1. In some instances, intervals may be taken as a factor in the creation of form. In movements I and II of this Sonata, it is to be observed that just about every formal plane is welded into a unified sonority by a characteristic interval, or combination of intervals, being used in an organum-like way or as parallel chords (on occasion thematically). In movement I in particular, this produces the "layered" effect of its cross-section. (Why this technique should be linked particularly to movement I is a question about which the chapter on "Meaning" gives information.)

2. Systematic expansion and contraction of intervals permeates nearly every part of movements I and II. This expansion and contraction has the character of a dynamic process, and coincides with the ebb and flow of the work's form and content. For example in movement I, there is a process of expansion throughout the exposition and the first half of the development section, while in the second half of the development and the recapitulation, there is one of contraction.

The main forms in which this manifests itself are summarized below:

a) Sequential entries of motives, or the melody notes themselves, systematically succeed one another in expanding or contracting intervals (e.g. in movement I the funnel-shaped progressions in bs. 208 - 217 and 239 - 247).

b) The voices open or close like a pair of scissors (e.g. movement I bs. 256 - 263).

c) Sections of themes periodically expand or contract their range (e.g. movement I, first subject bs. 32 - 40).

d) By a systematic expansion and contraction of scales, chromatic becomes diatonic, diatonic becomes pentatonic, pentatonic becomes motives based on fourths and fifths, the latter on occasion expanding to motives based on sixths - plus the reverse of all these (e.g. movement I, second subject bs. 84 - 105, and movement II, the chant bs. 48 - 56).

e) By a systematic expansion or diminution of the intervals of equidistant chords, chromaticism becomes whole-tone scale, whole-tone scale becomes diminished seventh chord, diminished seventh chord becomes augmented triad, and augmented triad becomes augmented fourth-chord, the latter evolving into the circle of fifths or fourths, or rather the fifth-chords and fourth-chords made from it - and again all this in reverse (e.g. the voice entries in movement I bs. 175-190 follow this principle).

To this process of interval expansion and contraction will be ascribed (in the chapter on "Meaning") a certain irrational significance, giving it a psychological emphasis, as being projections of the movements and dynamism of levels at the subconscious. This resembles the significance ascribed earlier to chromaticism, to illustrate which we might again refer to the Mad Scene in Boris Godunov. In movement III, by contrast, no trace of these processes is found - unlike in the turbulent first movement - the third movement radiating throughout an undisturbed gaiety.

3. Basic acoustical attributes, and also in part the images entrenched in us by classical harmony, mean that intervals have acquired a number of qualities which seem to have had an important influence on Bartók's usage.

Thus the perfect fifth, and its inversion the perfect fourth, due to its being one of the nearest harmonics, and having a strong individuality, is so pervasive in the entire Sonata, both vertically and horizontally, not to mention the compositional principle of answering at the fifth, that it seems almost superfluous for an analyst to dwell further on a fact so conspicuous. In essence, the sentence in the chapter on tonal organization which says that Bartók's tonal system fits comfortably the groundplan of the circle of fifths or fourths needs to be emphasized. The downward leaping fourth aspect is every bit as energetic here as the simultaneously sounding, or upward leaping fifth (in support of this we would refer to the themes of movement III bs. 19-20, 44-45, 176-177).

The sixth can be seen as an enlarged fifth, one self-expanded and overstrained by an extra degree. This is doubtless why in movement I, the sixths in the closing theme of the exposition, and the fugue subject have such an intense effect (bs. 105, 332). An important point is the fact that the sixth above the tonic is mostly climactic (it is enough to refer to these culminatory points: movement I, first subject b. 37, and the development b. 217). The sixth above the tonic is often treated in this way in classical melody types (e.g. the fugue subjects of Bach!).

The relationship between the "expansive" major third and the "anxious" minor third is a specially contrasted one. The major third tends, after all, to be associated with the "spacious" major, and the minor third with the "confined" minor triad.

