Whether or not we were once "merely" highly intelligent but automatically reacting animals - that communicated by talking - and that talking enabled us to cooperate well enough to build complex societies up to and including the AMP (African Method of Production) which is the basic organizing framework for modern civilization - we can leave aside for now. We have each of us witnessed ants, termites, bees, and wasps doing it, (complex social organization) and we do not image-in that on an individual basis these creatures are anything but "automatically reacting animals". (Not very much "questioning" going on in the hive, and consequently, not much possibility of psychological development)

There is no compelling reason to suppose that our collective anthropocosmos and the attenuated cosmos of these haplo-diploidy insects share any other exoteric organizing features in common - aside from our shared lunatical, intra-specific penchant for murder on a grand scale. So I'll leave off the speculative psycho-history and the hive-mind related epistemology of "Sarmoung" and "Beelzebub" (though the esoteric importance of this for "school" and "group" developmental processes may not be at all trivial given that we are Beelzebub's children's children's children) to refocus on the issue of conscious language.

Language is central to everything we do; therefore, a deep understanding of language is prerequisite to a deeper understanding of our "selves". It is language that has the power to make metaphors and analogies including "I" and "me". Since we know that natural language (along with a panoply of complex behaviours) is epiphenomenal to our genetically determined brain structure - and that language use itself is an automatic behavior - e.g., I am not now consciously selecting the words (still less the letters) that my individual keystrokes engender (or consciously governing the motor behavior behind the production of the individual keystrokes for that matter) nor will you be consciously reconstructing the phonemes and morphemes of which these written utterences are comprised. All very automatically and unconsciously, your organism will serve up the "meaning" of this writing (these symbols for phonetic events)

What if, instead of a metaphorical "I" embedded in the lexical field of subjective speech, (our nothingness) we instead had a system based on the direct apprehension of a material something; (that indefinable something that self-remembering helps us to sense and develop)

Could it be that by radically changing the lexical field upon which the automatic processes of language formation (illusory consciousness - sleep) occur, that the goal of metanoia could - at least in part - be obtained? Isn't this really what the "psychological" focus of the work is intended to accomplish?

In Beelzebub, Arousing of Thought Gurdjieff opines;

In his "law of three: and "the law of seven" (and by extension and application the table of the hydrogens) Gurdjieff presented the lexical field on which pharaonic Egypt (Khemit) based its "vocabulary" of objective construction. For purposes of this discussion, I am freely adapting the bewildering constructions of Egypt as overt expressions of the underlying mentality that produced them. (a la Schwaller de Lubicz) Kind of an inversion of Chomsky's premise that "language is culture and culture is language". At its heart, the core epistimology (lexical field) of Pharaonic mentality was based on a simple numerical progression beginning with the number 1 and elements that are all natural and real. In other words the direct response to the proportional laws of sound and form was the epistmological basis of the entire pharaonic cultural ouevre. Hearing goes directly to the emotional centre whose intelligence directly apprehends (and organizes) reality via harmony. It was by means of this mentality, this alien epistemology (lexical field) that the Egyptians constructed their cultural and technological complex (language-culture-technology)

It is quite simple: The inverse of every harmonic progression is an arithmetic progression, thus, 2,3,4,5 is an ascending arithmetic progression while the inverse series, 1/2, 1/3,1/4,1/5 is a descending harmonic progression. In music it is the insertion of the harmonic and arithmetic means between the two extremes in double ratios - representing the octave double - which gives the progression known as musical proportion, that is, 1,4/3,3/2,2. The arithmetic and harmonic means between the double geometric ratios are the numerical ratios which correspond to the tonal intervals of the perfect fourth and the perfect fifth, the basic consonances in nearly all musical scales.

