Applications of Numerical Qualities

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Laws of Three and Seven

The Musical Octave

Notes and Ratios

Chord Structure of the Octave

Circular Octave Relationships

Notes and Digits

Colors

Atomic Elements

Mathetical Principles of Classification

Gurdjieff's book contains a study of the musical octave. He had definite ideas or theories called Laws of Three and Seven. The Law of Three is only a restatement of the triangular conflict of thesis, antithesis, synthesis, or Creator, Preserver, Destroyer, or Isis, Osiris, Horus. The system of numerical complements displays this law.

Gurdjieff's Law of Seven points out that in the musical octave the notes increase in pitch fairly regularly except for the interval between the third and fourth notes, and between the seventh note and the first note of the next octave. A rough idea of this is given by writing the octave to show these irregular recessions.

do re mi fa irregular interval so la ti do' irregular interval

These recession points are important because they explain the nature of irregularities to be found within a harmonious qualitative system. For instance, irregularities in the qualitative structure of the atomic period table occur at these points and only at them.

These laws fit very well into the mathetical cosmology. The Law of Three corresponds to the phenomenon of digital complements. The Law of Seven provides the basis for further qualitative analyses. But what Gurdjieff did not state was that three and seven necessarily imply that there is a four somewhere and that there must be relationships among these triads, tetrads, and heptads and some common source for them.[top]

The simplest form of the octave is the ordinary musical one discovered so long ago by Pythagoras. Sound is caused by something vibrating. In the piano and string instruments such as guitars and violins, the vibration is provided by striking or rubbing the strings. In instruments such as drums or cymbals the vibration is produced by a large area of skin or metal. In wind instruments such as organs, trombones, and fifes, the air within the instrument is set into vibration.[top]

Notes and Ratios

Imagine an instrument like a guitar but with just one string. A hand reaches out and plucks the string. Instantly a musical note determined by various factors such as the material of the string, its tension, and its length is produced. The easiest of these factors for a player to control is the string's length. So, such instruments as violins and guitars are played by changing the length of the vibrating string. This is done by placing a finger on it. The shorter the string is, the higher is the pitch produced.

If a finger is placed upon the midpoint of the string, it makes a note which is called the octave of the note produced by the whole string. The relation of the octave (2:1) is the most basic relationship in music. A note that vibrates twice as fast (or half as fast) is essentially the same as the funda- mental or original note. For instance, if middle C vibrates at 264 cycles per second, then the notes at 528, 1056, 2112, ..., and the notes at 132, 66, 33, ..., cycles per second are also C.

Rather than using the number of cycles or vibrations per second, the ratio of these numbers will be used. The ratio is the simplified fraction that results when the number of vibrations of one note is divided into another. The larger number will be placed on top when written as a ratio like 9/8, or first when written as 9:8. When ratios are written thus they give the relationship between the number of vibrations per second. The relationship between lengths of string needed for the different notes is given by inverting these ratios. To produce a note with 9/8ths and many vibrations per second, a string is needed which is 8/9ths as long as the one which produces the fundamental is required.

Pythagoras showed that the harmonic mean between a fundamental
and its octave gives a very harmonious sound when played with the
fundamental note. Let a and b be any two. Then the harmonic
mean between a and b is given by this formula.

mean = (2 x a x b) / (a + b)

Thus the mean between a fundamental note (=1) and its octave
(=2) is found to be

(2 x 1 x 2) / ( 1 + 2) = 4/3 or 4:3

The arithmetic average produces another harmonious combination
with the fundamental. The arithmetic average between a notes and
its octave is

(1 + 2) / 2 = 3/2 or 3:2

Pythagoras discovered that the 4:3 ratio gave the fourth note of the seven-toned scale and that the 3:2 ratio gave the fifth note. So he now had the following notes of the scale identified.

do 1 or 1:1 re mi fa 4:3 so 3:2 la ti do' 2 or 2:1

From these results he was able to calculate the values of
the remaining notes by using two rules which apply to musical
notes.

a. To add notes, multiply the ratios together.
b. To subtract notes, divide the smaller fraction into the
larger fraction.