The character of the augmented fourth (diminished fifth) is determined by its perfect, indeed absolute, intertonality and acoustical rootlessness, which means it can be used to create the most extreme polar opposites (in the axis system the pole-counterpole relationship). It is this quality which elevates the polarity of the augmented fourth to being one of the central structural principles of this Sonata. Here we should add that Bartók was painfully sensitive to the 11th overtone, which at a distance corresponds to the tritone of the fundamental (and determines the Lydian sound of the overtone chord). ¹¹⁾

It should be noted that such characteristics of intervals can be exhibited clearly only in themes which are "12-tone" and completely chromatic in character, since their behaviour is free of the influence of the classical harmonic system. 

4. In the matter of interval symbolism, without doubt the most interesting find is provided by an analysis of movement II bs. 48-56, where - on the border between a microworld and a macroworld - the relationship between the minor third and the perfect fourth offers us a key to transposing between the chromatic and diatonic systems. It becomes clear that the minor third is inseparably associated with chromatic motive work, and the perfect fourth with diatonic. ¹²⁾ At the same time, it is the first clue in this analysis that points towards the formation of comprehensive unity in the Sonata. For the contrasting of "spacious" and "narrow" arises not as a method of treating in isolation the question of form, but constitutes an integral intellectual and logical unity with the programmatic conception of the work. 

5. Summed up, what has been said so far produces the following result: the parallel chords used by Bartók, together with consecutive intervals, characteristic melodic pitch ranges, a dogged use of particular intervals, and repeatedly recurring intervallic leaps employed as basic figures, form a self-contained system, which corresponds to this order of magnitude:

This arrangement is characterized by the fact that its components stand in a golden section relationship with one another (see in more detail the chapter on the proportions of the Sonata). For the sequence major second - minor third - perfect fourth - minor sixth - minor ninth produces, if counted in semitones, the simplest numerical relationships of the golden section:

(2 = major second, 3 = minor third, 5 = perfect fourth, 8 = minor sixth, 13 = minor ninth). This should provisionally be conceived as if, to build a house, only beams 2, 3, 5, 8 and 13 metres long were available. The builder is free to make use of any of them, but he cannot change their measurements. In this sequence, known as the Fibonacci numbers, each number is equal to the sum of the previous two ¹³⁾ (thus the sequence continues: 21, 34, 55, 89 etc.). A good example of this is the various changes of shape and compass of the main theme in movement III of the Divertimento:

Even the intervallic cells divide up according to the above progression of quantities. For example, 8 can only split into 5 + 3, the alternatives 4 + 4 and 7 + 1 being excluded from the system.

It is striking that the analyst hardly encounters any parallel major thirds or major sixths; it is almost as though there were a ban on consecutive major thirds and sixths. All the more frequent and widespread is the occurrence of consecutive minor thirds, perfect fourths, and minor sixths (see movement I of the Sonata). The major third has no melodic function worth mentioning. Naturally enough, though, the motivic role of the minor third is almost only too obvious:

The figures based on the minor third and perfect fourth outlined in paragraph 4 are also none other than basic elements of the system. Hence the relationship between them is accommodated to the golden section.

The notation of parallel major chords fits this principle for the most part (in movement I of the Sonata), in that the structural role of perfect fourths and minor sixths is emphasized:

This, it would appear, is the reason why, when Bartók writes a triad in a chromatic piece, it has its minor third "upwards" (in second inversion), and has its major third "downwards" (in first inversion):

The most characteristic Bartók chord was born out of a fusion of the two; this "major-minor" shape, made out of a minor third - perfect fourth - minor third (3:5:3), is well known:

If consideration is given to the fact that the pentatonic scale, in particular the la pentatonic, rests primarily on a framework determined by the perfect fourth, minor third and major second, then pentatony must be seen as the purest musical expression of the golden section.

Does this not mean that instinctively, in one of mankind's most ancient artistic utterances, form was given to the law of the golden section? The most ancient layer of folk music shows beyond doubt that the golden section is a law intrinsic to music, and not one applied from outside.

In this case, the major-minor configuration mentioned above is nothing less than an extension and magnification of the pentatonic l - s - m - r shape; its projection into a broader dimension:

6. a (alpha) type harmony - b (beta), g (gamma), d (delta).