Translation - The basic proportional structure which contains the axioms for our primary mathematical operations is also the basic proportional structure for the laws of music. Arithmetic proportion contains the law for addition and its inverse, subtaction, and describes the relationship which gives the natural series of cardinal numbers, 1,2,3,4,5,6,...etc. Geometric proportion contains the law for multiplication and its inversion, division, and describes the relationship which gives any series of geometric progressions. Addition and multiplication are mathematical symbols for patterns of growth. The harmonic mean is derived from a combination of the first two; it is formed by a multiplication of any two extremes (a,c) followed by the division of this product by their average or arithmetic mean (a+c)/2. For example, given two extremes, 6 and 12, the product of 6 and 12 = 72, the arithmetic mean between 6 and 12 is 9, and 72/9 = 8, so 6,8,12 is an harmonic proportion.

Each proportion has a number of characteristics that are peculiar to it. The arithmetic proportion shows an equality of difference, but an inequality of ratio, thus in the arithmetic proportion; 3,5,7 7-5 = 5-3 but 7/5 does not equal 5/3. The geometric proportion is the reverse of this - thus in the geometric proportion 2,4,8 4/2 = 8/4 but 4-2 = 2 does not equal 8-4 = 4.

The importance of the harmonic proportion is the fact that the inverse of every harmonic progression is an arithmetic progression. The progression 1,4/3,3/2,2 represents the frequencies of a fundamental, fourth, fifth, and octave. We then find the arithmetic and harmonic proportions between the string lengths 1 and 1/2 representing the division of the vibrating string in half which produces the octave increase in frequency. This gives the progression 1, 3/4,2/3,1/2 because the harmonic mean between 1 and 1/2 = 2/3, the musical fifth, and the arithmetic mean between 1 and 1/2 = 3/4, the musical fourth. In comparing these two progressions, we see an inversion of rations and a crossing of functional positions between the arithmetic and harmonic mean.

Musical harmony though directly accessible to the senses is esoteric because it develops out of a simultaneous inversion containing a simultaneity of addition and multiplication. (patterns of growth) The octave of a fundamental is achieved by the addition of the intervals: in string lengths the fifth plus the fourth equals the octave and the multiplication of the vibrational frequencies of the fourth and the fifth equals the octave (4/3 x 3/2 = 2). The combined effect of addition and multiplication produces the logarithm in mathematics. The Golden Proportion is the archetype for this form of growth.

Numbers considered as frequency ratios in a rising scale are equal to the string lengths for the descending scale. The law of musical harmony, when viewed from the idea of mediating proportion becomes the lexical field of the cosmos where simultaneous oppositional movements interact to create both sound and form. This numerical and harmonic principle can be represented geometrically. (Al-khemia)

The geometric mean corresponds to the formula b x b = a x c;

The harmonic mean corresponds to the formula b(a + c) = 2ac; i.e., the product of the sum of the extremes, multiplied by the mean is equal to two times the product of the extremes, or b = 2ac/(a+c). The geometric proportion is called the perfect proportion because it is a direct proportional relationship, an equality of proportion bound by one mean term. The arithmetic and harmonic medians work out this perfection through an interchange of differences in a play of alternation and inversion.

We can verify this in number progressions by examining a simple triangular array of numbers which crosses the geometric progression by 2 (horizontal) with the progression by 3 (diagonal). All the successive vertical numbers are to each other in the ratio of 2:3 which is the same as multiplying one term by 3/2 in order to obtain the term below. This successive multiplication by 3/2, the musical fifth is the method used for generating

the musical scale.

Thus:

1 2 4 8 16 32 64

3 6 12 24 48 96

9 18 36 72 144

27 54 108 216

81 162 324

243 486

729

In examining the table we can see that each square of four numbers, for instance 2, 4, 6, 3 contains within it two arithmetic progressions (2,3,4) and (2,4,6) giving us three sides forming the top of a square and one diagonal. We see in the same figure the harmonic progressions 2,3,6 and 3,4,6 giving three sides of a square, two of them overlapping with the first proportion, the other giving the fourth side of the square and the other diagonal.

