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Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle 1. According to Hungarian custom, the family name is followed, not preceded, by the baptismal name. This leads to a considerable amount of confusion in countries where the reverse custom prevails. In my study, the Western order is employed: Bela Bartok, rather than Bartok Bela, which is correct in Hungary.

In citing titles of Bartok’s works, I often use only the English titles as they appear in the comprehensive “List of Compositions” in Bela Bartok: A Guide to Research (Antokoletz 1997 [1988]: 5–43), but sometimes present the Hungarian titles as well, especially for major works such as Duke Bluebeard’s Castle (A kekszakallu herceg vara – literally The Bluebearded Duke’s Castle). In my study, the opera is variously referred to as Duke Bluebeard’s Castle, Bluebeard’s Castle, or Bluebeard.

Hungarian titles of scholarly publications are accompanied by English translations; those in French and German are given in the original languages. 2. Bartok gave opus numbers 1 to 21 to his early works. In 1898, he stopped assigning opus numbers to his compositions for six years. In 1904, with the Rhapsody for piano solo (Sz 26, BB 36a), Bartok began the numbering once again, giving opus numbers to compositions that he considered major works. In 1921, he assigned the opus number 21 to his Sonata for Violin and Piano No. in C-sharp-minor (I. Szonata hegedure es zongorara) (Sz 75, BB 84), dedicated to Jelly Aranyi (1893–1966), after which Bartok stopped using opus numbers for good. The Sz numbers refer to Andras Szollosy’s (1956) catalogue of Bartok’s mature works, “Bibliographie des oeuvres musicales et ecrits musicologiques de Bela Bartok,” in Bence Szabolcsi’s edition of Bartok, sa vie et son oeuvre. In Bela Bartok, Jozsef Ujfalussy (1971 [1970]: 400–430) revised that list with a few corrections.

In Bartok kompozicios modszere (Bela Bartok: Composition, Concepts and Autograph Sources), the Hungarian Bartok scholar, Laszlo Somfai (2000 [1996]) clarifies matters regarding active research into sources, and the methodology used in the complete edition (BB); see also, Somfai (1995). 2 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle mystery play (miszteriumjatek)3 the action is reduced to a minimum, with (internal) character development being the central dramatic process.

In Magyar szinpad (The Hungarian Stage), Bartok remarks on his opera as follows: It may sound peculiar but I must admit that the failure of my one-act play, Bluebeard’s Castle, prompted me to write The Wooden Prince. It is common knowledge that this opera of mine failed at a competition; the greatest hindrance to its stage production is that the plot offers only the spiritual conflict of two persons, and the music is confined to the description of that circumstance in abstract simplicity. Nothing else happens on the stage.

I am so fond of my opera that when I received the libretto [of The Wooden Prince] from Bela Balazs, my first idea was that the ballet – with its spectacular, picturesque, richly variegated actions – would make it possible to perform these works the same evening. I believe it is unnecessary to stress that the ballet is just as dear to me as my opera. (BBE 1976 [1917]: 406. ) Duke Bluebeard’s Castle was premiered on May 24, 1918, at the Magyar Kiralyi Operahaz (Hungarian Royal Opera House) in Budapest. On that occasion, Bartok declared: I set the miracle play Duke Bluebeard’s Castle from March to September 1911.

It was simultaneously my first stage [work] and my first vocal work. At that time conditions were not suitable for its performance, so that I showed it to count Miklos Banffy, and to conductor Egisto Tango only after the performance of The Wooden Prince. I am most grateful to them for sparing neither trouble nor pains in producing such a first-rate performance. Olga Haselbeck and Oszkar Kalman sang their parts so perfectly that the performance completely came up to my expectations. (BBE 1976 [1918]: 407. ) My primary purpose in this study is to investigate the pitch organization in Bartok’s opera by means of Aarre Joutsenvirta’s (b. 957) little-explored pitch-web analysis (1989). Put briefly, pitch-web analysis is a method of comparing the pitch content of different passages of music (tonal or atonal) so as to make explicit certain structural properties. A pitch-class set (PC set) is an unordered set of pitch-classes, without duplications, that are mutually related by one or more transformations (Forte 1973: 1, 3). Surface structures such as repetitions, octaves, and enharmonic spelling of individual notes are not properties of PC sets and do not influence the identity of the set.

For example, the pitch collection {0,3,7,12} contains the repetition of a pitch 3. In his writings, Balazs (1968 [1913]; and 1982 [1922]) referred to Bluebeard’s Castle variously as a “libretto” (libretto), “operaszoveg” (opera text), “drama” (drama), “damai jelenet” (dramatic scene), “miszteriumjatek” (mystery play), “szinpadi ballada” (scenic ballad) and “koltemeny” (poem). In the present study, Bluebeard’s Castle (1979 [1910]) will also be alternatively referred to in the generic terms given by Balazs; on the context of Balazs’s libretto, see Chapters 4. –4. 2, below. 1. Introduction — 3 class (0 = 12); hence {0,3,7,12} reduces to PC set {0,3,7} (Forte-number 3–11;4 Interval Class Vector [ICV] 0011105). In my study, the interval succession is shown as a sequence of numbers separated by commas and enclosed in brackets. Although interval succession provides useful information about the ordered relations between types of sets, a clearer and more concise representation is made possible by Joutsenvirta’s pitch-web square (Table 1. 1). 5 10 3 8 1 6 11 4 9 2 7 3 8 1 6 11 4 9 2 7 0 5 10 6 11 4 9 2 7 0 5 10 3 8 1 9 2 7 0 5 10 3 8 1 6 11 4 0 5 10 3 8 1 6 11 4 9 2 7 3 8 1 6 11 4 9 2 7 0 5 10 6 11 4 9 2 7 0 5 10 3 8 1 9 2 7 0 5 10 3 8 1 6 11 4 0 5 10 3 8 1 6 11 4 9 2 7 3 8 1 6 11 4 9 2 7 0 5 10 6 11 4 9 2 7 0 5 10 3 8 1 9 2 7 0 5 10 3 8 1 6 11 4 Table 1. 1. The 12 x 12 pitch-web square (Joutsenvirta 1989: 95). My analytic aim in this study is to demonstrate the existence and preponderance of set {0,1,4} in Bluebeard, clarified by means of Joutsenvirta’s (1989) pitch-web analysis.

It shows the role of the set in ordering the musical surface, without implying that the music exhibits a serial ordering. My claim is that, in Bartok’s Bluebeard, the different musical materials are unified by the ways in which set {0,1,4} becomes prominent in different sections (Example 1. 1, next page). I shall further propose that set {0,1,4} has supplanted the tonal system in Bartok’s opera, where temporary tonal centers are established by PC and intervallic centricity. In Tonality, Atonality, Pantonality, Rudolph Reti explains tonal centricity as follows: 4.

Allen Forte lists the prime forms of PC sets, and gives their inteval-class vectors (total interval content); for the latter, I hereinafter use Forte’s abbreviation, ICV (1973: 179–181; see also, the Appendix to this study). Forte lists 208 unordered sets in the twelve-tone system. Interval classes (ICs) indicate both the interval and its inversion (ibid. : 14, 20; see also, Cook 1987: 134, 326); the prime forms of sets are discussed in Forte (ibid. : 3, 5, 11–13); and ICVs of sets in Forte (ibid. : 15–18, 30–31). More explanation of PC sets is provided as this study goes on, especially in Chapter 2. (pp. 80–82). 5. The ICV (Forte 1973: 15–18, 30–31) is an array of six integers (mathematical notation) representing the IC content of a chord. The ICV exhibits the total interval content of the set. It is a practical and effective means to analyze the internal construction of a combination of PCs. The six numbered positions in the ICV stand for ICs 1 to 6. The numbers filling those positions show how many times that IC is represented in any set in the class. (Ibid. : 2–3; see also, Appendix. ) 4 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle

Example 1. 1. Set {0,1,4}. It is characterized by the fact that pitches, which by repetition and resumption become [prominent cornerstones] in the compositional course, can by this very quality assume a quasi-tonical role. This can help a group appear as a unit and can thus create form, even if the harmonic design of the group points to a different tonal path or to no [tonal] basis at all. (Reti 1958: 77. ) In Bartok’s opera, the unordered trichord set {0,1,4}, shown in Example 1. 1, has a fixed intervallic content, but not a fixed ordering.

It appears in foreground melodies and in long-term progressions, a point to which I return in Chapter 4. It is important to distinguish between ordered and unordered pitch intervals. Ordered pitch intervals are determined by subtracting the pitch number of the first pitch from the pitch number of the second pitch. This produces both positive and negative numbers, indicating ascending and descending intervals, respectively. The absolute value of the unordered pitch interval is used to determine the absolute size of an interval without regard to the order in which the pitches are presented. Forte 1973: 3, 60–61; and Straus 2000: 7–9. ) My claim is that the unordered set {0,1,4}, sometimes embedded in tetrachords {0,1,4,7}, {0,1,4,8}, and in the most typical Bartok chord {0,3,4,7}, also serve as links between certain themes, sections, and scenes. Those sets also establish largescale, organized structural coherence in Bartok’s opera (cf. pp. 195–196, Chapter 4. 3). On set-complex relations, Forte notes that: As sets are identified, and relations are revealed, it is usually not difficult to make judgments concerning significance. The most effective basis for these judgments is provided by set-complex relations. …] These relations, properly interpreted, will often point out sets of lesser significance, which would be more correctly replaced by a superset, or further decomposed into component subsets. (Forte 1973: ix. ) The statement above shows that a central concept of Forte’s brand of PC set analysis is “inclusion” (Forte 1973: 25). The concept of complete invariance – such as “inclusion under inversion” and “inclusion under transposition” – is discussed in Chapter 2. 4 (pp. 87–88). In the next quotation, Forte discusses the subset and superset relations determined by inclusion: 1. Introduction — 5

If all elements of a certain set X are also in a bigger set Y, then X is a subset, and Y is a superset of X. X is a proper subset of Y if the cardinal number6 of Y is greater than the cardinal number of X, and if every element of X is an element of Y. […] A set X is said to be a superset of a set Y if every element of Y is an element of X. X is a proper superset of Y if the cardinal number of X is greater than the cardinal number of Y, and if every element is an element of X. (Forte 1973: 211. ) In Duke Bluebeard’s Castle, the unordered, abstract PC set {0,1,4}, shown in Example 1. 1, is also found as a subset, e. . , in the tetrachord supersets {0,1,4,7} and {0,1,4,8}. Because no single method is sufficient to demonstrate the relations that I propose exist in Bluebeard, my concern is not with a mechanical application of pitch-web analysis alone. Rather, a synthesis of pitch-web (Joutsenvirta 1989) and motivic analysis (Cook 1987: 89–115, 151, 165) best describes the rapprochement between pitch structures and dramatic expression in the opera. Bluebeard’s Castle is especially interesting when studied from the viewpoint of pitch-web analysis, since it contains elements derived from various musical languages.

These include, e. g. , functional tonality, folk song modality, and symmetrical pitch combinations. I focus on harmonic constructs in terms of their intervallic properties, thus suggesting the secondary importance of traditional functional concepts. My analytical strategies produce results that are often strikingly different from those found in single-theory studies. I integrate various types of theoretical approaches, analytical techniques, and methodologies of Hungarian scholar Erno Lendvai, and of American theorists Elliott Antokoletz, Allen Forte, and Paul Wilson (who receives only limited attention).

Through those perspectives, my study gives a detailed analysis of the musical design and thematic material of Bluebeard’s Castle, in a way that combines, complements, and expands upon the studies of those theorists. Adding a new aspect to the field of Bartok analysis, I demonstrate the PC sets through application of the Finnish theorist Aarre Joutsenvirta’s (1989) savelverkko (pitch-web) analysis and two-dimensional savelverkkoruutu (pitch-web square), shown in Table 1. 1 (p. 3) and described in the author’s Introduction to Pitch-Web Analysis.

