Bartok Kodaly Workshop [PDF]

Zitiervorschau

SYMMETRIES OF MUSIC Ernő Lendvai

Introduction *Axis System Axis substitution Tonal reflections On the meaning of functions The relationship of the three functions Tonal ’ asymmetrical’and atonal ’ symmetrical’ elements Nature Symbolism *Golden section *Fibonacci series Harmonic Principles The pentatonic– chromatic system Alpha chords Models 1:2, 1:3, 1:5 1:2 model 1:2 model harmonic turns 1:5 model 1:3 model Complementary (annihilating) keys Equidistant scales Fourth chords Tonic– antitonic relations in pentatonic scales Omega chords The diatonic– overtone system Acoustic (overtone) chord Hypermajor and hyperminor Alpha inversion Authentic and Plagal Thinking DO system and MI system The complementary relation of the two systems Inverted relation of the two systems

Consonance and dissonance Openness and closedness Duality and synthesis The Quadrophonic Stage of the Music for Strings, Percussion and Celesta Polymodal Chromaticism Major triad — minor triad Polar chords (DI + MA tension) Major — minor — subminor On the ’ Tristan chord’ Function Diminished sevenths Substitute chords Remarks on Verdi’ s colour chords *Tonality: Modality: Atonality Summary The Ethos of the Axis System (tonal serialism) Pole– counterpole relations The symmetrical correspondences of the system Relations between parallel keys Polar major and minor chords Relations in content between substitute keys Relations in content between complementary (annihilating) keys Function Natural and modified keys Upper and lower spheres The potentialities of the axis system Overtone relations Wie lenkt’ich sicher den Kiel (modal serialism) Computer Language of Music The ’ Sprecher’Scene of The Magic Flute Appendix (Relative Solmization) Editors‘Postscript (Miklós Szabó– Miklós Mohay)

INTRODUCTION An analysis is justified only if it leads closer to the content of music and its authentic interpretation. Theoretical methods show their value in that –like keys –they help to unlock hitherto unintelligible connections and thus enable us to penetrate deeper into the secret of the composition. I have frequently discussed the question of whether relative solmization is a method or a conception? Is it suited to lead us beyond its educational purpose? If not, then its symbols proclaim their own emptiness. What is it that transforms the symbol into living material, the letter into explanation? Are we entitled to believe that a mere play with solmizating letters can be used to describe structure systems which classical theory cannot cope with? Whereas classical harmony is bound to seven-degree diatony, in the harmonic world of Romantic and twentieth-century music the chords move within the closed sphere of 12-degree chromaticism; accordingly, the former reflects a static way of thinking, while the meaning of the latter is determined by the relationship of harmonies to one another. For within the closed sphere of the fifth-circle it is as impossible to speak of fixed points of support –or ’ progress’–as it is nonsensical to call the distance covered on a sphere (or a circle) ’ progress’ . For this reason the late works of Verdi or Wagner have proved to be an impregnable fortress to classical theory: they stoutly resist all attempts at analysis. The effect of classical music lies in ’ functional’attractions, while that of Romantic music, in ’modal’or polymodal tensions. Modality is a relative system and it is to be analyzed most naturally through the devices of relativity. A classical melody is easily described with the devices of the figured bass (using degree-numbers and figures to indicate the arrangement of the chords). The use of the figured bass is derived from the diatonic system and is thus a completely unsuitable tool for the analysis of Romantic music. The most typical Romantic melodies exert their influence in quite a different manner! The sensory process undergone in our consciousness can be described as follows: For each successive chord we instinctively seek an answer to the question: which is the chord that would follow according to the ’ natural’logic of music? And this we compare with the chord that in fact replaces it. The meaning of the chord will be determined by the difference in tension between the two. The lifeblood of this music is relativity: the system of potential differences between the tonal elements, which we may with total justification call the system of modal tensions. Various pedagogical disasters have led me to recognize that Romantic music will remain the terra incognita –blank spot –of music theory, unless it is approached through the devices of relativity.

In our analyses, we treat the signs of relative solmization as mathematical symbols. (Readers unfamiliar with the principles of relative solmization, should consult the Appendix) Each of the 12 symbols designates a musical character, and if we recognize which sign represents light or darkness, which is accompanied by a rise or a descent, which embodies a materialistic and which a spiritual experience, why the content of one is expressionistic and the other impressionistic –if in other words, through the help of signs, we can differentiate between cold and warm colours, between positive and negative tension, if we know for example that the FI lifts high and the MA hides a painful feature –if we understand all of this, then, with no more signs than are necessary to cover the tones of the chromatic scale, we shall have conquered something of the realm concealed behind the notes. Kodály offers us a technique which covers all that one could wish an ideal theory to cover: (a) it is easily accessible to everyone –a grounding in musicology is not required, (b) it coincides with ’ live’ experience: its codes are directly perceptible, (c) it is suitable for examining (and understanding) phenomena comprehensively; one may without exaggeration say that it opens a whole musical universe. Kodály ’ s ideas can serve to find and interpret new facts in musicology –simultaneously leading us to a wider outlook on the history and organic development of music. (d) And above all, it gives a true image not only of the structure, but also of the content of music itself.

AXIS SYSTEM In European music, functional way of thinking has been established with the recognition of I-IV-V-I affinities:

The classical theory already distinguishes between primary and secondary triads –in so far as the I, IV and V degrees may be substituted by the relative VI, II and III degrees respectively:

Romantic music continues with this progression, naturalizing the upper relatives, too:

Axis system is none other than the recognition of the fact that the tonic A and E b (see sketch above) not only have C as a common relative, but also the F # or Gb degree. Likewise the common relative, between the subdominant D and A b is not only F, but also the B. Finally, G and C# (or Db) equally the common relatives of the dominant E and Bb degrees: Fig. 1

A backward glance at the past and the progress of European harmonic thinking gives evidence to the fact that the birth of the axis system was a historical necessity. As compared to the past, the advance consisted mainly in Bartók’ s having extended these functional affinities over the entire twelve-tone series –and with this, he brought the system to a ‘ close‘ . We call it the axis system because in this system the opposite poles –counterpoles (C and F#, for example) –are more directly attached to each other than the relative keys of classical harmony. Fig. 2

Naturally, the poles of the individual axes should not be regarded as diminished seventh chords –but as the kinship, the functional attraction among four different tonalities, similar to the relative major-minor keys in classical harmony (such as C major– A minor, or C minor– Eb major). In Bartók’ s Bluebeard’s Castle the relationship between the four poles is elucidated by the scenes of the opera: the ’ night theme’is counterbalanced by the ’ light theme’ , and the flower-garden by the lake of tears. As opposed to the ’ intellectualspiritual’dimension of the main branch, the ’ emotional’dimension is represented by the secondary branch: Fig. 3

Each movement of Bartók’ s Music for Strings, Percussion and Celesta follows the same pattern (see: p.59). The counterpole tension became the most fundamental constructional principle in Bartók and Kodály’ s works. Bartók’Allegro s Barbaro is based on the polarity of F# minor and C major: Fig. 4

The F tonic of the Divertimento turns into the B counterpole in the middle-part (b. 80). Likewise, the B tonic of the Violin Concerto is substituted for the F counterpole in the development (b. 115). Characteristically, Bartók organizes even twelve-degree "Reihe" themes according to the axis principle. In the famous Reihe of the Violin Concerto the 12 tones of the theme touch all degrees of the chromatic scale. Its opening, centre and close are roo-

ted in the A-D#-A counterpoles –crossed by the broken F# major and C major– minor counterpoles: Fig. 5

Since this Reihe appears as the ’ secondary theme’of the movement, conforming to the traditions of classical composition, a dominant significance is attached to it: in the B major basic tonality of the work the dominant is represented by the A-F #-D#-C axis. We frequently find axis harmonization in folksong arrangements, too. In Kodály ’ s Székely Lament, the tonal character of the main cadence is determined by the C fourth-step which is associated with the Gb major counterpole: Fig. 6

This Gb major is coloured by the relative Eb minor, and answered by the A major counterpole. Fig. 7

Axis substitution A pole can be replaced by its counterpole without any change in its function. The first strophe of the above folksong ends in A7, but the third strophe closes with the E b7 counterpole. Fig. 8

Especially in their variational works both masters exploit to the utmost the potentialities of axis-replacement: in the newer and newer variations they freely interchange chords belonging to the same axis (function).

Tonal reflections The frequent occurrence of tonal answers and tonal correspondences in Kodály ’ s and Bartók’ s music bear witness to a distinctly functional thinking. Chiefly those countermovement chord progressions are to be mentioned which rest not on ’ real’ , but on ’ tonal’reflections. In the axis system, the same function can be attained by a major second step in one direction, or by a minor second step in the other. (E.g. departing from the G dominant, we reach the subdominant either with a major second step downwards or a minor second step upwards: F or Ab). As this is pictorially reflected in the ’ bridge motif’of Bartók’ s Cantata Profana (’ Coming by a foot bridge saw’ ): Fig. 9

But this rule can be extended: by moving a major second, fourth, minor sixth or major seventh interval in one direction –or by moving a minor second, major third, fifth or minor seventh interval in the other direction, we arrive at the same function in the axis system. This principle permits much bolder counter-movements too –as the Entry of the Emperor and the Imperial Court from Kodály’ s Háry János indicates: Fig. 10

* Mention must be made of the apparent contradictions existing between the sevendegree and twelve-degree systems. (a) According to Riemann, the diminished triad on degree VII (B-D-F) is but an incomplete dominant seventh chord (G-B-D-F). Degree VII will have an independent function only if a major (or minor) chord is built upon it. In this case it acquires the significance of the SUBDOMINANT. Fig. 11

In classical music, the VII. degree major chord strives to resolve on III – consequently, it has a ’ secondary dominant’ , i.e. subdominant meaning. Fig. 12

The ambivalence between 7-degree and 12-degree music is conspicuous also in that the cadential sequence of chords

suffers a break between the F and B in the diatonic scale (a diminished fifth appears instead of a perfect fifth). Naturally, no such break occurs in the 12-note system. (b) According to Rameau, the Neapolitan sixth cannot be regarded as a real chord on the II. degree. In C tonality, the Neapolitan F-A b -Db is not a triad based on Db, but an altered IV. degree. This is why a ’ subdominant’sense has been attached to it. –In Romantic harmony, however, the minor second degree (Db) is granted an independent role and thus it receives a DOMINANT significance (see also pp. 92-93). It is as much as saying that there is no contradiction between traditional harmony and the axis system.

* On the meaning of functions Perhaps the most specific characteristic of the axis system is that the individual harmonic functions receive a symbolic meaning. In traditional music, too, subdominant was identified with the image of ‘ sinking‘ , and the dominant with that of ‘ rising‘– but there the various functions were dependent primarily on the circumvolution of chords (I-IV-V-I, for example). Bartók and Kodály, however, go much further and invest the individual functions with independent meaning: their content is determined by their absolute relation to the tonic axis. Examining the interaction between material and poetic content, a leading role can be attached to works with a plot or text. It can be assumed that it was in the opera genre that the ’ individuality ’of the three harmonic functions developed. For example, the twelve chimes from Verdi ’ s Falstaff (see Fig. 160 on p.86) could scarcely evoke —without the SUBDOMINANT atmosphere belonging to it —the secret mysterious shivering which the word "mezzanotte" (midnight) expresses. On the other hand, Verdi elevates us into the sphere of the DOMINANT when he wants to portray the ’ heavenly’beauty of his female characters, or the magic of nature. The Aida-theme itself — although we are in no doubt as to its tonality — does not even touch the tonic: it floats on the fifth (and major third) of the root. (See Fig. 171 on p. 91) On Mozart’ s opera stage, the mirror of the ’ inner’stage is the subdominant, and that of the ’ outer’stage the dominant. Two brief examples from The Marriage of Figaro. The outer action (which can be clearly seen ’ by everyone’ ) is accompanied by dominant-tonic harmonies; but when the characters begin to be ’ secretive’ , or speak ’to themselves’ , the function immediately changes to the subdominant.

Fig. 13

The more complex and refined individuality a Mozart figure has, the more powerful role is assigned to the subdominant function: the frequency of the subdominant is the sign of a rich ’ inner life’(as with Don Ottavio or Tamino). And conversely, the dominant and tonic gain the upper hand in the melodic world of his popular, peasant characters (like Leporello and Papageno). In Bartók’ s Bluebeard’s Castle this functional ’ sign-language’goes hand in hand with the libretto. The static pillars of the opera and the points of rest are based on the tonic. The subdominant function has a negative significance, it is reserved for the burning, shapeless passion; on the other hand, every positive ’ initiative’sets out from the dominant (the possibility of elevation also resides in it). The scene of Door VII — women of the past — is particularly illuminating in this respect, with its almost cosmic arrangement of ’ dawn– noon– evening– night’ . In accordance with the basic idea of the work, ’ night’is identified with the tonic F #, and ’ noon’ with the counterpole C. ’ Dawn’looks towards the rising dominant, and ’ evening’ b towards the sinking subdominant. The B major tonality of ’ dawn’and the D minor tonality of ’ evening’occupies a symmetrical position –whether in relation to F # or C. And again, in contrast to the major tonality of ’ dawn and noon ’ , the harmonies of ’evening and night’are minor in character. Fig. 14

Mention must be made of an interesting symbolism: the ‘ looking towards the past ’ face of the subdominant and the ’ looking into the future’aspect of the dominant – which is, on the one hand, closely connected with the subdominant character of the ancient six-four type folksongs and, on the other hand, with the T-D-D-T (T-T-D-T) cupola structure of the new-type Hungarian folksongs.