The above chord is found at every step in this form also:

This chord (which may be considered c major-minor seventh chord) is related to a number of others which are highly characteristic, and which as a group will be called a chords. First of all, also in the tonality of c, there is this common shape:

This sounds as if the earlier major-minor chord (g) had been extended by a lower minor third, the c# (the upper minor third in turn being omitted). On occasion, a minor third cell is joined also from the top:

All three configurations have in common that they are framed by a major seventh, that they consist of minor third (3) and major second (2) cells, and that they have the tonality of c; their tonality is provided by the prominently sounding perfect fourth (5):

Accordingly, the minor third cells of the a chords are always based round the core of a fourth:

These a type chords will be called individually b, g and d chords (b corresponding to a root position, g to a first inversion and d to a quasi second inversion). The large a chord shown above incorporates the b, g and d alternatives. Equally characteristic are these reduced forms:

The 4th alternative, e (bb -c-eb-f#-a), due to its unstable tonality is hardly used.

The basic principle therefore is the following: in order to establish a feeling of tonality, at least 2 notes are required, the simplest case being the root (e.g. c) and its fifth (g):

The note g here reinforces the c, whereas by itself the g has a dominant meaning (tonality can after all only be supported by means of the asymmetrical division of the tonal system; in the case of a division based on equidistance, it is not possible to know which note to keep to). If this principle is transferred to the relationships of the axis system, then it is found that the note g can be substituted by any note of the dominant axis without changing the c tonality:

(The first 3 alternatives are not new to us, since they correspond to the constituents of the dominant seventh chord).

Similarly, the note c can be replaced by any note of the tonic axis without the function being changed:

The only proviso is that the chord should be made up of two levels: that of the tonic and that of its defining dominant. The a chord is thus no more than the axis extension of the simple c-g relationship. In place of c we have the tonic axis, and in place of g, the dominant axis:

To build an even larger a chord, it is necessary that the pattern 2 + 3 + 3 + 3 be repeated periodically, as follows:

Given some latitude, figures in which 2 + 3 + 3 is repeated periodically can also be taken to be a chords, e.g. this:

The use of a chords is just as widespread in the music of Bartók as was for example the seventh chord in the classical style.

There remain two important groups left to discuss in the list of chords constructed following the golden section (GS).

7. Equidistant patterns.

In what follows, the whole-tone scale will have the symbol  w (omega - the letter farthest from a in the alphabet; this has been chosen deliberately, since these kinds of music are used by Bartók as contrasts, a being "ardent", w "neutral"). Because the 12-tone system only permits of two whole tone constructions:

any further patterns being inversions of these, then one w will be the dominant of the other. Like the w, the diminished seventh and the fourth chord have an equidistant shape. It should be noted that when fourth chords are placed together following the 2 : 3 principle:

then the tonic is the one lying a major second higher or a minor third lower than the other. Reduced to its essentials, we can see in the following cadential turn a form of this resolution:

The w chord is characterized by the number 2, the diminished seventh chord by the number 3 and the fourth chord by the number 5, because taken individually these chords are made merely out of the major second (2), the minor third (3) or the perfect fourth (5). None of them is in itself a GS structure, since the GS always expresses a relationship, a relationship automatically excluded by the equidistance. Even so, in their relationship to each other (2:3:5), as in their relationship to the complete Bartókian tonal system, these chords form a GS type (see §8). The augmented triad therefore - in Bartók's chromatic system - can only exist in the form of minor sixth + minor sixth (8 + 8).

8. Intertonal fourth models, minor third models and major second models (1:5, 1:3, 1:2).

The use of the proportions 1:5 and 1:3 in partnership is one of the oldest architectural formulas, used already by the Chaldeans (the remains of the city of Ur), and later playing a primary part in Gothic art. To anyone at all acquainted with Bartók's music, these formulas will be quite familiar: 

The first arises from the endless repetition of the proportion 1:5, the second similarly from the proportion 1:3. These two patterns, it should be known, appear in Bartók almost always as a pair of proportions harnessed together, since taxonomically they are correlative (5:3). In this analysis, the formula 1:5 will be referred to as the intertonal fourth model, and the formula 1:3 as the intertonal minor third model. These patterns are called intertonal because from the standpoint of 12-tone music they produce scales which have symmetry:

(In other words this represents a higher form of the equidistant division of the tonal system, the division here consisting not of individual notes, but of groups of notes.)