Subtracting fa from so gives the ratio of the value from do
to re.

re = so - fa = (3/2) / (4/3) = 9/8 = 9:8

From re to la are five notes, hence re plus the fifth gives la. la = re + fifth = 9/8 x 3/2 = 27/16 = 27:16

Subtracting the fourth interval from la produces mi.

mi = la - fourth = (27/16) / (4/3) = 81/64 = 81:64

And finally, mi plus a fifth interval gives ti, the seventh note.

ti = mi + fifth = (81/64) x (3/2) = 243/128 = 243:128

So Pythagoras defined all the notes in the seven-toned scale.

A more modern scale was developed by Zarlino, an Italian,
in about 1560. The great numerical simplicity of Zarlino's
scale carries with it more pleasing harmonies due to some principles
of physics and mathematics. The simpler ratios reflect a more
simple Fourier harmonic analysis of the composite. But the
differences between the two scales, where they exist, are very
small. Zarlino noticed that 3/2 was also the arithmetic average
between 9/8 and 15/8, and so he called ti 15/8 and calculated
the other values from this. Pythagoras was the one who first
discovered these ratios. Zarlino simplified three of them
into a more pleasing form. Both scales are listed here for
quick reference.

note Zarlino Pythagoras do 1:1 1:1 re 9:8 9:8 mi 5:4 81:64 fa 4:3 4:3 so 3:2 3:2 la 5:3 27:16 ti 15:8 243:128 do' 2:1 2:1

The tetraktys (1 + 2 + 3 + 4 = 10) provides the ratios for the octave (2:1), the fourth (4:3), and the fifth (3:2), all the uilding blocks for the octave. It must be noted, however, that this is not the scale used by musicians. The concert scale is based on a ratio between successive notes of the scale (both white and black keys on a piano) of the twelfth root of two. This enables all the musicians to sound well together with only slight disonnances.[top]

Chord Structure of the Octave

With Zarlino's scale there are three different, though similar, chords in the octave. These chords are combinations of three notes that sound especially well together. The first chord is the tonic chord and consists of do-mi-so. The second is the subdominant chord of fa-la-do'. Third is the dominant so-ti-re'. The corresponding ratios are listed

tonic subdominant dominant do 1/1 fa 4/3 so 3/2 mi 5/4 la 5/3 ti 15/8 so 3/2 do' 2/1 re' 9/4

If the first member of each of these chords is reduced to unity by dividing by itself and the other elements of the chord by the same amount, then all three chords will have the same set of ratios: 1:1, 5:4, 3:2. Probably the easiest way to visualize the musical octave with its recessions and chords is to divide a circle into ratios of the scale. Doing this produces an easily understood visual representation of the qualitative structure found in the musical octave. [top]

Circular Octave Relationships

On the preceding diagram are the whole octave (=0), the two recessions which correlate with the primary digits, and the seven notes that correspond to the seven descendent digits.

Each of the three chords in the octave produces a very
symmetric figure despite the fact that each chord spans a
recession. The tonic chord forms an isosceles right triangle,
the subdominant forms an equilateral triangle. And the
dominant chord forms an isosceles triangle. The three mandalas
(circular diagrams) which follow display these relationships.

Chords in the Circular Octave
[top]

The octave is divided into a group of three notes and then a set of four notes by the first recession. One octave is separated from the next by the second recession. This structure of the musical octave is thus identical with the structure of the stupa. This structural similarity is brought out in the following diagram.

1 recession at beginning 7 set of three 3 6 2 recession within octave 5 9 set of four 4 8

This arrangement indicates that as the pitch increases the relative weight of the corresponding number increases. The physics of the situation requires this increasing weight. In electro-magnetic vibrations, as the pitch or frequency increases, we have radio, radar, heat, light, x-rays, and so on until the densest waves - solid matter itself.

Weight, however, does not separate the digits out completely.
Indeed, by weight alone, some of the correspondences would be
doubtful.

do } re } 3 or 7 mi 6 fa 5 so 9 la } ti } 4 or 8

The first two correlations are determined by the elements since 3 is Air and 7 is Fire. Fire has the greater relative weight. Since Water and Earth have the same weight, the last two can be settled only by the study of color. The complete scale, therefore, is

recession 1 do 3 re 7 mi 6 recession 2 fa 5 so 9 la 8 ti 4

The primary digits are the recessions. Three starts the octave and four ends it. Each set is cyclic when place on the Hermetic square. The sequence goes circularly around the square: Air (=3) to Fire (=7) to Water (=6), back to Air (=5) to Fire (=9) to Earth (=8) to Water (=4).

