Twin Peaks or Not?
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Ernest McClain's "holy mountains" are surmounted with either single or twin peaks.
The first brick in any row is just one power of FIVE greater than the row below. For example, the mountain for Limit=60 has 25 or FIVE squared on top whilst D = 60, which has just one power of FIVE, times 12 (= 2^{2} times 3) is on the second row. It is the FORMULA for the Limit, in this case (22 times 3 times 5) =60, which places WHERE on the mountain D(= the Limit) will be: in this case (a) on the second row (because of 5^1) and (b) as the second brick in that row (because of 3^1).
The limit 60 has just one peak, the tone brick for b, value 2^{2} x 5^{2} =50, demonstrating a minor third of 5/6 to D=Limit.
This also shows the major third between b and G = 40 (2^{2} times 5^{1}).
The principle of raising D off the base of the mountain (due to FIVE) is opening up new types of tone (lower case)
but the lighter numbered bricks shown are not symmetrical with regard to D
symmetry being revealed by superimposing the brick pile upsidedown relative to D, leaving symmetries darkened.
But where will the top of a mountain be relative to this D brick? Does one simply have to calculate the whole mountain or can one know where the top will be just from the Limit's formula? The answer is yes.
The top of any mountain is reached at the highest power of FIVE which will "fit" within the Limit for D. In the case of a limit = 60, the largest power that will fit is 5^{2} = 25, rather than 5^{3} = 125 because the 2^{2} x 3 = 12 which can only be divided by one FIVE in addition to the FIVE already in the limit. This limiting power of 5^{2}(inherent in the 5 < 12 in the Limit = 60) needs to be fitted into the octave 30:60 and so it needs doubling to 50, the top brick. There cannot be a second brick on that row because any successor to the first brick MUST have 3^{1} in its formula and 3 times 25 is 75, too great for the Limit of 60.
So let's double 60 to a limit of 120:
Suddenly, there is a second brick whose value is 75. The first brick has been able to fit within 60:120 by a further power of 2^{2} = 4 times 25 instead of 50 in the limit 30:60. In the previous L=60 mountain, b and f created a near horizon in the tone circle, whilst in L=120 there is a hexagon of bricks surrounding the limiting D brick and two new symmetrical tones 25/24 different from the horizon. In the Bible, Moses was supposed to have died at 120, and here may be seen overlooking the promised land which, in this harmonic view, is Just tuning which harmonises the world of FIVE and THREE (milk and honey), so as to correct for harmonic errors and expand the tonal possibilities for ancient music making. Bible scholar Duane Christiansen called the single brick b (of L=60) the "Jonah bird" in which 25 looks forward to its twin at 120, a development that will complement Row 3 with Row 1 and shift the Pythagorean focus to row 2, the row of D=120 with two fifths of 80 and 90, paralleling the Platonic 6:8::9:12.
Those limits really useful in developing the system of Just tuning only have symmetrical tones in the first three rows of the mountains. As the limits are increased by doubling or trebling, the top of their mountains will rise up, but these can never be symmetrical tones. Therefore, using Harmonic Explorer and multiplying by TWO, one arrives at 240.
The limit 240 allows a new row 4 based upon 5^{3}=125 because the difference between that top row and the row of the D=Limit is 25, which fits within 48 = 2^{4} x 3^{1}. However, only 125 will fit the octave and so it is not doubled (to 250). Two new symmetrical bricks have appeared, turning the hexagon of 120 into the lozenge with c# and eb. eb is very important, often called the corner stone and consisting only of 2^{n}, and the lozenge type of shape shows that the Limit 240 has exhausted the potential of 3 times 5 = 15 as the basis for a limit. Additional doublings will create no more symmetrical tones. The limit would need more FIVEs or THREEs to generate a richer symmetrical tone field
But doubling again to 480, now fits 250 into the octave (as a first brick) and can then multiply 125 by THREE to make another brick on that row, n=375, again forming a Twin Peak as was the case with L=120, but now it can only fit 2 x 125 rather than 4 x 25, because the powers of FIVE are growing so quickly, and overtake the increase due to doubling. In the special case of D being at the top of the mountain, there will be no symmetrical tones around D but there will be a likelyhood of a twin peak because the top row has the same powers of FIVE as D.
This can be illustrated by setting the limit to 4 times 125 = 500:
The last limit, 480, can be seen below, as involved in this octave range of 250:500, which has no THREEs, just TWOs and FIVEs. Because four times 125 is the limit then the brick 3 x 125 = 375 can exist. 2 x 3 = SIX times 125 is impossible because as soon as the limit reaches FIVE times 125 => 625 (i.e. before SIX can be reached) a new row will be created with leading brick = 625 = 5^{4}.