Arithmetic

2-------->4

| / |

| / |

v / v

3/ 6

Harmonic

2 4

| / |

| / |

v / v

3/---------6

In Views From the Real World New York, February 20, 1924 Gurdjieff made the following remarks;

There is no compelling reason to suppose that our collective anthropocosmos and the attenuated cosmos of these haplo-diploidy insects share any other exoteric organizing features in common - aside from our shared lunatical, intra-specific penchant for murder on a grand scale. So I'll leave off the speculative psycho-history and the hive-mind related epistemology of "Sarmoung" and "Beelzebub" (though the esoteric importance of this for "school" and "group" developmental processes may not be at all trivial given that we are Beelzebub's children's children's children) to refocus on the issue of conscious language.

Language is central to everything we do; therefore, a deep understanding of language is prerequisite to a deeper understanding of our "selves". It is language that has the power to make metaphors and analogies including "I" and "me". Since we know that natural language (along with a panoply of complex behaviours) is epiphenomenal to our genetically determined brain structure - and that language use itself is an automatic behavior - e.g., I am not now consciously selecting the words (still less the letters) that my individual keystrokes engender (or consciously governing the motor behavior behind the production of the individual keystrokes for that matter) nor will you be consciously reconstructing the phonemes and morphemes of which these written utterences are comprised. All very automatically and unconsciously, your organism will serve up the "meaning" of this writing (these symbols for phonetic events)

What if, instead of a metaphorical "I" embedded in the lexical field of subjective speech, (our nothingness) we instead had a system based on the direct apprehension of a material something; (that indefinable something that self-remembering helps us to sense and develop)

Could it be that by radically changing the lexical field upon which the automatic processes of language formation (illusory consciousness - sleep) occur, that the goal of metanoia could - at least in part - be obtained? Isn't this really what the "psychological" focus of the work is intended to accomplish?

In Beelzebub, Arousing of Thought Gurdjieff opines;

"it was customary in long-past centuries on Earth for every man bold enough to aspire to the right to be considered by others and to consider himself a "conscious thinker" to be instructed, while still in the early years of his responsible existence, that man has two kinds of mentation: one kind, mentation by thought, expressed by words always possessing a relative meaning; and another kind, proper to all animals as well as to man, which I would call "mentation by form."Instead of viewing the legacy of pharaonic Egypt with an eye to heterodox psychological interpretation that would depend on a degree of philological and classical scholarship that would be confounding to most - let's instead look at something obvious, unmistakeable, and likely as not, - frustrating to many who have struggled with Gurdjieff's system - the law of three, the law of seven, where these came from, and what they are supposed to convey. It is here, I believe, that Gurdjieff sought to plant a specific seed in the mentations of contemporary humanity, a new set of "associations" by which the possibility of "mentation by form" might be reinstated.

In his "law of three: and "the law of seven" (and by extension and application the table of the hydrogens) Gurdjieff presented the lexical field on which pharaonic Egypt (Khemit) based its "vocabulary" of objective construction. For purposes of this discussion, I am freely adapting the bewildering constructions of Egypt as overt expressions of the underlying mentality that produced them. (a la Schwaller de Lubicz) Kind of an inversion of Chomsky's premise that "language is culture and culture is language". At its heart, the core epistimology (lexical field) of Pharaonic mentality was based on a simple numerical progression beginning with the number 1 and elements that are all natural and real. In other words the direct response to the proportional laws of sound and form was the epistmological basis of the entire pharaonic cultural ouevre. Hearing goes directly to the emotional centre whose intelligence directly apprehends (and organizes) reality via harmony. It was by means of this mentality, this alien epistemology (lexical field) that the Egyptians constructed their cultural and technological complex (language-culture-technology)

It is quite simple: The inverse of every harmonic progression is an arithmetic progression, thus, 2,3,4,5 is an ascending arithmetic progression while the inverse series, 1/2, 1/3,1/4,1/5 is a descending harmonic progression. In music it is the insertion of the harmonic and arithmetic means between the two extremes in double ratios - representing the octave double - which gives the progression known as musical proportion, that is, 1,4/3,3/2,2. The arithmetic and harmonic means between the double geometric ratios are the numerical ratios which correspond to the tonal intervals of the perfect fourth and the perfect fifth, the basic consonances in nearly all musical scales.

