PROPORTIONS
In analysing the
dimensions of the work, we must draw a distinction between proportions
of a dynamic nature, and those which are static. For while
those of a dynamic nature  like the "golden section" discussed
below, and the systematic diminution of dimensions  are always connected
to a directional tension of some kind, positive or negative, those static
in nature  below under this are listed symmetry and geometrical bisection
 are characterized by a harmonious balance of forces.
The golden
section is a formal element in the construction of this Sonata at least
as important as was the periodization of 8+8 or 4+4 bars in the Viennese
classical style.
In essence, the
golden section (sectio aurea) is a simple pair of proportions: a length
divided into two such that the proportion of the whole to the larger
section matches that of the larger to the smaller section.
^{6)}
If the whole length
is marked "A" and the unknown larger section "X", then the value of the
smaller section will be "A  X":
On this basis,
therefore, the equation of the golden section is:
A : X = X : (A
 X)
Since the result
of this equation is that "X = 0.618A", then we see the following formula:
A : 0.618A = 0.618A
: (A  0.618A)
or
A : 0.618A = 0.618A
: 0.382A
In practice, the
consequences of this for analysis are that 1. the larger section
of any length divided according to the golden section is equal to the length
of the whole ("A") multiplied by the proportional factor of 0.618
(i.e. X = 0.618A); 2. the value of the smaller section is found
by multiplying the length by 0.382 (i.e. 0.382A).
As an experiment,
let us find the golden section of movement I. Its overall length is 443
bars, therefore the longer section (based on the equation X = 0.618A) will
be 0.618 x 443 = 273 bars. Thus formally and tonally the golden
section coincides with the focal point of the movement, the statement of
the first subject in the recapitulation (b. 274).
The golden sections
tabulated below entirely fulfil the strictest geometrical requirements.
It is an easy matter to check the data with a sliderule. ^{7)}
The descriptions
"positive" and "negative", used to distinguish between the
two possible intersections, refer to the order of the sections: positive
= long + short and negative = short + long.
The unit of form
in which the golden section appears, acquires a tension character with
a plus or minus sign. The easiest way to imagine this is to regard symmetry
as a condition of balance (tensionwise neutral), since its focal point
is in the centre:
whereas the focal
point of the golden section in comparison to this has shifted in a direction
either positive or negative:
An analysis of
the Sonata reveals that a positive intersection is accompanied by
a tensing of the formal muscles, in essence an elevation or an increased
density of material, while a negative intersection is accompanied
by a waning of the dynamic forces, a subsiding.
The golden section
is only found together with certain types of themes, namely the "creeper"
type, motordynamic and polyphonically treated types, these by their very
nature not mingling easily with 8+8 i.e. 16 bar symmetry. So much so, that
the 8 bar melodies appearing in movement III in bs. 144159 and 379394
have involuntarily acquired the name "strophes", because they stand out
completely from their asymmetrical surroundings. The development section
of movement III will later allow us to examine more closely the staticdynamic
question.
As a general rule
the "left" and "right" sides of binary forms, as well as the antecedent
(Stollen) and consequent (Abgesang) of anapaest form ^{8)},
including the antecedents I and II that this contains, bring into conjunction
sections with positive and negative intersection:
As a result, the
two dynamic figures (complementing one another like a reflection) combine
to make a round static unit. The intersections "d" and "e", drawn into
the field of attraction of the central point "c" (see above diagram), mark
the unbroken arching of the form's tension curve:
In the majority
of cases, the node point "c" is itself the golden section of the length
"a  b" (a good example occurs immediately in the 17 bars that begin movement
I).
Evidence that
Bartók was preoccupied early on with the problems of the golden
section is provided by the clattering F# minor of the Allegro barbaro,
which results in units of
3  5  8  13
bars, exactly
the most characteristic and pure proportions of the golden section.
The proportions
listed are accompanied by only a few marginal notes, as the golden sections
and their combinations follow with almost seismographic accuracy the carefully
articulated formal and tonal thought  indeed, they prove to be absolutely
identical with it. If we are therefore aware of the formal and tonal position
of the units due to be analysed, of their function in the construction
of the work; in a word, their dependency, or derivation, their relationship
to neighbouring and related sections, their role of attraction or repulsion
 then this naturally provides an explanation of the inner structure and
the golden sections of the unit analysed as well.