The relationship of the three functions As opposed to the I-IV-V-I cadence of classical harmony, the tonic, subdominant and dominant functions are most powerfully represented by the three degrees which divide the fifth-circle into three equal parts –thus constituting an augmented triad relationship. For example C-E-Ab, in the sense of tonic, dominant and subdominant. This is, after all, nothing new. The movements in Brahms ’First Symphony follow each other in this way: C-E-Ab-C, in the sense of tonic-dominant-subdominant-tonic. The dominant secondary theme in Beethoven’ s C major (Waldstein) Sonata appears in E, and the subdominant slow movement in the C minor (Pathétique) Sonata in Ab major. The first movement in Bartók’ s Concerto for Orchestra rests on the five-fold appearance of the principal theme: F Db A F F

tonic subdominant dominant tonic tonic

(exposition) (first part of development) (second part of development) (recapitulation) (coda)

bar 76 bar 231 bar 313 bar 386 bar 488

A similar order of keys can be observed in the First Rondo for Piano: C tonic, E dominant, Ab subdominant, C tonic. The tonal arrangement in Movement I of Bartók’ s Sonata for Two Pianos and Percussion is as follows:

C E

tonic (exposition) bar 32 dominant (first part of development) bar 161,195 bar 232 G# subdominant (second part of development) C tonic (recapitulation, coda) bar 274 Each axis contains in itself a double attraction, a twofold dimension –depending on whether we contrast the pole with the counterpole, or the ’ main branch ’with the ’secondary branch’(see: Fig. 2 on p. 8) which means that the individual axes divide the fifth-circle into four equal parts. All in all, the tonal system created by the equal divisions of the circle of fifths matches the model of the axis system: Fig. 15

Therefore,

the

constituents

of

the

pole pole+counterpole = branch main+secondary branch = axis T + D + S axis = axis system It is easy to understand that — given three functions established by the distance divisions of the circle of the axis

axis

system

are:

(no dimension) (1 dimension) (2 dimensions) (3 dimensions) — only one tonal system can be fifths, and this is identical with system.

Tonal ’ asymmetrical’and atonal ’ symmetrical’elements It would be tempting to speak of ’ relativity theory ’ . The axis system preserves tonal — and indeed, functional — thinking, but at the same time it also satisfies the distance laws of twelve-tone music. It represents an order which results from the meeting of opposites. In this respect, too, the two Hungarian masters continue one of the most peculiar traditions of European music: they ’ re-live’ in a condensed form the historical progression which (with the advance of distance models and their gradually increasing conquest) finally led to the free and equal treatment of the twelve chromatic notes. It was exactly in the interaction between tonal ’ asymmetrical’ and atonal ’symmetrical’ elements that the harmonic world of European music became a dialectic lingual system. In the antagonistic features of the axis system, actually, the inherent contradiction between tonal and distance principles is reflected.

NATURE SYMBOLISM 'We are guided by nature in composition' Bartók GOLDEN SECTION Golden section (sectio aurea) comes about when the proportion of the whole to the larger part agrees with the proportion of the larger part to the smaller one. That is, the larger part becomes the geometrical mean of the whole distance and the smaller part. Practically, if the whole is considered as 1, the value of the larger section will be approximately 0.618 and that of the smaller section 0.382. Fig. 16

From a structural aspect Bartók's greatest works are the true representations of the golden section principle - frequently from the whole form down to the smallest formcells. For example, the first movement of Sonata for Two Pianos and Percussion consists of 443 bars. If 443 is multiplied by the key-number of the golden section (i.e. 0.618), we get 274, and indeed, b. 274 marks the 'centre of gravity' in the form: the entry of the recapitulation. The complete form of Sonata for Two Pianos and Percussion consists of 6432 eight notes. In compliance with the form construction of the work: slow-fast + slow-fast movements, its golden section - 3975 eight notes - touches the beginning of the second slow movement. The first movement in Bartók's Contrasts comprises 93 bars. Its golden section - 93times 0.618 - coincides precisely with the recapitulation (at the middle of b. 57). The first movement of Divertimento contains 563 triplet units (because of the varying time-signatures we have nothing to do with the number of bars), and its golden section - 563-times 0.618 - once more agrees with the recapitulation. At a cursory glance it may seem illogical that the change in tempo does not influence the position of section points. The contradiction, however, easily disappears if we realize that the 'heartbeat' of music is the metrical pulse, and not the absolute length

of time measured by the clock. In music, the progress of time is indicated by bar-lines or rhythmic beats - and the role of these metric impulses is more vigorous and emphatic than the space of time in which the music takes place. Subjectively time progresses more feverishly with fast beats, and more restrained in a slow pulsation. Golden section has two possibilities: a 'positive' and a 'negative' - depending on whether the larger part precedes the shorter, or vice-versa (long+short part provides a positive, and conversely, short+long a negative section): Fig. 17

A fine example of the extent to which golden section proportions may dominate melodic structure, is reflected by the 16 introductory bars of the first movement of Bartók's Sonata for Two Pianos (or to be more exact, bs. 2-17, since the organic life of the work is developed from b. 2 onwards). Fig. 18

This

is

what

happens:

Leitmotif in root position on the tonic (bs. 2-8) on the dominant (bs. 8-11) in inversion on the subdominant (bs. 12-17).* Similarly to classical period, positive and negative sections create questions and answers. Both in the larger and the smaller details questions and answers, that is positive and negative sections unite symmetrically - in such a way that the smaller form-units are finally merged in a positive main section. This is why the form is associated with a dynamic rise, from pianissimo to forte-fortissimo! We may draw the inference that positive sections are accompanied by a dynamic increase, the intensification or condensation of the musical material (i.e. question), and conversely, negative sections go hand in hand with a downward tendency, decline, lose of strength (i.e. answer, release). From these analyses it can be seen that the golden section meets, in every case, the most important turning point of the form. It goes without saying that golden section appears not as a mathematical rule - it were

useless to seek a mathematical logic in these structures. It may be added that in Bartók and Kodály's 'asymmetrical-polyphonic' way of writing, it is not the number of bars but the number of musical impulses that determines the proportions. To give a typical example, at the very end of the Fourth Quartet (from b. 365 on) it can be clearly perceived that 3 motivic 'waves' are answered by 5 motivic waves (bs. 365-374 and 375-385). Or in the Háry János Prelude, the 4-fold entry of the principal theme is responded by 6 theme-entries (regardless of the distance between the individual entrances). Accuracy controlled by perception and accuracy tested by calculation are two sides which must be kept distinct. I discovered golden section intuitively and empirically: my observations were founded on real experience. Where does precision end and inaccuracy begin? I would propose that: (a) The difference between the natural and tempered systems sometimes exceeds 1% and escapes detection. (b) The vibrato both of the singing voice and of the violin can approach 2% - and yet remains unobserved. (c) Since, in the formal structure of Bartók, golden section is the 'horizontal' manifestation of a 'vertical' (harmonic) principle, the same deviation is permitted in formal construction, too. Golden section is subject to three conditions. (a) It fulfils its task only if it can be perceived. (b) It appears as an organic element of musical dramaturgy. (c) It represents an idea (being the symbol of 'organic' existence). Golden mean is no more than an organic element of the musical dramaturgy and has a meaning only in relation to the contents of music. The point of our interest is first of all the 'effect mechanism' of the proportions: the attractive and repelling force functioning in the positive and negative sections, the interaction between the building elements.

* 'Two elements cannot be well coupled without the assistance of a third one, since the two can only be united by an intermediary link; but of all the links that one is the most beautiful which unites into a complete whole itself and the elements joined by it.' (Plato: Timaeus) The static quality, classic beauty of Parthenon in Athens may be attributed to the fact that into its forefront - between the base and the tympanon - exactly two squares can be inscribed. On the other hand, its dynamic quality - the floating effect which invisibly draws the building upwards - is due to the golden section proportions. We give the structural view after Zeising (see next page). Perhaps the most beautiful literary manifestation of the golden section principle is Dante's Divine Comedy. The golden section of the hundred Cantos comes at the end of the sixty-second Canto: this is the point where Dante parts with Virgil and

continues his way with Beatrice. From here (Purg.XXVIII) the language, the colour, the tone and even the atmosphere of the work change. The great physicist Einstein, too, pronounced in favour of the golden section. According to him, golden section provides a proportion which throws difficulties in the way of the bad and facilitates the development of what is good. Fig. 19

A pair of compasses can be seen in the Neapolitan museum which served for measuring the golden section in antiquity:

FIBONACCI SERIES The Fibonacci series covers the simplest golden section sequence which can be expressed in whole-numbers (the golden section of 89 being 55, and that of 55 being 34, etc.): 2, 3, 5, 8, 13, 21, 34, 55, 89 ... In it each number equals the sum of the two preceding numbers (that is, 2+3 =5, 3+5=8, 5+8=13, etc.). The sequence approaches nearer and nearer the proportion of the geometrical golden section i.e. the irrational key-number of the geometric mean: the square of every number is equal to the product of the numbers preceding and following it - with the difference of plus or minus 1. The Fibonacci series embodies the low of natural growth. In the fir-cone starting from the centre, a system of spirals runs in the right and left directions, in which the number of spirals always result in the values of the Fibonacci sequence: 3, 5, 8 and 13 spirals. Fig. 20

A similar setting can be seen on the sunflower, pineapple, chamomile, dandelion, marguerite, cactus, likewise in the arrangement of leaves on the stem, in the horns of some ruminating animals - to mention only the most characteristic examples. Bartók adopted this numerical series already at the beginning of his career, in 1911. The throbbing F # minor ostinato of Allegro Barbaro appears in groups of 3, 5, or 8 or 13 bars. Perhaps the most beautiful musical representation of the Fibonacci sequence is the first movement - the 'pyramid-fugue' - of Music for Strings, Percussion and Celesta (for a detailed analysis, see pp. 56-65). Golden mean is not merely an important factor of form and structure, but it gives a key - almost a solution to the question of tonal and harmonic structures as well. And this is no casual coincidence. The most elementary musical expression of the golden section principle is pentatony itself. When a child sings the simplest pentatonic phrases, he instinctively does nothing else than 'tunes' the motif to the proportion of the golden section, i.e. the geometric mean. Fig. 21

Calculated in semitones, the formula of the LA-SO-MI motif is 2+3=5; by adding also the upper DO to it, a distance of 8 semitones issues which shows a subdivision of 5+3 and 3+2+3. 'Folk-music is a phenomenon of nature... its formations have developed with the same organic freedom as other living organisms: the flowers, animals...' - Bartók once declared.

In the first movement of Sonata for Two Pianos and Percussion each new themeentrance rises one step higher in the Fibonacci sequence: Fig. 22

leitmotif 3+ 5 = 8 principal theme 5 + 8 = 13 secondary theme 13, 21 The harmonic structure of the exposition gives proof of a similar arrangement: each new harmony advances one degree in the golden section scale: Fig. 23

principal theme 2:3:2 its middle-part 3:5:3 secondary theme 5 : 8 closing theme 8 The closed quality is one of the distinctive marks of golden section forms. This world of form has a peculiar glow - a life-tension and an organic closedness - and this is a direct consequence of the fact that evidence of golden section structures is found only in living organic nature, but it is quite alien to inorganic nature, to the world of crystal forms. (See also Fig. 58-59 on pp. 37-38) *) For detailed analysis see: Ernő Lendvai: The Workshop of Bartók and Kodály, Editio Musica, Budapest, 1983, pp. 36-40

HARMONIC PRINCIPLES A) THE PENTATONIC– CHROMATIC SYSTEM

The

harmonic

types

of

the

chromatic

system,

that

is

(1) alpha chords (2) 1:2, 1:3, 1:5 models (3) chords of equal intervals are characterized by the fact that they unite the tension of the Fibonacci-models with the closedness of the twelve-tone system. The most characteristic chromatic melodies and harmonies obey the proportions of the Fibonacci sequence. Calculated in semi-tones: 2 means a major second, 3 means a minor third, 5 means a perfect fourth, 8 means a minor sixth, 13 means an augmented octave, etc. In reality, these numbers express proportion and not semi-tone steps.

(1) ALPHA CHORDS As Mozart or Haydn had employed primarily major and minor triads or as we had grown accustomed to seventh-chords in Romantic music, the use of harmonies built of Fibonacci intervals became just as common and predominant in the works of Bartók and Kodály: Fig. 24

We call this type of chord with the collective designation: the alpha harmony — the

various sections of which can be distinguished by letters beta, gamma, delta and epsilon. Chord alpha consists of two layers. In order to establish tonality, at least two notes are necessary: the key-note (C) and one of its overtones: i.e. the fifth (G) or the major third (E). In this simple case the G or E reinforces the C — although G and E has in itself a ’ dominant’significance. Type alpha has a strong tonal, even functional character. When, for example, in the main cadence of the ’ Recruiting Music’Kodály looks for a dominant alpha chord (before the tonic E major), he moulds its upper layer from the melody itself (B-G #-FD), and the lower layer from the diminished seventh chord C-D#-F#-A: Fig. 25

Thus type alpha is nothing less than the ’ axis’adaptation of the simple C-E-G or CG relation (this is why type epsilon rarely occurs owing to the absence of the fifth and major third –G and E –without which the tonal character of the chord is unsteady); the sole requirement is that the chord should be constructed of two layers (two ’axes’). Type alpha complies with two requirements. On the one hand it derives from pentatony. The intervals 2, 3, 5, 8 (or their octaves) sound together with every note of the chord. Fig. 26

This is how in one of Bartók’ s choruses the pentatonic DO-SO-LA-MI motif is transmuted into gamma and delta harmonies: Fig. 27

On the other hand, alpha harmonies are axis models and, as such, express the polymodal tensions of the axis system. (See: Fig. 32 on p. 27) Let us add: the tension of alpha harmonies may — most simply and effectively — be expressed by the symbols of polymodality. Fig. 28

Examples Fig. 29.

(2) MODELS 1:2, 1:3, 1:5 By the chromatic juxtaposition of intervals 2, or 3, or 5, ’ closed ’scales are produced (i.e. by the periodic repetition of the intervals we are taken back to the starting point). MODEL 1:2 is an infinite chain of minor and major seconds, e.g. C-C#-Eb-E-F#-G-A-Bb-C, MODEL 1:3 is an infinite chain of minor seconds and minor thirds, e.g. C-Eb-E-G-Ab-B-C, MODEL 1:5 is an infinite chain of minor seconds and fourths, e.g. C-C#-F#-G-C. Fig. 30

Model 1:2 Model 1:2 should be considered the ’fundamental scale’of the axis system. In the 12-note system three different 1:2 models may be established, in accordance

with the three functions: a tonic, a dominant and a subdominant. Every further form agrees with one of these models, e.g. in C tonality: Fig. 31.