Translation - The basic proportional structure which contains the axioms for our primary mathematical operations is also the basic proportional structure for the laws of music. Arithmetic proportion contains the law for addition and its inverse, subtaction, and describes the relationship which gives the natural series of cardinal numbers, 1,2,3,4,5,6,...etc. Geometric proportion contains the law for multiplication and its inversion, division, and describes the relationship which gives any series of geometric progressions. Addition and multiplication are mathematical symbols for patterns of growth. The harmonic mean is derived from a combination of the first two; it is formed by a multiplication of any two extremes (a,c) followed by the division of this product by their average or arithmetic mean (a+c)/2. For example, given two extremes, 6 and 12, the product of 6 and 12 = 72, the arithmetic mean between 6 and 12 is 9, and 72/9 = 8, so 6,8,12 is an harmonic proportion.

Each proportion has a number of characteristics that are peculiar to it. The arithmetic proportion shows an equality of difference, but an inequality of ratio, thus in the arithmetic proportion; 3,5,7 7-5 = 5-3 but 7/5 does not equal 5/3. The geometric proportion is the reverse of this - thus in the geometric proportion 2,4,8 4/2 = 8/4 but 4-2 = 2 does not equal 8-4 = 4.

The importance of the harmonic proportion is the fact that the inverse of every harmonic progression is an arithmetic progression. The progression 1,4/3,3/2,2 represents the frequencies of a fundamental, fourth, fifth, and octave. We then find the arithmetic and harmonic proportions between the string lengths 1 and 1/2 representing the division of the vibrating string in half which produces the octave increase in frequency. This gives the progression 1, 3/4,2/3,1/2 because the harmonic mean between 1 and 1/2 = 2/3, the musical fifth, and the arithmetic mean between 1 and 1/2 = 3/4, the musical fourth. In comparing these two progressions, we see an inversion of rations and a crossing of functional positions between the arithmetic and harmonic mean.

Musical harmony though directly accessible to the senses is esoteric because it develops out of a simultaneous inversion containing a simultaneity of addition and multiplication. (patterns of growth) The octave of a fundamental is achieved by the addition of the intervals: in string lengths the fifth plus the fourth equals the octave and the multiplication of the vibrational frequencies of the fourth and the fifth equals the octave (4/3 x 3/2 = 2). The combined effect of addition and multiplication produces the logarithm in mathematics. The Golden Proportion is the archetype for this form of growth.

Numbers considered as frequency ratios in a rising scale are equal to the string lengths for the descending scale. The law of musical harmony, when viewed from the idea of mediating proportion becomes the lexical field of the cosmos where simultaneous oppositional movements interact to create both sound and form. This numerical and harmonic principle can be represented geometrically. (Al-khemia)

The geometric mean corresponds to the formula b x b = a x c;

The harmonic mean corresponds to the formula b(a + c) = 2ac; i.e., the product of the sum of the extremes, multiplied by the mean is equal to two times the product of the extremes, or b = 2ac/(a+c). The geometric proportion is called the perfect proportion because it is a direct proportional relationship, an equality of proportion bound by one mean term. The arithmetic and harmonic medians work out this perfection through an interchange of differences in a play of alternation and inversion.

We can verify this in number progressions by examining a simple triangular array of numbers which crosses the geometric progression by 2 (horizontal) with the progression by 3 (diagonal). All the successive vertical numbers are to each other in the ratio of 2:3 which is the same as multiplying one term by 3/2 in order to obtain the term below. This successive multiplication by 3/2, the musical fifth is the method used for generating

the musical scale.

Thus:

1 2 4 8 16 32 64

3 6 12 24 48 96

9 18 36 72 144

27 54 108 216

81 162 324

243 486

729

In examining the table we can see that each square of four numbers, for instance 2, 4, 6, 3 contains within it two arithmetic progressions (2,3,4) and (2,4,6) giving us three sides forming the top of a square and one diagonal. We see in the same figure the harmonic progressions 2,3,6 and 3,4,6 giving three sides of a square, two of them overlapping with the first proportion, the other giving the fourth side of the square and the other diagonal.

