Various Tonal Systems and Phenomena in the 19th and 20th Centuries | Zeneművészeti Kar

Author: Zsolt Horváth

DOI: 10.5434/9789634902263/2

 

 

Abstract

This study – without claiming completeness – takes into account systems and phenomena in which the organising of pitches is different from that of functional tonality. A central topic is how the principle of symmetry prevails, along with possible interpretations of the notion of tonality. The starting point of the most important manifestations of symmetry is the equal division of the octave, from which the issues of the distance scales are inseparable. The study presents the interactions of diatonic scale and pitch-organization based on the principles of symmetry, which casts a new light on the phenomenon of polytonality. The study relies on important research published abroad during the last few decades – mainly in relation to Stravinsky’s music – but which may be lesser known in Hungary.

Keywords: “Ungar” scale, symmetry, polytonality

 

In the 19^th century, the decomposition of functional tonality began, which, by the 20^th century, lead to the coexistence of multiple, both tonal and atonal, systems. During the Romantic period, composers found several ways to innovate. New approaches included novel heptatonic scales, the equal (distantial) division of the octave, as well as symmetric arrangements. In the 20^th century, the possibility of polytonality, i. e. the simultaneous use of multiple diatonic scales, also arose, but this poses various questions from a theoretical perspective.

The explored systems and phenomena are tonal in the sense that they display some form of hierarchic ordering of pitches, as opposed to atonality. The study focuses on the discussed phenomena but does not provide overly complex analyses.

Although the investigated topics are not new, I attempt to highlight certain features of the presented pieces which have not been recognised widely and to draw attention to examples which are representative of a given phenomenon or are somehow unique.

I use the notation employed by Zsolt Gárdonyi and Hubert Nordhoff (2012) for diatonic and pentatonic scales and tertian chords. The diatonic scale can be represented as a sequence of perfect fifths. The scale composed of the white keys on the piano is the 0-diatonic scale, the fifth-chain starting one fifth higher is the +1-diatonic scale (alternatively, the 1#-diatonic scale), the version two fifths lower is the -2-diatonic scale (alternatively, the 2b-diatonic scale). Similarly, the five-element sequence of perfect fifths starting from C is called the 0-pentatonic scale, while the sequence A-flat–E-flat–B-flat–F–C is the -4-pentatonic scale. The scales discussed so far can be abbreviated in the following way: 0d, +1d, -2d, 0p, -4p. For harmonies, capital letters denote the major third and lower-case letters mark the minor third. The perfect fifth, minor seventh, and major ninth are denoted by numbers only. Compared to those, diminished or augmented versions are marked by b or # (e.g., C/E/G-sharp = C^5#, C-sharp/E/G/B/D = c-sharp^5b/7/9b)[1]. It is not necessary to denote the fact that a chord is incomplete (e.g., C/E/G/B-flat and C/E/B-flat can be both C⁷). The number of the inversion is denoted after the root note by default (e.g., C/F/A-flat = f^4/6), but with more complex chords, the number of the inversion precedes the root note (e.g., G/A-flat/C/E-flat = ²A-flat^7#). There can be separate notations for the half-diminished (^Ø or ^Ø7) and diminished harmonies (™^O and ^O™7).

 

Scales – Ungar Scale and Melodic Ungar Scale

In several movements of Via crucis, Liszt employs the Ungar scale, which is a natural minor scale with Ri and Si[2]. It was Lajos Bárdos who presented the occurrences of this scale in Liszt’s works and provided several examples to illustrate (1969). It is important to note that I consider the scale as a set of tones and do not mean it in the modal sense, that is, I do not consider any note to be the root note. Bárdos discusses the Mi mode of the Ungar scale separately, using examples from Liszt and folk music where the final note is Mi. However, while he builds the Ungar scale on La, the final note of his examples from Liszt’s music is not La invariably; instead, it is often Ri, as underlined several times by Bárdos.

The presence of the Ungar scale is especially apparent if the augmented second Do-Ri melodic step appears, which highlights its Hungarian character. In the first ten bars of Station XI, however, this feature is not emphasised and the scale is perhaps more difficult to notice. However, the set of notes in the excerpt, namely D, E, F, G-sharp, A, B, C-sharp, constitute a variation of the Ungar scale, the melodic Ungar scale (La, Ti, Do, Ri, Mi, Fi, Si). Of these, the instrument uses six notes (D, E, F, G-sharp, A, C-sharp), the B (=Fi, which makes it the melodic Ungar scale) is added by the male choir. The G-sharp serves as the pedal point. Above it, the c-sharp minor chord is transformed into d minor by stepping a half note upwards with each note in the chord (G-sharp → A, E → F, C-sharp → D), which is a technique often employed by Liszt.

Excerpt 1: The harmonic sketch of Station XI

 

The composer makes use of the characteristic of the scale which allows the construction of two minor harmonies at a minor second distance. The gradual expansion of the initial chord, which occurs as described above, causes a terrifying effect alongside the fortissimo dynamics, the use of ponderous blocks, and the staccato play, which represents the nailing to the cross with profound intensity. Furthermore, the resulting chords are also dissonant. The final two deserve special attention, namely the C-sharp/F/A augmented triad over the G-sharp pedal point (bars 7-9), followed by the d minor chord (bar 10). The former can be considered a tertian seventh chord (enharmonically reinterpreted as ²A^5#/7#), the latter, however, is clearly a harmony which foreshadows the 20^th century with the polar distance of d minor and G-sharp. In Ernő Lendvai’s terminology (1964), we could name it a d-delta.

In the closing unison, which represents the elevation of the cross, we can hear the basic form of the Ungar scale, with B-flat instead of B (C-sharp and E are not present here).

 

Symmetrical Phenomena

The second chapter of this study uses examples from one work by Franz Liszt and György Ligeti each. Bárdos (1976) cites two excerpts by Liszt in relation to this topic, namely the end of the Mephisto Waltz and the beginning of Preludio funèbre. We can consider the latter to be symmetric in the broader sense, due to the interval structure of the scale. The examples I provide represent the notion of symmetry in the same sense.

One of the main motifs in Liszt’s Via crucis is the Gregorian melody initiation of Sol–La–Do, which Klára Hamburger (2010) calls the “cross” motif, while demonstrating its occurrence in several of Liszt’s works. The hymn Pange, lingua, gloriosi by Venantius Fortunatus starts this way. It was Liszt himself who called it the “musical symbol of the cross” in relation to The Legend of St. Elizabeth. In Via crucis, Liszt uses another hymn by Venantius Fortunatus, namely Vexilla regis prodeunt. Its first notes are F–G–A–B-flat, which Liszt transforms into the “cross” motif by omitting a note (F–G–B-flat, first in bar 7 of the Introduction).

The “cross” motif appears as early as bars 2 and 3 of the brief instrumental introduction (F–G–B-flat). Before that, the work begins with a variant which can be noticed even at first hearing (D–F–G). The thematic connection is the result of the interval content, namely the major second and minor third. For the D–F–G sequence, only the position of the two intervals is inverted. This is the sense in which I describe the inversion of the motif.