All chords and models appertaining to the same axis constitute a functional unit. Fig. 32

Examples Fig. 33

1:2 model harmonic turns **) If we combine a major seventh and a subminor chord — lying a minor third higher — we get the 1:2 model. This combination became current among the Romantics. Fig. 34

In Act I of Tristan, the ’ inner’ world is visualized by the ’ Tristan chord’(F subminor), while the ’ outer’world by the chorus of the Sailors –at all times appearing in the form of a ’ D major seventh’chord. Fig. 35

Model 1:5 Model 1:5 represents another typical axis sequence: it rests upon polar relations. Fig. 36

Fig. 37

Model 1:2 may be split up into two 1:5 models. Fig. 38

In the Mikrokosmos piece From the Island Bali both left and right hands play 1:5 models, which together create a complete 1:2 model. Fig. 39

The closing chords unite these four elements. Fig. 40

In the final chord two triads are merged: the difference of six accidentals between the two triads create a polar tension and the two triads fuse in a perfect alpha harmony. Fig. 41

1:3 model While models 1:2 and 1:5 have a powerful ’ tonal’ character, the 1:3 model ’annihilates’tonality due to its augmented triad structure. For instance, the C-E b-EG-Ab-B 1:3 model comprehends the following triads:

The ’floating’quality of model 1:3 was already recognized by Liszt and Wagner. Fig. 42

Examples

Fig. 43

Complementary (annihilating) keys A 1:3 model can be created by uniting a major and a minor triad (the latter lying a major third lower). In this case one triad neutralizes the other since their notes combine in an atonal 1:3 model. This is why such triads express a contrast in their content as well. Fig. 44.

The idea of ’ annihilation’goes back to Romantic models. When Wotan in the great Monologue of the Valkyrie prophesies the Twilight of Gods, his words "Das Ende! Das Ende!" evoke E major and C minor –which tonally destroy each other. Fig. 45

A peculiar manifestation of the ’ annihilation’idea is Webern’ s famous Reihe –in which two 1:3 models are merged:

Fig. 46

A change from major to the complementary minor (E majorC — minor) results in a ’negative effect’ , it is associated with gloomy, or even, oppressive and irrational experience. (See Fig. 45 above) And conversely, a change from minor key to the complementary major (e.g. C minor— E major) creates a ’ positive’impression, it is inspired by enthusiasm and serenity – like the E major episode at the end of Beethoven’ s Piano Concerto in C minor. In the chorus The Aged, Kodály translates the life-and-death symbolism of the piece into the language of music by means of complementary keys: G major and Eb minor – then Bb minor and D major: Fig. 47

The entire tonal plan of Bluebeard’s Castle is built up of such complementary relations. F# minor is the key of ’ night’and C major that of ’ light ’ . C major can be destroyed by means of the Ab minor key –thus the latter is associated with ’ death’ b # symbolism. On the other hand, the night’ s F minor can be defeated by B major – thus it became the symbol of ’ love’ . The four triads together include every degree of the chromatic scale: ’light’ :

C-E-G

—’death’:

Ab-Cb-Eb

—’love’: ’night’: F#-A-C# Bb-D-F The basic tonality in Kodály’ s Psalmus Hungaricus — A minor — is equivalent to weeping, imploration, despair, curse. The dénouement of the action, on the other hand, takes place in the complementary key: Db = C# major. It is the task of D b major to absolve from the weight of the curse: ’ From you he removes your every burden ’ .

And that of C# major to become the key of elevation and apotheosis: ’ In honour Thou wilt raise him on high!’ Complementary keys may appear in a hidden form, too. The "Sündenqual-motiv" from Parsifal exerts a tormenting effect because the authentic sequence (moving authentically on the circle of fifths) is coloured by complementary chords: F major and Db minor in b. 1; Db major and A minor in b. 2, etc. Fig. 48

(3) EQUIDISTANT SCALES Closed sequences may, however, be created by simpler means –by composing equidistant scales from the Fibonacci intervals 2, 3, 5, 8: from major second intervals from minor third intervals from perfect fourths from minor sixth intervals

(2) (3) (5) (8)

whole-tone scale, diminished seventh, fourth chords, augmented triad

can be established.

Fourth chords**) Owing to the folksong inspiration, strikingly frequent is the theme formation and harmonisation with fourth chords. ’ The characteristic fourth-accumulation in our ancient melodies spurred us on to the forming of fourth chords: we have projected here the horizontal succession into vertical simultaneity’(Bartók: The Influence of Peasant Music on Modern Music, 1920). This is how Bartók transforms the fourth melody into a fourth harmony in the Violin Concerto: Fig. 49

Tonic–antitonic relations in the pentatonic scale Thus, fourth chords in Bartók and Kodály’ s compositions derive from folk-music, and in keeping with the folksong inspiration, also in the connection of fourth chords the inner laws of pentatony assert themselves. To mention the most important one: of two fourth chords which are placed at a distance of a minor third (3) or major second (2) from each other, the ’ tonic’model is always the one which lies a minor third lower or (which means the same) a major second higher than the other. We call one of them the ’ tonic’ and the other ’antitonic’model: Fig. 50

In Fig. below, the ’ tonic’model is represented by fourth-degrees C-F-B b, and the ’antitonic’by fourth-degrees Bb-Eb-Ab. Fig. 51

The tonic-antitonic attraction originates in the SO-RE’ MI (DO-SO’ LA) cadence so frequent in folk melodies: the SO-RE holds the tension, while the MI corresponds to the tonic resolution: Fig. 52

It deserves a special attention that this time we are faced with a two-function system (and not a three-function one, as in classical harmony). Incidentally, chords based on the SO-LA-DO-RE structure have a floating, soaring effect since the tonic and antitonic (RE-LA and DO-SO) relation supports not the lower, but the upper note.

Omega chords In my analyses, the letter omega indicates the whole-tone scale. I have deliberately chosen the letter farthest from alpha because Bartók himself used them oppositely. Alpha is ’ tense’in character, omega is ’ dissolved’–and this quality becomes apparent in that the whole-tone scale, as opposed to alpha and pentatonic structures, contains not one single perfect fourth (nor a perfect fifth) without which the tonal character of the chord becomes unstable. In the 12-tone system two omega scales can be distinguished (6+6 notes), they are mutually ’ geometrical dominants’of each other (C-D-E-F #-G#-A# and Db-Eb-F-GA-B): w1 and w2. Fig. 53

Kodály’ s chorus Fancy concludes with a complete omega harmony –depicting the peal of bells: Fig. 54

The omega harmony — due to its fluid character — lends itself particularly well for tonepainting (landscape painting). We give a Kodály example: Fig. 55

Because of the contrast, the direct confrontation of the alpha and omega tonalities is very effective. In the third movement of Music for Strings, Percussion and Celesta, the central ’ light’theme —accompanied by the high-pitched cymbal —also includes this duality. The motif is ’ centralized’around the C note. Depending on whether it occurs in root position or inversion, Fig. 56

it can be accompanied by omega chords (Fig. 53 on p. 35) or by alpha chord: through the periodic repetition of diminished triads, a closed alpha structure is created: Fig. 57

* In practice, the Fibonacci models — i.e. alpha and axis harmonies, 1:2, 1:3, 1:5 models and equidistant scales — merge into each other. In Fig. below, the hunting ostinato is quoted from the chromatic first movement of Bartók’ s Sonata for Two Pianos and Percussion: Fig. 58

The entire chromatic scale is included in the twelve notes of the ostinato. A polecounterpole — relationship exists between the opening and the closing notes (upper part A and Eb, lower part F# and C), forming together an axis arrangement: F#-A-CEb. The upper part is composed of the A-B-Db-Eb-F-G omega scale (its formula being 2+2+2+2+2+2), while the lower part of the complementary omega scale –F#-G#-BbC-D-E. The two parts progress in parallel minor thirds (3). Motivically, each part is built of minor sixth (8) elements: the upper part of augmented triads A-F-D b and BG-Eb, and the lower part of augmented triads F #-D-Bb and G#-E-C (8+8+8). The harmonic character of the ostinato is defined by gamma chords (3+5+3) and 1:3 models: Fig. 59

The two 1:3 models can be fitted chromatically. Thus all of the structural elements are ’ Fibonacci formulas’ .

B) THE DIATONIC-OVERTONE SYSTEM ACOUSTIC (OVERTONE) CHORD The basic form of the diatonic system is the so-called acoustic scale (DO-scale with FI and TA), e.g. C –D –E –F# –G –A –Bb –C and the acoustic harmony, for example, C major triad with natural seventh B b, acoustic fourth F#, and major sixth A (’ pastoral’sixth) –which is called acoustic since its notes originate in the natural overtone series: Fig. 60

Examples Fig. 61

Other examples see: Fig. 87 b on p. 50 and Fig. 107 on p. 65. The acoustic character becomes even more effective if it is the result of an expansion. The hunting-fugue in Bartók’ s Cantata Profana becomes so explosive since the minor third changes into major and bursts with an acoustic fourth (MA-MI-FI): Fig. 62

The perfect fifth frame of the theme and its two ’ points ’—the acoustic fourth at the half-close (FI) and the acoustic MI-TA step at the full close — only enhance the natural atmosphere of the scene. The acoustic chord, with the exception of one tone, contains a whole-tone scale (omega) as well: Bb-C-D-E-F#. Therefore, the acoustic chord can easily be coloured by the omega scale: Fig. 63

The acoustic harmony is familiar in Kodály’ s music too –although he had an affection for the brighter Lydian modality: Fig. 64

In Kodály’ s music there is often very little difference between the acoustic harmony and the polar harmony. For example, in Fig. above the first chord is the combination of Bb seventh and E seventh. The acoustic chord of Fig. 84 b consists of the C major and F# major counterpoles. Now we try to find a new path in deducing the acoustic harmony. Let us start from our observations See: p. 77 that the relative of the C major is A minor –and that of the latter is F# subminor: Fig. 65

The tones have a symmetrical distribution –around the virtual RE (=D) symmetry center! If we combine the tones of the three chords we obtain an acoustic harmony. In both cases FI and TA are determinants of character. And what is evident again: FI and TA are exact reflections of each other in relation to the RE symmetry center. The acoustic scale became a static ’ colour chord’because it lacks the two sensitive notes that characterize the major scale: instead of FA and TI (F and B) FI and TA (F # and Bb) notes occur. What the spectrum of rainbow-colours is in optics, is the natural overtone scale in music. (The term ’ acoustic scale’ comes from me – 1947).

Hypermajor and hyperminor Whenever Bartók or Kodály intends to endow the acoustic harmony with more ’light’ , the minor seventh of the acoustic chord is raised to major seventh. Fig. 66

We shall call this chord type (major triad with major seventh) the hypermajor harmony: Fig. 67

The hypermajor embodies the most solemn sound-type in Bartók and Kodály’ s music (the opening chords of Psalmus Hungaricus and Budavári Te Deum are hypermajor harmonies) –it has become the characteristic concomitant of apotheoses. The origin of the well-known ’ Bartók signature’also goes back to the hypermajor: Fig. 68

The hypermajor owes its light not only to its major character and major seventh, but primarily to its consisting of two perfect fifths (see Fig. 67 above). In the most pictorial effects the hypermajor harmony merges with the acoustic fourth (FI): Fig. 69

The hypermajor has a counterpart: the hyperminor. Its construction is a minor triad with major seventh (the Eb-G-B augmented triad adds considerably to its individual colouration): Fig. 70

’Your leitmotif’–wrote Bartók in a letter to Stefi Geyer. It is to be found all over where the text speaks of pain and passion (see: Fig. 70 b above). By inverting Bartók’ s’ leitmotif of love’ , a peculiar kind of chord arises –which in Bartók’ s works is associated with the symbolism of death: ’ Eros turns downwards his torch!’ Fig. 71.

It appears whenever desire is fulfilled and, as a consequence of the fulfilment, passion ceases. Through the inversion the augmented triad of the hyperminor (E b-G-B in the foregoing example) moves to the bottom of the chord: A-F-C #; this is the source of the neutralizing effect. The final chord of Bluebeard’ s Castle is also a hyperminorinversion, the whole opera terminates in this symbol of death:

Fig. 72

Alpha inversion If it is true that the diatonic system is merely a mirror-image of the chromatic system,***) then ’ diatonic’sound-types can be produced by inverting the layers of the alpha harmonies: Fig. 73

The diatonic impression is the direct result of the alpha-inversion being ruled by fifths, major thirds and minor sevenths (i.e. the closest overtones) that are precluded by the alpha harmonies. Oddly enough, the harmony with a major third above the root and with a minor third below it, evokes the most ’ opened’impression: Fig. 74

And to bring to an end the interconnections: the alpha-inversion carries in itself the seed of the acoustic harmony as well: Fig. 75

By exchanging the C and F# notes, a polar relationship can be effected (C and F #

seventh chords, or C and F# ninth chords). The recapitulation theme in Kodály’ s Háry Prelude (Fig. 84 b on p. 48) may be interpreted equally as a C acoustic or an F # acoustic tonality. This 4-note polar formula started gaining ground already in Romantic music. The tonal nucleus of Station 10 in Liszt’ s Via Crucis, or the fortissimo explosion introducing the great ensemble of Act III in Verdi’ s Otello, allows the following interpretations: Fig. 76

* Let us summarize the basic types of the two harmonic systems: CHROMATIC-PENTATONIC SYSTEM Pentatony, Fibonacci-models Alpha chords Models 1:2, 1:3, 1:5 Equal-degree harmonies from intervals 2, 3, 5, 8

DIATONIC-ACOUSTIC SYSTEM Overtone chord, acoustic scale Alpha-inversion Hypermajor, third-tower Equal-degree harmonies from fifths, major thirds and minor sevenths

*) It includes subminor chords, too (see. p. 77) (ed.) **) Title given by the editors. ***) (see pp. 48-50)

AUTHENTIC AND PLAGAL THINKING The most characteristic feature of Bartók and Kodály’ s technique is the dual treatment of material. In my analytical studies, I have used the terms ’ pentatonic’and ’ acoustic’(system) to describe the two aspects of their music. Systematically, the two basic types are mirror images of each other. If, in the most general way, we have defined the ’ chromatic— pentatonic’technique as obeing the rules of the Fibonacci series, then the diatonic overtone system is none other than the exact counterpart — systematic inversion — of the chromatic rules. Pentatony has its source in Eastern folk music; accordingly it is of melodic origin. The overtone (acoustic) system, on the other hand, is rooted in Western traditional music and is therefore of harmonic origin. The overtone system is controlled by the laws of physical consonance. (Harmony is perfect when the closest overtones are merged in it.) In the major triad our ears register the most simple arithmetical proportions. A major chord represents an order based on the simultaneous sounding of notes –thus it is vertical in construction: it has a’ spatial’extension. Pentatony, on the other hand, is of melodic origin. And since melody presupposes tones following each other in ’ time’ , it has a horizontal, linear extension. The primary distinguishing mark of pentatonic cultures is the descending DO-LA minor third and LA-MI fourth (as a cadence). The derivation of DO-LA and LA-MI cannot be traced back to the laws of physical consonance. Just the opposite. Pentatony reflects a peculiar tension (which could well be termed ’ life tension ’ ) and is justified by the organic –physiological –disposition of our ears. This implies that while the overtone system suggests arithmetical proportions, the pentatonic system owes its tension-character to the most simple geometric progression –which governs organic development of ’ natural’ growth.