It is possible to attempt formalising this situation, which then gives insight to how the rows and their lengths operate within these mountains:
 The number of FIVEs in any Limit = D, establishes its row whilst its position is defined by its powers of THREE.
 The Limit's row exhausts the FIVES of the limit, leaving the remaining SUBPRODUCT of TWOs and THREEs (in that Limits composition) to form any rows above.
 Bricks on rows above can be filled by powers of FIVE not in the limit providing they fit within the said subproduct's magnitude (as is generally true within rows).
 The number of bricks on a row result from a similar filling of that subproduct of TWOs and THREEs, first by powers of THREE, each increased by doubling so as to sit within the Limit's octave range.
This defines the form of the mountain with
 a continuous left hand ascent, where powers of FIVE increase but the powers of THREE are zero and can suffer no further leftwards development without resulting in a fractional result, divided by THREE.
 a jagged righthand descent, where only the powers of TWO in the limit can allow further powers of THREE to initiate new bricks using 3/2, which often needs to be brought down into the octave by a further division by TWO.
 a flat base below which the powers of FIVE would be negative as 1/5, again causing a fractional result.
This reveals the mountain to be an island of rationality in a sea of fractional ratios, held above that sea by the magnitude and composition of the local divinity, D=Limit. The island itself exists because of the rules of successively tuning the powers around D = Limit by symmetrical fifths horizontally and symmetrical thirds diagonally upwards from left to right. In the South Pacific, the island metaphor would probably be king.
Lattice for Musical Intervals
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Musical Intervals generate a unique world of phenomena. Here I saw the polarity between successive and simultaneous to form a Dyad for the World. These two develop into the more familiar modality of, on the one hand scales and melody and on the other, Tonality or key defined by the often shifting tonic. The third factor, the first balanced manifestation of the monad, Musical Intervals, is the Invariance of that world which can only be discovered through experience. This is the subject of Ernest McClain's famous book, The Myth of Invariance.
Moving to the row of four, the Tetrad, its ground (the world of Fact being to the left) is musical instrumentality. It is (invariance which blends with modality) to produce a performance in time and (invariance blending with tonality) which produces composition. Tonality itself, along the right hand extreme representing Value, becomes Harmony. This means that the motivational dyad for this tetrad is instruments <=> harmony whilst the operational dyad is composition <=> performance.
In the Pentad, the world of the musician is the quintessance and the arising of a selfmaintaining ipseity. The food is the audience whilst the 'god' who consumes is the tradition, these two forming the outer limits of the musician's world. The inner limits define the range over which that essence class can move so that, at the least, a musician is a player whilst at the most they are a composer.
At this point one faces further challenges in the systematic progression. A printable version to go further is provided, to fill in the terms, and the resources panel gives access to Systematics online and to books. There is an empty lattice to print: SystematicLatticeSEVENBLANK.jpg.
Modalities of the Lattice Progression
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GESTURES OF WILL IN MANIFESTATION
In a previous article I used the systematic lattice to explore the wholeness of ancient civilisations relative to their concerns such as astronomy and musicology, a fairly immiscible pair of terms. Substances are said to be immiscible if in some proportion, they do not form a chemical solution and J.G. Bennett proposes that in the Dyad, or twoterm systematic, the two terms are incommensurate; that is they cannot divide into each other but instead create a static field as is found with force fields that are dyadic between poles but within which complex behaviours can be acted out.
PLEASE READ: Anthony Blake's essay Language of Will at the DUVersity.org website, if you have not already. His lattice progresses horizontal numbers of terms of the same systemic attribute from the Monad of ONE amd increasing in number according to what is then a Triangular Number in its sum, as below.
SO: The Systematics Lattice is a triangular array in which each row grows by one term from a single term, this representing an authentic whole. To comprehend the lattice or to engage in constructing one, it is important to grasp the component processes which are, in the language of will, gestures or moves used in a “game” of understanding.