Let us see an
example. In movement I bs. 2  18, the handling of the leading motive ^{9)}
can be considered a unit, equally from the thematic, dynamic and formal
standpoints. In brief it divides up as follows:
Theme 
rectus 
(tonic) 

rectus 
(dominant) 
Theme 
inversus 
(subdominant) 
The most practical
way of counting will be to take triplet units (3/8), in which case the
whole section consists of 46 units, the golden section being 28
(because 46 x 0.618 = 28). This result matches our expectations, as the
rectus
sections contain exactly this number of units:
Now let us extract
from this set of relationships the rectus section, which takes up
the first 28 units. Geometrically the positive intersection should fall
in the first third of the 18th unit (because 28 x 0.618 = 17.3), and in
fact this position coincides exactly with the second appearance of the
theme in the dominant:
Now let us consider
individually the tonic and dominant sections of the theme.
Both units are clearly divided in sections by the cymbal stroke,
furthermore the tonic section is split in a positive direction,
the dominant in a negative one:
(because 17.3
x 0.618 = 11, and 9.6 x 0.382 = 3.6). The positive and negative intersections
are here so intertwined with each other, they arch like the rising and
falling of waves:
The crests of
the waves link to form a positive intersection (the double line in the
diagram). The contrary image of this positive intersection is delineated
by the "inversus section", i.e. where the entry of the tamtam gives
it a negative (minus) sign:
Following from
this, the rectus and inversus sections (positive and negative
intersections) complement each other mirrorwise, while as an overall form
they combine in a positive intersection, signifying that the general
progression is an upward moving one:
Measurements carried
out on even the smallest formal units prove to be equally persuasive. Let
us take as an example the dominant section: in the part extending
as far as the cymbal stroke, the long eb note of the motif falls
at the positive intersection, and in the part following the cymbal
the side drum entry falls (symmetrically) at the negative intersection:
Likewise, the
positive
intersection of the part of the tonic section extending as far as
the cymbal, falls at the third entry on c# on the timpani, and the
symmetrically falling negative intersection of the part following
the cymbal is at the side drum stroke.
From these positive
and negative intersections linked in pairs arises a continous fluctuation
on a large and a small scale, one whose various undulations eventually
meet with a positive (plus) sign, hence producing a powerful
dynamic
elevation:
It is characteristic
of this complex, "potential" form, that the golden section always lies
at the most important point in the divided formal unit, and that the longitudinal
waveformation caused by the intensification and relaxation in the occurrence
of intersections ("nodes") can be fully felt.
The golden
section is the framework within which beauty and life touch one another
at the deepest level. The golden section is a symbol of life, a formula
taken from nature as it were, contrary to everything we know to be disorganized,
distorted, and lifeless. One of the most beautiful of its symbolisations
is the pentagram, every proportion of which without exception conforms
to the golden section: ^{10)}
6) Or
the larger section is the geometrical mean of the overall length and the
smaller section.
7) A
question the reader may justifiably ask is why the golden section is measured
by bar numbers (or note values) and not to time. In the author's opinion,
the metrical beat is music's heart beat, and as such, ranks higher than
the measurement of time.
8) See
further text regarding the connection between anapaest form and Barform
(translator).
9) The
1st bar, as it were, opens an endless perspective onto "timelessness";
the organic life of the work has been counted as beginning only from the
2nd bar.
10)
Some data regarding the practice of the golden section: in its most beautiful
form, the golden section finds expression in Gothic architecture. The Lancer
of Polykleitos, as has been demonstrated, is in every proportion constructed
to golden sections. Fibonacci, when studying the fertility of rabbits,
came into possession of an arithmetical progression, every unit of which
stood in a golden section relationship with its neighbouring unit, each
number being at the same time equal to the sum of the two preceding numbers:
2, 3, 5, 8, 13, 21, 34 etc. The spirals going in one direction or another
on the stalks of flowers also follow the numerical proportions of the above
progression. In the opinion of Sainte Lague, the golden section is the
characteristic of living forms, while it is lacking in crystalline and
inorganic forms, precisely in connection with their pentagonal
qualities (the calculation 0.618 involves the root 5); think of the large
role played by pentagonal radii in the organisms of plants and primitive
organisms.
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