Schoenberg’s annotations from June 26, 1923, include two machine-typed sheets entitled ’Zur Notenschrift’ (On Notation). The composer here sketches his ideas about a new notation for atonal music and remarks in passing:
Bartók’s and Krenek’s procedures are inadequate and pedantic. They lead to contradictions, consequently cannot be applied consistently, and make a demand on the memory that is unjustified and illogical. The only possible task of any pictorial notation is to make its effect purely through picture, without using abstractions of any kind.
Schoenberg’s criticism obviously refers to Bartók’s use of altered degrees in cases, in which there seemed to be no need for them. Stevens gives a revealing example. He quotes a passage from the Allegro barbaro (1911), ‘in which sharps and double-sharps are employed in order to notate a melody lying entirely on white keys’ (Ex. 1).
Ex. 1 Allegro barbaro (1911), mm. 27–30.
Schoenberg called his new procedure of musical composition ‘Komposition mit zwölf nur aufeinander bezogenen Tönen’ (‘Method of Composing with Twelve Tones Which are Related Only with One Another’). As there are no altered notes in this method, it makes no difference, whether notes on black keys of the piano employ sharps or flats; the choice is based on clarity in a given context. Double-sharps and -flats may be altogether abandoned.
Did Schoenberg in 1923 assume that Bartók’s music—like his own—was based on a twelve-note-scale? His comment on one example (from Bartók’s Bagatelle Op. 6 No. 10) in the last chapter of the Harmonielehre seems to indicate that he did. ‘It is the chromatic scale that seems to be responsible for chord sequences of this kind.’ Harmonielehre was published in 1911. Later Schoenberg came to another conclusion. When writing his notes on notation in 1926 he had obviously realized that Bartók never abandoned diatonic notation.
In the years around 1920, the notation of atonal music occupied Bartók’s mind. In an article entitled ’Das Problem der Neuen Musik’ (The Problem of the New Music), published in Melos, Vol. 1 No. 5 in April 1920, he discussed the very subject that was bothering Schoenberg:
In conclusion, just a word about our notation. It was created on the basis of the diatonic system, and—exactly for this reason—it is utterly unfit for the written reproduction of atonal music. The accidentals, for instance, mean an alteration of the diatonic degrees. Here, now, it is not a matter of alteration or non-alteration of the diatonic degrees, but of twelve semitones of identical value. Furthermore it is rather difficult to observe consistency in the method of the notation; for instance, one often hesitates whether to pay attention to an easier legibility in the vertical or in the horizontal sense.
It would be desirable to have at one’s disposal a notation with twelve similar symbols, where each of the twelve tones would have a comparably equivalent symbol, in order to avoid the necessity of notating certain tones exclusively as alterations of others. Meanwhile, however, this invention awaits its inventor.
Later on, Bartók’s explained that there was a time when he thought that he was ‘approaching a species of twelve-note music.’ ‘Yet even in the works of that period the absolute tonal foundation is unmistakable,’ he concluded.
In the notation of his works from this period, such as the two violin sonatas (1921, 1922), there are no signs of a tendency to abandon diatonic writing in favor of twelve-tone notation. The only aspect of traditional notation Bartók did abandon was the use of key signatures, the absurdity of which ‘in certain kinds of contemporary music’ he wanted to demonstrate, as he wrote in 1945, in his Bagatelle Op. 6 No. 1 (1908) for piano. This demonstration, the point of which was the simultaneous use of different key signatures for left and right hand (Ex. 2), has been a riddle for analysts trying to decide in which key the piece actually was.
Ex. 2 Bagatelle Op. 6 No. 1 (1908), mm. 1–5.
‘After carrying the key signature principle ad absurdum in the first piece, I dropped its use in all the other Bagatelles and in most of my following works as well,’ Bartók explained. Nevertheless, he repeated the procedure in some other pieces, such as ’Lullaby,’ ’Harvest Song’ and ’Counting Song’ (Nos. 11, 33 and 34 of the 44 Duos for two violins) and ’Melody against Double Notes,’ ’Crossed Hands’ and ’Playsong’ (Nos. 70, 99 and 105 of Mikrokosmos for piano).