In measuring PC intervals, I follow Forte in my use of closed Modulo 12 (Mod 12) arithmetic (also called Clock Math or Modulo Math), to which Joutsenvirta’s ideas are closely related. The common thread between Forte and Joutsenvirta is their reliance on theoretical concepts and systems that were developed mainly to deal with music generally labeled as “atonal. ” By contrast, Lendvai, Antokoletz, and Wilson are Bartok specialists par excellence. Those theorists’ groundbreaking ideas were formulated independently of each other, and each focuses on a different phase of the analytical spectrum.

An integrated analytic methodology yields fruitful results, when applied to local and global tonal centers of Bartok’s opera (Chapter 4. 3). I examine how Bartok displaced the functional connections of traditional tonality and created a new 6. Cardinal numbers, or cardinals for short, are numbers used to denote the size of a set. It means the number of elements in a set, discounting repetitions. It denotes quantity, not ordering. (Forte 1973: 3, 12, 19, 209. ) 6 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle sonic universe in Bluebeard’s Castle.

My examination will show how Bartok used polymodal scale combinations (Lendvai 1968) to create a twelve-note chromatic environment with a single fixed point of reference (Wilson 1982; 1992; and 1993). This involves examination of the use of pitch-cell interval content (Antokoletz 1984: 16, 78–137), the main analytical tool for which is Joutsenvirta’s pitch-web square (1989: 95; see also, Table 1. 1, p. 3). Through the analytical process, thematic and formal coherence can be established in Bluebeard’s Castle, because the preponderance of particular intervals in the structure of set {0,1,4} automatically assures a certain homogeneity.

Necessarily related sub-questions are also engaged, concerning other important structural formations that Bartok incorporated in the opera, including chordal settings and phrase structures (Chapter 4). First, however, the structure of that set and the manner in which it is employed require special consideration. Forte’s (1973) set theory operates with particular reference to three sequences of numbers: 1. The collection representing the twelve different PCs, numbered from 0 to 11, where C is designated 0. 2. The collection representing the six different ICs, numbered from 0 to 6. 3.

The sequence of prime form PC sets, which total 224, or 208 if the sets of cardinality (total number of PCs) 0, 1, 2, 10, 11, and 12 are excluded. Forte’s way of mapping certainly draws attention to various pitch-class and interval similarities among sets. Yet, a more differentiated means of comparing particular events in a musical piece is needed, if translating pitch, via PC, into number is to be justified. According to Joutsenvirta (1989: 1), to make comparisons means to observe the possible degrees of differences and similarities, so that, in the end, their relative musical significance may be evaluated in specific contexts. Figures 1. –1. 4 display sets {0,1,4}, {0,1,4,7}, {0,1,4,8},and {0,3,4,7} through Joutsenvirta’s 12 x 12 pitch-web square (Joutsenvirta 1989: 95; see also, Table 1. 1, p. 3). Those sets share certain characteristics; moreover, each replicates itself in inversion in Bartok’s opera. The designation “set” signifies that the pitches may occur in vertical (harmonic) or horizontal (melodic) configurations, or in both. 1 4 0 Figure 1. 1. Set {0,1,4}. 1 4 7 0 Figure 1. 2. Set {0,1,4,7}. 1. Introduction — 7 8 1 4 0 Figure 1. 3. Set {0,1,4,8}. 4 7 0 3 Figure 1. 4. Set {0,3,4,7}. In this study, the sets shown in Figures 1. 1–1. 4 serve as the point de depart.

They have idiomatic differences from Antokoletz’s starting point, which is based on intervallic cells (for more details, see especially, pp. 73–75, Chapter 2. 3). According to Antokoletz (1984: 89–92), the musical language Bartok uses in Bluebeard is a fusion of tonal writing, with an emphasis on old-style Hungarian peasant songs7, with a deliberate use of dissonance. In “Bartok’s Bluebeard: The Sources of Its Modernism,” Antokoletz (1990: 79) argues that the opera is based on the shifting relations among semitones, and on the various transformations of those relations, both of which appear to be associated with special dramatic situations.

Antokoletz (1984: 89–92) takes Bluebeard as an illustration of Bartok’s use of dissonant intervallic cells for dramatic purposes, especially the X-cell, shown as set {0,1,2,3} in Figure 1. 5 and as used in Example 1. 2 (next page). 1 0 2 3 Figure 1. 5. Set {0,1,2,3}. The excerpt from Bartok’s Bluebeard’s Castle, shown in Example 1. 2 (next page), is a prominent foreground statement of X-cell (Figure 1. 5). It occurs in the “Torture Chamber” scene, the first of the doors to be opened by Judith. The text at this point reveals that Judith has just noticed blood in Bluebeard’s castle: “A te varad fala verzik” (Your castle’s walls are bleeding). I take Antokoletz’s (1984: 89–92; and 1990) analysis, discussed briefly above, as a point of departure for my own. My fundamental concern is to analyze pitch relations, not on a larger-sized scale, but in terms of smaller-unit PC sets, to show how Bartok incorporates dissonance into the music. My assumption, however, is the same as Antokoletz’s: the opera develops toward “increasing dissonance” (1990: 79). 7. Old-style folk songs are discussed in Chapter 3. 5. 1. 8. For a detailed analysis of the “First Door” scene, see Chapter 4. 4. . ld major and minor modes, and even of the church modes, while the chromaticism of the German school deliberately leads away from tonality. The advantage of polytonality is that it still leaves room for stylistic possibilities such as neo-classicism, and for music based on folk song. Bimodality can actually lead to panchromaticism, from a melodic point of view: an ascending Lydian scale and a descending Phrygian scale within the same octave produce the aggregate of twelve notes, with one note occurring twice and another three times. BBE 1976 [1943]: 367. ) 63. Erich Kapst (1972) discusses the primacy that tonality had for Bartok, focusing on pentatonic, modal, whole-tone, chromatic, and especially polymodal-chromatic combinations in the composer’s works; for more on this issue, see Karpati (1982). 32 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle […] the study of folk music […] freed me from the tyrannical rule of the major and minor keys. The greater part of the collected treasure, and the ore valuable part, were in old ecclesiastical or old Greek modes, or based on more primitive (pentatonic) scales. […] It became clear to me that the old modes, which had been forgotten in our music, had lost nothing of their vigor. […] This new 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. (BBE 1976 [1921a]: 410. )

Bartok’s statement above emphasizes a very important aspect of his harmonic language: it is not subservient to the traditional hierarchy of tonic–dominant functions (T–D), nor does it necessarily incorporate other harmonic relations that characterize art music of the Western tradition (Helm 1953;64 Travis 1970;65 and Starr 197366). By contrast, Bartok’s own tonality is often best conceived “non-harmonically”, as musicologist Eino Roiha describes: We may briefly say that we comprehend a composition melodically when the harmonic cadence-feeling does not play any role in our mind.

We perceive tone-relations, but there is no latent feeling of harmony, which in its own way strengthens and leads the tonal process and its organization. (Roiha 1956: 41. ) Tonality, as the term is used most broadly in Bartok’s “Harvard Lectures”, refers to the domination of a single tone over a section of music; all other tones resolve to it, either in actuality or potentially (BBE 1976 [1943]: 365). For Bartok’s music, developments taking place in new art music proved less decisive than the influence of peasant music (see pp. 168–169, Chapter 3. ); in the latter, he found new kinds of tonal structures, which he could further develop and shape into his unique personal style (ibid. : 367; and Gillies 198267). Milton Babbitt (1949: 378), writing on the String Quartets, suggests that Bartok was a “traditionalist” who employed “generalized functional tonal relationships”, 64. Helm (1953) examines thematic relations and gives a brief overview of the movements, and concludes that tonality is present, but not in the traditional sense of tonic, dominant, and subdominant functions. 65.

Roy Travis (1970) attempts to explain the pitch structures of Bartok’s Fourth String Quartet in terms of traditional tonality, by taking a Schenkerian approach to analyzing the piece. His voice-leading graphs of foreground, middleground, and background levels are intended to show the contrapuntal prolongations of structural harmonies (scale-steps I, IV, V, I). However, Travis replaces Schenker’s crucial concept of the tonic triad as Ur-Akkord with that of a “dissonant tonic” sonority. 66. Starr (1973) attempts to join certain features of structure and prolongation.

He merges the familiar methodological approach of Schenker-Salzer-Travis with more recently established concepts of structure based on axes of symmetry. 67. Gillies (1982) advances a theory of Bartok’s tonality and modality based on the composer’s “Harvard Lectures” (BBE 1976 [1943]), the principles of which are especially applicable to the late works. Notation (e. g. , E instead of F-flat) is essential for discovering the tonal and modal bases of musical passages, modulation, and pitch hierarchy. Among other means of establishing a tonal center, the author discusses encirclement/half-encirclement (leading-tone motions). . In search of a system — 33 even though the “exclusive employment of unique, internally defined relationships leads to a considerable sacrifice of tonal motivation”. (Also see, especially, Monelle 1970;68 and Nordwall 1972. 69) In Die Tonalitat im Instrumental-Schaffen von Bela Bartok, Peter Petersen (1971: 15–30) presents almost everything that Bartok ever said about pitch organization and tonality, with the exception of a few rather sketchy analyses the composer made of some of his own works (BBE 1976 [1943]). 0 According to Petersen, Bartok recognized two components in tonality: a controlling pitch, and a system of relationships. The latter could be predetermined (based on modes, pentatonic structures, etc. ) or constructed from all twelve pitches available in the chromatic system. As to the latter, Bartok sometimes created chromatic fields by using melodic mirror-inversion, as in Mikrokosmos No. 144, “Minor Seconds, Major Sevenths” (Sz 107, BB 105, 1926–1939); in the Sonata for Two Pianos and Percussion, by contrast, twelve-tone pitch groups are repeated in ostinato fashion.

Petersen (1971) also demonstrates that Bartok’s later writings and music both develop in the direction of a new concept of tonality, which Peterson identifies as having three principal ingredients: 1. Tonality is no longer based on the traditional diatonic hierarchy, but on the total chromatic resource as afforded by polymodality71. 2. Dominating pitches are no longer determined by their position in the scale, but by a variety of other criteria. 3.

The configuration of given chords no longer indicates their tonal function per se, partly because their very structure is equivocal, and partly because this structure may be no more than the result of horizontal motion. As a conclusion, Petersen remarks that “ist Bartoks Tonalitat insgesamt eher durch das Linear–Polyphone bestimmt als durch das Akkordisch–Homophone” (Petersen 1971: 179). Malcolm Gillies (1983), in his study of the sketches of the Sonata for Solo Violin, argues that new insights into Bartok’s use of tonality and modality are to be gained from three sources: 68.

Raymond Monelle (1970) gives descriptive analysis of various passages of the String Quartets, differentiating those features that are classically derived from those that are radical, “astringent,” and originating in territories removed from those of Western art-music. 69. In Bela Bartok: Traditionalist / Modernist, Ove Nordwall (1972) includes biographical and autobiographical materials, in addition to a study of the numbering of Bartok’s works in the context of their relationship to traditional and modern features of other composers; of special interest to this study is his examination of tonal relations in Bluebeard. 0. Bela Bartok Essays (BBE) were compiled and edited by the leading expert on Bartok’s music and writings, Benjamin Suchoff (1976). Included therein are Bartok’s writings on musical folklore, book reviews, polemics, autobiographical statements, brief analyses of his own music, and discussions of other music and musicians. 71. The term essentially designates two different meanings in the case of Bartok and Messiaen. In Technique de mon langage musical, Messiaen (1966 [1944]: 61–63) uses “polymodality” only in reference to his own theoretical concoctions, the “modes of limited transposition”. 4 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle 1. the composer’s ethnomusicological writings; 2. his writings on wider musical issues; 3. the compositional process: from sketch to revised published score. According to Gillies, the sketches suggest that notation is one clue as to how the composer produced tonal centers (e. g. , by half-step encirclement of a given tone). One of the most important concepts in Bartok’s harmonic thinking was the idea of polytonality, polymodality and bimodality.