There is no phenomenon quite analogous to complementaries in music. However, there are two interesting items: first, the chords and the associated digits; secondly, the relationships among the Fire and Earth sets of digits.

The tonic chord contains do-mi-so. The corresponding digits are 3-6-9; 6 and nine are complementary, and 3 is neutral. The subdominant chord is fa-la-do; the digital equivalents are 5-8-3; 5 and 8 are complementary, 3 is again neutral. The dominant chord is so-ti-re' with corresponding digits 9-4-6; 9 is complementary to both 4 and 6. Thus the presence of complementaries leads to a complete and harmonious tone combination.

The mathetical relationships between the Fire numbers and the Earth numbers appear in the harmonious pairing of tones.

do 3 ----| re 7 ----|---| mi 6 ----|---|---| fa 5 ----| | | so 9 --------| | la 8 ------------| ti 4

The ratios between the pairs of notes connected above are all 4:3 in harmony with the relationships:

3 corresponds to 5 7 corresponds to 9 6 corresponds to 8 and vice versa.[top]

The relationships between the digits and colors are easily discovered. Pythagoras reduced the musical octave to mathematical law in the Sixth Century B.C., but it was only around 1916 - 1920 that a German chemist named Ostwald reduced color to some sort of law. Color is a very complicated affair compared to sound, and even yet all the many difficulties in color theory have not yet been settled.

Ostwald was the first to point out that there are two entirely different kinds of color. The first kind is called unrelated color, the second kind is called related color. Unrelated colors are those of the spectrum as seen through a prism. In order to observe these colors, we must make a prism catch a beam of light and throw the prismatic colors into a slightly darkened room. Related colors are those that we see in normal conditions in our every day environment. The difference between these two kinds is that unrelated colors are never mixed with any sensation of white or black, whereas related colors always have some degree of white or black or both.

A simple application of the numerical sequence just developed to the spectrum of colors will fail because this would ignore the most basic classification of colors. White, black and gray do not appear in the spectrum of a prism, but these colors are important and can not be overlooked.

The analysis of colors in terms of the digits and the stupa depends on finding a group of three and a group of four. Ostwald has done most this analysis. He pointed out there are to main types or subgroups of related colors. There are three achromatic colors which the tiny rods in the eye respond to, and four primary chromatic colors which are perceived by the cones on the retina of eye. The three achromatic colors are white, gray, and black. These colors give no sensation of a hue. The four chromatic colors are red, yellow, green, and blue. These are called the primaries because these produce no sensation of any mixing. But orange, purple. turquoise, brown, or other hues produce a mixed sensation.

Painters say that the only primaries are yellow, blue, and red because they form green by mixing yellow and blue together. However, green is a primary color because visually it is a single color and does not resemble yellow or blue at all. A green light passed through a prism does not break up but remains green. Physicists, on the other hand, say that the primaries are red, blue, and green because they can form a yellow colored light by combining or running together red and green rays of light. But again, yellow does not resemble the red and green which form it. The only colors that both a painter and a physicist would call primary are red and blue - and this is some consequence. The eye, however, identifies four distinct chromatic colors which mix to form other ones but which themselves are not a result of mixing.

Perhaps the most interesting thing about color is that
it has a very definite system of complementation fully analogous
to the additive complements presented in the last Chapter.
In color all such complementaries combine to form gray just
as all such numbers add up to 3 or 13.

In primary colors the sets of complementaries are:
white vs. black
red vs. green
yellow vs. blue.

Just as there is no complement to 3, there is no complement to gray. A simple way of testing for complements uses the after-- image effect. Stare intently at a colored area for about 40 seconds, and then shift your eyes to a white area. You should then see a faint after-image in the complementary color.