The intertonal fourth model shows an axis tonality:

i.e. the notes c-g belong to the tonality of c, and the notes f# and c# to the tonality of f#. It is always given a tonal significance by Bartók:

(in the first example the tonic intertonal fourth model of c is heard, in the second the dominant intertonal fourth model of F major ¹⁴⁾). In fact, it is used in two ways, as a 1:5:1 form and as a 5:1:5 form:

The intertonal minor third model¹⁵⁾ plays a similar part in the chromatic system that tetrachordal outlines do in the diatonic system; each is the exact taxonomical transposition of the other:

In other words, if the pentatonic scale l-s-m-r-d-l

is translated into the language of the chromatic scale, then we arrive at this characteristic scale:

(The intertonal minor third model will be discussed in greater detail in the analysis of movement I of the Sonata, b. 135.)

Still remaining is the 1:2 formula type, the "intertonal major second model", which can further be connected to the GS along with the 1:5 and 1:3 models:

This scale can be treated as the tonic scale in Bartók's chromatic system. Its tonality can be conceived as consisting of the ingredients of an axis:

i.e. the tonal resting point is always the lower note of the minor second interval in the scale, or the upper note of the major second interval (in this case the notes c-eb-f#-a). ¹⁶⁾ The example below shows how the axis system is connected to the laws of the GS ¹⁷⁾ :

and shows also that the concept "axis" does not mean the diminished seventh chord, but the relationships and connections in their complex entirety of the 4 various tonalities concealed in the intervals of the diminished seventh chord. This means that each of the three axes creates individually a GS system, and that transfer from one GS system (axis) to another can only take place through a violent infringement of the laws of the GS. This is why the main functional notes do not adhere to the first, fourth and fifth degrees of the diatonic scale (I=T, IV=S, V=D), but (as has been shown) are in a major third relationship to one another; transposing a GS system (axis) by a major third thus brings about a radical cancellation and complete disintegration of the previous system (axis), since the GS makes no allowance for the major third.

In the tonal system it is possible to put forward three intertonal major second models corresponding to the three functions:

The tonic intertonal major second model is given a special significance by the fact that it contains  within  itself all   the tonic (C, Eb , F#, A)  major,  minor and  seventh  chords, the   b,   g, d chords, as well as both tonic intertonal fourth models (i.e. this model can be made by geometrically fitting together 2 intertonal fourth models):

Apart from their common origin, chords of the GS type are linked together by many threads. Let us try to sum them up in a single scheme (at least their tonal forms). In the diagram below, each example is made up of the same notes. In the 1:2 model these notes are formed into a scale (top right hand example). The two "levels" of the a chord are arranged in the tonal c-g relationship (upper left hand example). In the 1:5 model the "axis" pole-counterpole relationship is shown (lower right hand example). The lower left hand example illustrates the connection between the axis system and the GS system. This last example is best thought of spatially, as if the notes c-a-f#-eb separated by a minor third (3) were on the upper rim of a cylinder, and the notes g-e-c#-bb were on the lower rim, the height of the cylinder (the distance between the two rims) being a fourth (5). Each model is written in its tonic form:

The axis system in fact is the only means whereby the GS laws can be reconciled with the characteristic number 12 of the tonal system. 

9. Chromaticism and diatonicism. 

To form an accurate definition of Bartók's use of chromaticism and diatonicism is not an easy task, since it is not possible to identify its indications by the concepts "narrow" and "spacious", both being equally applicable to the laws of chromaticism.