Arithmetic

2-------->4

| / |

| / |

v / v

3/ 6

Harmonic

2 4

| / |

| / |

v / v

3/---------6

In Views From the Real World New York, February 20, 1924 Gurdjieff made the following remarks;

Last time we spoke a little about the Law of Three. I said that this law is everywhere and in everything. It is also found in conversation. For instance, if people talk, one person affirms, another denies. If they don't argue, nothing comes of those affirmations and negations. If they argue, a new result is produced, that is, a new conception unlike that of the man who affirmed or that of the one who denied.

This too is a law, for one cannot altogether say that your former conversations never brought any results. There has been a result, but this result has not been for you but for something or someone outside you.

But now we speak of results in us, or of those we wish to have in us. So, instead of this law acting through us, outside us, we wish to bring it within ourselves, for ourselves. And in order to achieve this we have merely to change the field of action of this law.

What you have done so far when you affirmed, denied and argued with others, I want you now to do with yourselves, so that the results you get may not be objective, as they have been so far, but subjective."

## 14 comments:

Now, after a computer analysis of three decades of hit songs, Dr. DeWall and other psychologists report finding what they were looking for: a statistically significant trend toward narcissism and hostility in popular music. As they hypothesized, the words “I” and “me” appear more frequently along with anger-related words, while there’s been a corresponding decline in “we” and “us” and the expression of positive emotions.

A New Generation's Vanity

does this make your generation and the one under you more conscious?

In all ancient society the #3 and #7 were seen as powerful spiritual numbers for the 1 or the I could not stand without others factors.

Seven is a magical number, as it combines the cosmic numbers of man (three) and woman (four) in the Bamana creation Story.Amerinias we now know was an area of intersect between the humans coming our of Africa, and then the travel East to West - Afroeurasain and Eurasain. It is one reason Turkey made them dark and attacked them.

Tenet 7:

True insight does not issue from specialized knowledge, from membership in coteries, from doctrines or dogma. It comes from the preconscious intuitions of one's whole being, form one's own code.

This I somehow don't see. Can any musician (able to add) help me out...

"in string lengths the fifth plus the fourth equals the octave"

Tomas look here http://books.google.com/books?id=SWPASkffNlUC&pg=PA84&lpg=PA84&dq=%22in+string+lengths+the+fifth+plus+the+fourth+equals+the+octave%22&source=bl&ots=VGTgjFp9Hb&sig=RHAIY6DuvdtH5kOghbHADDoak70&hl=en&ei=jfa2TfLBDJKC0QHA29HxDw&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CBUQ6AEwAA#v=onepage&q=%22in%20string%20lengths%20the%20fifth%20plus%20the%20fourth%20equals%20the%20octave%22&f=false

Tomas I'm not a musician, but the string length argument works if you invert the fourth:

2/3 + 4/3 = 6/3.

So a fifth up but a fourth down.

For anybody who isn't obsessed with this stuff: An octave is a factor of 2 in frequency, a fifth (a shift by seven keys on a piano, including black keys) is a factor of 3/2 in frequency, and a fourth (a shift of 5 piano keys) is a factor of 4/3 in frequency.

String lengths vary by the reciprocal of frequency, so an octave down makes the string twice as long; an octave down makes it twice as long. A fifth up means a string 2/3 as long. A fourth down would be a string 4/3 as long.

The sum of string lengths 2/3 and 4/3 to give 2 is what I gave kind of telegraphically up above: the string lengths corresponding to the notes a fifth up and a fourth down, added together, give the string length for the note an octave down.

gah! "an octave down makes the string twice as long; an octave

upmakes the stringhalfas long." Sorry, I know how frustrating typos are when you're trying to follow some other guy's math.hmm. Why would you go fifth up and fourth down. I am not musician at all, so I will stick with simple numbers.