In Twentieth Century Harmony, composer and theoretician Vincent Persichetti summarizes those terms as follows: Polytonal writing is a procedure in which two or more keys are combined simultaneously. If only two keys are sounded, the specific term bitonality may be used, but polytonality has come generally to imply the use of more than one tonal plane at the same time. The scales that form the different tonic centers may be intervallically identical or contrasting, traditional or synthetic. (Persichetti 1961: 255. )

In Polymodal Chromaticism, Lendvai (1980) points out that the basic texture of Bartok’s music remained true to tonality, which he expanded to include chromatic polymodal structures and starkly dissonant chordal combinations. (For more on Bartok’s adaptations of traditional tonality, see R. Honti 2006: 698–703; cf. also, pp. 58–60, Chapter 2. 2. 1. ) Bartok, Stravinsky, and Karol Szymanowski (1882–1937) are among those who perfected the art of writing in several keys at once: a procedure that charges the harmony with competing tonal centers, while retaining a differentiated musical space.

Bartok explains: I used the term “bi-modality”, a little-known designation. There are, however, two other very frequently used terms, I would almost say “slogans”: atonality (or twelve-tone system) and pantonality (if only two parts are concerned, bi-tonality). […] Polytonality means the use of different diatonic keys in music of two or more parts, each part in a special key. The pioneers of polytonality used to present their system as the opposite of atonality: the former doubling, tripling, or quadrupling tonality, the latter abolishing tonality, or purporting to do so. …] Polytonality exists only for the eye, [that is to say, only] when one looks at such music. But our mental hearing 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. […] Our hearing cannot perceive two or more different keys with two or more different fundamental tones, as such; it will simplify matters by reducing the maze of keys to one principal key. (BBE 1976 [1943]: 365–366. In The Craft of Musical Composition, Paul Hindemith gives a similar opinion on the matter: The game of letting two or more tonalities run along side by side and so achieving new harmonic effects is, to be sure, very entertaining for the composer, but the 2. In search of a system — 35 listener cannot follow the separate tonalities, for he relates every simultaneous combination of sounds to a root – and thus we see the futility of the game […] polytonality is not a particular principle of the composition. (Hindemith 1942 [1937]: 156. )

As will be shown later (Chapters 4. 3–4. 4), bitonality and/or polytonality alone cannot explain the pitch structures of Bluebeard’s Castle. In that opera, the chord formations, the avoidance of traditional tonal patterns of resolution, and tendency toward bimodal textures and dissonant voice-orchestra relationships give Bartok’s tonality a broader, less easily defined meaning. To continue the present discussion: Bartok’s tonality is based upon a new system, which made possible the adaptation of the diatonic modes of peasant music to a free use of the twelve notes.

The new compositional devices derive from what Bartok calls “polymodality”,72 by which term he designated the simultaneous use of two or more of the Church modes, based on a common pitch, to derive extensions from the basic diatonic collection (BBE 1976 [1943]: 363–364; see also, Antokoletz 1984: 2, 27–28, 51–66, 204–270). Bartok’s complex and variegated approach to polymodal interaction in his works is best described in the following comment by the composer, to which I shall return at several points in this study: In our works, as well as in other contemporary works, various methods and principles cross each other.

For instance, you cannot expect to find among our works one in which the upper part continually uses a certain mode and the lower part continuously uses another mode. So if we say our art music is polymodal, this only means that polymodality or bimodality appears in longer or shorter portions of our work, sometimes only in single bars. So change may occur from bar to bar, or even from beat to beat in a bar. (BBE 1976 [1943]: 370. )

In his analysis of Bartok’s Fourth Quartet, Colin Mason (1957) asserts that polymodality, based on simultaneous modes on a common tonic, is subordinate to symmetrical formations in the pitch organization. Nevertheless, tonal implications are gradually fulfilled in the course of the work, which gradually grows into the key of C. Mason draws the following, general conclusion about the composer’s harmonic style: […] he [Bartok] had already discovered, in the Piano Sonata and the first Concerto, what was to be his final, “tonal” solution of the problem of total chromaticism, in polymodality (i. . , a multiplicity of simultaneous modes on a common tonic). In these two works, he had used this polymodality only within a restricted and to some extent uncharacteristic linear neo-classical style. 73 Then in the Third Quar72. On polymodality, see also, Karpati (1971). 73. What, then, is Bartok’s relationship to neo-classicism? It is certainly quite different from Stravinsky’s. To speak of contemporary composers in terms of classicism and romanticism, as if they could be essentially either in an age that is itself neither, may seem futile.

The terms are, however, useful for historical reference and for a shorthand way of drawing attention to analogies, however incomplete. In addition, “neo-classicism” forces the issue on one’s attention. The term is 36 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle tet, he had tried to apply it to his most advanced expressionist style, and later, in the Fifth Quartet, and thereafter, he was to find in it the means of restoring to, and controlling in, his music, the lyrical richness and freedom of harmony of his early period. Mason 1957: 195. ) On the chance that Mason has too narrowly defined Bartok’s solution to the “problem of total chromaticism”, it is perhaps a good idea to quote the composer himself: One point, in particular, I must again stress: our peasant music, naturally, is invariably tonal, if not always in the sense that the inflexible major and minor system is tonal. (An “atonal” folk music, in my opinion, is unthinkable. ) Since we depend upon a tonal basis of this kind in our creative work, it is quite self-evident that our works are quite pronouncedly tonal in type.

I must admit, however, that there was a time when I thought I was approaching a species of twelve-tone music. Yet even in works of that period, the absolute tonal foundation is unmistakable. (BBE 1976 [1928]: 338–339. ) In itself, the fact that a definite tonality, which arranges movements and musical works around a single center, disappears from music, is neither a positive nor a negative development. It is inevitable: music has tended in this direction since Beethoven. (Putz 1968;74 and Seiber 194575. The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments. According to Persichetti: Atonality is a term loosely applied to music in which a definite key feeling has been weakened or lost, and to music in which no key gravitation ever existed. Atonal writing is the organization of sound without key establishment by chordal root relationships; but tone combinations or areas may form an atonal equivalent of tonality. …] Atonal movement is often linear but may produce vertical combinations of mixed intervals, compound harmonies that are free from the power of an overbearing tonic. (Persichetti 1961: 261. ) intended to evoke a specialized way of writing and a certain attitude towards music, but does not necessarily have any relation to other or wider meanings of the word “classicism”. Bartok is not neo-classical in this sense, nor does his espousal of eighteenth-century principles make him any the more so. However, there are phases in his development for which the labels “classical” and “romantic” are useful.

In great music, the two are balanced, according to (1) whether the composer has a feeling or idea, then finds the notes to express it; or (2) whether he trusts in his own musical material. How that balance is struck is one measure of the importance of Bartok’s life and work: perhaps the means he found for achieving equilibrium will be the means by which composers of the future will find it, too. 74. Werner Putz (1968) relates Bartok’s use of motivic development to that of Beethoven, and points to the folk-like materials within the contrapuntal contexts as the basis for both composers’ means of expanding inherited compositional designs. 5. In The “String Quartets” of Bela Bartok (1945), the Hungarian-born composer and cellist Matyas Seiber (1905–1960) approaches Bartok’s string quartets as his most representative works, suggesting an analogy to Beethoven’s development in his quartets. 2. In search of a system — 37 Bartok’s absorption of folk music properties into modern music of an increasingly atonal tendency led to works that are not tonal per se, but rather what Bartok described as “quite pronouncedly tonal in type” (BBE 1976 [1928]: 338). As will be shown, Bluebeard’s Castle falls into the latter category (Chapters 4. –4. 4). Bartok felt that his usage of dissonance had moderated somewhat by the year 1910. In September of that year, Bartok wrote to Frederick Delius (1862–1934) in response to the Parisian composer’s criticism of the “arbitrary” dissonance in his Suite No. 2: Since the piano pieces, I have become more “harmonious” again, so that I know no longer need the contradictory accumulation of dissonances to express the feeling of a mood. This may possibly be a result of my giving way more to the influence of folk music. Quoted in Demeny, ed. 1971: 105. ) Delius was not the only one to notice the “atonal” characteristics of the composer’s earlier works. In “Bartok, Schoenberg, and Some Songs”, Adelan Collins points out that: The Five Songs […] are not only worth study for their own sake, but are also of great value as a key to the understanding of a certain amount of Bartok’s later work, and even to the nature of atonic [sic! ] music in general. There are, of course degrees in atonality; and these songs are by no means wholly atonic.

But one harmonic device is so quickly displaced by another that the same rapid adjustment of the point of view from moment to moment is required from the listener as in the understanding of any purely atonic piece of Schoenberg. (Collins 1929: 177. ) Schoenberg, of course, went on to make the transition from atonal to twelve-tone composition, which Joseph N. Straus aptly summarizes: The twelve tones are arranged in series, which can be inverted, reversed, and transformed in all the ways that are familiar from contrapuntal writing in the tonal tradition.

Certain rules of the musical syntax are then adopted. Unlike the rules of tonal harmony, these rules have an a priori character: they are laid down in advance, as willed constrains on the composer’s practice. (Straus 2000: 144. ) Carl Dahlhaus (1928–1989) reminds us that Schoenberg found the word “atonal” offensive and that the reason why he had renounced tonality, as stated in the composer’s Harmonielehre (1911), was to enable the “emancipation of dissonance” (1987 [1968]: 120–121).

The specific distinction between consonance and dissonance, the division of intervals into those two contrasting groups, was based on compositional technique, argued Schoenberg, not on the unalterable nature of the consonance/dissonance dichotomy itself. Schoenberg preferred the term “pantonality” over atonality to describe his compositional method of using the twelve tones, and he referred to movable tonics and 38 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle fluctuating harmonies (Dahlhaus 1987b [1978]: 80). 6 Reti (1958) describes pantonality as a distinct form of tonal organization, and the final and lasting result of the modernist experiments. Reti writes in this context of a “tonality, which does not appear on the surface but is created by the ear singling out hidden relationships between various points of a melodic or contrapuntal web” (ibid. : 65). Reti further suggests that any pitch-class can function as a tonic, and a piece may keep an indefinite number of tonics in play without canceling their primary function, either as the focus of melodic organization, or as the reference point against which the harmonies must be read (ibid. . Thus, the basic principle behind composition with twelve-tones is the even distribution of PCs such that none of them dominates the others. As Schoenberg clarifies, “the construction of the set of twelve-tones derives from the intention to postpone the repetition of every tone as long as possible” (1984: 246). This is called a twelvetone row, or series, which is a group of PCs (usually the 12 chromatic PCs), set {0,1,2,3,4,5,6,7,8,9,10,11},77 placed in a particular order to be so used in a composition (Figure 2. 3). 2 7 0 5 10 3 8 1 6 11 4 9 Figure 2. 3. Set {0,1,2,3,4,5,6,7,8,9,10,11}.

Schoenberg (1984: 127) took pains to minimize the differences between tonal and twelve-tone music, stressing that in the evolution from diatonic to chromatic there had been no break with or divergence from natural musical laws. In some of his writings, Schoenberg refuted the idea of “natural law” in music, as in the preface to his Harmonielehre (1978 [1911]: 8–9). There he referred to rules traditionally called “laws of harmony”, such as the avoidance of parallel fifths. These Schoenberg considered mere practices, not “natural laws”, but he did not question the idea that certain universal laws existed in music. 8 In The Listening Composer, Perle remarks that: 76. It is clear that this new twelve-note method of organization rescued music from the tempting dangers of the freedom it had so recently gained. Indeed, the new rules were stricter than the old rules of tonality. They are comparable in strictness with the advanced canonic techniques of tonal music, and often draw upon that style; for example, in the usage of mirror and retrograde structures. (Perle 1962: 7–8, 148–149. ) 77. Forte’s “List of Prime Forms and Interval Class Vectors” (see the Appendix) does not contain this set.