Excerpt 2: The cross motif and the inversion of the position of its intervals

 

Liszt composed the paraphrase of the hymn in D-Aeolian mode[3]. Starting the inversion from D creates modal unity between the beginning and end of the movement.

In the first part of Station IV, the originally linear idea reappears vertically. These two musical dimensions are not yet interconvertible in the Romantic period, only in the 20^th century. As so often, Liszt finds the path of radical innovation. What is more, the starting chord is none other than the combination of the thematic core and its inversion. This harmony is named by Bárdos as a “pseudo-pentatonic” chord (Bárdos, 1976, p.94). It is “pseudo” because it is not represented as F/G/B-flat/D-flat1/E-flat¹ but as F/G/B-flat/C-sharp¹/D-sharp¹[4]. Although the ascending melody is a good reason to use the sharp notation (Bárdos, 1976), and a diminished sixth, which carries significantly larger melodic tension, is present instead of the perfect fifth (B-flat¹–D-sharp¹ instead of B-flat¹–E-flat¹), the tonal uncertainty allows for the interpretation of the chord with D-flat and E-flat.

Excerpt 3: The initial chord of Station IV

 

Excerpt 4: The initial chord of Station IV with enharmonic notation, and the symmetry of the chord

 

There is a curious passage in Station XII, between “Consummatum est” and the “O Traurigkeit” choral. The soprano sings brief, two-note motifs, which creates a scattered effect. The first four distinct notes are repeated with harmonies. The set of notes in the first four bars are the following:

Excerpt 5: The set of notes in the passage (Station XII, bars 69-76)

 

The notes are built from thirds and seconds, just as at the beginning of the Introduction. In this case, the seconds surround the third, and one of the seconds is a minor second. The frame is thus a diminished fifth, inside which the symmetry is not perfect. By contrast, the other four notes, connected to the first four, constitute perfect symmetry in order.

Excerpt 6: Symmetry in the set of notes[5]

 

The end of the investigation has not been reached, however. The key signature indicates the key of the choral. The scale is a mixture of the 2b Ungar and melodic Ungar scales, with the omission of La (G) and Mi (D).

Excerpt 7: The set of notes in the passage in light of the Ungar and melodic Ungar scales

 

As a result, Liszt’s favoured scale is also present beside the modern, symmetric arrangement (although not at the most frequent D = La pitch).
During the piece, the “cross” motif reappears in various ways. Careful analysis shows that this motif advances the main idea the furthest and is the most progressive. It is no surprise that it is sounded at Jesus’s death, the dramatic low point.
The harmonic sketch of the section from bar 67 of Station XIV, with the lyrics “Ave crux” (“Hail to the cross”), is the following[6]:

 

Excerpt 8: The harmonic sketch of Station XIV, bars 67-75

 

Between the consonant chords, namely D major and (enharmonically) b-flat minor, one of the strongest tonal contrast can be found: their roots are a third apart and they do not have any common notes; in this case the notes of the two chords constitute a scale of the 1:3 model[7] (in this respect, the example could also belong to the chapter about distantial phenomena). Furthermore, the interim bars also use this scale exclusively, which, in the end, dominates for eight bars[8].

Excerpt 9: The set of notes in the passage[9]

 

Despite the considerable contrast, the harmonic sequence is bound together by a special force, namely the underlying symmetry. The second couple of chords are the respective inversions of the first two (D, ⁶B^5#)[10]. In the end, the sequence is inverted back into itself.

Excerpt 10: Symmetry in the passage[11]

 

The harmonies which can be heard lend an extra-terrestrial nature to the passage, which beams the ambience of transcendence, of having overcome all worldly feelings and thoughts. The effect is even more pronounced due to the simplicity of the preceding Gregorian melody and the modesty of the neo-modal harmonies of “Amen”.

In his analysis of Webern’s Cantata No. 1, Ligeti explains that one of the most important organisational principles in the tempered chromatic space is symmetry. In addition, he also argues that symmetry in music should often be “disturbed” with asymmetry. He writes as follows: “... complete order would be boring, and disorder would be pointless; genuine organic nature is derived from apparent disorder which, however, contains order inside.” (Kerékfy, 2010, p.100) He also touches upon the perspective of the form with respect to periodising tonal music, but his main topic comprises cases of the symmetric arrangement of the sound space. The author gave the lecture on which the study is based in 1959. Although it had been some years prior, in 1955, when Ligeti’s choir piece titled Night was composed, the described principles can be observed in several of its sections, even though it is dominated almost completely by diatonic and pentatonic scales.

The choir piece can be divided into three sections, which is marked by textual partitions, changes in structure, and, most importantly, shifts in tonality. The first section uses the 0-diationic scale[12], the second employs the -6-pentatonic scale[13] with the exception of the soprano in bars 56-61, which will be discussed in further detail. In the third section, the 0-diationic scale is represented only by a C major chord, which can be interpreted as a (symbolic) recapitulation in the tonal sense. Consequently, the largest possible tonal contrast is realised in the middle section. However, the contrast is dissolved subsequently at a higher level, which is also true for the soprano melody.

In the second section (from bar 43), the initial harmony is a complete pentatonic chord with symmetric arrangement and A-flat¹ as the axis of symmetry. The A-flat is also one of the notes which can be considered as the axis of symmetry for the 0-diationic scale, alongside D. Directly before the shift (bars 39-42), the climax is marked by a four-time repetition of the following bar:

Excerpt 11: Night, bar 39

 

The bar is unique because the middle chord is symmetrically arranged with respect to A-flat¹, the first and last chords are mirrored images of each other with the same note as the axis of symmetry. As a result, the three chords are inverted around A-flat¹:[14]

Excerpt 12: Symmetry in bars 39-43[15]

 

Despite the sizable contrast, the A-flat establishes the relationship between the two sections as the common axis of symmetry. However, asymmetry is also present due to the rhythmic structure of the prosody.

The contrary is true for the next great formal boundary (bars 64-65). In the middle section, pentatonic chords descend gradually. From bar 59 on, the chord is sounded in six voices by alto, tenor, and bass singers. The axis of symmetry is D, serving the same purpose for the 0-diationic scale, which returns with the closing C major harmony. Although the closing chord is not arranged symmetrically, the evidence discussed so far implies that the composer followed the introduced principles of construction.

The relation of the melody of the soprano towards the underlying pentatonic chord reflects the same principles (bars 56-61). Here, the contrast is also great: the common element is the C-sharp only, otherwise the -6-pentatonic scale is completely alien from the tetratonic scale (Antal, 2008). The latter can be interpreted as part of the scale of the 1:2 model (B, C-sharp, D, E, F, G, A-flat, B-flat). Even without this interpretation, the bitonal nature of the passage is still evident.

However great is the distance between two layers of the composition, there is a hidden connection in the background, which is subtler than the discussed characteristics but contains asymmetry, as well. In the soprano melody, the lowest and highest notes (B¹, F²) have a symmetric position around D. The first, shorter phrase (bar 56) highlights D, while the second (bars 59-61) emphasises the diminished fifth framework. By the beginning of the second phrase, the pentatonic chord reaches the arrangement which is exactly symmetrical to D.