In ’Lullaby’ for two violins (Ex. 3), the key signature of the upper staff consists of two flats, Bb and Db (!) and that of the lower staff of one sharp, the F#. The first violin only uses three adjacent notes, Bb, C and Db, the three first notes of either the Dorian or the Aeolian pentachord on Bb; the second violin, in turn, uses five notes of the Dorian or the Aeolian mode on E.
Ex. 3 44 Duos for two violins (1931) No. 11 (’Lullaby’), mm. 1–6.
Both parts have an independent final note as in a bitonal structure, and the impression of bitonality is further strengthened by the dissonant quality of the interval between the final notes: the tritone. We know that Bartók, at the time he wrote the duos, did not believe in the possibility of bitonality or polytonality anymore, because he thought that the ear tends to find one center even in most complex chords. In the so-called ‘Harvard Lectures’ that he gave at Harvard University in February 1943 Bartók put it this way:
… polytonality exists only for the eye when one looks at such music. But our mental hearing again will select one key as a fundamental key, and will project the tones of the other keys in relation to the one selected. The parts in different keys will be interpreted as consisting of altered tones of the chosen key.
In this case, the fundamental key would be E-Dorian or E-Aeolian, because of the weight of the note E as the final note of the lower staff. The resulting 8-note scale (Ex. 4) is a synthetic one. It cannot be reduced to any combination of two diatonic scales, although both parts in themselves are purely diatonic. But if we accept a contradiction between what we hear and what we see, we can say that the tonal structure of the piece, as notated, is bitonal and consists of a diatonic mode on E and of another on Bb. Both parts are independent; they do not share a single note—a complementary relation not rare in Bartók’s music.
Ex. 4 Scale of the ’Lullaby’
The most important notion contained in ‘Harvard Lectures’ is the notion of bimodality or polymodality, which Bartók defines as simultaneous use of two diatonic scales with a common fundamental tone.
Polymodal combinations consist either in complete diatonic scales or in incomplete ones, such as tetrachords and pentachords. In ‘Harvard Lectures’ Bartók gives three examples of the polymodal use of pentachords: Lydian and Phrygian, major and minor, and the two forms, ascending and descending, of the upper pentachord of the melodic minor scale. When discussing the simultaneous use of the Lydian and the Phrygian pentachord, as in the second theme (Ex. 5) of the first movement of the Sixth Quartet (1939), he points out the major difference between a chromatic and a polymodal scale:
As the result of superimposing a Lydian and a Phrygian pentachord with a common fundamental tone, we get a diatonic pentachord filled out with all the possible flat and sharp degrees. These seemingly chromatic degrees, however, are totally different in their function from the altered chord degrees of the chromatic styles of the previous periods. A chromatically-altered note of a chord is in strict relation to its non-altered form; it is a transition leading to the respective tone of the following chord. In our polymodal chromaticism, however, the flat and sharp tones are not altered degrees at all; they are diatonic ingredients of a diatonic modal scale.
Ex. 5 String Quartet No. 6 (1939), 1st mvt., mm. 81–85 (vn. 1).
The difference between chromaticism and diatonicism is based on the fact that chromaticism is a harmonic principle, whereas bimodality or polymodality is a melodic one. In polymodal music, harmonies are derived from the melody, not the other way around as in classic and romantic music. The modal way of using the diatonic scale ‘brought freedom from the rigid use of the major and minor keys, and eventually led to a new conception of the chromatic scale, every tone of which came to be considered of equal value and could be used freely and independently,’ as Bartók remarked in his ‘Autobiography’ from 1921. It seems that he became conscious of the possibilities of polymodal composition in the 1920s at the latest, and this consciousness is reflected more and more clearly in his notation, which in his later works meets the requirements of polymodal thinking in great detail. There are only a few cases in which exceptions are made from the general rule that enharmonic notes cannot be treated as each other’s equivalents as they are in twelve-note music.