In Joutsenvirta’s pitch-web analysis (1989: 6), the 12 chromatic PCs are called “3 x 4 kromaruudukko” (“3 x 4 chroma-square”). 78. On Schoenberg’s compositional technique, see also, Wellesz (1989a [1926]); (1989b [1926]); (1989c [1926]); and (1989d [1926]). 2. In search of a system — 39 The achievement of such a change of register through a sequential progression is a familiar procedure in the music of the “common practice”. The significant distinction is that where Berg subdivides the registral span into equal, i. e. cyclic, intervals, his tonal predecessors subdivide it, in changing register through sequential transference, into the unequal intervals of the diatonic scale. […] The qualitative transformation in the language of music, which we have experienced in our century, has a long prehistory. Beginning with Schubert, we occasionally find normal diatonic functions questioned in changes of key that progress along the intervals of the whole-tone scale, or the diminished seventh chord, or the augmented triad. An even more radical example of a cyclic progression in a tonal composition is […] from Wagner [Die Walkure, Act III]. Perle 1990a: 96. ) Later Perle observes that: If [Alban] Berg departs so radically from tradition, through his substitution of a symmetrical partitioning of the octave for the asymmetrical portioning of the major/minor system, he departs just as radically from the twelve-tone tradition that is represented in the music of Schoenberg and Webern, for whom the twelvetone series was always79 an integral structure that could be transposed only as a unit, and for whom twelve-tone music always implied a constant and equivalent circulation of the totality of pitch classes. Perle 1990a: 98. ) Bartok (BBE 1976 [1920c]) explains the principal difference between his concept of harmonic dissolution and that achieved by composers such as Schoenberg, Webern, and Berg. Bartok talks of contemporary music striving toward atonality, which is a gradual development stemming from tonality.

On twelve-tone music, Bartok remarks as follows: […] the decisive turn toward atonality began only when […] the need was felt for the equality of rights of the individual twelve tones of our dodecaphonic mode: when the attempt was made to avoid arrangement of the twelve tones according to certain scalar systems or to attribute to the individual tones greater or lesser value […], so that use could be made of the individual tones in any optional combination horizontally as well as vertically irretrievable to any scalar system.

It is true that certain tones in this combination also gain by this procedure a relative predominance; this difference of importance, however, is not based on a certain scale pattern but is the outcome of the occasional combination. […] The possibilities of expression are increased in great measure, incalculable as yet, by the free and equal treatment of the twelve tones. (BBE 1976 [1920a]: 455. ) Bartok looks at Schoenberg’s (1978 [1911]) Harmonielehre regarding this development, saying that atonality came from the need for equality among the twelve tones. 0 Bartok gives examples of adjacent, dissonant-note sonorities, and suggests 79. Perle’s original emphasis. 80. Bartok (BBE 1976 [1920c]) gives examples of new types of chords (comprised of fourths and mixed intervals). He also discusses briefly some principles of organizing atonal material, and ends with a statement regarding notation for symbolizing the twelve tones equally. 40 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle a more varied approach to atonality than is found in the dodecaphonic methods of Schoenberg.

He proposed the retention of certain features of the tonal system, such as triads and scale constructions, claiming that, as isolated occurrences, they need not imply a sense of tonality (BBE 1976 [1920c]; see also, Bardos 1974: 138). Bartok never made exclusive use of dodecaphony, and made clear his intention never to sacrifice tradition merely for the sake of innovation. In his view, the proper use of twelve-tone methods was to supplement and enrich traditional musical thought. The most obvious musical embodiment of those ideas is Bartok’s Three Studies for piano (Op. 8/3, Sz 72, BB 81, 1918), which is undoubtedly one of his most radical creations (on the latter, see Stevens 1964: 125;81 Dille 1965;82 and Orvis 197483). Bartok’s music is by no means atonal; rather, the term tonality means something quite different in his case. The main problem lies in the tension between two apparently opposite ideas. Comparing Schoenberg and Bartok, one finds the basic difference to be that Bartok used a system combining two or more modal segments, based on a single fundamental pitch.

That system enabled Bartok to use all twelve pitches of the chromatic spectrum. (BBE 1976 [1920c]: 455. ) What is unique about the situation, is that the use of the special twelve-tone set in Bartok’s works is analogous to the pre-compositional assumption of the traditional major and minor scales of tonal music, shown in Figures 2. 4 and 2. 5, respectively. 84 (For a different view, see Karpati 1969b; 2004 [1963–1964]; and 1966. )85 11 4 9 2 7 0 5 Figure 2. 4. Set {0,2,4,5,7,9,11}. 2 7 0 5 10 3 8 Figure 2. 5.

Set {0,2,3,5,7,8,10}. 81. Stevens (1964: 125) remarks that Bartok’s Studies deal with specific pianistic problems not engaged in the etudes of Chopin, Debussy, Stravinsky, or Prokofiev. Although they encompass a smaller range of expression, they are technically and tonally beyond anything found in his earlier piano works. 82. Dille (1965) discusses the personal connections between the two composers, and the bearing those had on their subsequent exchanges of opinion on artistic issues; see also, Demeny (ed. 1976: 262). 83.

In Technical and Stylistic Features of the Piano Etudes of Stravinsky, Bartok, and Prokofiev, Joan Orvis (1974) discusses the technical difficulty of performing Bartok’s Etudes, which are described as displaying atonal principles. 84. Forte’s (1973: 179–181) “List of Prime Forms and Interval Class Vectors” does not contain these sets. 85. Karpati (1966) relates Bartok’s harmonic language to the twelve-tone idiom and to some of Stravinsky’s techniques, concluding that Bartok provided a synthesis of those divergent trends. 2. In search of a system — 41

Like Schoenberg, Bartok emphasizes the “natural” evolution of atonality: The music of our times strives decidedly toward atonality. Yet it does not seem to be right to interpret the principle of tonality as the absolute opposite of atonality. The latter is much more the consequence of a gradual development, originating from tonality, absolutely proceeding step by step – without any gaps or violent leaps. (BBE 1976 [1920c]: 455. ) Schoenberg (1975 [1947]: 166) justified his own innovations by appeals to (German) tradition, but spoke disparagingly of the “static treatment of folk song” (ibid. 165). 86 Many other composers and theorists also saw art music and folk music as basically incompatible, and the merging of these styles as a serious problem that could not be conceptually resolved (Schoenberg 1984: 161–184). Bartok felt obliged to respond, saying that Schoenberg “is free from all peasant influence and his complete alienation to Nature, which of course I do not regard as a blemish, [but] is no doubt the reason why many find his work so difficult to understand” (BBE 1976 [1921e]: 326; see also, Demeny 1946: 27–28). Most western treatises on harmony, from Renaissance times (e. . , Zarlino) to the mid-twentieth century (e. g. , Hindemith), seek to codify explanatory principles of chord construction and progression. It is commonly acknowledged that these were most lastingly codified in the theoretical works of Jean-Phillipe Rameau (1683– 1764). In his Traite de l’harmonie reduite a ses principes naturels (1971 [1722]), Rameau, as others before and after him, started from the naturally occurring acoustical phenomenon of the overtone series (as demonstrable on any vibrating body) (Example 2. 1). 87 Example 2. 1. The natural overtone series.

Schoenberg (1978 [1911]), following that theoretical tradition, explains dissonances as more distant overtones, which in principle are as easy to understand as consonances. According to Schoenberg, the method of composing with twelve tones was the fruit of striving for a deeper logic. Still, for both Schoenberg and his pupil, Anton Webern, the materials of music originated in the natural phenomenon of the 86. Schoenberg (1975 [1947]) elsewhere in the article voices his belief that the use of folk music in compositions inevitably produces incoherence and stasis. 87.

In the Introduction to the Psychology of Music (1954: 17–20), the Hungarian pioneer of experimental psychology, Geza Revesz (1878–1955), offers a comprehensive overview of research in both psychology and acoustics. Revesz, as many acousticians have done (e. g. , Helmholtz), remarks that, in addition to the overtones (Example 2. 1, above), there are also “combination tones”, which can be heard when two notes sound together. Both overtones and combination tones confirm the “natural” primacy of the (major) triad. 42 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle vertone series. The notes of the diatonic scale are found among the lower overtones, and the chromatic notes as higher overtones. In 1930, in an essay later published in his The Path to the New Music, Webern regarded the development of Western music over the centuries as “the ever-extending conquest of the material provided by nature” (1975 [1960]: 17). He saw no reason why the quarter-tones and other microtones among still higher overtones would not someday be used (on Bartok’s use of microtones, see Nordwall 196588).

Viewing the overtone series as a continuum, Schoenberg saw no essential differences between intervals that formerly were differentiated as consonances and dissonances: What distinguishes dissonances from consonances is not a greater or lesser degree of beauty, but a greater or lesser degree of comprehensibility. In my Harmonielehre, I presented the theory that dissonant tones appear later among the overtones, for which reason the ear is less intimately acquainted with them. This phenomenon does not justify such sharply contradictory terms as concord and discord.

Closer acquaintance with the more remote consonances – the dissonances that is – gradually eliminated the difficulty of comprehension and finally admitted not only the emancipation of dominant and other seventh chords, diminished seventh and augmented triads, but also the emancipation of Wagner’s, Strauss’s, Mussorgsky’s, Debussy’s, Mahler’s, Puccini’s, and Reger’s more remote dissonances. (Schoenberg 1984: 216–217. ) Schoenberg proposed that harmonic theory be based “not on the seven tones of the major scale, but on the twelve [tones] of the chromatic scale”, offering a brief outline as an initial step in that direction (1978 [1911]: 387).

Bartok described the contrast between works based on an atonal system and those fixed in a concept governed by tonal centricity as follows: To point out the essential difference between atonality, polytonality, and polymodality, in a final word on this subject, we may say that atonal music offers no fundamental tone at all, polytonality offers – or is supposed to offer – several of them, and polymodality offers a single tone. Therefore our music, I mean the new Hungarian art music, is always based on a single fundamental tone, in its sections as well as in its whole.

And the same is the case with Stravinsky’s music. He lays stress on this circumstance even in the titles of some of his works. He says, for instance, “Concerto in A”. The designation “major” or “minor”, however, is omitted; for the quality of the third degree is not fixed. (BBE 1976 [1943]: 370–371. ) Applied to Bartok’s music, therefore, the term tonality refers to the way in which a given tone acts as a focal point for a constellation of vertical sonorities and pitch relationships, based loosely (at this point in his career) on the language of tonality, which he had inherited as a composer orking in the early twentieth century. 88. Nordwall (1965) discusses significant differences between the original version and the published edition of the Sonata for violin, the finale of which provides the clearest example of Bartok’s use of microtones. On the same topic, see also, Bartok’s letter to violinist Yehudi Menuhin (in Demeny, ed. 1976: 696–697). 2. In search of a system — 43 When in the 1920’s Bartok looked back to what he had written ten years earlier, he re-avowed his unswerving adherence to tonality. 9 Bartok spoke of the Three Studies, in a letter to von der Null: “Even in the Studies there are firmly held, prominent centers of sound (masses of sound at the same pitch), as a consequence of which, regardless of anything else, an effect of tonality is evoked” (quoted in Null 1930: 58). 2. 1. 4. New chromaticism Certain theorists at the beginning of the twentieth century, most notably Schoenberg and his followers, developed an understanding of chromaticism that was intended not only to pave the way to new compositional methods, but that also had the potential to shed new light on the music of their predecessors.

Viewed in many quarters as a radical who had broken away from the natural laws, which had governed music for centuries, Schoenberg insisted all his life that, in creating the method of composition with twelve tones, he was simply carrying forward an evolutionary process that linked him with previous generations of German masters (for more on Schoenberg, see pp. 37–42, Chapter 2. 1. 3). Bartok remarks on his own new chromaticism as follows: As to the general characteristics, exactly the same can be said about my melodies as what I said already concerning the chromatic folk melodies.

That is, the single tones of these melodies are independent tones having no interrelation between each other. There is in each specimen, however, a decidedly fixed fundamental tone to which the other tones resolve in the end. The main difference between the chromatic folk melodies and my own chromatic melodies is to be found in their range. The former consists exclusively of five, six, or at most seven half tones, which correspond to a range of about a fourth.