DO system and MI system In Western musical cultures the feeling of tonality is created by the triad conson-ance: the ’ con’ -sonanceof the third and fifth. It was the prelude and folk choruses of Mussorgsky’ s Boris Godunov that first made me feel that there exists a musical

culture radically different from European harmonic thinking. The archaic quality of these melodies resides in their six-four structure. Fig. 77

In the Hungarian folk song below the keynote and tonal resting point is the C. Both the harmonic and the tonal meanings of the quoted melody are determined by the ’ MI pentatonic’structure. This harmony is not F minor –but it is a scale based on C as a root: Fig. 78

And this six-four harmony (C-F-Ab) is not the second inversion of the root-position F minor triad; consequently, it is not a secondary formation but a musical element equal in rank with the root-position triads. A distinctive feature of melodies on MI as keynote is that the root is frequently reinforced with a ’leading-note’step: the Phrygian FA ® MI turn (Fig. above). – NB: the Phrygian scale is a MI scale! The basic step of the plagal ’ six-four’system is the LA ® MI cadence, while the basic step of classical harmony is the authentic V– I dominant – tonic SO ® DO cadence. Perhaps nothing demonstrates better the relationship of Western and Eastern ways of thinking than the fact that (in relation to the RE symmetry center) the authentic SO ® DO cadence of classical music and the plagal LA ® MI cadence, are precise mirror images of each other:

the DO and its reflection, the MI, can alike serve as the tonic fundamental note: Fig. 79

The overtone system follows the law of free-fall (delineated in the fifth-circle, it points towards thesinking direction) while the pentatonic system obeys the principle of tension and, as such, acts against the law of free-fall. If the dominant-tonic cadence, characteristic of Western music, is represented by the G7 ® C chords, then we can take it for granted that the ’ mirror image’of this formula (related to the RE symmetry center)gains an Oriental colouration. This cadence is well-known to us from the Prelude of Act III of Tristan. The sensitive notes TI and FA play an important role in both instances. But while in the V-I cadence theTI-DO leading note step comes to the fore, the oriental colouration originates from the FA-MI ’ leading note’step. Fig. 80

As in classical harmony, the TI ® DO leading-note step affirms the tonic DO keynote, so in this style of expression, the FA ® MI step reinforces the MI tonal keynote. Not to mention that, in relation to the REsymmetry center, the TI ® DO and FA ® MI leading-note steps occupy a symmetrical position – and they move symmetrically as well:

The notes of the acoustic scale are also symmetrically arranged –around the RE symmetry-axis (the mirror-image of FI being TA). (See: Fig. 214 on p. 110)

Degree RE constitutes the symmetry center not only of the pentatonic scale, but of the major and minor scales as well (where upwards and downwards from the RE, each interval has its exact mirror image):

The relative major and minor keys therefore show an inverted relation to each other – as compared to the RE. (See the relationship of C major and A minor in Fig. 183 on p. 96). The unison melody at the opening of Wagner’ s Tristan — the Sailors’ Song — appears in two basic forms: one accompanied by a SO pedal point –and the other by a MI pedal point. In the former it is the SO-DO, in the latter it is the LA-MI that determines the tonal quality of the song –thus the first takes on a ’ major ’and the second a ’minor’character. Fig. 81

Let us bring also degrees DI and MA into the above relation –creating a polar tension. The symmetrical position of the DO ® DI and MI ® MA steps —in comparison to RE — is again evident:

The basic scale of Bartók’ s chromaticism is, as we know, the 1:2 model. If we add degrees DI+MA to the notes of the above example, model 1:2 is created: Fig. 82

While DI and MA appear as expressive ’ tensional’elements, FI and TA appear as impressionistic ’ colour’elements, the former taking on a dynamic, the latter a static character.

The complementary relation of the two systems The harmonies of the chromatic and diatonic systems are ’ Janus-faced’harmonies. They form contrast in unity. The two harmonic spheres complement each other to such an extent that the 12-note scale can be separated into a ’ golden section’ b sequence and an acoustic sequence (D — Phrygian step! — and B, as chromatic degrees, require a chromatic interpretation). Fig. 83

The characteristic intervals of the two harmonic worlds complement each other like the positive and thenegative of a photograph:

PENTATONIC SYSTEM ACOUSTIC SYSTEM minor third (3) major third fourth (5) acoustic fourth minor sixth (8) fifth and major sixth In the Háry Prelude, Kodály opposes the MI scale to the DO scale: the C=MIpentatonic theme of theexposition returns with DO character in the recapitulation: Fig. 84

Inverted relation of the two systems But the relationship between the two systems is multidimensional. The elements of the pentatonic and acoustic systems not only complement but also reflect each other: through the inversion of the 8, 5, 3, 2 steps, acoustic intervals are produced, and viceversa: PENTATONIC SYSTEM minor sixth ( 8)................ fourth (5)........................ minor third (3)................. major second (2)..............

ACOUSTIC SYSTEM major third fifth major sixth natural seventh

In Bartók’ s Cantata Profana, too, D represents the tonal key-note. The beginning scale and closing scaleof the work are each other’ s mirror images –note for note. Fig. 85

The opening of the work rises out of MI pentatony (D-F-G-Bb-C framework) — while # the framework of the closing theme is DO pentatony (D-E-F -A-B) — and both scales are coloured by degrees FI and TA. The closing scale will therefore take the shape of an ’ acoustic’scale (DO-RE-MI-FI-SO-LA-TA-DO), whereas in the opening theme we find a diminished fifth — TA — instead of a perfect fifth.

In his Harvard Lectures Bartók himself made reference to the MI scale in question.* The basic tonal idea of Bartók’ s Cantata Profana had already been realized by Liszt in his Via Crucis –its form built on the ’ reflecting’relationship of the DO and MI systems. The beginning and end of the work are alike rooted in the D keynote. But while the former develops from MI pentatony (D-F-G-B b theme of six-four character) –the latter shows its exact inversion, thus closing the work with a DO scale. Fig. 86

Bartók’ s Sonata for Two Pianos and Percussion reflects a similar structure: the principal theme ofMovement I fills out the MI pentatonic frame: C-E b-F-Ab, while the principal theme of the final movement gains a firm DO character (C major chord!) –being at the same time a perfect acoustic (overtone) scale. Fig. 87

In Bartók’ s and Verdi’ s music alike, the direct confrontation of the two systems form the ’mainmasts’ofthe work. In Bartók’ s Sonata for Two Pianos and Percussion, the dramatic metamorphosis takes place on the border-line of Movements II and III. The end of the second movement is a six-four chord based on C (C-F-Ab), the beginning of the third movement, however, is founded on a C major chord –with an acoustic

scale. In Beethoven’s Missa Solemnis, "sepultus est" is associated with a six-four chord based on C (C-F-Ab-C), whereas "et ascendit" with a C major harmony (cf. Fig. 79 on p.45).

Consonance and dissonance It follows from the very nature of the acoustic system that its formations are consonant: owing to theovertone relations only consonant intervals take place in it; whereas elements of the Fibonacci system are precisely those regarded as dissonant by classical theory. In the acoustic system the most powerful dissonance is embodied in the Fibonacci intervals 2, 3, 5, 8 (thatis: minor subdominant harmonies). Within the pentatonic system, however, this is also inversely true! In melodies based on the 3, 5, 8 structure, the most vigorous dissonance coincides with the appearance of the perfect fifth. The opening theme of Kodály’ s Háry János (Fig. 84 a on p. 48) rests upon the C=MI pentatonic scale (C-Eb-F-Ab-Bb) in which the perfect fifth marks the point of highest tension (TI degree!) and this C-G dissonance calls for a resolution (as in classical harmony, the fifth-dissonance is also to be treated heavily: the accentuated G note occurs in a strong metrical position). We give a typical folksong example, too: Fig. 88

Openness and closedness The most direct experience (the most profound at the same time) is the duality existing between the ’closed’world of chromaticism and the ’ open’sphere of the acoustic system. The closed Fibonacci-models produce a tense and dynamic effect; the open overtone harmonies create a balanced and static impression. Also the melodic lines follow this principle: in the closed world of the chromatic system, the themes are most naturally joined with circular, rotary motions –whereas in the open diatony, with straight scale-lines (see Fig. 105 on p. 63). It evolves from the dynamic nature of the chromatic system that its formations are in a continuous progress of expansion or contraction –hence the frequency of ’ scissor’ themes, ’ funnel’ motifs, canons in ever-widening distances, ’ fanshaped’ progressions, and so on. In the diatonic technique, however, there is no trace of these

pulsating progressions; acoustic harmonies are characterized by stability and constancy. The duality of openness and closedness is also expressed in the number relations. The acoustic system rests on whole numbers (multiplying the fundamental tones vibration by 2, 3, 4, 5, 6, 7, 8, etc., we obtain an overtone scale). On the other hand, the keynumber of the pentatonic system is irrational –just like the p of the circle (about the role of the geometric mean in Bartók’ s music, see chapter Nature Symbolism). Also the 3:5:8 proportion is only roughly accurate. Finally, let us raise the question: how can we pass over from the plagal six-four system to the authenticmajor-minor system? There are two possibilities. (a) By means of a tonal answer, we can change the E-A-C six-four chord into the A-C-E root-position triad. The unison principal theme of Psalmus Hungaricus is a model of this procedure: Fig. 89

NB., in the middle of the melody (bs. 5-6) the antitonic C-G-D, and the modal dominant G major emerge: within the theme these represent the ’ tension function’ . (b) But the change is even more effective if the six-four chord based on E (E-A-C or E-A-C#) is resolved by the triad of the same name: i.e. E major or E minor. In fact, a similar change happens to the melody of the Bartók's Second Quartet, too, at the very end of the first movement: Fig. 90

* The question here is why DO and MI are the most static points of the two tonal

systems. As we know, tonality can be established only through the asymmetrical division of the tonal system. If the tones of pentatony are arranged in a fifth-order (see: Fig. 214 on p. 110), the most perfect asymmetry is realized in the outer degrees: in DO and in MI. The individuality of DO pentatony is decided by the fact that it can be exclusively built up of perfect fifths. The character of DO pentatony is therefore the most ’materialistic’ . On the other hand, that of MI pentatony is the most incorporeal and disembodied, as not a single perfect fifth can be built on the MI root. Consequently, the DO scale is ’ material’ , while the MI scale is ’ spiritual ’(abstract) in character, suggesting an ’ inner’world. This dualism was already realized by Wagner. The Parsifal-bells ’ sound differently’ in the first and in the third act. The four-note motif C-G-A-E gains a ’ C’meaning in the former and an ’ E’meaning in the latter: Fig. 91

This difference in character was already recognized by Verdi in Otello.The inaccessible figure ofDesdemona is painted with MI-centered pentatony, whereas Jago’ s down-to-earth materialism is depicted with DO-based pentatony. It is remarkable that the notes of the two scales are identical. Fig. 92

On the other hand, degree RE — as symmetry center — proves to be the most unstable point in the scale: symbolically speaking, within the pentatonic scale the RE represents the point of ’ atonality’ , which is why it has such a floating effect. One single pentatonic harmony (the opening chord of the opera!) is enough for Verdi to set the ’ storm music’of Otello whirling. The secret of the effect is that the chord is founded on RE; the RE-character is further emphasized by the LA-DO-MI-SO meaning of the G-Bb-D-F notes: Fig. 93

In our 12-degree system (and this is well shown by the external look of white and black keys on our keyboard instruments, or by our system of notation), — besides degree RE — there is to be found one more symmetry center –and this is the tritone of RE: the ’ SI’degree (in case of C tonality, the G#=Ab note). Thus chromatic and diatonic systems are phenomena which have developed not unrelated to each other but they represent the two sides — negative and positive aspects — of the same musical cosmos. They affirm and deny, exclude and at the same time presuppose each other. They form contrast in unity.