Excerpt 13: Symmetry and asymmetry in bars 56-64

 

The idea of contrasts dissolving at a higher level is also present in the text. At the first formal boundary, the thickness of “countless thorns”, which evokes suffering and the passion, is opposed to the subsequent “silence”, which represents auditive absence and emptiness. At the same time, a new unity is achieved as early as the first utterance of “countless”, which, after being separated from “thorns”, begins to refer to “silence”. After the impersonal, even inhumane “silence” (which requires no human presence), the melody of the soprano cries out in pain, which is undoubtedly the most personal moment of the piece. This utterance makes the “silence” personal at an instant (“My silence”). The relation offered by the symmetry is less regular here, which is in accordance with the lines which express the individual’s loneliness and torturing separateness.

 

Distantial and Akin Phenomena

In the broader sense, distantial phenomena, that is, equidistant (distantial) or alternating divisions of the octave, can be considered symmetric. This is because symmetry in music can be interpreted as periodic repetition, as Nóra Keresztes also notes (2017). Consequently, this chapter could also belong to the previous one, but I have decided to include only truly symmetric phenomena there.

Distantial phenomena are discussed by several authors, including Lendvai (1994), who invented the term, as well as Gárdonyi and Nordhoff (2012). In his study on the Second Tableau of Petrushka (1987) and in his monograph about Stravinsky (1996), Taruskin explores in depth how extensions of the mediant relationship in the Romantic period lead to the emergence of circles of major and minor thirds, which later evolved into the whole tone scale and the octatonic scale. The latter is of major importance for Bartók and Messiaen as well. Lendvai, the theoretician who provided the most comprehensive picture of Bartók’s works, calls it the scale of the 1:2 model (1994), while Messiaen employs the term “mode 2 of limited transposition” (1944).

The sequence of thirds, which is capable of defining tonality, creates an axis. The tritone can also form an axis, either separately or as part of the axis of minor thirds. In the Second Tableau of Petrushka, the C–F-sharp polarity does not allow a role for A and E-flat. Here, the C–F-sharp polarity is connected to the octatonic scale as well as to the whole tone scale: the tonal plan is C–D–E–F-sharp (Taruskin, 1987, 1996), while in local passages its inversion can be observed (C–B-flat–G-sharp–F-sharp, ([49]+2-6) and the last three bars[16]).

In practice, composers employ the axis of minor thirds[17] and the scale of the 1:2 model the most. The two may appear together as well as separately.

From the perspective of diatony and functional tonality, the distantial arrangement of the sound space has an atonal effect. However, it is precisely due to their organised nature and distinct order why such phenomena can be considered tonal.

According to Klára Hamburger, the downward diminished fifth is a sort of musical fundamental cell of Via crucis, “the symbol of Christ’s passion”, which is present in every station except the Introduction. It is present melodically either by itself or together with other notes, while it also plays a role in the construction of chords (Hamburger, 2010, p. 74). The tritone, the interval which divides the octave into two equal parts, has an especially apparent distantial character when it is actually completed to the octave.

The first example for this is Pilate’s solo at the end of Station I. The tonality is the Ti-mode of the harmonic minor scale. The frame of the melody is provided by the B–B octave, while the F is emphasised as a repeated phrase-starting note.

Excerpt 14: End of Station I

 

Another example is the end of Station XIII, which refers back to previous material. At the end, the beginning of Station VI is repeated, with the exact re-emergence of the first two phrases (the frame is the F¹–B diminished fifth) and the third phrase bends chromatically towards F.

Excerpt 15: End of Station XIII

 

A more complex example can be found at the beginning of Station XII. Both baritone solos have the ambitus of D¹–G-sharp. The first begins with the thematic core itself, which is the repetition of the downward diminished fifth, and is completed with the notes from the Ungar scale (D¹–C-sharp¹–B-flat–A–G-sharp) (Hamburger, 2010), while the second has diatonic scale (D¹–C-sharp¹–B–A–G-sharp). Between the solos, the upper voice of the instrument completes the D¹–G-sharp diminished fifth to the octave. The upper tritone echoes the preceding Ungar scale (D²–C-sharp²–B-flat¹–A¹–A-flat¹), while the lower introduces a new distantial phenomenon, namely the scale of the 1:2 model (A-flat¹–F-sharp¹–F¹–E-flat¹–D¹). The chords are mostly distantial, that is, augmented or diminished[18]. It is noteworthy, however, that it is the A-flat note which first gets a relatively stable minor sixth chord (bar 6), then a minor chord (bar 7)[19], which reinforces the structural role of the D²–G-sharp¹(A-flat¹)–D¹ tritone axis.

Excerpt 16: The realisation of the distance principle in Station XII (bars 2-11)

 

The harmonic sequence exerts an atonal effect in the traditional sense, which is appropriate for Jesus’s words, who is dying on the cross. In the broader sense, it cannot be considered atonal due to its exceptionally conscious arrangement.

In The Carnival of the Animals by Camille Saint-Saëns, the tonality of the second movement (“Hens and Roosters”), which also contains onomatopoeic elements, is more difficult to disentangle than it is the case with other movements, since it surpasses diatonic, that is, functional and neo-modal solutions as well as Romantic arrangements such as third-related relationships. Regarding its form, the first section is repeated variedly (A and A’, 1-[1]-1 and [1]-[1]+4), followed by a short medium section with opposing characteristics, which incorporates legatos as well as the staccatos of the first section and also includes a clarinet solo resembling the crowing of a rooster (B, [1]+5-[2]+4). Finally, the initial material returns without imitations (A’’, from [2]+4, with overlap), finished by a fortissimo G major chord.

The tonality of section A might be interpreted as the dominance of a harmonic G minor scale. The only foreign note is B, which serves as a light ornament (appoggiatura). The continuation, however, does not support this interpretation, which is why it is more reasonable to characterise the section with the scale of the 1:2 model. The incomplete axis of minor thirds (C–A–F-sharp), which is built on the lower notes of the half notes, can also be observed.

Excerpt 17: The set of notes in section A

 

It is impossible to interpret the altered repetition the same way since another scale of the 1:2 model with two missing notes dominates clearly. The C-sharp–A-flat downwards augmented third is an interesting detail, which comes from the consistent notation of A-flat.[20]

Excerpt 18: The set of notes in section A’

 

In section B, the imitation of chromatically ascending melodies with minor third ambitus can be heard (viola: B→D, 2^nd violin: F→A-flat, 1^st violin: D→F), which creates a compete axis of minor thirds. After the imitation is finished, only three axis notes remain, namely the D¹, A-flat¹, and F². Of these, the first gets a chromatic appoggiatura (C-sharp¹), the tonal role of which becomes apparent imminently. With the third crow of the rooster (from [2]+2), the third possible scale of the 1:2 model emerges. The C-sharp grace notes are precursors; the scale is completed with B-flat in bar 4 of section A’’.