In the beginning there is nothing, that is no recognition of wholeness. This is our own transcendence, in that we are independent of any need to understand that wholeness. In our gallery of gestures this is an empty frame, perhaps filled with noise.

When a wholeness is identified then it has the property of the number one, the Monad, in which there are many things and relations that somehow belong to this greater whole. The whole can be identified as Unitive, in the sense that the whole holds together many relations and terms but there is only an implicit order, not yet understood. Its symbol is the circle forming a boundary for all that the monad contains.

To enter into understanding, there has to be a fundamental cut into two parts and these parts must represent a strong duality or dyad operating on the elements within the whole. For example in cosmic structures, the dark and light matter might be a good cut. This action of division, similar to cell division for the whole modadic cell, is a creative act because it will define everything that emerges in the higher order terms later in any progression. Its symbol is the circle with a line across its diameter.

The division of one into two is the gesture of divergence from a single term into two terms, repeated throughout the lattice. There is a left hand and a right hand path, each leading to a new term related to the preceeding term. Thus, the creative act creates two out of one and brings into existence a relationship (or connective) between two terms – and this is repeated in all of the new relationships made on the horizontal level between terms and this new world of relationship between terms will have this first relationship as its source just as every term in the lattice will have the initial monadic term as their source. This advent of relationships between terms represents consciousness which requires difference or, possibly, is created by difference as a cosmic energy. Its symbol is an upward facing equilateral triangle connecting each of three terms at each apex, the monad being above.

The transition, in level, from two terms to three terms introduces a third term that is a blending of the two dyadic terms above. The archetypal gesture of blending is what naturally occurs everywhere except on the boundary edges of the triangular lattice. The symbol of blending is a downward facing equilateral triangle with two terms above and one blended term below.

Where terms are on the outside edge, left or right of the lattice, they mutate into a new term relevant to the systemic attribute of the level they are entering. The left side by convention describes the development of the quantitative aspect of the whole and the right hand side the qualitative or intensive side of the whole.

Some blended terms appear in the centre of the lattice, when a layer is odd in number, in which case a line of concreteness or incarnation, within the whole, is being recognised. This appears as an achievement of the whole within its manifestation whilst it is actually an achievement within our understanding of the structuring of will within the whole.

Each layer, containing an additional term, comes under the framework of the systemic attribute for that number of terms. JG Bennett identified these within his systematics.

The layers have no two dimensional structure but their connectives (i.e. relationships between each pair of terms), have the property of being generated explicitly between the terms on that level, or of already existing as the relationships directly above nonadjacent terms, in the lattice. That is, those relationships between terms found in the two dimensional graphic for a given system are to be found in all of the relationships (horizontal lines) between the terms on that level and above. Those relationships above the current level will may need to be adjusted (mutated) within the two dimensional form.
Bennett's opus The Dramatic Universe introduced his Systematics plus a set of philosophical categories that interact because Systematics was his evolved technique for Understanding, itself a category of Will quite different to Knowing. Bennett uses the dyad FACT and VALUE as quite fundamental to the world of consciousness and a triad FUNCTION BEING and WILL that were different to Gurdjieff's AFFIRMING, DENYING and RECONCILING. These can be shown as:
Note how the relationships beg to become clearer between terms whilst staying within the frame of the wholeness itself. This is the holistic property of this technique which I hope to show in later papers connects us to Phenomenology in which the parts reveal the whole and the whole reveals the parts.
This Lattice can be extended as a basis for understanding wholeness without particular thought for Bennett's originational terms, as in:
Obviously there are steps here in transforming Function into Properties, Being into Systems and Will into Media. These are worthwhile meditations that can reveal much about the world. In the Tetrad, Chaos and Order provide a strong dyad that can be mediated by instrumentality and direction. There is much about traditional Systematics at www.systematics.org and a blank template for printing out is available here for download.