Its probably silly, but what I am wondering about is the word "plus" concerning the string lengths.

frequency: 1, 4/3, 3/2, 2 -> I am octave up.

string length: 1, 3/4, 2/3, 1/2 -> I am octave up.

frequecy: multiplication, 4/3*3/2 = 2 ->ok

string length: addition, 3/4 + 2/3 != 1/2 -> nok; I can multiply them 3/4*2/3 = 1/2; I can add the logarithms log(3/4) + log(2/3), but the same I can do with frequency, no?

so, why is it said "plus" regarding the string lengths? As I said, probably just silly....

Yes there is something to musicans playing by tones and space, unconsciously. Played Conga and African drums for years 1's 3/4 4/4 6/4 6/8 - Mongo never knew how to write music and his grandfather was a slave in Cuba. And it was three drums that gave us bass, second and top, creating a rocking flow under the voice of a lead singer and chorus, before Rock and Roll. Thanks Ken for the article - preconscious zombies the terms Westerners use is amazing.

btw I forgot - The 5th scale around the C E G was written about by the great Paul Robeson as the universal folks music scales. He was quite a conscious person, yet; my nation has not demanded his story on screen

Tomas,

Why lengths rather than frequencies? I guess because string lengths are much older than frequencies. The Egyptians, Pythagoreans, &c, couldn't measure frequencies of musical notes. (You can do pretty much the same numerology either way, but the string length observations have been around much longer.)

"Plus" means they're literally just stretching out strings end-to-end and measuring the total length. Why? As far as I can see, just to point out some pretty numerology. Even thousands of years later, when we see pretty numerology (atomic ionization energies in the 20th century, chemical compound compositions in the 19th), we tend to find something interesting behind it.

If I add the string lengths a fourth up and a fifth up, as you say, I get 9/12 + 8/12 = 17/12. Doesn't look pretty.

But if I add the string lengths a fourth

downand a fifth up, I get 4/3 + 2/3 = 6/3 = 2. Looks pretty.I can make it pretty in frequency too. I go a fourth

upand a fifth down, 4/3 + 2/3 = 6/3, and the sum of frequencies is an octave up. Sum of frequencies is even meaningful acoustically: if you play tones at 4/3 of f and 2/3 of f together, there is a beat tones at 4/3 + 2/3 = 6/3 (and at 4/3 - 2/3 = 2/3 too). So in that sense a fourth and an inverted fifth sounded together really do make an octave.Still, though, I think the answer to "why a fifth up but a fourth down" is: just to make the numerology pretty. (And there's some excuse for playing fast and loose with inversion: in music inversions and octave shifts sound kind of compatible with the originals.)

Tomas,

Maybe you're asking "Why add string lengths at all? What does that prove? What does it mean physically?" As far as I can think of offhand, nothing much. Still, simple ratios tend to mean something. There is the beat frequency thing, which the Pythagoreans, and the Egyptians they stole these ideas from, and whomever the Egyptians stole them from, couldn't nail down without oscilloscopes.

Or, frankly, you can make workable mechanical oscilloscopes, and even electical ones might not look like much after being buried for 10,000 years, so who knows? Maybe the proto-Namibians had a whole recording industry, Departments of Acoustics at their universities, the whole nine yards. And they, well they lost funding as it were, but some kind of folk wisdom about beat frequencies survived. Who knows?

nice

yes and thats why people with absolute hearing hears a diferent tone for the same note thus C+ is not the same as D- on the pianoscale it is the same place to play. And D- is 4 down and C+ is 5 up .

And so music reveals the relation between harmonie and processes. In the magical triangle the first line is a progression by 2 thus 1.2.4.8.16(1+6=7).5(32 3+2=5) again the sequence you've found 1.2.4.8.7.5.1 as shown in the vortex and the torus. Dawnwords we have the 3 bud all mumbers above 9 becomes 9 (18) (27)

Karel, welcome to the spot. There are further morsels along these lines http://subrealism.blogspot.com/2008/02/zairajah.html

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