The most general distinctive feature of Bartók's chromaticism is that it obeys the tonal laws of GS, while his diatonicism avoids them. The best measuring instrument for identifying chromaticism seems to be the minor third; and motivic material in which the minor and major third appear as equal partners is in almost every case diatonic:

For this reason chromaticism in Bartók tends to gravitate towards scales which are minor in character, and Bartókian diatonicism towards scales which are major in character (in paragraph 3 mention was made of the "anxious" quality of the minor, and the "expansive" quality of the major).

The most spacious diatonicism in Bartók, and the extreme example of the composer's diatonicism, is the acoustic chord or scale:

which the present author has called acoustic because its notes are derived from the natural harmonic series:

To avoid unnecessary repetition, the reader is advised to turn to the introductory section of the analysis of movement III, without which the following cannot be understood. There it emerges that there is a close connection between the GS system and the acoustic system, the one being the exact taxonomic inversion of the other. It is amazing how things spontaneously "drop out" of the whole system. Instead of the minor third - perfect fourth - minor sixth characteristic of the GS system (3 - 5 - 8), the acoustic scale has the major third, Lydian (augmented) fourth and major sixth. The two systems complement each other to the extent that the chromatic scale can be split into the GS scale and the acoustic scale. This means that to approach Bartók's genius as a creator of musical material is to realize the latent possibilities and natural attractions themselves operating in the music. The two systems represent the two sides, or double profile, of a single connected whole.

(The c# and b, being chromatic intervals, belong to the chromatic system.)

Theoretical confirmation is also conferred on the fact that - and this can hardly be overstressed - the basic formula of diatonic harmony, in contrast to chromatic harmony, is the fifth and the major third (the nearest natural harmonics!). Also traceable to a construction based on fifths and major thirds is another common form found in Bartók's diatonicism: the type of major chord made of thirds whose alternate steps rhyme with one another at the fifth:

One of the patterns cut out from this is Bartók's well-known "signature" (d-f#-a-c#), which is also dominated by the sound of fifths and major thirds. 

Mention must also be made of a third type. The a chord, as we know, owes its characteristic flavour to the fact that its upper and lower levels (T and D) are in GS relationship to one another:

If the upper and lower levels are exchanged, however, then we immediately arrive at a "spacious" (diatonic) effect:

in this way giving voice to the major third and the fifth; we inevitably hear these relationships:

which in principle are excluded from the GS system, since there can be no fifths or major thirds in the a chord. (Only the above two relationships can be conceived between two diminished seventh chords.) As, in connection with the a chords, we made a distinction between b, g and d, so here use can be made of the most varied patterns of the basic type. And to make the connection complete, the acoustic chord turns out to be a version of this latter type:

Often the root is ambiguous, both C and F# being found:

(e.g. at the beginning of movement III of Music for Strings, Percussion and Celesta). We might describe as "over diatonic" the triad which has a major third above and a minor third below (e.g. in the muted trumpets at the end of the 2nd Violin Concerto):

This is why the minor triads of the 2nd subject of movement III are also diatonic (from b. 44).

Since Bartók's chromaticism is based on the division of the tonal system along equidistant (i.e. symmetrically) principles, then tonality must rest in principle on 2 notes at minimum - the pole and its counterpole; in other words, it always presupposes virtually the complete system. Connected with this is the fact that Bartók's chromaticism is a closed system - the "axis system" - the laws of GS also making it self-contained. By contrast, his diatonicism - in extreme instances - is adapted to the overtones of a single root; in other words, it is taxonomically open ended. The one is perhaps best symbolized by the "circle", the other by the "straight line". (It is characteristic that the axis system can only be depicted in a circle.) Interestingly - and this will be touched on in the chapter "Meaning" - the axis system goes most happily with thematic material "moving in a circle", while diatonicism usually goes with scalar or, in other words, "straight" themes. There is also another reason for this, however. In diatonic music, chords are constructed from the bottom upwards, a requirement not found in chromatic music, where scales and chords are constructed both upwards and downwards following the same principles; thus in diatonicism there is a root note, but in chromaticism there is a central note. In the chromatic system, everything can be inverted without the central note losing its significance. Bartók's "mirror" passages themselves show us how chromaticism can ignore the requirements of the harmonic series, concepts like "up" and "down" in this context completely losing their meaning. For example, in the recapitulation of the (2nd) Violin Concerto, the tonality of the first subject is B, despite the fact that the tonic B major, as a result of the inversion of the theme, is "stood on its head", and our ears, which are conditioned to feeling the relationships of the harmonic series, feel that the music is in e minor (the central b is highlighted also by a pedal point):

The effect of the harmony heard below the central b, together with its denial of the harmonic series, is as if objects in the physical world were to suddenly cast their shadows towards the light.

