonisation noted by Gurdjieff.
The relations and patterns that lead to the octave can be appreciated and explored in different ways. For example, the four denying forces are equally spaced, whereas the spacing of affirming and denying forces change: as one ‘breathes in’, the other ‘breathes out’. Contemplating these patterns can be worthwhile.
The tail-eating pattern of six constructors used above is not the only possibility. Nor is the octave structure found within it. Other possibilities can be found, and can throw further light on the octave itself.
 The Book of Jubilee, a Primer of Saros Philosophy. https://thebookofjubilee.wordpress.com/
 Chapter 39 of “Beelzebub’s Tales To His Grandson”. An electronic version of Chapter 39 is here.