Excerpt 19: The set of notes between [2]+2 and [2]+7

 

We step out of this framework one bar before [3], when the high notes (together with the antecedent) create the chromatic sequence of C-flat–B-flat–C–D-flat–D. The transition to the closing bars can be considered diatonic with a modal flavour, as the piece proceeds from F to the final G major chord. The latter continues with a cadential step towards the beginning of the next movement, which is set in C minor, thus reinterpreting itself as dominant.

As for the execution of distantial principles, it can be observed that the scale of the 1:2 model and the axis of minor thirds are present both together and separately. There is some potential significance in relation to the fact that two notes are missing from the first two model scales. At first, the scale is related to harmonic G minor, while in section A’, the diatonic character is prevalent (B-flat⁷, then a Lydian turn). As a result, modernity is present in the tonality rather as a flavour, and the movement fits organically into the whole composition. Besides the tone of the sounds and the manner of play, the effect of disorganisation, which reflects the henyard exceptionally, is also elevated by the loose connection to functional music and diatony as well as by the lack of a tonal centre. However, the analysis shows that in the background, there is indeed order, which has been thought through carefully.

At the end of Francis Poulenc’s cycle titled Un soir de neige, we find a significant date: 24-26 December 1944. Both Paul Éluard’s poems and the movements of the cycle were created during and influenced by the war. In them, the winter represents passing and death (Ákos Erdős, 2013).

It is rare to observe the distantial arrangement of tonality in such a clear way as in the third movement, with the title Bois meurtri. If the compositional core idea had to be pointed out, it would be the tritone, which plays a key role both melodically and tonally, while it is also crucial regarding the harmonies. The movement has three sections (1-[2]-1, [2]-[4]-1, [4]-), and its form can be described by the letters A B A’, that is, it contains a modified, and also shortened, repetition. The beginning of A and A’ is the f-sharp minor chord, while the middle section starts with a polar-distance c minor harmony. In other words, the key of the global tonal plan is the tritone. It is hardly an exaggeration if we consider it the musical symbol of passing and death.

The tritone is also present as a sequence of notes several times. Although the beginning of the movement is harmonically inclined, the melody with tritone ambitus in the soprano, performed in the first two bars, will have motif-related significance later.

Excerpt 20: The motif of Soprano I (bars 1-2)

 

The melody reflects the scale of the 1:5 model (F-sharp, C-sharp, C, G), which automatically implies that it also fits the scale of the 1:2 model[21]. After four bars, we hear a freely diminished and transposed permutation:

Excerpt 21: The first version of the motif (bars 6-7)

 

A further diminished version of this permutation can be found after [1] in bar 3:

Excerpt 22: The second version of the motif ([1]+3)

 

In the middle section, another version appears, and the step of fourth is continued as a sequence:

Excerpt 23: The further development of the motif (from [3])

As for tonality, the global tonal plan discussed above is contained in a concentrated form in the very first change of harmony, reflecting the organic arrangement. The two tritone-distance minor chords determine the F-sharp–C tritone axis and, simultaneously, the F-sharp–A–C–D-sharp axis of minor thirds, while they fit into the following scale of the 1:2 model:

Excerpt 24: The set of notes, the axis of tritone and minor thirds (bars 1-2)

 

The sequential repetition a major second lower (bars 3-4) determines the E–B-flat tritone axis and, simultaneously, the E–G–B-flat–C-sharp axis of minor thirds, while another scale of the 1:2 model is sounded:

Excerpt 25: The set of notes, the axis of tritone and minor thirds (bars 3-4)

 

In theory, the sequence would be continued by the D–F–G-sharp–B-flat axis of minor thirds and the D–E-flat–F–F-sharp–G-sharp–A–B–C model scale:

Excerpt 26: The set of notes and the axis of minor thirds (bars 5-9)

 

This is what occurs with two exceptions in chords. The harmonies (exceptions in bold) are the following: „b^Ø7” C-sharp^9b „G^3/4” b-δ[22] g-sharp^5b e-sharp^O™7. In the bass, the entire axis of minor thirds can be heard. The two “foreign” notes (bass: C-sharp, soprano and tenor: G) are the result of transition, and the two highlighted chords are also in a position without particular emphasis. The solo of the alto (bars before [1]) employ only the D–E-flat–F–F-sharp–G-sharp–A–B–C scale. We may conclude that it is indeed the discussed scale and the axis of minor thirds which dominate.

The continuation (from [1]) has similar tonality. The material in the first four bars return a major second higher so that the sequential repetition arrives back at the original pitch and the initial model scale at the same time. Subsequently, the same logic is applied when the notes and the axis of minor thirds drop by another major second (thus it reflects the tonality of bars 3-4). Notes which do not fit this scheme are present in one chord only (A and D-sharp, at the end of [1]+2), again as part of a transition. There are even connections between chords with polar distance
(e.g., C-sharp^4/6 with G and G⁷, the last two chords of [1]+5). The second of these turns towards C minor with G⁷, f⁷ and f^5/6,[23] and it becomes apparent with the sounding of c minor harmony that the closing G major chord has a dominant function in the half cadence in C minor.

In summary: the principle of distance is reflected not only in sequences of limited length and the global tonal plan but also in the intermediate level until the second bar before the second rehearsal mark, due to the consequent descent of whole tones.

After the functional preparation, the first five bars of the middle section are characterised by romantic harmonies with strong functional logic, which provides contrast against preceding passages. The key is C minor with a subdominant digression to F minor in bar 5[24]. After the end line, a connection occurs with a common third (f-E), after which the distantial principle returns. Besides dynamics, the ascending sequence also contributes to the gradual intensification, while three elements of the axis of minor thirds which initiated the movement participate in dominant-tonic progressions. The keys in question are A minor, C major, and E-flat minor ([3]+2-[3]+4), which is centred around C symmetrically and provides tonal contrast by omitting the most important note in the movement, namely the F-sharp.

Excerpt 27: The harmonic sketch of the sequence ([3]+2-[3]+4)

 

The bars before the repetition are harmonically characterised by a disguised freely chromatic mixture again, this time ascending. At the same time, the gradual dynamic intensification is continued[25]. The bar preceding the recapitulation brings back the principle of distance, with notes exclusively from the E-sharp–F-double-sharp–G-sharp–B–C-sharp–D incomplete scale of the 1:2 model.

From [4], the first four bars of the movement are literally repeated, followed by the continued descent of axes of tritones and minor thirds as well as scales of the 1:2 model, major second by major second[26]. This time, the cadence does not continue the whole tone descent; instead, it changes back to the G–C-sharp tritone axis. The final harmonic step highlights the tritone again, due to the tenor and the lower bass.