To clarify the principles of polymodal notation let us consider some examples from Bartók’s music. One early example of polymodal writing is the secondary theme of the 2nd movement of Bartók’s First Quartet (Ex. 6). Bartók’s notation has been called ‘thoroughly perverse,’ because the theme is ‘clearly in A major going to A minor with an answering phrase in C sharp minor.’
Ex. 6 String Quartet No. 2 Op. 7 (1909), 2nd mvt., mm. 73–78 (vn. 2 and va).
This description may correspond to what might be called the ’perceived’ structure of the theme, but it certainly does not correspond to its notated structure. The literal reading of Bartók’s notation is essential to the understanding of its meaning. It is easy to rewrite the passage in such a way that it corresponds to the above description of its ’perceived’ structure (Ex. 7). In this case, the tonal context would be lost, the structure of the movement misunderstood, and the intonation of the melody quite different.
Ex. 7 Notation of the ’perceived’ structure of the above passage.
Bartók’s intention with the chosen notation becomes quite clear when we compare the final score with the sketches. The sketch BH:39 of the Bartók Archives, Budapest, contains the following version:
Ex. 8 String Quartet No. 2 Op. 7, 2nd mvt., mm. 73–78 (sketch BH:39).
Between the sketch and the final version, there is one significant difference: the second quarter note of m. 76 has been changed from D-natural into Ebb. Bartók obviously made this change, because he wanted to treat the Ebb as the Phrygian 2nd degree of a polymodal key in Db. The notated structure of the passage consists in the polymodal combination of Db Phrygian and its inversion, the Db Ionian (Ex. 9).
Ex. 9 Db Phrygian/Ionian.
By combining these two symmetric modes on the same fundamental tone we get an eleven-note scale (Ex. 10), every single note of which is in use in the melody in question.
Ex. 10 Db Phrygian/Ionian combined
In the sketch, there is a D-natural in the accompanying voices that Bartók did not change into an Ebb. The cello-part, in fact, uses both intonations in measure 76 (Ex. 11). As there must be a reason for this (remember, Bartók became famous for the ’photographic accuracy’ of his folk-music notations), it most likely is contextual and probably due to moving from one tonal region into another. Symmetric, crystalline structures, closed in themselves, do not allow evolution. In order to move ahead, symmetry must be broken, and this is precisely what is happening in measures 76–87: the notes of Db Phrygian are replaced, one by one, by their enharmonic equivalents of C# Phrygian, of which D natural is the 2nd degree.
Ex. 11 String Quartet No. 1 Op. 7 (1909), 2nd mvt., m. 76 (vcl.).
In early works, such as the First Quartet, one cannot expect to find the technique of polymodal composition fully developed, and one should not be surprised if there are details of pitch notation that do not support the polymodal explanation. ‘In our works, as well as in other contemporary works, various methods and principles cross each other,’ Bartók assured in ‘Harvard Lectures.’ Yet, in the long run, polymodal thinking is reflected more and more faithfully in his notation. Consider e.g. the first six measures of the Third Quartet (Ex. 12), a crystalline structure with a melody lying above a cluster chord (C#–D–D#–E). The notes of the melody and the accompanying chord form a 13-note scale on C# (Ex. 13), in which the F double-sharp and the G natural must not be understood as two different spellings of the same pitch but as independent modal degrees, the augmented Lydian fourth and the diminished Locrian fifth.
Ex. 12 String Quartet No. 3, 1st mvt., mm. 1–6 (vn. 1).
Ex. 13 The 13-note scale of Ex. No. 12.
This construction is a perfect example of the Hungarian master’s sense of symmetry since the Locrian and the Lydian modes are inversions of each other (Ex. 14).
Ex. 14 C# Lydian (ascending) and C# Locrian (descending).
It might be interesting to notice that exactly the same structure is used at the beginning of the second movement of the Divertimento, where the composer asks the second violins to play an augmented seventh, instead of an octave, as a double stop (Ex. 15).
Ex. 15 Divertimento, 2nd mvt., mm. 1–5 (vln. 2).
There is another situation in which Bartók writes an octave instead of an augmented seventh required by the polymodal construction. The final measures of the first movement of Music for Strings, Percussion and Celesta (1936) uses a mirror voice leading from A to Eb and back in both directions.