Mathematically also, the GS system and the acoustic system are founded upon a certain antagonism, namely that the acoustic system (harmonic series) numerically is a ratio of whole numbers, while the key number of the golden section is an irrational number (0.618...; mathematically the GS proportion 13:8:5:3 is 13:8.034..., and 8:4.944..., and 5:3.090...). This means that the GS rests on geometrical proportions; the acoustic system on mathematical ones.

Also reversed in chromaticism and diatonicism is the relation between consonance and dissonance. Thus chromaticism is at its most unfettered, and most neutral, when the divisions according to the equidistance principle take it to the furthest point (a good example being the "resolution" of movement IV of Music for Strings, Percussion and Celesta, bs. 215-234, where the tonal system gradually clots into complete chromaticism). To sum up, in the diatonic system everything relates to a single note, while in the chromatic system everything relates to the whole system, expression being given in the latter to almost a "change of field" for the complete tonal system.

If the reader here encounters a concept of chromatic and diatonic not quite in agreement with that in general musical use, then he must blame it on the fact that at the present time, there are no technical terms available which are more suitable. Even so, it is certain that the greatest contrast found in the music of Bartók, and which is consciously exploited to the utmost, is the duality between the chromatic and the diatonic, this being the purest musical equivalent to the idea of tension and resolution. The relationship between Bartók's chromaticism and diatonicism is like that between the Inferno and Paradiso of Dante: taken alone, each is but a part of the Whole, and can reveal only one side of life. Only together can the chromatic and the diatonic form an ideal unity and a comprehensive world concept. 
 
 

11) The leading motive (bs. 2-5) of movement I is of this type.

12) The chromatic fugue subject (its characteristic interval the minor third) of Music for Strings, Percussion and Celesta acquires a diatonic shape (its characteristic interval the fourth) in the final movement. The same can be seen in the Divertimento movement III and Microcosmos No.64 "Line and Point", versions a) and b):

13) Which in fact results from the law of the golden section.

14) The latter a-bb-d#-e model can be broken down into the fifth a-e and the fourth bb-d#, i.e. into the tonalities of  a and d#; both are the Bartókian dominant of the tonic F.

15) Two intertonal minor third models can be so combined that they produce the complete chromatic scale e.g. d-f-f#-a-bb-c# + e-g-g#-b-c-eb. Equivalent to the intertonal minor third model scale is the g-type chord structure below (5:3:5:3:5:3):

This chord structure will be called g+g in this analysis. The 1:3 model can be constructed not only from three minor thirds, but also three fourths, e.g. a-d + c#-f# + f-bb. In the piano pieces "A bit Tipsy" and "See-saw", the 1:3 model arises as a synthesis of e minor + Ab major: e-g-b + c-eb-ab = c-eb-e-g-ab-b.
 

16) On the basis of the 1:2 model, it is possible to straightaway examine the different intervals regarding their tonal significance. The interval root of the major second, perfect fourth, minor sixth and major seventh is always the upper note, that of the minor second, major third, perfect fifth and minor seventh the lower note. In the case of the minor third, augmented fourth and major sixth both lower and upper notes can be considered the interval root, since they lie on the same axis.

17) The reader will find an interesting example of the fact that the pentatonic scale and the a chord are related, the one easily being transformed into the other, in the choral piece Senkim a világon (I’ve no one in the world). At the repeat of the line "csendes folyóvíznek csak zúgását hallom" (”I can hear only the soft murmur of the river below”; translated by Miklós Hernádi) the pentatonic scale (db-bb-ab-f) is projected into the d  chord (ab-f-d-c-a).