The distantial principle is incorporated in the construction of several harmonies. The French sixth chord is sounded twice; what is more, both times with the same notes ([4]-1 and [5]+1, with F-double-sharp notation the first time). In the first case, it is the dominant of F-sharp, in the second, it progresses another way. The French sixth can be constructed from two tritones in the distance of a major third, which is obvious at first ([4]-1) when the women’s and men’s choir separate, or in a reverse way, from two major thirds in the distance of a tritone ([5]+1). Chords which are characteristic of axes of minor thirds and tritones as well as scales of the 1:2 and 1:5 models, are also present. These include the quasi b-delta (the “real” one would be, from below, F-sharp/B/D/F) in bar 5[27], the g-delta ([1]+2, 5. eighth), the C-sharp major six-four complemented with an acoustic fourth ([2]-2, 5. eighth), and the chord with the entire scale of the 1:5 model ([4]-1, 4. eighth).

The tonal structure of the whole movement is in accordance with Ernő Lendvai’s theory on axis systems (1994). This study does not attempt to evaluate whether elements of the main axis of minor thirds in such an axis-based tonality may be called tonic, and if so, what explanations could be offered for this, because this would require the extensive analysis of Lendvai’s theoretical work, thus exceeding the limitations of the present work. Consequently, I consider functional characteristics only in relation to local harmonic connections.

The main elements of the tonality are provided by the F-sharp–A–C–E-flat axis, and, within this, by the F-sharp–C tritone. The beginning of the movement and that of the repetition lead us to conclude that the F-sharp minor enjoys priority, which is further corroborated by the deceptive nature of the cadence (see below).

Elements of the E–G–B-flat–C-sharp axis are present on multiple occasions as functional local dominants, as discussed previously: in the C minor part and its preparation, in the axis sequence ([3]+2-[3]+4), and in the preparation of the repetition. The latter is especially interesting because besides C-sharp^5/6, the other two chords can be interpreted as having a root of G or C-sharp: the C-sharp and G are supported by the major third in the French sixth and by the fifth (as lower fourth) in the 1:5 chord. Here, the similar function of notes in tritone distance is necessarily true. The last three bars in the movement are not so clear, however. The g-sharp^Ø7 and the following French sixth could be continued with an f minor chord as a perfect authentic cadence (which would have a rather flat effect, however). In this sense, the end may be considered a deceptive cadence, what casts doubt on the dominant function of the ending chord, at any rate.

Among elements of the D–F–G-sharp–B axis, it is possible to interpret f⁷, f^5/6 ([2]-1), and the f minor chord ([3]+1) as subdominants for C minor. The latter fits into Lendvai’s observation (1964), namely that the subdominant is related to the emotional sphere the most. This is perhaps the most heart-breaking moment in the movement as it depicts the lyrical self’s suffering.

Using distance-based arrangements, Poulenc creates an exceptionally coherent musical construction in this movement. Although listeners perceive the perfection of the structure at least subconsciously, the high quality of the composition is also evident from the movingly suggestive expression of pain, hopelessness, and death in the piece.

Besides the distance-based division of the octave, an axis can also be created from intervals which do not divide the octave equally. If the composer chooses the perfect fifth, the possibility of augmenting or diminishing the axis arises, as opposed to distance-based divisions, precisely because the pillars do not repeat for each octave. Tonal phenomena and systems in relation to distance-based divisions are employed mostly to offer something novel instead of the fifth-based organising principle, which has its roots in harmonics. However, a fundamentally novel result is achieved here, instead of turning back to the old practices.

To determine such phenomena, the unambiguous articulation of the pillars is required. It can be observed several times in The Soldier’s Tale that the sequence of perfect fifths plays a tonality-structuring role[28], which is especially apparent in The Soldier’s March. In my doctoral thesis (2015), the reader may find a comprehensive analysis of the entire piece. Now I offer a summary of the analysis about the first movement from my thesis, with slight modifications and extensions. I also emphasise that the resulting tonal order is related to distantial phenomena.

In the movement, the starting point is represented by the G–D–A–E axis of perfect fifths, which corresponds to the strings of the violin. In the ostinato of the double bass from [1], the D is sounded as a downward fourth, which is an inversion to play a cardinal role in the tonal plan. Furthermore, the A/E fifth represents a separate layer, which provides the focal sounds of the thematic material for the cornet and the trombone ([1]+2-[3]). The axis of fifths includes two clashes of major second (the G/A and D/E at a distance of two fifths). The D and E provide support for G and A as harmonics (although the former as a lower fourth), which is why momentarily it is the polarity of G and A which plays a key role in the tonal structure (the D/E clash is ornamental only). The G cannot be regarded as tonic in the original sense, but it has priority as the base of the axis of fifths, while it is also unique in its continuity (throughout almost the entire movement). By contrast, the A is reinforced by solutions such as the repetition by the bassoon at the end of the first major formal section (from [3]) or the subsequent cadence (A–B–A pendulum, cornet, [3]+2-[3]+3), which is the linear representation of the major second polarity at the same time. With the introduction of the B major chord, the axis of fifths begins to expand (G–D–A–E–B).

From [1] until here, there was a uniform tonal framework due to the fact that the notes were selected from the +2-diationic scale only. The D-sharp from the B major chord disrupts this. Immediately thereafter, the diatonic March-theme (from [1]+2) is played in a distorted and chromatically compressed modification (cornet and trombone, [3]+4-[3]+6; clarinet, [4]) (White, 1976), which brings about the emergence of the chromatic scale through the compression of the scale steps and of the distance between voices[29].

Excerpt 28: The March-theme and its distortion

 

The axis of fifths expands further (G–D–A–E–B–F-sharp) as the cornet and the clarinet repeat the cadence sequences a fifth higher (E–F-sharp–E, [4]+1 and [4]+3). As a result of the reinforcement through the cadence, the E breaks out from its previous secondary role. This was prepared by the clarinet by revolving around E and D-sharp and halting at D-sharp, which is the leading tone of E ([4]-[4]+1).

From [5]+1, the focal points inside the axis of fifths change. It is worthwhile to mention that the D is transferred to the lower position, while the G and D of the ostinato are taken over with chromatic alternate notes by the violin and the B–F-sharp fifth is filled polymodally (bassoon, [6]-1: D-sharp, and clarinet, [6]: D).

From [6]+4, the original ostinato returns. At the same time, the E–F-sharp–E cadence is played, while the accompaniment of the initial theme also contains G-sharp, which results in a bitonal clash. In the first genuine tutti of the movement (from [8]), both the ostinato and the March-theme become richer. The violin imitates the G and D notes of the double bass in the upper fifth (twelfth) which is implied by the axis of fifths, supplemented by consequent ninths (=two fifths) and a relatively free inner voice. At the beginning and end of phrases, the accompanying voice of the trombone seeks consonance; in the middle, dissonance.