The subject of this circular fugue movement, as it appears at the beginning on the violas (Ex. 17), is a combination of the lower pentachords of the Lydian and the Locrian mode on A (Ex. 18).
Ex. 17 Lydian and Locrian modes on A.
Bartók’s notation follows this construction with a painful precision in all the transpositions. When the subject appears in inverted form after the climax of the movement (mm. 64–68), it quite logically uses the upper pentachords of the respective modes.
The mirror structure of the last three measures of the movement finally fills out the complete polymodal field. The first violins use the lower pentachord of the Locrian mode and the three lowest notes of the lower pentachord of the Lydian mode, the second violins the upper pentachord of the Lydian mode and the three highest notes of the upper pentachord of the Locrian mode. There is only one detail that disturbs the perfect balance: the lowest note of the lower part should actually be D-sharp. If this orthography is correct (that I cannot say, because I have not been able to consult the manuscript), there must be a reason for it. Did Bartók, in this structurally very important moment, want to prevent the impression that the violins play out of tune? In my opinion, this explanation would be acceptable.
The concept of bimodality (or polymodality) seems to explain some aspects in Bartók’s notation, which does not ‘make its effect purely through picture’, as Schoenberg required. Instead, Bartók used notation in such a way that the system behind the music, its basic tonal and modal conception, becomes as clear as possible. The concept of polymodality does not, however, solve all problems of Bartók’s notation and melodic style. In ‘Harvard Lectures’ Bartók also speaks of a ‘new chromaticism’ and enumerates some of his compositions in which this new chromaticism is supposed to be found.
The difference between a polymodal melody and a chromatic one, however, does not become quite clear through Bartók’s casual description. How to tell them apart, in my opinion, remains a problem.
(Lecture given at the ESTA conference, Budapest 1996)
A. Schoenberg, ’On Notation,’ Style and Idea. Selected Writings of A. Schoenberg, ed. L. Stein (Berkeley and Los Angeles: University of California Press, 1984), p. 351.
 H. Stevens, The Life and Music of Béla Bartók, rev. ed. (London–Oxford–New York: Oxford University Press, 1964), p. 117.
 A. Schoenberg, ‘Composition with twelve tones (I),’ Style and Idea. Selected Writings of A. Schoenberg, ed. L. Stein (Berkeley and Los Angeles: University of California Press, 1984), p. 218,
 A. Schönberg, Harmonielehre (Wien/Leipzig: Universal Edition, 1911), p. 469.
 B. Bartók, ’The Problem of the New Music,’ Béla Bartók Essays, ed. B. Suchoff (London: Faber & Faber, 1976), p. 459. NB! The words ‘alteration’ and ‘non-alteration’ are misprinted in the source as ‘alternation’ and ‘non-alternation.’
 B. Bartók, ’The Folk Songs of Hungary,’ in Suchoff, op. cit., pp. 338–339.
 B. Bartók, ’Introduction to Béla Bartók Masterpieces for the Piano,’ in Suchoff, op. cit., p. 433.
 See I. Oramo, ’Die notierte, die wahrgenommene und die gedachte Struktur bei Bartók. Bemerkungen zu einem Problem der musikalischen Analyse,’ in Studia Musicologica Academiae Scientiarum Hungaricae 24 (1982), pp. 439–449.
 Suchoff, op. cit., pp. 365–366.
 Ibid., p. 367.
 ‘These degrees have absolutely no chordal function; on the contrary, they have a diatonic melodic function.’ Ibid., p. 376.
 Ibid., p. 410.
 D. Gow, ’Tonality and Structure in Bartók’s first two string Quartet,’s The Music Review 34 (1973), p. 261
 Suchoff, op. cit., p. 378.
 Cf. I. Oramo, ’Modale Symmetrie bei Bartók,’ Die Musikforschung 33 (1980), pp. 450–464.
 Suchoff, op. cit., pp. 376ff.
 Cf. however M. Gillies, Notation and Tonal Structure in Bartók’s Later Works (New York & London: Garland Publishing, Inc., 1989).