My own melodies generally have at least eight half tones and cover, in some cases, the distance of an octave or more. […] The working [sic! ] with these chromatic degrees gave me another idea, which led to the use of a new device. This consists of the change of the chromatic degrees into diatonic degrees. In other words, the succession of chromatic degrees is extended by leveling them over a diatonic terrain. (BBE 1976 [1943]: 381. ) 89. The period of 1920–1926 represents a beneficial crisis in the creative life of Bartok.

In those years, he struggled to find a new, more balanced style, and composed relatively few works (in 1925 he wrote nothing at all). (Bartok Jr, ed. 1981a: 175–242. ) In 1926 – as in the years between 1908 and 1911 – he wrote a considerable amount of piano music: the Sonata for Piano (Sz 80, BB 88, 1926), Szabadban (Out of Doors) (Sz 81, BB 89, 1926), Nine Small Pieces for Piano (Sz 82, BB 90, 1926), the Piano Concerto No. 1 (Sz 83, BB 91, 1926) and the first pieces of the Mikrokosmos series.

In those compositions his ideal of style, which in earlier years he had found in the work of Beethoven, was now enriched by the strictness of form, counterpoint, and linear technique characteristic of the music of J. S. Bach. (Demeny 1946: 29–30; Brelet 1955; and Breuer 1975a. ) In his extensive concertizing during the years 1920 to 1926, Bartok established relations with concert bureaus and broadcasting corporations worldwide, with foreign artists and Hungarian musicians living abroad, and with violinists and conductors.

In 1928–1929, he traveled even farther abroad, giving concerts in the United States as well as in the Soviet Union. (Bartok Jr, ed. 1981a: 255–285. ) 44 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle In his dissertation, Technical Bases of Nineteenth-Century Chromatic Tonality: A Study in Chromaticism, Gregory Proctor (1978: 149) argues that nineteenth-century composers began to move away from thinking in terms of asymmetrical, diatonic tonal space, and toward conceptions of tonality based on chromatic, symmetrical space.

Schoenberg and his followers continued that line of thought, with new speculations about musical pitch space. For Schoenberg, chromaticism performed an essential function in the articulation of large-scale tonal forms: Whether something be a principal or subordinate idea, introduction or transition, episode, bridge, connecting link, embellishment, extension, or reduction, whether independent or dependent, and, further, at which moment it begins or ceases to express one of these formal characteristics – all this is possible for masters of form to make manifest through harmony. …] The degree of relationship allows a graduated removal of individual parts away from the tonal center, according to the degree of their meaning: more remote digressions can thus be characterized differently from ideas that are closely related. (Schoenberg 1984: 278. ) On problems of form arising from the relativized, non-hierarchical concept of tones as found in 12-tone music, Bartok states the following: The fear that atonal compositions would present a shapeless mass as a consequence of relinquishing the symmetric scheme founded on the tonal system is unjustified.

First of all, an architectonic or similar scheme is not absolutely necessary; the construction of the line born out of the different degrees of intensity that are inherent in the tonal succession would be completely satisfactory. (This structural method is somewhat analogous to that of works written in prose. ) Secondly, the atonal music does not exclude certain exterior means of arrangement, certain repetitions (in a different position, with changes, and so forth), the previously mentioned progressions, [the] refrain-like reappearance of certain ideas, or the return to the starting point at the end. BBE 1976 [1920c]: 455. ) Finally, Webern (1975 [1960]: 22–23) discussed pitch space most often in terms of the “conquest of the tonal field”. Although on occasion he used this expression in discussing general compositional space, more typically his phrase seems to mean “the full explanation of the natural resources of the overtone series”. That, for Webern, was the basic historical principle that had led to the equalization or relativizing of all twelve pitches of the chromatic scale. . In search of a system — 45 Even in some early works, significantly in the Fourteen Bagatelles, First String Quartet90 and Duke Bluebeard’s Castle91 – in which traditional triadic structures still contribute to the musical work – Bartok’s innovative use of the diatonic scale led to a new conception of the chromatic one. In his introduction to the published score, the composer escribed the Fourteen Bagatelles in this way: “A new piano style appeared as a reaction to the exuberance of the Romantic piano music of the nineteenth century; a style stripped of all unessential decorative elements, deliberately using only the most restricted technical means” (BBE 1976 [1945]: 432). Bartok was clearly aware of the importance of those pieces in his compositional development: “The Bagatelles inaugurate a new trend of piano writing in my career” (ibid. : 433).

Many Bartok scholars have emphasized the importance of the Bagatelles in the wider context of fin-de-siecle modernism. 92 Halsey Stevens (1964: 41–42), for 90. In the early mature works of 1908–1909, Bartok’s new resources appear only in given movements or works, rather than being synthesized or transformed into a unified style. In terms of the Germanic style, the First String Quartet, like Strauss’s Elektra, is historically transitional in its interaction of triadic harmonies with chromatic melodic lines that unfold according to non-functional voice-leading patterns. Batta, ed. 1999: 594–599. ) With the disappearance, in the early part of the twentieth century, of the traditional triad as the basic harmonic premise, greater importance was placed on the interval as a primary means of harmonic and melodic integration. Both his First String Quartet and Strauss’s Elektra epitomize late Romantic music on the threshold of a new chromatic idiom. However, while Strauss never crossed that threshold, Bartok’s First String Quartet was only the beginning of his new chromaticism. BBE 1976 [1910]; and Antokoletz 1984: 14. ) 91. In Bluebeard’s Castle, purely triadic passages, as at the opening of the fifth door (74/19–23), coexist with passages of markedly atonal character (for a detailed analysis of this scene, see Chapter 4. 4. 8). The dissonant music that accompanies the opening of the first door (30/1–31/6) gradually yields to triadic harmonies once the shock of seeing such horrors has eased for Judith (see Chapter 4. 4. 4).

The B-flat pedal serves as an underlying tonal anchor during the opera’s most shatteringly dissonant passage, the orchestral climax at the very end (136/1–137/10), directly before the return of the F-sharp pentatonic melody in 138/1–139/12 (discussed in Chapter 4. 4. 10). Standing in contrast to passages like these are monuments of almost serene triadic repose; for example, as occurs when a D-major triad (54/1–59/10) accompanies the treasure chamber scene (see Chapter 4. 4. 6). 92. In early 1908, Bartok composed his Fourteen Bagatelles, two of which (4. Grave” and 5. “Vivo”) were obviously folk-song arrangements. He viewed those as experimental, containing many a device from folk music made to function in a new way. In the piano piece in question, Bartok has begun to engage with transforming vocal styles of folk music into instrumental melodies. The Bagatelle No. 1, written in two keys simultaneously, shows aspects of parlando style. The right hand plays in four sharps throughout (C-sharp minor), and the left hand is written in four flats, in Phrygian C-minor (or the F-minor scale, starting on C).

The listener is made aware of the fact that the two melodies are in different keys since one is an ascending line, the other descending. Other Bagatelles in the set, especially numbers 10 and 13, also show combinations of two different tonalities. In the tenth, Bartok uses a technique that might be termed successive polytonality. One section of the piece, in which broken chords are combined in a striking way with an upper part, was quoted by Schoenberg in the last chapter of his Harmonielehre (1911). Bagatelle No. 13 consists of an E-flat minor ostinato in the bass.

The upper part arpeggiates D-major, G-major, and other chords. The piece as a whole has two tonal planes: the E-flat minor ostinato is interrupted from time to time by a similar A-minor ostinato, the two tonalities pressing ever closer on each other’s heels. In this piece, Bartok only combines tonalities that are distantly related (in this case, a tritone apart). The tritone relation is also characteristic of Stravinsky’s early uses of bitonality, notably in Petrushka (1911). The famous Petrushka-chord is a combination of the broken triads of C-major 46 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle xample, points out that Bartok was ahead of his contemporaries: “The piano music of 1908 shows experimentation with bitonality, dissonant counterpoint, chords in intervals other than thirds, somewhat before the works of Stravinsky and Schoenberg in which these devices first came to light. ” They mark an important transition, from Bartok’s youthful style, to the individualism that became characteristic of his mature style, displayed in his later works. One of the few recorded statements made by Bartok about his compositional method occurs in his third “Harvard Lecture”, in the context f a discussion of twentieth-century chromaticism: Bimodality led towards the use of diatonic scales or scale portions filled out with chromaticized degrees, which have a totally new function. They are not altered degrees of a certain chord leading to a degree of a following chord. They can only be interpreted as the ingredients of the various modes used simultaneously and at a given time, a certain number of seemingly chromaticized degrees, [some] belonging to one mode, other degrees to another mode. These degrees have absolutely no chordal function; on the contrary, they have a diatonic-melodic function.

This circumstance is clearly shown if the degrees are picked out and grouped into the modes to which they belong. (BBE 1976 [1943]: 376. ) The composer’s comments in that lecture suggest that he viewed and used chromaticism in three basic ways (BBE 1976 [1943]: 376–383): 1. as restricted bimodality: the juxtaposition of two modal forms based on one fundamental tone, but retaining their separate modal obligations; 2. as modal chromaticism: elements of modes are mixed with chromatic degrees having no chordal basis or function, yet retaining diatonic melodic functions; 3. s new melodic chromaticism: single tones are independent, but there is one fundamental tone to which the other tones eventually resolve; such is the case in Arabian and Dalmatian folk songs, but Bartok’s melodies usually have a wider range (tessitura) than those. The uniqueness of Bartok’s case lies in the fact that modality led him to a loose kind of tonality in which a melody (formal section, movement, or whole composition) can begin and end with a clear statement of a mode, but may exhibit alteration and diversity of modality along the way (BBE 1976 [1921c]: 61; Oramo 1977b: 21–56; 1980; and 1982a).

Bartok considered modal chromaticism, “as we will call this phenomenon henceforward, to distinguish it from the chordal chromaticism of the nineteenth century,” to be the most distinctive feature of the new Hungarian music (BBE 1976 [1943]: and F-sharp major. The effect is particularly striking because keys whose tonics lay a tritone apart have notes in common, e. g. , the notes F (E-sharp) and B in the case of C-major and F-sharp major. Sequences of chords a tritone apart are altogether characteristic of Russian music. (Ujfalussy 1971 [1970]: 85–94; and Lindlar 1984: 24–25. ) 2. In search of a system — 47 76). Moreover, polymodality can lead to interesting symmetrical formations. For example, although the C-major scale is not symmetrical in itself, when it is combined with Phrygian mode on C, the result is a symmetrical scale. Lydian may be combined with Locrian, and Aeolian (natural minor) with Mixolydian, so as to obtain similar results (see Example 2. 2, below). For simplicity’s sake, the modes in the example are shown as having a common final. In practice, however, it is inconsequential if any two, “overlaid” modes have the same final, as long as they together produce the desired symmetrical scale.

Example 2. 2. Combinations of modes. Together with Dorian and Mixolydian modes on the same note, a combination of Lydian (strongly characteristic of certain Slovak melodies) with Phrygian mode proved particularly useful, especially in Bartok’s rapidly developing notion of what he called “polymodal chromaticism”. Bartok (BBE 1976 [1943]: 367) explains: “Just as the two types of the minor scale can be used simultaneously; two different modes can be used at the same time as well. ” Such a combination is shown in Example 2. 3.