The A–B–A pendulum ([9]) is followed by the distorted version of the March-theme, which has already been foreshadowed. Invariably, Stravinsky starts out from the original motif form, which means an ascension after a descending tetrachord, and supplements it with repetition and alternating notes. The ambitus of the melody is compressed to a mere perfect fourth (F-sharp–C-sharp, clarinet, bassoon, and trombone [10]-[10]+6, clarinet and cornet [11]-[12]+5). The outline of the melody, with special notation for the second augmentation, is the following:

Excerpt 29: The outline of the melody

 

The above occurs against the ostinato, which is joined by the violin (from [10]) and the bassoon (from [11]). From [10] in the score, this is the only thing which represents the axis of fifths, while the chromatic upper layer becomes detached. By contrast, Stravinsky’s suite-version for clarinet, violin and piano as well as the piano arrangement are exceptional examples for the tonality-structuring significance of the axis of fifths, including the G–D–A–E starting point. From [10] to [13] in the entire score, the E is present in an insignificant role, while the A is not present at all. As opposed to this, the piano excerpt from [11] contains, alongside the ostinato and the melody, the A¹/E² fifth (furthermore, the accompanying voice of the trombone is printed in cue size one octave higher for four bars), while at [7] in the suite (which corresponds to [11] in the main piece), the violin changes to the A/E fifth. The substitution depicts the G–D–A–E axis of fifths perfectly.

The expansion of the distorted March-theme in tutti, alongside the emphasised final note of D-sharp before [13], or E-flat in the piano version, causes a conflict which disrupts and contrasts the prevailing order. It is important to note that all this follows directly from the antecedents. Until then, D/G have formed mostly a fourth, with additional fourths based on them from [13]. Consequently, a darkening occurs measured on the pillar of fifths[30], and the axis of fifths expands downwards for the first time. The main notes are D, G, C, and F, which are joined by alternate notes B-flat and E-flat as a descent of further two fifths. The effect is made even more powerful by the fact that between [1] and [13], besides the chromatically transformed March-theme, there has not been a lower note on the pillar of fifths than G.

The final section resembles the beginning. Like the axis of fifths, it contains the major second polarity. At the beginning of the movement, the upper, leading voice moves around G at first[31], starting from the lower major second, and then the bass arrives at G ([1]), thus establishing the priority of the note. The two closing harmonies are a major second apart (f-sharp²–e^5/6). At the end of the movement, this material returns twice in a row, in an increasingly shortened form, a fifth lower, which means that it adjusts to the downward expansion of the axis of fifths.

In the initial G–D–A–E axis of fifths, it is not difficult to notice the strings of the violin. The instrument symbolises the soldier’s soul, which is why a deep connection emerges in the movement between the tonal system and the content of the piece, which is based on a Faust-like folk tale (Taruskin, 1996). The distortion of the diatonic and carefree March-theme brings about the reversion of the axis of fifths, while the return of the beginning of the movement a fifth lower adjusts to the disruption in the final section, thus foreshadowing the soldier’s demise and the devil’s final victory.

 

Polytonality in Stravinsky’s Music[32]

When discussing polytonality, Igor Stravinsky’s music is especially worthwhile to analyse. As with many other movements at the beginning of the 20^th century, he is considered among the initiators of polytonality. Alfredo Casella (1924) has argued that Sacre[33] was the first polytonal masterpiece, while Roman Vlad (1967) has pointed out that many refer to the Second Tableau of Petrushka as the beginning of the phenomenon, in particular the famous superimposition of the C major and F-sharp major chords. Others have connected polytonality to Béla Bartók’s music, among others. Lajos Bárdos, for example, has written an entire chapter about this in a study (1974). At the same time, the idea has met significant criticism over the past decades, especially in relation to Stravinsky’s music. Arthur Berger (1963), Pieter C. van den Toorn (1983, 2003), and Richard Taruskin (1987, 1996, 2011) have all questioned its relevance based on octatony, either partially or fully. This chapter attempts to explore the validity of the concept through some examples from Stravinsky’s music.

For a start, it is worthwhile to quote Stravinsky himself about Petrushka (Craft and Stravinsky, 1987, p.136-137): “I had conceived of the music in two keys in the second tableau as Petrushka’s insult to the public, and I wanted the dialogue for trumpets in two keys at the end to show that his ghost is still insulting the public.” This is a clear statement for bitonality. The author of the perhaps most renowned monograph on Stravinsky, Eric Walter White regards it as a central tool of the first two periods, namely the Russian and Neoclassical periods, and mentions it in the foreword of his book that he has the composer’s approval (White, 1976).

By contrast, it is interesting to listen to what Bartók proposed in his second Harvard lecture (Tallián, 1989, p.170-171), especially because it also concerns Stravinsky:

“Perfect and real atonality doesn't exist even in Schonberg's works, because of that unchangeable physical law concerning the interrelation of harmonies, and their relation to their fundamental tone. […] The same phenomenon appears when one deals with so-called polytonal music. Polytonality exists only for the eye, 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. […] 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-sharp. It sounded queer, and the misled public said, ‘Oh, this is a very interesting, very modern and daring music.’ Such artificial procedures have no value at all. […] 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.”

It seems that Bartók is referring to an oral or written statement made by Stravinsky[34]. If it is true, it contradicts the quote about Petrushka, which has already been mentioned and was uttered later. Either way, Stravinsky’s public comments about his music need to be taken with a grain of salt[35]. As for orthography, there are other possible explanations besides denying the polytonality of his music. Taruskin (1996) compares sketches for The Nightingale with the piano arrangement, which predates the score for the opera, and he finds that the former displays white-key diatony and black-key pentatony, while the latter contains B-sharp instead of C and E-sharp instead of F consistently, not to mention double sharps. Taruskin argues that the notation is misleading on purpose to make imitation for other pianists more difficult.

Another important thing to mention is that, according to Taruskin’s (1996) report, Stravinsky conducted experiments while composing the second act of The Nightingale on how to resolve the problem of simultaneous contrasting of white and black keys. This implies that in the composing process polytonality was a starting point for the composer.

Before analysing concrete examples, it is important to note that the significance of octatony in Stravinsky’s music, besides that of diatony, cannot be questioned any more. Van den Toorn (1983), and after him, Taruskin (1987, 1996) refer to the three possible octatonic scales with the following numbering:

Excerpt 30: Octatonic scales

 

Now let us see some concrete examples from the perspective of the polytonal surface. The easiest case is when two separate diatonic layers are part of the same octatonic scale, such as in the Petrushka chord.

Excerpt 31: The third octatonic scale and the Petruska chord

 

It is possible for a diatonic and an octatonic layer to interact with each other based on their common elements[36]. One possibility is that one of the tertian harmonies, which are constructed on the lower notes of the octatonic half note steps, is supplemented diatonically. According to van den Toorn (1983), the third octatonic scale and the C major scale interact in the Second Tableau of Petrushka, the notes of the C major chord serving as common elements.