The comparison of the qualitative structure of color with the qualitative structure of the digits easily shows that the three achromatic colors are associated with the Fire digits and the four chromatic colors are associated with the Earth digits. Gray is obviously equivalent to 3; white then is 7 and black is 6. The most basic of the chromatic primaries are red and blue since these are basic to both systems of color theory. Then, by complementaries, yellow is 9 and green is ascribed to 8. This agrees with Birren's dictum that yellow (=9) and blue (=4) are the basic set of complementary colors - 4 and 9 are the two square numbers on the earth square.

All other colors are combinations of these seven primary ones. Any mixed or composite color can be numbered by using the law of decimal octaves. For instance, pink is a mixture of red (=5) and white (=7), so pink = 75. Maroon is a mixture of red (=5) and black (=6), hence maroon = 65. Brown is a mixture of orange (itself a combination of red and yellow) and black, so brown = 695. Pastel colors are formed by using white; dark colors show the presence of black. And the cool, refined tints sometimes called cream colors derive from gray.

The Ostwald color solid shows all the different combinations of both the chromatic and achromatic colors. This solid is in the shape of two cones put together by their bases. The equator of the solid represents the purest possible spectrum colors arranged circularly. The north pole is white, the south pole is black, and the axis between these poles shows all the shades of gray from almost white near the top to almost black at the bottom.

On the surface of the top cone are those colors formed by adding white to the pure hues around the equator. Similarly, the dark colors or shades lie on the bottom surface. In the interior of the volume are the colors that are mixtures of pure hue and both white and black (or gray) depending on the varying proportions of hue, white, and black. Ostwald Color Solid and Digits [top]

This section shows the connection between the qualitative
structure of the digits and the atomic periodic table.
The ideas used are straight-forward applications of previous
results. However, the subject is more complicated and
for the general reader might be tedious.

Note> | recess. | do | re | mi | recess. | fa | so | la | ti |
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

TI | H | ||||||||

DO | He | Li | Be | B | C | N | O | F | |

RE | Ne | Na | Mg | Al | Si | P | S | Cl | |

MI | Ar | K | Ca | Sc | Ti | V | Cr | Mn | |

FA | Fe, Co, Ni | Cu | Zn | Ga | Ge | As | Se | Br | |

SO | Kr | Rb | Sr | Y | Zr | Nb | Mo | Tc | |

LA | Ru, Rh, Pd | Ag | Cd | In | Sn | Sb | Te | I | |

TI | Xe | Cs | Ba | La | * | Hf | Ta | W | Re |

DO | Os, Ir, Pt | Au | Hg | Ti | Pb | Bi | Po | At | |

RE | Rn | FR | Ra | Ac | ** |

J. Emerson Reynolds, in his presidential address to The (British) Chemical Society in 1902, discussed the then-known elements of the chemical periodic table in light of a fresh concept. He considered the atomic elements as points on a vibrating string. He discovered that it would take two strings, vibrating at different rates or periods to produce the periodic table. One such string would give the regular seven-member octaves of elements. The other string would produce the inert elements and metallic triplets such as iron, cobalt, and nickel. This concept, together with Gurdjieff's doctrine that the recessions of the octave caused various irregularities and deviations was used to analyze the periodic table as basically a musical system of octaves and notes as the table shows. This scheme classifies the elements into nearly one and a half octaves of octaves of elements.

According to Reynolds, hydrogen (=H) is the very last element of an octave rather than the first of an octave of which six elements are unknown. From this all else flows. The universal rule is that alternate octaves of elements are more similar just as alternate musical notes are more harmonious.

The recession between hydrogen (=H) and lithium (=Li) gives rise to the inert element helium (=He). Then in succeeding octaves are two more inert gasses, neon (=Ne) and argon (=Ar). The recession between mi and fa (manganese = Mn and copper = Cu) causes another irregularity here appears the relatively stable magnetic metallic triplet of iron-cobalt-nickel. Inert gasses and metallic triplets alternate as the rule demands throughout the rest of the table. The next recession between ti and do (rhenium = Re) and gold = Au) causes a further irregularity - not at the end of the octave - but at the recession point within the octave. This recession point contains the two octave set of lanthanides. According to the alternation rule, there is a similar set of actinides in the next alternate position.