* Duality and synthesis*) I would like to attempt here an intepretation of Bartók’ s dual world, his ’ yang-yin ’ technique, in terms ofan equation, contrasting some special elements encountered at every step in Bartók’ s compositions. This interpretation is particularly applicable to the construction and content of the Sonata for Two Pianos and Percussion. First, ‘Inferno’movement chromaticism golden-section system closed world circular pattern of melody presence of central tone rhythm with strong ending uneven metre asymmetries

Third, ’Paradiso’movement diatony acoustic system open world straight pattern of melody presence of fundamental tone rhythm with weak ending even metre periodicity

C major end F# minor beginning demoniac world serene world, festive and playful instinctive existence intellectual existence organic logic love– hatred perfect understanding– irony tension freedom from tension emotional nature sensuous nature inspiration thought experience knowledge, solution feminine symbols masculine symbols dependency on fate law, order, form permanent change validity at all times augmentation– diminution stabilized forms occurrence existence process in time extension over space origin– development– conclusion division finite: circular motion infinite geometric nature mathematical nature (key figure to golden section: (key figures to overtone system: irrational figure) integrals) The principle of duality meant the possibility of synthesis for Bartók. He was not only aware that F# and C were counterpoles but also knew what it meant to get from darkness into light; he does not only teach us to see dialectic opposites in chromaticism and diatony but also shows the way from Inferno to Paradiso. *) from the essay with the same title (ed.)

THE QUADRPOHONIC STAGE OF THE ’MUSIC FOR STRINGS,

PERCUSSION AND CELESTA As far as concentration and closeness of form are concerned, none of Bartók ’ s works surpasses the opening movement of the Music: the famous ’ pyramid-fugue’ . Its closeness, resembling a circle, is already manifest in the key structure. The entrances of the fugue-theme are based on the circle of fifths. Starting from the central A note and the ’middle’viola part, the entrances progress in two directions –up and down – round the circumference of the fifth-circle, till they meet at the centre of the movement on the opposite side (the ’ counterpole’Eb): Fig. 94

then continuing their progress, they find their way back to the starting-point: from E b to A. The dynamic line also follows this arc-form: the movement is based on a singlesweep crescendo-decrescendo: starting from pianissimo, it rises steadily to attain the climax, the fortefortissimo –and then step by step sinks back to pianopianissimo (this is why the movement is also known as the ’ scissor-fugue’or ’ fan-fugue’ ). In addition, the entrances become more and more frequent up to the culmination, and from there they become rarer and rarer: the first five theme-entrances are presented one by one, entrances 6-7 appear in canon, and entrances 8-11 simultaneously (in a narrow sequence); while after the culmination the same process is reversed. This means that the pyramid form is also effected in the condensation and thinning-out of the material. In fact, the fugue-theme itself is inverted from the peak of the pyramid, and proceeds in mirror fashion: Fig. 95

Faced with such a degree of concentration, we can be justified in asking whether this is merely a technical stunt or whether, on the other hand, the visible form itself represents the projection of the poetic conception. As far as my own impressions are concerned, I would define the place of the fugue in the work as follows. The opening movement is born in the spirit of the Bartókean ’creation’idea. Bartók evokes the elementary explosions at the movement’ s central point in order to create the transition from chaos into a dialectically articulated world. At the moment of culmination, the swirling, shapeless material –a resound-ing chaos –is organized into ’ intelligible’pairs of antitheses (questions and answers): the material separates into mutually complementary elements; set against the homogeneous, impersonal whirling of the first part, it is precisely this dialectical separation of light and shade that signifies the appearance of personal elements and individuality in the work: Fig. 96

In the qualitative transformation of the material, a significant role is played by the double sound stage: the double orchestra. The piece engages two string orchestras, between which the piano-celesta-harp group and the percussion are placed. Thus the arrangement of instruments not only polarizes the tonality (as the title vividly expresses, from the resounding drum to the ethereal celesta), but through the stereophonic effect of the strings on the right and left polarizes the musical ’ stage’as well. The introductory part (the exposition) of the fugue takes place in the acoustical area to the right of the centre; the movement, however, comes to an end on the opposite, left-hand side of the stage. The concepts of left and right even in ancient philosophy were identified with the ’ inner’and ’ outer’worlds. On modern stereo stages (’ sonic stages’ ) this identification has actually become the rule! The special content of

’ right’and ’ left’may be connected with the asymmetrical construction of our body –in particular with the fact that our heart is on the left side. A stereo record-player easily persuades us how completely the character of the movement would change if the orchestras on the left and right were exchanged. (My art history teacher once put slides of Giotto’ s fresco, The Mourning of Christ, and one of Rembrandt’ s landscapes became in the projector in ’ mirror-view’ , reversing the left and right sides –in order to illustrate how the change radically altered the effect, mood and message of the picture; for example, the landscape impressionistically open or intimately personal in character, depending on whether the tree came to the right or left of the picture.) All this coincides with our previous observation that the shapeless swirl in the first half of the movement contains impersonal, while the clarification in the second half, personal elements. This is how the fugue-theme rises from the depths to the heights: to the dream-like swaying of the final part –and from the ’ outer’(right-hand) stage to the ’ inner ’ (left-hand) stage. (The progression from the right to the left corresponds to the Eastern way of thinking –similarly to the pentatonic system of the movement.) The foregoing reveals that the severity of the composition reflects not the laws of formal logic but those of organic development. This is all the more evident in the formation of the proportions, for these follow not the principles of classical symmetry but the laws of natural growth. For example, if each branch of a tree grows a new branch every year, but the fresh branches grow their first young branch two years later, the number of branches shows an annual progression as follows: 2, 3, 5, 8, 13, 21, 34, 55, 89 The 89 bars of the fugue are divided by the climax into 55 + 34 sections. The removal of the sordino divides the first part of the movement in a 34 + 21 ratio, while the second part of the movement is chopped by the renewed con sordino effect into 13 + 21 bars –with sharp contours. The exposition ends in b. 21, and even the final 21 bars of the movement show a 13 + 8 proportion. The section– points –like the nodes of a longitudinal wave –are attracted towards the centre (in keeping with Bülow ’ s Beethoven analyses, the movement must be completed by a rest bar): Fig. 97

The form of the movement thus creates one single wave-arc. The centre of the movement has the task of bringing about the metamorphosis –that transformation which is the basis of every real dramatic action. The form constitutes

one single ’magnetic field’ : crossing the centre the poles of attraction are exchanged! So the appearance of the ’ counterpole’–at the climax –also has a fairly essential function of content. The transformation of the material is also indicated in the other movements by the appearance of the counterpole –by this means the four movements, collectively as well as individually, are enclosed in a polar circle: Fig. 98

MOVEMENT I beginning and end: A, middle point: Eb (56). MOVEMENT II beginning and end: C, middle point: F# (263). MOVEMENT III beginning and end: F#, middle point: C (46). MOVEMENT IV beginning and end: A, middle point: Eb (83). In order to illustrate the dual plan of the Second movement, let us place the themes of the exposition and recapitulation side by side. The movement springs to its feet with an irritated reflex. Observe how (with a tigre-like gesture) the second orchestra cuts into the theme-entrance of the first! Fig. 99

(And vice versa, from b. 10 the first orchestra into the second.) It would scarcely be possible to bring about this effect with mono sound! The parts ’ bite into one another’ savagely. (The collisions and sharp clashes of the parts also determine the character of the exposition in what follows.) On the other hand, in the principal theme of the recapitulation the instrumental groups of the two orchestras unite. The rapid, tearing motions of the exposition pass into a balanced ’ rocking’ : Fig. 100

this is why the role of the timpani alters (see: Fig. above). The sharpest contrast is nevertheless produced by the closing theme. Its entry in the exposition is equivalent to ’ conflict’ : the broad flashing lines and flashes of lightning (accompanied by side-drum crescendos and crackings of the bass) lead to a ’wrestling’of the parts: the various instrumental groups struggle resolutely with one another –without arriving at a result (b. 141). The essence of the exposition is that its plot remains unsolved. Not so in the recapitulation! The closing theme of the reprise is meant to bring about fulfilment: Un poco largamente (b. 466, taking the place of the previous wrestling!). After the unresolved, unaccomplished exposition, the secco clatter and high tension ’spark-discharges’ of the development ensue with the certainty of a physical reaction. Behind the string-rending staccatissimos and murderous excitement of the rhythmic flashes, there once more stands the fugue-theme –note for note: Fig. 101

At its every step, the score of Music for Strings, Percussion and Celesta betrays that the composer’ s inner hearing was stereo. What is more, Bartók was acquainted with principles which the pioneers of modern stereo recordings did not begin to develop until the early sixties. In the principal theme of the second movement, the left group of strings is taken over by that on the right (see Fig. 99 on p. 59), while in the principal theme of the fourth movement, the right orchestra is answered by the left, in accordance with the fact that –as opposed to the energetic short-tempered impulses of the beginning of Movement II –the mood of the finale is relieved and joyous. We have already experienced the crucial importance of the fact that the two orchestras are united at the recapitulation of the second movement (likewise the sound becomes ’ centralized’in bs. 74 and 114 of the finale). Melodies appealing to ’ emotions’–as the secondary themes of Movement III – come forward consistently from the left, whereas thoughts of ’ spiritual’content come from the right. It is not even conceivable otherwise: at the end of Movement IV, where the fugue-theme returns in ’ diatonic’form, the tune is heard from the righthand stage –the spiritual quality of the thought is in this way significantly extended. Further it can be observed that the ’ impressionistic’character goes hand in hand with the spatial polarization of the tonality; and conversely, the more ’ expressionistic’the character of the music, the more the external space loses its importance and the tonality becomes homogeneous –mono-sounding. This in itself conceals exceptional possibilities! E.g., at the climax of Movement I –when all our attention is focused on the inner dynamics and tension –Bartók suddenly transforms the polarized sound into ’mono’sound, that is, he makes the two orchestras play the same parts. The

reverse is just as effective; but as opposite laws apply to the ’ chromatic ’and ’diatonic’techniques, in the diatonic world this also comes to pass the other way round: the homogeneous sound becomes a ’ stereo’sound at the climax –in much the same way as when we have ascended a hilltop, the landscape all at once opens up before us. The opposition of the left and right often produces the sensation of ’ here ’ and ’ away’(the music of the next movement offers an interesting example of this). Ferenc Liszt also writes about this symbolism in a poetic letter (Florence, 1839) on Raphael’s painting Saint Cecilia. ’ The painter places Paul and John on the left of the picture: the former is deeply absorbed in himself, the outer world ceases to exist for him; behind his giant figure immense profoundities are lurking. John is a man of "attractions" and "feelings"; an almost feminine face looks out at us. On the other hand, Augustine on the right of Cecilia, maintains a cool silence ... he abstains even from the most sacred emotions –constantly fights against his feelings. On the right edge of the picture stands Magdalene in the full splendour of her worldly finery; her whole bearing suggests worldliness, her personality radiates a sensuousness somewhat evocative of Hellas... Her love stems from the senses and adheres to visible beauty. The magic of sound captivates her ear faster than her heart is possessed by any supernatural excitement.’ Movement III is a magic nature music. Its form again rests on symmetry –on a fivepart ’ bridge form’(the order of the themes being A-B-C-B-A): on the one hand, the dirge melody of the first and last sections, and on the other, the alluring siren music of the second and fourth sections rhyme with each other; the sharp flashes of the third theme mark the centre of the bridge. To put it another way, the form of the movement delineates a spacious cupola, rising from the sobbing dirge-melody (A) up to the ethereal siren-song of the secondary theme (B); then an undulation –stirring up the whole orchestra –prepares the midpoint: the ’ light’effects of the climax (C), (see: Fig. 56 on p. 36) in order to lead back to the starting point, in reverse order of the themes (melody B and finally, A). Erich Doflein believes the xylophone rhythm at the beginning of the movement to have been inspired by the wooden drum of Japanese No dramas. That Bartók resorts to such sound-effects not for their own sake is proved by these very pages of the score. From theme 1, the sobbing lament song, the ’ fume’of a gong-stroke rises to the ethereal clear dolce-melody of the celesta-violin (theme 2); in the recapitulation, however, since the order of themes is reversed, the previous dolce-melody is suddenly stopped by the ’ snapping’of the strings (produced by slapping the strings against the fingerboard), and leads back to the dirge. And whereas the dirge-melody and its nocturnal F# tonality is deepened by the shuddering sound of the timpani (i.e. the lowest drum effect), a high-pitched cymbal effect indicates the centre of the movement –and the key of light: C. Fig. 102

As in the previous movements, the peak of the cupola (the counterpole of the movement) also transmutes the action in its content –and this is movingly expressed by the recapitulation of the secondary theme: Fig. 103

The most essential effect often escapes the attention of performers: this theme reappears in canon, and from the ’ imitating’part of the canon (cello) Bartók requires a more intensive dynamism than from the ’ leading’violin: the cello is piano, the violin pianissimo. Thus the effect arises: the melody becomes a recollection, a memory image: with the help of the canon, it shifts in space and time (attention and mind are divided into two) –it takes place on a divided double-stage and, owing to the stressed imitating part, a stronger light is thrown on the more distant stage. We point this out because in Bartók’ s recapitulations, canon melodies of slow space usually play the role of ’memory’ , remembrance, reminiscence. Here the reminiscence effect is enhanced by something else. The violin is heard from the left, and the imitating cello from the right side of the stage. As on modern stereo stage, the left is associated with ideas of ’ inside and here’–while the right with ideas of ’ outside and far’ . The’memory’character makes us realize why the tune must end with the break of the strings (a strong pizzicato so that the string rebounds off the fingerboard). As a consequence of the crack, the basses groan and the dirge-melody returns. The finale contains the poetic solution of the work. The solution lies in the fact that the leitmotif of the work, the ’ closed’fugue-theme –which hitherto occurred in a narrow a chromatic form –reappears towards the end of this movement in a wide diatonic form: in the ’ open’sphere of the natural overtone scale (see Fig. 107 on p. 65).