Excerpt 32[37]: The interaction of the third octatonic scale and the C major scale

 

The second, more frequent possibility for the interaction is based on the Dorian tetrachord[38]. The octatonic scale contains four of these[39], one of which is completed to a diatonic scale. In the Ritual of Abduction movement of Sacre, the third octatonic scale interacts with the A-Dorian scale. The sustained chord (C/E/G/B-flat/D-flat/E-flat) belongs to the third octatonic scale, the melody in the discant is A-Dorian. The common tetrachord of the two scales is A–G–F-sharp–E (in descending order):

Excerpt 33[40]: The interaction of the third octatonic scale and the A-Dorian scale

 

In the first tableau of The Wedding (from [11]+3), the interaction of the second octatonic scale and the +3-diatonic scale can be observed. The upper part of the first and third piano and the soprano solo are in the +3-diatonic scale (from above B–A–G-sharp–F-sharp–E–D–C-sharp–B). The upper tetrachord of this is completed to an octatonic scale (B–A–G-sharp–F-sharp–F–E-flat–D–C) by the bass solo, reinforced by the instruments[41]:

Excerpt 34: The interaction of the second octatonic scale and the B-Dorian scale

 

It is also possible to interpret the connection between two diatonic scales which are a tritone apart as an interaction. In the 3. piece of the Cat’s Cradle Songs, two tetrachords of the first octatonic scale are complemented to become complete diatonic scales:

Excerpt 35[42]: The interaction of the first octatonic scale with the 0-diatonic and the +6-diatonic scale – Cat’s Cradle Songs, 3.

 

Van der Toorn presents the interaction of the first octatonic scale with the F-Dorian and, subsequently, the B-flat-Dorian scale in the Introduction of Sacre (from [10], and from [10]+4, respectively):

Excerpt 36[43]: The interaction of the first octatonic scale with the F-Dorian and B-flat-Dorian scale

 

On the contrary, the passage at rehearsal mark 9 cannot be explained in such a way. It is worth looking at the excerpt from [8] to [10]. From [8] to [10], the alto flute, following the bassoon, has clear D-modality. The first three bars contain an unambiguous interaction between the first octatonic scale[44] and the C–D–E–F–G–A hexachord, with the D–E–F–G Dorian tetrachord as the common element. In the interaction of the motif on the flute, the G-sharp, A-sharp, and C-sharp belong to the octatonic scale, which can be interpreted as part of the G-sharp–A-sharp–B–C-sharp tetrachord and is completed diatonically (to a pentatonic scale) by D-sharp ([8]+2-[8]+5). In bar 4 ([8]+3), however, the alto flute changes with F-sharp to a major third mode, which does not interact directly with the still prevalent first octatonic scale due to the lack of a common tetrachord. From [9], only the oboe joins the alto flute with an F-modal pentatonic scale, and these remaining two voices cannot be described as an interaction, either. At the same time, it is the clearly articulated root notes of the mode, namely the D and F, which cause the passage to remain in the framework of the minor third relations, which is characteristic of octatony. The D and F are elements of the first octatonic scale, which provides continuity between the preceding (before [9]) and subsequent (after [10]) first octatonic scale [45].

Excerpt 37: The material of the alto flute, the flute and the oboe

 

Taruskin does not consider the final section of Chant dissident (Quatre chants russes, 4.), which contains no bar lines, to be polytonal; instead, he argues that it is a “linear composing-out of a single, static, eminently tonal concluding chord, the sum of the three finals” (Taruskin, 1996, p.1198). This seems to be an oversimplified explanation if we uncover the connections between voices. The two voices of the piano are in bitonal opposition, while relations between other voices constitute tonal unity, which provides coherence for the whole, resulting in a finely balanced and complex system. The vocal part is polymodal by itself, but its combination with the the left hand can also be regarded polymodal in the broader sense, the E being the fundamental note. The notes played by the right hand are clearly continued by the vocal part.

Excerpt 38: Chant dissident, scales of the closing section

 

The emblematic chord of the Augurs of Spring in Sacre – alongside subsequent bars – is widely described as another classic example of bitonality, interpreted as being the combination of an E-flat dominant seventh and an F-flat major chord.

Excerpt 39: The Sacre chord

 

Even Peter Hill (2004) prefers the polytonal interpretation, which includes E major, E minor, C major, and E-flat dominant seventh chords from [14], even though he considers the concept in relation to Sacre outdated.
In addition, Dmitri Tymoczko (2002) proposes that the initial chord contains the notes of the complete harmonic G-sharp minor scale. However, van den Toorn (Tymoczko and van den Toorn, 2003) rightly notes that the G-sharp minor scale is out of the question if the initial chord is not analysed in isolation. He also highlights the significance of the constant elements, namely the E-flat dominant seventh, the D-flat–B-flat–E-flat–B-flat ostinato, and the uniquely doubled E, that is, the purely octatonic layer. It must be added that from a theoretical point of view, the initial harmony is more adequately interpreted as a F-flat/G/B-flat/D-flat/E-flat octatonic harmony[46] + other notes. White (1976) also refers to the root-note nature of E-flat, which is corroborated by Hill’s analysis (2004), namely that the F-flat in the bass (which is present in both the score and the two-piano version instead of E) is resolving twice into E-flat ([16] and [28]). In addition, the 3b key should also be taken into account. The lower E note of the octatonic harmony provides major seventh tension and a basis for the diatonic tertian chords, the exact specification of which is of secondary importance. This is corroborated by the e minor broken chord of the bassoon from [14], which is not present in the two-piano version.

The above is also helpful in determining the closing chord ([72]-1) of the short movement titled The Kiss of the Earth (The Oldest and Wisest One). Using the notation of the two-piano version:

Excerpt 40: The closing chord of The Kiss of the Earth (The Oldest and Wisest One)

 

The upper layer can be separated well from the lower in this case as well. Written in a framework of major seventh, the C/E-flat/F-sharp/A-flat/B octatonic chord[47] assumes the upper position, while the lower harmony is a diatonic one built on C – which provides the major seventh tension –, this time using fifths instead of thirds.

In the Second Tableau of Petrushka, ambiguity is found in the passages where the white-key layer plays the G major chord and escapes the third octatonic scale, which contains the Petrushka chord (right hand on the piano from [49]+10 and [50], first clarinet from [49]+14)[48]. Here, Taruskin permits and even prefers dramaturgically the bitonal interpretation, whereby the G major arpeggio is regarded as the dominant of the C major harmony, but only if ”we bear it in mind that the keys in question were chosen from among the circumscribed and historically validated wares of the time-honored (and specifically Russian) octatonic collection” (Taruskin, 1996, p.748-749). He considers the Petrushka chord to be an active polarity and not a static constellation.

The C sharp chord becomes independent as it is preceded by two anti-tonic chords (Taruskin, 1996): d minor (from [60]) and b^Ø7 (first from [50]+4 and especially from [59]+6), both with the presence of a black-key layer. It must be added that the latter also becomes independent occasionally, although less frequently. An example for this is the tonal sequence inside the +6-pentatonic scale, with the adjustment of the third element to the scale (from [50]+4, repeated), thus making use of pentatony completely.

Excerpt 41: Voice of the left hand from [50]+4

 

Another example is provided by the scale sequence in the two bars before rehearsal mark 60. Both examples create maximum tonal tension before the tutti play of the theme. Indeed, the movement progresses from octatony to the separation of white and black keys gradually, but the end result is more than a mere interaction.

