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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
Laws of Three and Seven
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]
Musical Octave
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
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Notes and Digits
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.
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Ostwald Color Solid and Digits