Gurdjieff's Law of Seven has been amply confirmed since there are irregularities from a straight 7 x 7 system only at those points indicated by the recessions of vibrations in the musical octave. Arranging the periodic table in this way makes very apparent its ordered character and shows its dependence on the musical octave.

The structure of the atomic table has thus been shown to be identical with the qualitative structure of the digits. Hence numbers can be applied to develop a precise structure of the atomic table.

There are ten octaves of elements shown by the ten horizontal lines comprising the table. For example, the second octave contains lithium (=Li), beryllium (=Be), boron (=B), carbon (=C), nitrogen (=N), oxygen (=O), and fluorine (=F). These ten octaves are grouped together into great octaves. A complete great octave has an octave of octaves of elements. Hydrogen by itself forms one great octave. The next seven lines compose the second great octave. Gold (=Au) starts part of the third great octave. The law of decimal octaves provides the means to assign an accurate qualitative numerical equivalent to each element.

Everything on the periodic table is matter, so the digit in the unit's position is 8, the number of matter.

There are three great octaves, so in the ten's position will be 0 for hydrogen, 1 for the second great octave, and 2 for the third great octave. This triad of numbers was chosen since hydrogen is the source of matter just as zero is the source of 1 and 2.

There are ten small octaves, shown here horizontally, in the hundred's position will be the digit connected with the note at the left edge of the table.

Finally, each small octave has seven elements, so in the thousand's position will be the most specific number, the one connected with the note at the top of the table.

These rules for calculating the classification number
for the elements depend only on the general results of the
mathetical analysis of the musical octave and the law of
decimal octaves. Several of the mathetical atomic numbers
are given here as examples.

hydrogen = H = 4408
oxygen = O = 8318
silicon = Si = 5718
gold = Au = 3328
radium = Ra = 7728

The recessions are also associated with elements. The
classification numbers of the "irregular" elements - inert gasses,
triplets, and the lanthanides and actinides - are based on the
recession numbers.
The inert gasses and metallic triplets occur at the beginning
of the octaves and so have the number 1. The lanthanides and
actinides occur at the recession with the octave and so
have the number 2. For example

helium = He = 1318
neon = Ne = 1718
argon = Ar = 1918
xenon = Xe = 1418

Members of the triplets are distinguished from each other
by prefixing 3, 6 or 7.

iron = Fe = 31518
cobalt = Co = 71518
nickel = Ni = 61518

osmium = Os = 31328
iridium = Ir = 71328
platinum = Pt = 61328

The lanthanides and actinides each consist of two octaves
within the inner recession denoted by a 2. To show which of the
two octaves the element belongs in we use the digits 1 and 2.
the number for lanthanium = La is 6418. From this the other
numbers follow.

first octave of lanthanides
cerium = Ce = 312418
praseodymium = Pr = 712418
neodymium = Nd = 612418
promethium = Pm = 512418
samarium = Sm = 912418
europium = Eu = 812418
gadolinium = Gd = 412418

second octave of lanthanides
terbium = Tb = 322418
dysprosium = Dy = 722418
holmium = Ho = 622418
erbium = Er = 522418
thulium = Tm = 922418
ytterbium = Yb = 822418
lutetium = Lu = 422418

The numbers for the actinides are constructed in the same way from the number for actinium, which is 6728.

Many of the elements have from one to ten isotopes or variant forms. These could possibly also be numbered by using one or two more decimal places to the left. The subatomic particles might fit into the zeroth great octave which now contains only hydrogen. [top]

At the point a few general words about the mathetical principles of organization will help one use mathesis in his own field of study. Any time seven is found as an organizing principle, the results from the study of the octave may be applied directly. Look for a set of three, a set of four, and either recessions or strong differences between the two sets. If there is any phenomenon corresponding to complementation use it to check results. Irregularities ought to come only at the recession points of the octave. This is another good check and was used to good advantage in first studying the periodic table. If things come in fours, then use the Hermetic elements and the Hermetic square. On the other hand, if there is a Hegelian triadic pattern of thesis, antithesis, and synthesis, check the meanings of the digits and the pattern of complementaries.

The ten digits form a remarkable filing system, a system which is not arbitrary but true to the cosmos. [top]