The following summary can be offered. Mostly, we find diatonic layers well separated in register and timbre. We cannot fully agree with Bartók’s argument that one key is always heard as the central key, and the other is perceived as a modification in relation to this (Tallián, 1989). While in Petrushka, it is possible to interpret occasionally the notes of the G major scale as alternating notes for the F-sharp major (although the reverse is possible too, see [50], [50]+1), and during six bars from [59]+6, the interpretation with the alternating notes is plausible, Stravinsky takes the independence of separate layers too far overall. Historically, the starting point of polytonal phenomena is octatony in his music, but during the composition of certain pieces, he sometimes approached from polytonality. This is apparent in Petrushka and is evidenced by written records for The Nightingale (Tarushkin, 1996). Let us not forget Stravinsky’s own words about Petrushka (“in two keys”) and the quoted dramaturgical justification. In the final section of the Chant dissident, octatony is out of the question.

We may conclude from Bartók’s words (Tallián, 1989, p.170-171) that the mechanical use of polytonality, for example “a hackneyed-sounding melody in C and a very hackneyed accompaniment in F-sharp”, does not lead to a valuable artistic result. However, if the composer creates a strong connection between both successive and simultaneous processes and layers, the various tools lead to a uniform tonal framework. With Stravinsky, this is most often but not exclusively executed through the participation of octatony. Therefore, polytonality is a valid concept in Stravinsky’s music, if we use it with the necessary precaution.

The most important ideas throughout the chapters were the following. Even if it is concealed, some form of artistic order can always be observed. What evokes an atonal effect from the perspective of diatony and functional tonality cannot be regarded as atonal viewed from the 20th-21st centuries. The structure of the tonal system might create an organic unity with the poetic content, as in The Carnival of the Animals or The Soldier’s Tale. The fact that in a masterpiece we can find the organic unity even under a heterogeneous surface is illustrated by polytonal examples.

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References

• Antal Laura (2008): Formák és kifejezésmódok Ligeti György vokális művészetében. Doktori értekezés. [Forms and Ways of Expression in György Ligeti’s Vocal Works. Doctoral thesis.] Liszt Ferenc Zeneművészeti Egyetem, Budapest.
• Bárdos Lajos (1969): Liszt Ferenc „népi” hangsorai. In: Harminc írás. [Franz Liszt’s Folk Scales. In: Thirty writings.] Zeneműkiadó, Budapest. 98-126.
• Bárdos Lajos (1974): A Bartók-zene stíluselemei az Egyneműkarokban és a Mikrokozmoszban. In: Tíz újabb írás. [Elements of Bartók's Musical Style in the Choral Works for Equal Voices and Mikrokosmos. In: Ten new writings.] Zeneműkiadó, Budapest. 5-106.
• Bárdos Lajos (1976): Liszt Ferenc, a jövő zenésze. [Franz Liszt, the Musician of the Future.] Akadémiai kiadó, Budapest.
• Berger, Arthur (1963): Problems of Pitch Organization in Stravinsky. Perspectives of New Music, 2. 1. 11-42.
• Casella, Alfredo (1924): Tone-Problems of To-day. The Musical Quarterly, 10. 2. 159-171.
• Craft, Robert & Stravinsky, Igor (1987): Beszélgetések. [Conversations.] Hungarian translation: Pándi Marianne. Gondolat, Budapest.
• Dahlhaus, Carl & Eggebrecht, Hans Heinrich (1985, ed.) (Editor of the Hungarian edition: Boronkay Antal): Brockhaus Riemann zenei lexikon. [Brockhaus Riemann Dictionary of Music.] (1th edition, 3 volumes). Zeneműkiadó, Budapest.
• Erdős Ákos (2013): Óbudai Kamarakórus honlapja. [Website of the Chamber Choir of Óbuda.]
  http://kamarakorus.hu/mediatar/forditasok-es-muismertetok/francis-poulenc-un-soir-de-neige-paul-eluard/
• Gárdonyi Zsolt & Nordhoff, Hubert (2012): Összhang és tonalitás. [Harmony and Tonality.] Hungarian translation: Terray Boglárka. Rózsavölgyi és Társa, Budapest.
• Hamburger Klára (2010): Liszt Ferenc zenéje. [The Music of Franz Liszt.] Balassi kiadó, Budapest.
• Hill, Peter (2014): Stravinsky: The Rite of Spring. Cambridge University Press, Cambridge (UK).
• Horváth Zsolt (2015): Stravinsky 1913 és 1920 között írott műveinek harmóniai és tonális vonatkozásai. Doktori értekezés. [Harmony and Tonality in Stravinsky's Works Composed between 1913 and 1920. Doctoral thesis.] Liszt Ferenc Zeneművészeti Egyetem, Budapest.
• Kerékfy Márton (2010, ed.): Ligeti György válogatott írásai. [Selected Writings of György Ligeti.] Rózsavölgyi és Társa, Budapest.
• Keresztes Nóra (2007): A funkciós tonalitás felbomlásának folyamata. Doktori értekezés. [The Dissolution Process in Functional Tonality. Doctoral thesis.] Liszt Ferenc Zeneművészeti Egyetem, Budapest.
• Lendvai Ernő (1964): Bartók dramaturgiája. [Bartók’s Dramaturgy.] Zeneműkiadó, Budapest.
• Lendvai Ernő (1994): Szimmetria a zenében. [Symmetries of Music.] Kodály Intézet, Kecskemét.
• Messiaen, Olivier (1944): Technique de mon langage musicale. [The Technique of My Musical Language.] Leduc, Paris.
• Tallián Tibor (1989): Bartók Béla írásai 1. [The Writings of Béla Bartók 1.] Zeneműkiadó, Budapest.
• Taruskin, Richard (1980): Russian Folk Melodies in „The Rite of Spring”. Journal of the American Musicological Society, 33. 3. 501-543.
• Taruskin, Richard (1987): Chez Pétrouchka: Harmony and Tonality „chez” Stravinsky. 19^th-Century Music, 10. 3. 265-286.
• Taruskin, Richard (1996): Stravinsky and the Russian Traditions. A Biography of the Works through Mavra. University of California Press, Berkeley, Los Angeles.
• Taruskin, Richard: (2011): Catching Up with Rimsky-Korsakov. Music Theory Spectrum, 33. 2. 169-185.
• Tymoczko, Dmitri (2002): Stravinsky and the Octatonic: A Reconsideration. Music Theory Spectrum, 24. 1. 68-102.
• Tymoczko, Dmitri & van den Toorn, Pieter C. (2003): Stravinsky and the Octatonic. The Sounds of Stravinsky. Music Theory Spectrum, 25. 1. 167-202.
• van den Toorn, Pieter C. (1983): The Music of Igor Stravinsky. Yale University Press, New Haven, London.
• Vlad, Roman (1967): Stravinsky. Transl. Frederick and Ann Fuller. Oxford University Press, London.
• White, Eric Walter (1976): Stravinsky. A zeneszerző és művei. [Stravinsky. The Composer and His Works.] Hungarian translation: Révész Dorrit. Zeneműkiadó, Budapest.