Example 2. 3. Two types of minor scale used simultaneously. In Table 2. 1, the notes of the scales shown in Example 2. 3, above, are translated into pitch-class integers: 48 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle Integers Minor1. Minor2. Integers 0 C 10 Bb 8 Ab C 0 7 G Bb 10 9 A Ab 8 11 B G 7 0 C A 9 10 Bb B 11 8 Ab C 0 7 G Bb 10 9 A Ab 8 11 B G 7 0 C C 0 Table 2. 1. Two types of minor scale, with corresponding integers. Bartok describes how he combined Phrygian and Lydian modes to create a olymode comprising all twelve tones of the chromatic 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 tonally different in their function from the altered chord degrees of the chromatic styles of the previous periods [of musical style]. A chromatically altered note of a chord is in strict relation to its non-altered form; it is a transition leading to […] 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. (BBE 1976 [1943]: 367. ) Bartok’s statement above provides a clear illustration of bimodality through the combination of two heptatonic modes to create a twelve-note chromatic scale. Of the seven letter-names, only two are used once, those on which the Phrygian scale and Lydian scale forms coincide (Figures 2. 6–2. 7). 5 7 0 10 3 8 1 Figure 2. 6. Set {0,1,3,5,7,8,10}. 11 4 6 9 2 7 0

Figure 2. 7. Set {0,2,4,6,7,9,11}. C is encircled by half-steps B (Lydian) and D-flat (Phrygian); G by F-sharp (Lydian) and A-flat (Phrygian). The chromatic, Phrygian, and Lydian scales are shown in Table 2. 2, along with their respective integers. 2. In search of a system — 49 Scale Chromatic Phrygian Lydian Pitch-Classes C–D-flat–D–E-flat–E–F–F-sharp–G–A-flat–A–B-flat–B C–D-flat–E-flat–F–G–A-flat–B-flat C–D–E–F-sharp–G–A–B Point0 C C C Integers {0,1,2,3,4,5,6,7,8,9,10,11} {0,1,3,5,7,8,10} {0,2,4,6,7,9,11} Table 2. 2.

Chromatic93, Phrygian, and Lydian scales. Bartok has reflected on such modal combinations: If we examine these two modes […] we will see that the upper halves of both modes are in exactly the same relation as the upper halves of the two minor scale types. So we can declare our example shows an extension of the above-described methods of old composers to the lower of the scale. […] But not only different modes can be superposed; the same can be done with the common major and minor scale or, to be more exact, with a major and minor pentachord.

As a result, we will get a triad with a doubled third: one minor, the other major. (BBE 1976 [1943]: 366. ) Bartok (BBE 1976 [1943]: 376) asserts, “In modal chromaticism, when the modal forms are intermingled, these diatonic melodic functions are still maintained. ” He also mentions that, in his and Stravinsky’s music, the fundamental key (tonality) cannot be called major or minor, because the quality of the third degree is not fixed. By the way, much mischief was done in the worship of polytonality or bitonality.

Some composers invented a hackneyed-sounding diatonic melody in, let us say, C, and added a very hackneyed accompaniment in F. […] Incidentally, much of Stravinsky’s music, and also of my music, looks as if it is bitonal or polytonal. Therefore, the pioneers of polytonality used to regard Stravinsky as one of their fellow polytonalist. Stravinsky, however, deliberately denies this circumstance, even in such exterior features as orthography. (BBE 1976 [1943]: 367. )

The diatonic and chromatic scales are all based on the assumption of a hierarchy of intervals, proceeding from the essential nature of musical tones themselves; the latter must not be disregarded if music is to result from the composing, literally the “putting together”, of tones. The composer explains: The anhemitonic pentatonic scale [Figure 2. 8, next page]94, with its peculiar leaps because of the missing second and seventh degree [sic! ], is the very opposite of the chromaticized scale used, for instance, in Wagner’s music.

So we took it – quite subconsciously – as the most suitable antidote for the hyperchromaticism of Wagner and his followers. (BBE 1976 [1943]: 364. ) 93. It is based on the constant intervallic symmetry of the half-step. 94. Bartok explained the features of Hungarian folk art-music in numerous writings throughout his life (e. g. , BBE 1976 [1921c]; and 1981[1924]: 12–23). Here I refer frequently to the pentatonic mode. By pentatonic, I mean the particular mode that Bartok discovered in Hungarian folk song, set {0,3,5,7,10}.

Other modes are discussed in Chapter 3. 5. 1 (pp. 155–157). 50 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle 5 7 0 10 3 Figure 2. 8. Set {0,3,5,7,10}. Varieties of scale in Bartok’s output include the pentatonic, the whole tone, ecclesiastical modes, major and minor, as well as composite forms of modal chromaticism. Along with pentatonic melodic structures, Bartok called modal chromaticism “a main characteristic of the new Hungarian art music” (BBE 1976 [1943]: 376; also see, especially, Antokoletz 1984: 27–28, 33–50).

Bartok refers to modal chromaticism as that which is most prevalent in his work, which he distinguished from the chordal chromaticism of the nineteenth century. Bartok gives an interesting account of this discovery: Let us make the statement that chordal chromaticism in folk music is absolutely inconceivable; first, because folk music, apart from some exceptional areas, is monophonic music – music in unison, all over the world; and secondly because it represents, even in Western European art music, a more or less – if I may say so – artificial development, standing on a “higher” level, a level which cannot be expected in rural music.

Nor is modal chromaticism possible in folk music, because this style again would presuppose a polyphonic structure of two or more parts, which does not exist in rural music. (BBE 1976 [1943]: 376–377. ) Bartok further developed new structural (that is, non-embellishing) types of melodic chromaticism, in which earlier modal obligations are dispensed with, though allegiance to a focal note is retained. Within that framework, Bartok applied his theory of tonal axes as the basis of tonality. His melodic chromaticism has folk parallels, in that it differs from European chromaticism based upon alteration.

On this subject, Bartok remarks as follows: […] very few areas where melodies, or even a melodic style, exist look as if they are based on a genuine chromatic system. Then what kind of chromatic system can it be? The single degrees generally are at a half-tone distance from each other; thus, they cannot be regarded as ingredients of various modes. As a matter of fact, they are as much independent tones as are the single degrees of the diatonic scale, and they have no interrelation except their relation to the fundamental tone. For all these chromatic scales have a fixed fundamental tone.

In any case, their chromaticism very much resembles that of the new chromaticism. […] Such a chromatic style exists in Arab areas of Northern Algeria and in Dalmatia (a district of Yugoslavia, on the Adriatic cost). (BBE 1976 [1943]: 377. ) In the Dance Suite, one finds a new synthesis of folk song and composing techniques. Bartok explains his use of melodic chromaticism in that work: 2. In search of a system — 51 My first “chromatic” melody I invented in 1923; I used it as the first theme of my Dance Suite. This music has some resemblance to Arab melody. …] This kind of melodic invention was only an incidental digression on my part and had no special consequences. My second attempt was made in 1926; on [that] occasion, I did not try to imitate anything known from folk music. I cannot remember having met [with] this kind of melodic chromaticism deliberately developed to such a degree in any other contemporary music. (BBE 1976 [1943]: 379–380. ) Antokoletz describes the integrative process in Bartok’s music, which lies just beneath the surface of his varied melodic and harmonic constructions.

He writes: While the diatonic extensions themselves appear as one or another of the church modes in authentic folk melodies, the octatonic extensions represent abstract formations of the original non-diatonic folk sources. In addition, in certain instances in Bartok’s music, whole-tone scales may be understood as abstract extensions of one or another of the folk modes. All these extensions, whether or not they can be found among the authentic peasant melodies (the completed octatonic and wholetone scales cannot), are exploited both melodically and harmonically by Bartok as pitch sets, that is, as divorced from traditional tonal functions. Antokoletz 1984: 204. ) Bartok referred to the principle of diatonic “extension in range” of chromatic themes, as well as the reverse, i. e. , chromatic compression of diatonic themes. This bi-directional principle governs his “organic” technique shaping of melodic and harmonic materials. Bartok describes that technique as follows: We have mostly the impression that we are dealing with an entirely new melody. And this circumstance is very good indeed, because we will get variety on the one hand, but the unity will remain intact because of the hidden relation between the two forms [of chromaticism]. …] If, perhaps, you object that this new device is somehow artificial, my only answer will be that it is absolutely no more artificial than those old devices of augmentation, diminution, inversion, and cancrizans [crab inversion] of themes; in fact, cancrizans seems to be even much more artificial. (BBE 1976 [1943]: 381. ) Bartok mentions other examples that show his “extension in range” technique: A rather surprising circumstance has been discovered in connection with the compression of diatonic into chromatic melodies.

I discovered it only six months ago when studying the Dalmatian chromatic style, consisting of independent chromatic melodies. […] The chromatic melodies of this style are, as a matter of fact, nothing other than diatonic melodies of neighboring areas, compressed into a chromatic level. There are irrefutable proofs for this theory, which, however, I will not enumerate now, but in one of my later lectures. This theory offers a very easy explanation of the queer major-second distance between the two parts.

The compression simply works in two directions: in the horizontal direction for the melody, and in the vertical direction for the intervals, or distance between two 52 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle [adjacent] parts. Evidently, the major or minor thirds distance, usually met with in two-part singing, is compressed into the unusual major-second distance. (BBE 1976 [1943]: 382–383. ) Whole-tone pitch patterns, surprisingly, play an insignificant role in Bartok’s Duke Bluebeard’s Castle, while fourth-chords (quartal chords) are much more in the foreground.

That fact suggests that Bartok, at least at that time, seems to have been influenced more by folk music than by the ideas of Schoenberg (see Chapters 4. 4–4. 4. 11, below). Bartok comments: When I first used the device of extending chromatic melodies into a diatonic form, or vice versa, I thought I invented something absolutely new, which never yet existed. And now I see that an absolutely identical principle exists in Dalmatia since Heaven knows how long a time, maybe for many centuries.

This again proves that nothing absolutely new in the world can be invented; the most unusual-looking ideas have or must have had their predecessors. (BBE 1976 [1943]: 383. ) Unlike Schoenberg, Bartok fuses dense chromatic counterpoint with the modal material of peasant tunes and the transparent textures of Debussy. Schoenberg’s music, by contrast, is tonally and rhythmically free from the influences such sources. A distinctive compositional signature for Bartok’s contrapuntal style consists of the superposition of melodic lines, each in a different key.

What is to be immediately noted here is that Bartok used homophonic pitch relations to expand his harmonic palette to all twelve-tones of the chromatic spectrum, while still remaining tonal. With that, my discussion of analytic approaches to Bartok’s music is complete. In sum: tonal theorists subsume resultant chromaticism within a pitch-centered, diatonic background; in contrast, atonal theorists cut across the diatonic strands to conceive pitch cells or motifs that move more properly in a chromatic twelve-pitch space (see, e. g. , Agawu 1984;95 and Parks 198196).

Both analytic approaches, however, obscure 95. Writing about Bartok’s work for piano, Eight Improvisations on Hungarian Peasant Songs (Op. 20, Sz 74, BB 83, 1920), Kofi Agawu (1984) discusses previous approaches to analyzing the piece, starting with Richard Parks’s atonal approach to the music of Bartok. Agawu notes, as we have done, that Bartok advocated a tonal basis as a starting point for analysis. Agawu goes on to mention Babbitt’s advocation of approaching the works of Bartok from both tonal and atonal standpoints. Babbitt never put that principle to use in analysis, however.

But, as Agawu mentions, Antokoletz did, by emphasizing the parallels between principles of folk-music construction and the harmonic and contrapuntal procedures used in the composing of art music. Agawu analyzes the Improvisations using the analytical systems of Schenker, Forte, and Lendvai (though Agawu finds fault with various aspects of Lendvai’s system). 96. Richard Parks (1981), in his analysis of Bartok’s piano piece “Kvartok” (“Fourths”) from the Mikrokosmos, attempts to use atonal set theory to make sense of that brief, so-called tonal work.

Parks first determines its form: nine sections defined by changes in thematic material. Parks breaks the harmonic material down into tetrachords and performs a PC-set analysis, which reveals Bartok’s economical use of tetrachords. Only five different sets are used throughout: {0,3,5,8}, {0,1,5,6}, {0,1,6,7}, {0,2,5,7}, and {0,1,5,8}, each of which contains at least two perfect fourths. Parks then investigates the overall tonal plan, which he sees to comprise two key centers a third apart, E-flat and G. Parks also notes the ways in which the five sets are distributed over the nine sections, and . In search of a system — 53 the individuality and coherence of perceivable diatonic strands: the first, by reducing chromaticism to diatonicism, the second, by picking apart diatonicism to show how it fits within the chromatic universe. 2. 2. Lendvai’stheories97 Presented next is a brief examination of theories advanced by Bartok analyst, Erno Lendvai (1925–1993), whose unique contributions to the analysis of pitch and proportions in Bartok’s music has generated one of the most long-lasting debates in the field.