The transformation from closeness to openness is already revealed by the principal theme: the ’ circular’melodic lines of the first movement are here extended to ’straight’scale-lines: Fig. 104

Bartók’ s closed chromaticism can be represented by the symbol of the ’ circle ’ , while his open diatony can be seen in the symbol of the ’ straight line’ . The themes also become assimilated to these emblems: the chromatic system is most naturally combined with the circular, whereas diatony with the straight melodic line (scaleline). The opening and closing themes of the Sonata for Two Pianos and Percussion, Music for Strings, Percussion and Celesta, and of the Fourth String Quartett are shown here: Fig. 105

In Dante’ s Divine Comedy, the symbol of the Inferno is also the circle, the ring, whereas his Paradise is symbolized by the straight line, the arrow, the ray. The concentric circles of the Inferno narrow till they reach the Cocytus –the circles of Paradise, however, expand into the infinite Emphyreum. In the Comedy we frequently come across the transformation of the circle into straight line, and vice versa. The poet approaches, for instance, the denizens of the Purgatorio in this way: ’ You, who are bent by life, keep circling to straighten out again’(Purg. XXIII); or later on, looking into the light-river: ’ Into roundness it seemed to change its length ’(Purg. XXX). How characteristic of Bartók’ s simplicity that when the diatonic fugue-theme returns, he is satisfied with a unisono melody (on the G string) –the artistic solution is achieved virtually without the assistance of technical means. Even when repeated, the melody is coupled only with simple major-triads, through which the sound becomes

solid and solemn, like an organ –signifying that the fugue-theme which was born out of the resounding chaos of the first movement, through the piercing humour of the second movement, and the spell of nature in the third, has finally arrived at its poetic fulfilment. But what does this ’ openness’actually mean? The hypnotic effect of the first movement is the result of the fact that — during the progress along the fifth-circle — at every moment, in every phase of the circumvolution, we are necessarily aware of the positions the theme occupies in relation to the centre. Bachofen’ s mythological analyses call our attention to how deeply and indelibly the ritual act of ’ going round ’ has its roots in human nature (the excitement of ancient circus-games or of modern horse-races, Dante’s journey through the rings of hell or the lovers of the Magic Flute going round the circles of the ’ fire and water ordeal’would produce quite a different impression on us should we disrupt this outer framework of the action). In the closing movement all this happens differently. Here each new episode opens up before us with the result that the material of the former section ’ bursts open’ : Fig. 106

What at first appears as a sort of ’ montage’or ’ mosaic’form, is in reality a conscious constructional principle, thus constituting a striking contrast to the singlearched ’ wave-form’of the first movement. —If our earlier observations were correct, we should expect to find the decisive turn again after the central theme: in the unexpected ’ change of scene’of b. 114 (a change in key, too, because instead of the expected C major, it switches over to the F# counterpole!). (See: Fig. 176 on p. 93). The diatonic entrance of the fugue-theme marks the most beautiful ’ opening ’of this kind. This is already reflected in the scale of the melody: its notes are derived from the natural overtone sequence (C-D-E-F#-G-A-Bb-C) – that is, the theme is introduced not as a melody but rather the projection of a harmony: Fig. 107

This is the source of the pervasive clarity, the hymnic floating of the theme and, as mentioned earlier, its open spatial effect. Over the theme — again from the right — a piping major-sixth organ point particularly underlines this effect (the major sixth in Bartók’ s tonal world may justly be called the ’ pastoral sixth’ ). This theme is the key to the comprehension of the work. Is it not conspicuous how vividly this tune — this unisono melody — is pervaded by the metrical pulsation of ancient hymns, suggesting a text, the Hellenistic sense of form: infinite in its asymmetry, but at the same time, clear in cadence and lilt –like the conscious revival of the famous Seikilos hymn, Fig. 108

of that Seikilos ode which is simultaneously both a drinking-song and an epitaph, wisdom and love of life –a balance between life and death.

POLYMODAL CHROMATICISM Should the modal character and — from a birds-eye ’ view — the organization of the twelve solmizating letters be summed up in a few lines, it might run as follows: The twelve degrees of the chromatic scale can be arranged in three groups. First: the most peculiar feature of the system is that its center is being marked by a ’Black Hole’ . The center —the zero point —coincides with the point of ’ atonality ’ . # Up- and downwards from the RE or SI (D or G ), every degree occupies a symmetrical position. (E.g., on the piano keyboard — related to D or G # — each interval has its exact mirror image). Symbolically speaking, an element that is symmetrical to everything represents the state of ’ physical death’ . The symmetrical division (the repeated bisection) of the circle of fifths — around the RE symmetry center — results in the ’ tension’points of the system. (See first diagram of Fig. 214 on. p. 110) Thus, if we separate these symmetry points from the chromatic scale, we arrive at the RE– SI and TI– FA tritones –creating the so-called ’ sensitive notes’of the major and minor scales (in the major: B and F –in the minor key: D and G#). However, if we omit these 4 notes from the circle of fifths, the basic scale of the ’chromatic system’is obtained: in our terminology, the 1:2 model. Second (in fact, this should be put in the first place): the most static pillars of our tonal system are constituted by the DO-MI-SO and the relative LA-DO-MI triads (i.e., C major –and A minor belonging to it). The third group contains typical modal colours: The upward luring DI and FI produce a chiaro effect, whereas the MA and TA an oscuro one. But while DI and MA (C# and Eb) appear as tense dynamic elements, FI and TA (F# and Bb) emerge as static colour elements. DI and MA suggest a ’ major’ and ’minor’tension, respectively. On the other hand, FI and TA determine the character of the ’ acoustic’ (overtone) scale. ( Summary on pp. 110-112)

MAJOR TRIAD — MINOR TRIAD I suppose it was Verdi’ s Otello which made me realize that the basic concepts of our music theory (and even the elementary particles and atoms of classical harmony: such as the images of major and minor chords fixed in our minds) called for a reappraisal, a better approach, a new interpretation. Let us set out from the Db major triad (C# in fact) which crowns the first act of Otello. Can we manage to apprehend the essence of the phenomenon if this chord is interpreted, on the basis of our classical knowledge, as a ’ major chord of degree I’ ? In this case precisely the most important experience would be lost: the description of the unique ’ elevation’which causes the stars to light up at the end of the act ("Vien... Venere splende") and which raises the lovers into heaven. (See Fig. 109 on p. 67) If the tonic E major is regarded as a DO-MI-SO chord and the C # minor as a LA-DOMI chord, the C# major harmony will then necessitate a LA-DI-MI interpretation! Relative solmization makes a tangible distinction between the two kinds of major triad: the E and C#. The DO-MI-SO and LA-DI-MI triads represent quite dissimilar tonal qualities, different musical ’ characters’ . The principal idea of the work, the ’ kiss theme’ , into which the end of the act flows, leaves no doubt about the basic tonality. It takes place within the compass of the 4 # E major and C# minor; then after a definite cadence — with a few linking notes — it opens out in the C# major chord. Fig. 109

Both are ’major’chords but because of the DI degree, the LA-DI-MI harmony is much brighter and more exalted than the simple major triad. The DO-DI rise produces the illusion of an elevation. This is what makes the stars vibrate, and arouses the feeling of emotional fulfilment. Fig. 110

The difference of 3 key-signatures between E major and C # major denotes the same modal tension that has been called ’ axis tension’in the music of Bartók and Kodály. (Chords based on identical axes –for example C, A, E b, F# –show a difference of 3, or 3+3=6, accidentals.) The famous ’ light chord’in Haydn’ s Creation oratorio blazes up with the same DODI rise! The meaning of Eb major is DO-MI-SO, C minor is LA-DO-MI, C major is LA-DI-MI! what is more, Haydn doubles the effect, by another 3-sharp elevation swinging over from C major to A major. The question suggests itself: what happens in the case of a descent of 3-flat keysignatures? For example, when C major is exchanged by Eb major. Fig. 111

In this case DO major is transformed into MA major (C major into E b major). The character of the Eb major chord is determined by the MA degree — the melody itself expresses the MI-MA decline (E-Eb theme-opening). Thus the cited melody became the leitmotif of ’ dream and sleep’in Wagner’ s Valkyrie. (In b. 3 a similar change is enacted.) The difference of 3 accidentals between C major and E b major expresses once more an ’ axis tension’ . The state of deep sleep (daze caused by poison) in Mozart’ s Così fan tutte, is b symbolized also by MA major tonality: B major following the G major key: Fig. 112

Or let us take a MINOR triad, for example, an E minor chord, and replace it by the key lying 3 signatures lower –that is, by G minor. If we consider E minor as a LADO-MI chord, the G minor will have a DO-MA-SO meaning, and its character will be determined by the sombre-sounding MA degree. Fig. 113

The recapitulation of the Dies irae in Verdi’ s Requiem makes a cruel impression because the dominant B7 leads — not into the expected E minor — but into G minor: DOMA-SO chords! Fig. 114

Verdi’ s most ethereal musical formula is produced by replacing the expected A minor tonic by F# minor: FI-LA-DI (Fig. 115). Its airy and sublimated effect arises from the upward-luring FI and DI degrees. The music reaches, as it were, a point beyond earthly gravitation. Fig. 115

If a triad (either major or minor) is placed a minor-third degree lower, the DO rises to DI — and the sound becomes brighter. ’ ’ When the transposition is carried out in the opposite direction, the tone ’ darkens’ , because the MI sinks to MA (e.g. in the case of a C major — Eb major change, or A minor — C minor change, the E note is modified to Eb). The DO-DI change implies a rise of 3 sharps, the MI-MA change a descent of 3 flats.

An organic part of the cheerful atmosphere in the 2. episode from Kodály ’ s Dances of Marosszék is that the theme, recurring four times (bs. 108, 120, 132, 144), moves by minor-third steps downwards (that is, in each case it rises by 3 key-signatures) — so that, in passing around the whole C#(Db)– Bb–G– E axis, it becomes brighter and brighter. As opposed to this, the dark and disquieting effect of sequences moving upwards on a minor-third ladder, was already well-known to the Romantic masters. The call-todeath tone at the beginning of the Tristan Prelude would certainly dissolve without this minor-third rise (4-4 bar sequence). In the "Liebestod" theme this process is only accelerated: the melody moves upwards through minor-third degrees –so that, in passing around the whole Ab-B-D-F axis, it becomes darker and darker. Fig. 116

The direct opposition of the two kinds of sequence is particularly effective — as it happens in Bartók’ s Violin Concerto: in this way does the first obtain a ’ joyful‘ , the other a ’ struggling’character: Fig. 117

The refrain-melody of Dances of Marosszék acquires a fervently passionate or a triumphant character according to which colouring element gains the upper hand: the triad a minor third higher (on the first appearance of the theme), or a minor third lower (on the last appearance of the theme): Fig. 118

Tristan’ s desire motif —in the course of the Prelude —emerges in three basic forms: EXPOSITION MIDDLE PART REPRISE

F major — D major F major — B major F major — Ab major

and thus exhausts the complete F-D-B-Ab axis. Fig. 119

The exposition involves an elevation of 3 key-signatures, the recapitulation implies a descent of 3 key-signatures –while in the middle part, there is no change in tension (a difference of plus 6 = minus 6 signatures is established between F major and B major and, as a result, the two cancel out each other’ s opposite tension). In Beethoven’s Missa Solemnis a breath of fresh air flows over the orchestra when in the Gloria movement authentic (V-I) steps alternate with LA-DI-MI effects (B – A b, Db–Bb, Eb–C, F– D major chords): Fig. 120

In Grigory’ s dream-narration from Mussorgsky’ s Boris Godunov the text itself indicates this ascension: ’ Long flights of stairs led me to the top of a high tower ’ . The vision is evoked through harmonic sequences deepening by minor thirds: Fig. 121

Casting our net a little wider now, we note that (related to the DO-MI-SO and LADO-MI triads), the DI suggests a ’major’character, MA suggests a ’minor’character. In my previous investigations (between 1947-1974) I worked out almost twenty theoretical deductions of the axis system. Nevertheless, I have so far overlooked a possibility: namely that relative solmization could be the most simple means of ’making visible’ the attractions functioning within the system.

POLAR CHORDS (DI+MA TENSION) On an elementary level, then, two key-words govern Romantic harmony: DI and MA. However, the most characteristic axis tension is realized in polarity, in the relation of the farthest keys: in the distance of 3+3, that is, 6 key-signatures (e.g. in the connection of A major and Eb major –or A major and C minor). It follows from the above that a polar tension-character can be brought about by modifying DO to DI and, simultaneously, MI to MA. Let us take the simplest relation: C major and its relative A minor. If we replace MI by MA in the C major chord, and DO by DI in the A minor chord, the result will be a DO-MA-SO and a LA-DI-MI triad. Fig. 122

The difference between the 3 flats of C minor and the 3 sharps of A major is 6 accidentals. (A chord based on LA calls to mind a minor triad — whereas a chord based on DO suggests a major triad. In Figs 122-123, however, it is LA major and DO minor that appears!) Tristan’ s death motif is, in fact, a materialization of this polar idea: half-close cadence: A major full-close cadence: C minor

= LA - DI - MI, = DO - MA SO.

Fig. 123

The formal turning-point of Isolde’ s Liebestod ("wundervoll und leise") reflects a similar opposition: C# minor and A# major. Fig. 124

In Bach’s St. John Passion, the contrast between the soldiers and the Virgin Mary is depicted by D minor and B major harmonies: dark DO-MA-SO and bright LA-DI-MI chords. As B major appears, the scene is suffused by an almost celestial light: ’ and at the cross of Jesus stood his mother...’ Fig. 125

At the moment of Otello’ s suicide, this order is reversed: the LA-DI-MI comes first, then, with a sudden fall of six key-signatures, the DO-MA-SO chord (D major and F minor). Fig. 126

But let us return to the relative C major and A minor keys, and modify DO to DI and MI to MA! The chord thus created condenses the A 7 and Eb7chords polarly (tritonic relationship). Fig. 127

Our next example is the appearance of Parsifal in the flower-garden scene. The basic tonality is represented by C major and the relative A minor (C-E-G and A-C-E). –If we replace DO by DI and MI by MA, the continuation of the theme: the A-E b-G-C# polar formula will then automatically spring from the C-C # (DO-DI) and E-Eb (MIMA) steps, Fig. 128

again combining A7 (A-C#-G) with Eb7 (Eb-G-C#=Db). NB: A and Eb mark the ’extreme’points of the fifth-circle, see diagram of the axis system.

This polar character is demonstrated with an almost sensuous force by Fig. 129. At the beginning of Act III, Desdemona and Otello greet each other: Desdemona is still unsuspecting — but Otello is already tormented by the thorn of jealousy. In keeping with this, Desdemona stays harmoniously within the sphere of the tonic E major — C# minor, while the same melody suffers a polar distortion by Otello: he deforms the DO to DI and the MI to MA (E-E# and G#-G turn). Fig. 129

Without precedents of this sort it would be hard to understand how the combined use of DO-DI and MI-MA became regular in Bartók’ s music —as we can see for instance in the "Marcia" theme of the Sixth String Quartet. Fig. 130

And finally, we must mention the ’ reciprocal’case of the polar DI+MA tension. Let 7 us take a D major and an F chord as an example. The meeting of the two results in a polar effect –as the DO-MI (D-F#) notes of the first chord become DI-MA (D#=Eb and F) notes in the second. Fig. 131

A reverse effect can be created when the two chords are interchanged: in this case the chord bearing polar tension is followed by a sudden resolution. In the Insurrection scene of Verdi’ s Don Carlos, the distorted situation is solved by an F7-D major turn: the forbidding figure of the Grand Inquisitor towers above the crowd and with a single gesture forces the rebels to their knees. The effect speaks for itself. Fig. 132

Systematizing what has been said above, we can distinguish essentially three kinds of major, and three kinds of minor triads. Shifting the MAJOR triad by a minor-third interval, the DO changes into DI, or the MI into MA — according to whether we move downwards or upwards. In this manner three kinds of major character come into being: the three types are to be distinguished from each other by the solmizating symbols DO-MI-SO, LA-DI-MI and MA-SO-TA, respectively. Fig. 133

If the operation is performed with MINOR triads, three kinds of minor character will be obtained which can be differentiated by the symbols LA-DO-MI, FI-LA-DI and DO-MA-SO. Fig. 134

That the problem in question is not a theoretical, but a ’ live ’one is well exemplified by Verdi’ s Falstaff (Act II, scene 2) where A major = DO - MI - SO, F# major = LA - DI - MI, C major = MA - SO - TA. Fig. 135

Liszt’s Rákóczy March (Rhapsody XV) is an ’ embodiment’of Fig. 134: the melody

itself has an A minor signification (=LA-DO-MI) –accompanied by F # minor (FILA-DI) and C minor (DO-MA-SO): Fig. 136

It may seem paradoxical but late Romanticism discovered a new kind of harmonies: the ’minor’key (LA key) that has a major third (DI) –and inversely: the ’ major ’ key (DO) key that has a minor third (MA).

MAJORMINORSUBMINOR — — The axis system developed from the classical major– minor tonality in two ways: from the correspondences between relative and parallel major–minor chords. In the simplest case, the parallel mode of C major is C minor –and its relative key A minor. This combination carries within itself the germ of a significant chord-type which became a true emblem of Romantic music (suffice it to recall the well-known ’ Tristan chord’—see Fig. 149 on p. 81): Fig. 137

Since this chord proves to be the ’ relative’of the minor harmony, we call it the subminor chord. In the Parsifal-motif this relative connection takes shape as follows: Fig. 138

To make a distinction between the three basic types, in our harmonic diagrams we shall indicate seventh chords with a major character by capital letters; seventh chords with a minor character by small letters; and seventh chords founded on a diminished triad by a circle under figure 7 (its seventh-note being a minor-seventh interval). Fig. 139

In the example above, between C major and C subminor a ’ polar’tension is established because C major and Eb minor — the latter inherent in C subminor — show a difference of 6 accidentals (similarly to keys facing each other in the circle of fifths). This is why Bartók and Kodály preferred the use of the G subminor –rather than the conventional dominant seventh G-B-D-F: Fig. 140

In the refrain-melody of Isolde’ s monologue (Act III), the polar tension of D major and D subminor manifests itself! Fig. 141

* From a dynamic point of view, in the movement of the major, minor and subminor chords the simplest relationship is created by the relative key connections, i.e. by a change

from major to the relative minor, or from minor to the relative subminor, or from major to the relative subminor (for example, from C major to A minor, from A minor to F # subminor, or from C major directly to F# subminor) — and vice versa. Fig. 142

A change in the direction of the minor or subminor involves an increase in tension, Fig. 143

whereas a change in the opposite direction results in a lessening of tension. Fig. 144

But it is worth mentioning that the change in the relative minor direction is what we discern as a ’ natural’connection, while the change in the other direction creates in a certain sense an ’ unnatural’ , violent effect. Thus did the melody of Fig. 142 become a’ curse’motif, and the harmonic motion of Fig. 144 the symbol of the ’ dead’swan. In Liszt’s oratorio Via Crucis the death of Jesus (Consummatum est –XII. station) produces a paralyzing effect because F# minor is followed by the relative A major. Fig. 145

A ’modal’change is considerably more powerful than a relative change, that is from major to the minor of the same name, or from minor to the subminor of the same name, or from major to the subminor of the same name for example, from C major to C minor, from C minor to C subminor, or from C major immediately to C subminor) –and vice versa. Fig. 146

It should be noted this time, too, that the parallel subminor-minor-major change is what we fell to be ’ natural’(it virtually passes unnoticed if we close Bach’ s minor # fugues with a major triad). The C major episode in Kodály’ s Dances of Marosszék owes its serene nature-atmosphere precisely to these ’ natural’ , positive connections, i.e. from major always to the relative minor, and from minor to the parallel major. The harmonies pass through the entire axis-circle in this way: Fig. 147

And conversely, the parallel major-minor-subminor change creates a passionate, short-tempered, dramatic impression! In the middle-part of Liszt’ s "R.Wagner, Venezia" (from b. 31) we hear only pure

triads, and yet, these triads receive a tragic character since they move in the ’negative’’ unnatural’direction, i.e. from minor always to the relative major, and from major to the parallel minor. Fig. 148

It is just as one expect: Isolde’ s "Liebestod" follows the same harmonic thread (see Fig. 188 on p. 99).

On the ’ Tristan chord’ In vain do we challenge classical theory: what is it in the Tristan chord that ’ cuts to the heart’of the listener? Instead of theoretical considerations, we must follow our healthy musical intuition. We instinctively know about the F minor triad that it represents the relative key of Ab major, that is, if Ab major = DO-MI-SO, then F minor = LA-DO-MI, creating the most fundamental musical relationship. And if we unexpectedly exchange the MI note with the painful dark-coloured MA note, we at once perceive the tension that almost disrupts the chord. Fig. 149

The MA change is well demonstrated by the Tristan-quotation of the Mastersingers. Fig. 150

The ’ basic’form of the Tristan-motif, its typical occurrence is that which starts with b an A note (as at the end of the Prelude). And if Wagner starts the work with an A

note after all, then the polar tension created by the DI+MA notes also becomes tangible. Fig. 151

The subminor may function as a tonic, too. Fig. 150 makes clear that the Tristanchord: F subminor appears as the relative key of the tonic Ab minor –quasi as a deceptive cadence! The ’ F major’character of the initial step in the overture (A-F) and the B major end of the opera reflects a ’ counterpole’relationship. On the other hand, the final chord of the opera: B major represents the relative major of the ’ basic key’(= Ab minor) and, as such, it suggests a ’ death’symbolism. Perhaps nothing shows the axis-tension of the Tristan chord better than its resolution, which can be performed — according to the axis system — in four different ways, in four directions: towards E, or Bb, or Db, or G. Fig. 152

In Parsifal, too, the love symbol is represented by F subminor, i.e. the Tristan-chord.

The emotional ’metamorphosis’comes about when a Db root gets below the tense Tristan-chord, and this Db — as if by magic — changes the meaning of the F subminor at once: the F-Ab-Cb-Eb notes are transformed into overtones (major third, fifth, seventh, ninth) and, as a consequence, start ’ floating’sensually: Fig. 153

This is how a tense harmony is transformed into ’ colour’ . The D b bass functions like an acoustic ’ prism’which changes the sense, the colour, the atmosphere of the Tristan-chord. Since we perceive the former basic notes as overtones, their effect will be incorporeal and disembodied. What these and similar solutions may have meant to Kodály is revealed by the following Psalmus-quotation. At the moment subsequent to the dramatic dénouement (’ From you he removes your every burden’ ) the Tristan-chord melts similarly into the Db ninth-chord: Fig. 154

FUNCTION With the aid of relative solmization, the tonal relations can be further simplified. We place the relative connections one beneath the other (e.g. C major, A minor, F # subminor) –and the modal connections side by side (e.g. A major, A minor, A subminor). Fig. 155

The columns of A major, C major E b major are separated by a difference of 3 accidentals from each other. The ’ air pressure-difference’ thus created is also expressed by the fact that the character of the left column is determined by degree DI, and that of the right column by degree MA (C # and Eb notes, respectively). Only one more link is required — either vertically or horizontally — to bring the axis to a close. These ’ ascending, descending and collateral’relations are valid for the other two functions: the dominant and subdominant, too. We will turn again to Verdi, quoting the Ave Maria of Otello: Fig. 156

We note that the upper organ point (Eb) remains common to every chord — which is why we meet tonic and dominant chords only, the former marked below the line, the latter above it. The dominant chords are easily dealt with, three variants appearing altogether (the V. degree: Eb major and Eb minor, and the relative of Eb major: C minor). Of the tonic chords, likewise the parallel connection (Ab major– Ab minor) and the relative connection (Ab major– F minor) prove to be the most direct. These three chords provide the conditions for the F subminor as well (see Fig. 157 a). The F minor and Ab minor harmonies, however, are to be connected not only ’ inwards’(by means of F subminor) –but also ’ outwards’ , in the direction of F major and C b major: the parallel of F minor being F major, and the relative key of Ab minor being Cbmajor: Fig. 157

F major and Cb major not only polarize the sound (they are counterpoles), but also create a homogeneous tonal unit: Fig. 158

The continuation of Fig. 138 on p. 77 (Parsifal theme) rests on the F # major–C major pendular movement.

Fig. 159

The 12 bell-strokes from the night-scene of Verdi’ s Falstaff serves as a model example for the subdominant. (As the F note of the bell remains unchanged in each chord, only tonic and subdominant harmonies are to be found.) All the permutation are exhausted by Verdi. Fig. 160

* As stated before in the axis system it is not degrees IV and V that most potently represent the subdominant and dominant functions –but the degrees which divide the circle of fifths into three equal parts. The wide emotional range of the love-duet of Otello and Desdemona is attributable to the fact that the ’ depth’of the subdominant and the ’ height’of the dominant are strained to the utmost (Fig. 161). As compared to the tonic G b major, the negative tension of the subdominant can be enhanced if the IV degree C b major (B major) is replaced by its upper minor third degree: D major. — On the other hand, if the positive tension of the dominant is to be enhanced, we must put in the place of the dominant V degree (Db major) its lower minor-third degree: Bb major. In this way, the tonic Gb, the D subdominant, and the Bb dominant come into an augmented triad relation. Fig. 161

At the end of Act III, the augmented triad cadence C-A b-E almost bursts the framework of tonality. Fig. 162

* It appears from the diagram of the axis system that — in the case of C tonality — a dominant– tonic cadence may assume the following forms: Fig. 163

(1) The fourth step upwards corresponds to the classical VI—cadence (G 7® C). (2) The major second step upwards (Bb7 ® C) is a typical modal dominant cadence. (3) The III. degree dominant (E7 ® C) became current among the Romantics. (4) The minor second step downwards (Db7 ® C), by its peculiar colouring, is called the phrygian dominant. Theoretically, these four forms can readily be traced back to the structure of the axis system. Nevertheless, the clue to the individual character of each form of resolution again lies in the modal quality of the theme.

Let us begin with the ’modal’dominant: Bb7 ® C After the dominant Bb major seventh we would expect an E b major tonic –and if (instead of Eb) C major surprisingly emerges, this means that the place of the expected DO basic note (Eb) has been occupied by DI (note E)! In this way, the tonic has also become sharpened and more elevated. Fig. 164

Bartók’ s Violin Concerto is introduced by a pendulum of tonic (=B major) and modal dominant (A7) chords. Fig. 165

In Act I of Otello, the Jago– Roderigo duet becomes more and more tense due to the use of modal dominants. Fig. 166

Thus in each subsequent chord, instead of DO, DI comes; this is in perfect harmony with the text, which speaks of ranks (of the difference between the envied captaincy and the subaltern rank of ensign); it is as though Jago were standing on tiptoe, the words "mio!" are amplified by the DI degrees. By the way, perhaps these Romantic models help us better understand why Bartók and Kodály preferred the use of the modal dominant to the dominant fifth-degree (in which the influence of folk-music naturally played a considerable role).

The inverse procedure of the modal dominant is the III. degree dominant (majorthird step downwards): E7 ® C the dominant seventh chord is succeeded not by the usual A major (DO-MI-SO) — but by C major: MA major chord. Fig. 167

At the deepest point of Otello (Fig. 162 on p. 87), the end of Act III, the impact of the E majorC —major close is the more shocking since the place of the expected MI degree is taken by MA (C note). This is the reason why the III-I cadence is so heavily laden with death-symbolism. In Il Trovatore it appears when Leonore admits she is ready to die: III. degree G major followed by Eb major tonic: Fig. 168

Isolde’ s death in Wagner’ s Tristan is accompanied by a whole chain of III-I cadences. The keys follow one another in axis order. Fig. 169

Let me refer to the Gloria of Beethoven’ s Missa Solemnis. The heroic F major dominant in "Qui sedes ad dexteram patris" is followed by a MA major tonic (D b), which is why the "Miserere nobis" causes anxiety and disquietude (F ® Db = III-I cadence). In Verdi’ s Requiem, a III-I cadence casts a shadow of doom upon the recapitulation of the "Dies Irae" theme: after the B major dominant a G tonic emerges — hence the shock effect (see Fig. 114 on p. 69)! The III-I cadence may also figure as an onomatopoeic ’ colouring’element. At the start of the second scene –in Act II –Mistress Quickly attempts to imitate: to parody Falstaff’s deep, full-throated voice. The change of tone occurs through a III-I cadence: E7 ® C (see Fig. 135 on p. 76).