Lendvai proposed several unique concepts by which to understand Bartok’s musical language, but scholars are anything but united in assessing the significance of his theories. Decades of scholarly discussion have yet to produce even an approximate re-definition of Lendvai’s terms. (See, e. g. , Karpati 1975 [1967];98 200099; concludes by briefly relating Bartok’s method in “Fourths” to some of his other pieces, e. g. , the Fourteen Bagatelles (1998 [1908]). Parks argues that atonal theory provides a suitable method for analyzing and understanding Bartok’s music. 7. Chapters 2. 2–2. 2. 3 are based on my article “Bartok’s Harmonic Language” (R. Honti 2006). 98. In Bartok’s String Quartets, Karpati (1975 [1967]) discusses fundamental theoretical principles underlying Bartok’s general musical language, along with information regarding the sources of Bartok’s compositional tools: folk influences, monothematicism and variation, polymodal chromaticism, tonality, polytonality, and the phenomenon of mistuning (possibly Karpati’s most peculiar topic; see, e. g. 1972 [1971]) and 2004 [1995]). Karpati (1975 [1967]) includes a detailed chronological chart of Bartok’s various musical activities (folk-music research, composition, etc. ), reviews the evolution of his string quartets, and discusses Bartok’s forerunners and contemporaries. In addition to providing historical, thematic, motivic, tonal, scalar, phrasal, and structural details of the quartets, the author reviews the work of other scholars responding to the theories of Lendvai; see also, Karpati (1991 [1976]). 9. Especially pertinent in this regard is Karpati’s (2000) article, “A Bartok-analitika kerdesei: Meg egyszer Lendvai Erno elmeleterol” (“Questions of Bartok Analysis: Once More about Erno Lendvai’s Theories”). 54 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle Szentkiralyi 1976;100 and 1978;101 Bachmann & Bachmann 1979;102 Somfai 1981b; Howat 1977;103 1983a;104 and 1983b;105 and Agawu 1984. As mentioned above, the present chapter deals with Lendvai’s theoretical claims, particularly his approach to Bartok’s handling of pitch, which he views through what he calls the “tonal axes system”, discussed in Chapter 2. 2. 1, below. (Lendvai 1964: 16–24; 1971: 1–16; and 2000 [1971]: 283–288. ) In his entry in The New Grove Dictionary of Music and Musicians, Ian Bent points out that: […] the work of Erno Lendvai on the music of Bartok should be mentioned, not so much for its theory of a tonal axis system as for its locating of proportion in musical structure.

Lendvai sought to demonstrate the presence of the Golden 100. Andras Szentkiralyi’s dissertation (1976) presents an analysis of Bartok’s Second Sonata for Violin and Piano (Sz 76, BB 85, 1922) based on Lendvai’s theoretical principles. This piece can be considered as a landmark in Bartok’s development of technique. It is at once a consolidation of his former practice and a starting-point for further explorations. The two movements are mature examples of very characteristic moods, the freely rhapsodic, and the stylized dance-movement of elaborate and violent rhythms.

The violin part is purely melodic, the piano part almost entirely percussive, using what may be called chord-clusters. Free from the complexities of counterpoint, it serves as a particularly clear example of the chief features of Bartok’s technique. Szentkiralyi explores the tonal structure of the piece in terms of alpha–formations, distance models, the acoustic (overtone) scale, set {0,2,4,6,7,9,10}, and the pentatonic scale (e. g. , set {0,2,4,7,9}), and demonstrates the structural relation of these tonal structures to the Fibonacci Series. 101.

Szentkiralyi (1978) bases his arguments on Lendvai’s theories. He asserts that Bartok’s compositional techniques cannot only be the result of intuition, but rather of conscious, systematic, and logical thinking. Szentkiralyi outlines a conglomerate of diverse principles (based on folk modes, romantic chromaticism leading to equal divisions of the octave, etc. ) then focuses on Lendvai’s usage of the Golden Section, Fibonacci Series, and Axis System as the basis of an analysis of the first twenty measures of the Second Sonata for Violin and Piano. 02. Tibor Bachmann’s and Peter J. Bachmann’s (1979) “An Analysis of Bela Bartok’s Music through Fibonaccian Numbers and the Golden Mean” is an application of Lendvai’s theories. The article explains the Golden Section and its relation to Fibonacci Series, and relates those to Bartok’s interest in nature. Bachmanns show the relation of Fibonacci Series to the anhemitonic scale and the pentatonic chord, a minor triad with a minor seventh, specifically in second inversion, showing the numbers 2, 3, 5, and 8.

Bachmanns also give examples from Duke Bluebeard’s Castle, which show durational values of various passages. 103. In “Debussy, Ravel, and Bartok: Towards Some New Concepts of Form”, Roy Howat (1977) explores proportional relations in the musical forms of those composers, based on an expansion of the principles of the Golden Section ratios established by Lendvai in his studies of Bartok’s music. Howat applies his method in a comparative analysis of Debussy’s Reflets dans l’eau (1905), Ravel’s Oiseaux tristes (1905), and Bartok’s Music for Strings, Percussion and Celesta. 04. Much has been written on the Golden Section, but one should approach the issue cautiously. In “Bartok, Lendvai and the Principles of Proportional Analysis”, Howat (1983a) concentrates on that proportion in Bartok’s music, and on Lendvai’s claim that Bartok organized many pieces around the Golden Section. Unlike Lendvai’s, Howat’s theory is more expansive, and attempts to link together intervallic structures in harmony, tonality and melody, the use of rhythm and meter, and the organization of forms in terms of large- and small- scale proportion. 105.

In Debussy in Proportion: A Musical Analysis, Howat (1983b: 6, 15, 22, 170, 187) evaluates the musical relevance of proportional analysis and gives a wide-ranging discussion of technical issues, in response to critical challenges to Lendvai’s methods. 2. In search of a system — 55 Section and of the Fibonacci Series […] in Bartok’s compositions. (Bent 1987 [1980]: 369. ) Lendvai’s study of Bluebeard’s Castle, in his book Bartok’s Dramaturgy: The Stage Works and the “Cantata Profana”, is one of the cornerstones of his harmonic theories about Bartok’s music (1964: 65–112).

That study was revised and condensed for the major chapter on Bartok’s opera in The Workshop of Bartok and Kodaly (Lendvai 1983: 219–245); the chapter in question contains Lendvai’s appraisal of the theoretical foundations of Bartok’s music, but analyses of individual scenes have been omitted. Lendvai’s Poetic World of Bartok also contains a brief analysis of the opera (2000 [1971]: 19–40). Substantial portions of Lendvai’s (1960–1961) and Kroo’s (1962) original researches into Duke Bluebeard’s Castle remain untranslated, so those who do not read Hungarian cannot appreciate their work.

The opera constitutes not so much a new departure as a consummation on a higher level. Bartok’s later style does not display a single significant element that was lacking when he composed his opera, a fact emphasized by Lendvai (2000 [1971]: 21): “In Bluebeard, Bartok’s style sprang at once into existence. ” Lendvai views the various levels of Bartok’s music as integrated by means of several interlocking systems. The latter are illustrated in his groundbreaking study, Bela Bartok: An Analysis of His Music (1971). 106 The three pillars of his theories are as follows: 1.

The Golden Section (GS) as governing principle in both pitch and formal organization. 107 106. Lendvai’s (1971) relatively short study gives a comprehensive English presentation of his theories, including tonal principles based on the AS, formal principles based on the Golden Section and Fibonacci Series, the use of chords and intervals derived from the Golden Section as well as the chromatic, acoustic (overtone), or diatonic system. Reference is made to ancient Greek, Gothic, and Renaissance mathematical models, and a discussion given of the philosophical significance of those principles.

For a summary of Lendvai’s views on Bartok’s music, see Szentkiralyi (1976: 163–166). 107. In “Bartok and the Golden Section”, Lendvai (1966a) evaluates Bartok’s use of the proportio divina, and explains how its use brings about a fusion of Bartok’s two-fold view of life: as a bonding of biological being (body) with the intellectual power of logic (mind). The Fibonacci Series is a number series, named after a 13th-century Italian mathematician, that is derived from a recursive formula in which each subsequent element is the sum of the previous two, (e. g. , 1, 2, 3, 5, 8, 13, 21, etc. . The Fibonacci Series, though infinite, has the following property: the ratio between neighboring pairs of numbers increasingly approximates the Golden Mean (0. 618033… ), with the accuracy increasing the further one progresses along the number series. The Golden Mean, or “Golden Section”, can be understood as the point on a line that divides it into two segments, such that the ratio between the segments is the same as that between the longer segment and the entire line. (See also, Bardos 1974: 98–102; Howat 1983b: 1–4; and Kuokkala 1979; and 1992: 26, 206. So far, however, no evidence has emerged that Bartok consciously planned his works in the ways Lendvai suggests (Howat 1983b: 6, 15, 22, 170, 187). Another striking, if contrived, image of symmetry is produced when an ascending scale based on the intervals of the Golden Section is superimposed over a descending acoustic (overtone) scale, set {0,2,4,6,7,9,10}. The second “pillar” in the list above, the axis system (AS), been dealt with by various theorists; e. g. , Szentkiralyi (1976), Wilson (1992: 6–8, 203–208), and Solomon (1973). 56 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle . The axis system (AS) with its complement in the acoustic (overtone) scale. 3. Chords are based on (1) the harmonic paradigm of the ? -chord, and its ? , ? , ? , and ? segments; and (2) on alternating-distance scales that divide the octave, most importantly those constructed in 1–2–1 (alternating minor- and major seconds), 1–3–1 (alternating minor seconds and minor thirds), and 1–5–1 (alternating minor seconds and perfect fourths). In his “Introduction to Bartokian Forms and Harmonies”, Lendvai (1968) shows that Bartok, in his early thirties, devised his own method of integrating all the elements of music.

Lendvai argues that one basic principle – the Golden Section, i. e. , X:1 = (1–X):X – governs scales, chordal structures and their respective melodic motifs, as well as durational proportions between the movements of a multi-movement work, main divisions within a movement (such as exposition, development, recapitulation), and even the phrases within sections of movements. By means of that principle, Lendvai describes many of the scales, harmonies, and formal structures of Bartok’s compositional style. Thus, the pentatonic scale, the ? -chords and the modal scales all belong to this system. See also, Bardos 1974: 19, 98–102, 107, 133, 164, 203, 239–240. ) The Fibonacci Series and the Golden Section principle form such a large part of Lendvai’s theoretical writings that they have attracted the attention of other theoreticians (e. g. , Kramer 1973108). Lendvai is also concerned with symmetrical systems, which he believes are one of the few features common to both Eastern and Western European music. That concern is evidenced in his development of the axis theory and its deployment in Bartok’s music (R. Honti 2006: 698–706; see also, pp. 58–59, Chapter 2. 2. , below). Many of Lendvai’s calculations have subsequently been discredited; for example, his measurements of the Golden Section (see, e. g. , Howat 1983a; and 1983b: 6, 15, 22, 170, 187; see also, Oramo 1982b). I, too, find some of Lendvai’s work questionable. Still, one suspects that his theories have been negatively criticized mostly due to the absolute lack of any comments by Bartok regarding the Golden Section (GS) or Fibonacci Series (FS). The Golden Section can be found in the harmonies, in which the structure of the intervals sometimes mirrors the Fibonacci sequence.

Such questions cannot be fully resolved, however, without consideration of musical temporality. Harmonies create an acoustic space; but what about durational, that is, temporal space? The Fibonacci series and its related sequences (e. g. , the Lucas series) have long attracted the attention of composers and theorists; but their application to pitch structure has not yet been fully understood. In my opinion, neither the Golden Section nor the Fibonacci Series designate a unified and objective framework through which to view pitch organization in Bartok’s oeuvre. 08. Jonathan Kramer (1973) discusses application of the Golden Section to pitch structure by several composers. In discussing Bartok’s Music for Strings, Percussion and Celesta (first and third movements), Kramer “corrects” some of Lendvai’s findings, but generally accepts them as valid, on the condition that they be understood as approximations. 2. In search of a system — 57 Lendvai (2000 [1971]: 291) claims that Bartok’s chromatic system is based on the Golden Section and the Fibonacci Series, but that claim is questionable.

Some organic formations in nature have been shown to grow according to Golden Section proportions, at least ideally (e. g. , shells of certain sea creatures, foliage and fruit, trees, etc. ). Lendvai and his followers (see, e. g. , Szentkiralyi 1976; and Bachmann & Bachmann 1979) have spent a considerable amount of energy identifying such proportions (often multiply nested) in the durational and rhythmic parameters of Bartok’s music. Their work in that area rests on two assumptions: that the concept of GS in the temporal domain is similar to the one in space, and that we hear and listen linearly.

In his L’Evolution creatrice, Henri Bergson (1859–1941) spoke of a “natural geometry of the human intellect” in order to explain people’s resistance to noticing the development of things in time (1941 [1909]). That may well be true, but an investigation of “natural geometry” lies outside the scope of the present study, and will not be pursued. If music is endowed, either consciously or unconsciously, with the same ratios that nature occasionally displays, that might validate the claim that music is linked to the natural world.

But it does not mean that one should overlook fundamental flaws in Lendvai’s analytic methods and calculations based on the Golden Section. Here, however, a full-blown critique of his theories is not necessary. For my purposes, it suffices to sketch and summarize those characteristics of Lendvai’s system that relate most closely to the music in question. 2. 2. 1. The tonal axis system Lendvai explains Bartok’s method of composition according to what he calls “the tonal axis system”. What exactly does that mean? The answer is rather complex.

Lendvai defines a historical tradition for Bartok’s axis system (AS), and elaborates on it at length (1983). 109 The key point for Lendvai is Bartok’s tonal axis (TA). The latter may be viewed as a method of assigning functional harmonic significance to the pitch-classes (PCs) operating within his particular diatonic and chromatic scales. Lendvai, in the course of his investigation, concludes that Bartok’s art represents a logical continuation and, in a certain sense, conclusion of the development of European music.

The circles of harmonic development are fused into a single, coherent, closed unit in Bartok’s sound system: the tonal axis (TA). (See also, Lendvai 1947; 1971; and 1993. ) 109. Lendvai (1983) analyzes many of Bartok’s major works according to the composer’s theories. Lendvai’s discussions have a common thread: the principle of duality. The latter is represented in chromatic (Golden Section) versus acoustic (diatonic) systems, in the principles of the polar-axis system, as well as in more musically abstract juxtapositions, such as instinctive/intellectual, masculine/feminine, emotional/sensuous, etc.

Lendvai’s ultimate aim is to demonstrate by means of analysis how these opposing principles are integrated in Bartok’s music. 58 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle Before discussing that system in detail, it is appropriate to consider in a general way the term “tonality”. Traditional tonality depends upon scale and chord relationships for its organization. The composer Vincent Persichetti remarks on tonality: The tonal meaning of an isolated chord is indefinite; it may be a crucial, or an ornamental chord of many keys, or it may belong to no key.

When surrounded by other chords its meaning may be restricted to a single tonality, to two, or more wavering tonalities; or if it has atonal intentions, the fact can be made obvious. Tonality does not exist as an absolute. It is implied through harmonic articulation and through the tension and relaxation of chords around a tone, or chords base. A particular style or period is not always limited to a predilection for a single kind of tonality. Twentieth-century music makes use of many degrees of tonality and employs many means for establishing them. Persichetti 1961: 248. ) To make the concept clear, consider a circle of fifths110, which is a general presupposition for the development of Bartok’s harmonic system. In classical harmony, the cycle of fifths F–C–G–D–A–E–B (notes in the C-major scale) corresponds to the functional series S–T–D–S–T–D, respectively. If continued, that circle returns to its starting point F, producing a diminished fifth between B and F. However, if the circle is extended by all perfect fifths (here, B to F-sharp, etc. ), it will eventually cover all twelve chromatic tones.

In that case, the scheme of the axis system (AS) becomes apparent, as shown in Figure 2. 9. Figure 2. 9. The Axis System (Lendvai 1971: 2). The question arises, can the axis system be considered as a functional system in the true sense of the word? Lendvai believes so. In his article (1957), “Einfuhrung in die Formen- und Harmoniewelt Bartoks”, Lendvai points out that the pitch axis 110. The term “Circle of Fifths” was coined by the German composer and theorist, Johann David Heinichen (1683–1729), in his treatise Der Generalba? in der Composition (1967 [1728]).

Heinichen’s visualization of the circle is somewhat hard to decode immediately, because the major-key (odd numbers, ma) and minor-key (even numbers, mi) cycles are interleaved. Heinichen did not anticipate today’s convention of representing minor keys by lower-case letters, but the letters associated with even numbers should be read that way; e. g. , A (a) is the relative minor of C Major (“relative” because it has the same key signature as the latter). 2. In search of a system — 59 (PA) may be viewed as a method of assigning functional harmonic significance to PCs. 111 Lendvai’s axis system (Figure 2. ) affirms the presence of tonic–subdominant– dominant chord functions, if not traditional ones, among the tonal relationships in Bartok’s music (Table 2. 3 and Figure 2. 10). Axis Tonic Subdominant Dominant Pitch-Class C–E-flat–F-sharp/G-flat–A F–A-flat/G-sharp–B–D G–B-flat–D-flat/C-sharp–E Point0 C F G Set {0,3,6,9} {0,3,6,9} {0,3,6,9} Forte-number 4–28 (3) 4–28 (3) 4–28 (3) ICV 004002 004002 004002 Table 2. 3. The Tonal-Axis system. 0 3 6 9 Figure 2. 10. Set {0,3,6,9}. 112 This threefold functioning in the classical sense (T, S, D, etc. ) is not typical of the small-scale harmonic progressions of any of Bartok’s mature music.

Still, Lendvai’s axis system, in which those three functions are fulfilled by the three diminished seventh groupings, is undeniably operative in the composer’s music. In essence, it functions as an organizing principle of tonal levels or centers in movements or large sections thereof, as a structure in which all 12 notes are strictly related to one center. (Lendvai 1964: 16–24; 1971: 1–16; and 2000 [1971]: 283–288. ) For example, Bartok’s augmented fourth/diminished fifth is based almost entirely on an octatonic collection formed by joining two {0,2,3,5} sets six semitones apart (Figure 2. 1). 0 2 5 3 Figure 2. 11. Set {0,2,3,5}. The relations of the AS (Figure 2. 9, previous page) differ from those of traditional tonality in that they are essentially symmetrical and relative; that is to say, 111. Lendvai’s (1957) article also contains two examples from Music for Strings, Percussion and Celesta as well as a chart showing the formal middle point of the movement (according to the Golden Section proportions), where the tonality has moved from the opening key of A (which also closes the movement) to its polarized area at the tritone (E-flat). 112.

Milton Babbitt (1986) has revealed the startling persistence of {0,3,6,9} and {0,4,8} symmetry as governors of centricity in music constructed according to Stravinsky’s technique of hexachordal transposition and cyclic permutation (“rotation”). 60 — Principles of Pitch Organization in Bartok’s Duke Bluebeard’s Castle their definition depends solely on context (Gervers 1969113). Taking into account how those properties affect Bartok’s harmonic language, Lendvai’s (1971) polar axis system is founded on the following premises: 1. A tonic (T) is interchangeable with its parallel mode. 2.

Any major or minor triad built on one of the pitches belonging to the circle of minor-third relations with the tonic (T), can be substituted for the other, such that no change of function is considered to take place; the same holds for the remaining functions (S, D), each determined by its respective circle of minor thirds and resultant scales and harmonies. Ujfalussy (1969), in “The Basic Harmonic Conception of Allegro barbaro and Bartok Scales”, seems to have been the first to discover the role and significance of harmonic and tonal relationships of the common third in Bartok’s music.

The piece starts with the tonal duality of F-sharp minor–F-major revolving around the melodic A, the axis-note in Allegro barbaro. 2. 2. 2. The ? -chord structures It is time to consider the characteristic features of Bartokean ? -chord structures. The most innovative aspects of that harmony become part of the sound world of the more mature Bartok: (1) the distinctive use of major-minor chords, or set {0,3,4,7}; and of (2) chords derived from the overtone series. (Lendvai 1957: 114; see also, Karpati 2004 [1977]. 14) On the major-minor third, Bartok remarks that it […] is very interesting to note that we can observe the simultaneous use of major and minor third even in instrumental folk music. Folk music is generally music in unison; there are areas, however, where two violins are used to perform dance music; one plays the melody and the other plays accompanying chords. 113. Hilda Gervers (1969) discusses the Five Songs (Sz 61, BB 71, 1915–1916) in terms of their formal and tonal properties, pointing up the expressionistic style of the texts. The analyses are based on Lendvai’s polar-axis system rather than classical key relations.

Gervers notes the scarcity of solo song in the composer’s overall output, and relates the Five Songs to Bartok’s later stylistic development. (See also, Demeny 1946: 22–23; and Lindlar 1984: 103–104. ) 114. In “Alternative Structures in Bartok’s Contrasts”, Karpati (2004 [1977]) demonstrates that Bartok, by using systematic chains of fifths with a common third, arrived at a method reminiscent of 12-tone serialism. The Contrasts for violin, clarinet and piano (Sz 111, BB 116, 1938) was commissioned by Benny Goodman (1909–1986), and represents Bartok’s only music featuring solo clarinet (Demeny, ed. 1976: 605; and Lindlar 1984: 93).

In his article, Karpati (ibid. ) discusses triads with major-minor structures, e. g. , set {0,3,4,7}, as dual or alternative structures, by which he means that they still can be justified separately, and, when appearing together, preserve their original modal content. Karpati discusses the meaning of this structure also in the context of Null’s as well as Lendvai’s theoretical concepts (e. g. , Lendvai’s 1:3 models). Karpati explains that if the two equivalent kinds of third appear within the stable frame of a fifth, then one arrives logically at the dual root and fifth situated around the stable third.

In Contrasts, the motif of the Lydian fourth becomes equivalent to the dual third-structure of sound. See also, Karpati (2004 [1995]), “Perfect and Mistuned Structures in Bartok’s Music. ” 2. In search of a system — 61 And rather queer-sounding chords may appear in these pieces. (BBE 1976 [1943]: 369–370. ) Bartok at first placed the minor third on top; e. g. , C–D-sharp/E-flat–E–G, set {0,3,4,7} (C = 0). After a time, the chord appeared consistently only in inversion, stacked as minor third, perfect fourth, and minor third.

Often a minor seventh is added above the root: C–D-sharp/E-flat–E–G chord with B-flat, set {0,3,4,7,10} (Figure 2. 12). 4 7 0 10 3 Figure 2. 12. Set {0,3,4,7,10}. The function remains unchanged when the C-major mode comprised by the above chord is replaced by the parallel E-flat-major: A-minor – C-major-minor – Eflat-major. Those substitution chords may also be employed in major-minor form, which brings the system to a close, because the parallel keys of A-major / F-sharpminor and E-flat minor / G-flat-major conjoin enharmonically, F-sharp = G-flat. (See Figure 2. 9, p. 58, Chapter 2. . 1. ) The major-minor chord has a number of synonymous forms. Lendvai (1964: 33) refers to that harmony as ? , i. e. , the alpha chord: C-sharp–E– G–B-flat / C–E-flat–F-sharp–A. That chord is shown in Example 2. 4 (no. 1), along with harmonies derived from it (subsets): the beta chord, ? and its variants ? 1, ? 2 (nos. 2–4); the gamma chord, ? and ? 1 (nos. 5–6); the delta chord, ? , ? 1, ? 2 (nos. 7–9); and no. 10, the epsilon chord, ?. (See also, Bardos 1974: 71–76, 105. ) ? ????? ? ?? ? ?? ? 1 2 ??? ? ?? ? 8 3 ? ? ?? ? 9 4 ? ? ?? 5 ???? ? ? ? ?? ? ? ? ? 6 7 ????? ?? ?

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