* In short, the modal dominant– tonic cadence has a positive meaning, while the III-I cadence gives a negative impression. From the above it follows that a contrary effect can be attained if the same key has been prepared for by the lower major second degree (modal dominant), or the upper major third degree (III). Two B b major themes from the Masked Ball suggest themselves here –the first introduced by an A b major, the other by a D major dominant. Renato’s first aria is the manifestation of a friend’ s love and devotion. But b the very same B major key, following right after the dramatic turn: at the exposure of Amelia (in the second act) has an infernal impact, expressing scathing sarcasm. Fig. 170

If the dominant and tonic chords are exchanged, their meaning will also be reversed! In the case of the I-III turn:

the E7 will be related to the A major key –which lies 3 sharps higher than C major. Thus the III. degree will produce the effect of ascent (a ’ light’effect –chiaro). On the other hand, in the case of the modal dominant following upon the tonic I. degree,

the Bb7 will be related to the Eb major key – which lies 3 flats lower than C major. The modal dominant will thus produce the effect of descent (a ’ shadow’effect –oscuro). Here are some examples for both. The lofty effect of the Aida theme may be attributed to the fact that the tonic is succeeded by a III. degree dominant. Fig. 171

In contrast to this, the dramatic turning point in Rigoletto: the blast of "La sua figlia!" is evoked by a modal dominant, subsequent to the tonic (F tonic and then E b dominant). Fig. 172

For Beethoven, the major-third change became the means for expressing ’ mystic absorption’(III-I) or ’ transubstantiation’(I-III): Fig. 173

To

sum

up,

modal dominanttonic — cadence tonicmodal — dominant cadence III. degree dominanttonic — cadence tonicIII. — degree dominant cadence

the

has a ’ positive’ , has a ’ negative’ , has a ’ negative’ , has a ’ positive’

meaning. These regularities may also be extended over the relationship of the subdominant and tonic (or dominant and subdominant) functions. We may consider it a general rule that

a major-third step upwards a major-third step downwards a major-second step upwards a major-second step downwards

produces a ’ positive’ produces a ’ negative’ produces a ’ positive’ produces a ’ negative’

effect. We still owe the intepretation of the Phrygian dominant (minor-second step downwards), for example

In the Phrygian dominant — just as in the classical V. degree seventh — the leading ’ ’ role is played by the sensitive notes TI and FA: the leading note TI pulls towards the root (DO), and the FA towards the MI. Fig. 174

The sensitive notes TI and FA halve the octave; therefore they are interchangeable — without any change in their relationship. Thus TI and FA are common notes in both the V. degree dominant and in the Phrygian dominant; as we see from the example above, the third and seventh note of the G7 is identical with the seventh note and third of the Db7 dominant. In the Recruiting Music of Háry János, Kodály exchanges the dominant seventh (G7) with its counterpole (Db7) — and in this way, the Phrygian dominant is reached. Fig. 175

We quote a typical Bartók example, too: this resolution is reserved by Bartók for a sudden ’ change of scene’ : Fig. 176

An interesting combination of the Phrygian dominant and V. degree dominant can be observed in Wagner’ s Mastersingers, at the appearance of the night-watchman.*) Fig. 177

* It is easy to survey these interrelationships –if we realize that from the DOMINANT G seventh chord four different TONIC degrees can be reached:

We quote the final scene: ’ ecstasy of immortal love’(” l’ estasi d’ un immortale b amor” ) from Verdi’ s Aida. A major stands for the V. degree dominant in bs. 3-4, D major for the Phrygian dominant (b. 7), and F major for the III. degree dominant (b. 8). In the middle of b. 4, also the modal dominant emerges for a moment: B minor (!). The glorification (’ I see the heaven opening’ , bs. 5-6) is associated with a LA-DI-MI b rise: B major. Fig. 178

DIMINISHED SEVENTHS In his late style, Verdi accords a singularly large scope to the diminished seventh chords. These diminished seventh chords, however, turn out to be not real diminished sevenths - but such major seventh harmonies (chords of DO-MI-SO-TA pattern) whose DO root ’ overstrains’to DI. A classical example of this DI-change is the "Barrabam!" exclamation in Bach ’ s St. Matthew Passion: instead of the tonic D major, the threatening D #-F#-A-C diminished seventh emerges. Fig. 179

The diminished seventh C-Eb-F#-A accompanying Desdemona’ s murder begins as a modification of the B seventh chord (B-D #-F#-A) and ends polarly: in an F seventh chord (C-Eb-F-A). Fig. 180

Authentic cadences occur at every step in Verdi’ s music, like this:

It is, however, a well-tested practice to replace one or another of the major seventh chords by diminished sevenths. In our example (the stretta from the opening scene of Falstaff), almost every chord is replaced by a diminished seventh - without veiling or overshadowing the S-D-T-S-D-T (subdominant-dominant-tonic) meaning of the sequence. Fig. 181

The diminished seventh belongs to the family of distance chords: it divides the tonal system into four equal parts. Hence it follows that any diminished seventh can supplant four different major-seventh harmonies. In the storm scene of Otello, Verdi exhausts all combinational possibilities: Fig. 182

Through the chromatic lowering of any note of the diminished seventh chord a major seventh can be achieved (e.g. from the C#-E-G-Bb diminished seventh the C-E-G-Bb major seventh). If the next third is also lowered, we shall obtain a minor seventh (CEb-G-Bb) - and lowering the subsequent third, a subminor chord emerges (C-Eb-GbBb). Reversing the process, the harmonies start widening systematically (subminor-minormajor-?). Verdi takes advantage of this possibility in the tonal construction of the "Credo". In our tonal system three different diminished seventh chords ( constructed:

) can be

(1) a ’ tonic’ : by raising the root of the C seventh chord, a C #-E-G-Bb diminished seventh is obtained, (2) a dominant ( , B-D-F-G#), and (3) a subdominant (

, F#-A-C-Eb).

(Every other form agrees with one or other of the above formulae.) As we see, the ’tonic’diminished seventh contains a DI and TA note. However, the root of the major and minor keys is represented by DO and LA, respectively.

SUBSTITUTE CHORDS The harmonies of ’ Grand Romanticism’ frequently obtain some sort of ’background’meaning - a secondary, transposed sense - and this occurs whenever a major triad is replaced by the minor chord lying a major-third higher (e.g. C major substituted by the E minor chord). Or, on the contrary, when a minor triad is replaced by the major chord lying a major-third lower (e.g. A minor substituted by the F major chord), Fig. 183

that is: the DO-MI-SO major triad is substituted by MI-SO-TI - or the LA-DO-MI minor triad by FA-LA-DO. For example, in the Rigoletto-Gilda duet, the substitute chord (E minor replacing C major) reflects Gilda’ s spiritual purity: the substitute chord invests the melody with a ’sublime’emotional content. Fig. 184

As opposed to this, the tragic weight and baleful atmosphere of Kodály ’ s Psalmus Hungaricus lies in that the melody of A minor origin is inserted into an F major harmony - right at the beginning of the work. (See: Fig. 67 on p. 40) Owing to the upward-pointing TI degree, the tonal quality of the MI-SO-TI substitute chord is ’ positive’ , while that of the FA-LA-DO substitute chord is ’ negative’ , owing to the downward-pointing FA degree. (Note the ’ outward’and ’ inward’acting forces functioning

in TI and FA, respectively.) Distinction must be made between a FA-MI and a MI-FA step, as is suggested by the opening of the hymn Ode to Joy in Beethoven’ s Choral Symphony. The spiritual transparency of the song originates in the MI-FA initial notes.**) On the one hand, MI-SO-TI became the ’ transcendental’unearthly element of late Romanticism. On the other hand, the FA-LA-DO lends the music some sort of emotional ’ depth’or weight. From the notes of the C major (or A minor) scale, six perfect triads can be formed: besides C major and A minor, the 1-sharp G major and E minor - as well as the 1-flat F major and D minor (cf. L. Bárdos: Modal Harmony). Therefore the ’ highest’minor:MI-SO-TI (E minor), and the ’ lowest’major:FA-LA-DO (F major) are identical with the ’ positive’and ’ negative’substitute chords, respectively. The ’major’substitute chord (F major) intensifies the ’ minor’character of the theme, while the ’minor’substitute chord (E minor) intensifies the ’ major’character of the melody. In the former, the F-C perfect fifth reinforces the minor third (DO) of the root (A) - while in the latter, the E-B perfect fifth reinforces the major third (MI) of the root (C). It is enough to glance at Fig. 183 to see that the two kinds of substitution (MI-SO-TI and FA-LA-DO) are again each other’ s reflections - precise mirror images - in comparison to RE. Now we can complement our previous observations (pp. 78-81): the change from major to the minor substitute chord is perceived as a ’ positive’ : elated turn (e.g. from C major to E minor) - and conversely, the change from minor to the major substitute chord produces a ’ negative’ downward-pulling effect (from A minor to F major, for example). But at the same time, the negative substitute chord makes a ’ natural ’impression while the ’ positive’substitute chord creates an ’ unnatural’effect. From these facts it emerges that negative or positive effects may also be piled up! It pertains to the dolorous atmosphere of the Parsifal theme quoted in Fig. 42 on p. 30 that two negative effects - parallel major-minor and substitute minor-major turns alternate with each other: C minor, A b major, Ab minor, Fb (=E) major, E minor, C major, C minor. If we reverse this sequence, then parallel minor-major triads alternate with positive (=minor) substitute chords. The theme thus created is well-known to us from Tristan (NB: it stems from the opening cello-motif of the opera: A-F-E-Eb-D):

Fig. 185

Let us think back to the beginning of Parsifal; where C minor substitutes for the tonic Ab major. The ’ elevated’ , immaterial tone of the melody emerges from here: Fig. 186

In the ’ death motif’of Tristan, however, the tonic C minor was overshadowed by the heavy Ab major (see: Fig. 123 on p. 73). Let us remain within the sphere of Ab major and C minor: the substitute chord of Abmajor is C minor, the substitute chord of C minor is Ab major. In other words, if we treat the A b major as a DO-MI-SO chord, the meaning of C minor will be MI-SO-TI. On the other hand, if we consider C minor as a LA-DO-MI chord, the meaning of Ab major will be FA-LA-DO. Combining Ab major with C minor a symmetrical chord - the so-called ’ hypermajor’ harmony is produced (see: pp. 40-41). Its construction is: major triad with a major seventh - Ab-C-Eb-G. It adds considerably to the peculiar quality of these chords that the positive substitute chord falls within the range of the dominant, while the negative substitute chord falls within the range of the subdominant. Fig. 186 (in Ab major) receives a dominant tinge through the positive C minor - while Fig. 123 (its tonic being C minor) assumes a subdominant shade through the negative Ab major opening chord. From Fig. 183 it emerges that the relationship of C major and E minor is determined by the C-B step, A minor and C major is determined by the A-G step,

F major and A minor is determined by the F-E step, i.e., by the melodic turns: Fig. 187

In a minor key, the deceptive cadence (VI. degree chord appearing instead of the tonic I. degree) may play the role of the substitute chord - especially if it loses touch with the tonic I. degree and becomes independent. The key to Isolde ’ s "Liebestod" b also lies here: in b. 3 the C major deceptive cadence emerges as the negative substitute chord of Eb minor! Fig. 188

The duet of Rigoletto and Gilda "Piangi, piangi, fanciulla piangi" brims over with sorrow by appearing not in the expected F minor, but in D b major - after the dominant C (Db = substitute chord and, at the same time, a deceptive cadence - which loses touch with the tonic F minor). In the minor key, the I. degree relates to the deceptive cadence (degree VI) - as degree IV relates to the Neapolitan chord. Thus, in the minor key there exists an analogy between the Neapolitan chord and the deceptive cadence. In East European folk music the ’ leading notes’ FA-MI and TA-LA play a determining role, calling forth a ’ plagal’(subdominant-tonic) relationship and, at the same time, a substitute relationship (F major-A minor and B b major-D minor). If we invert the above chords - around the RE symmetry center - an ’ authentic’(dominanttonic) cadence is produced by means of the TI-DO and FI-SO ’ leading note’steps: Fig. 189

In Wagner’ s "Opus Metaphisicum" (as Thomas Mann called the Tristan), these chords suggest a high aspiration - an abstract "Gothic" passion - like in Scene 3 of Act I, in the opening three bars: Fig. 190

Under certain conditions, negative substitute chords may suggest a specifically Eastern, whereas positive substitute chords a specifically Western way of thinking. (a) It is a rule in melodies with a six-four character and MI-LA-DO structure that the ’ tension-chord’(the dissonant ’ changing chord’of the tonic) is represented by FA-LA-DO - and this comes about when the MI root is raised to the FA degree (through a Phrygian step!). Fig. 191

The ominous atmosphere of Desdemona’ s leitmotif in Act IV is determined by the G#-C#-E six-four chord - its dissonant ’ changing chord’being A-C #-E (=a negative substitute chord): Fig. 192

(b) As stated above (Fig. 186 on p. 98), MI-SO-TI is the transfigured, unearthly element of late Romanticism. In Verdi’ s Requiem the beginning of the work (in A minor) is laden with oppressive death-symbolism. But the very same A minor theme at the recapitulation receives a transcendental character - since it reappears as the ’positive’substitute chord of F major: now it makes visual the perpetual light. Fig. 193

Remarks on Verdi’s colour chord Let me refer to the modification of the cadential I six-four chord. In almost every significant major-theme of Otello, the I. degree six-four chord is coloured by its substitute chord; the E major cadence, for instance, by the G # minor harmony. At the end of the ’ kiss-theme’ , a D# note appears instead of the usual tonic E. Fig. 194

Verdi exchanges the SO-DO-MI six-four chord for the positive SO-TI-MI substitute chord: we hear a TI note - in place of the accustomed DO (or, at least, a SO-TI-MI chord colours the dominant). This relationship is even more unequivocal in the motto theme of Falstaff: ’ We’ ll cuckold him - neatly!’(A minor-F major cadence). Fig. 195

In Verdi’ s music this most complex (and perhaps most beautiful) ’ colour chord’ (Fig. 194 above) is composed of a major triad, and two additional elements: the seventh note (A) ranging below the fundamental note, and the major sixth (G #) ranging above it. Between these two notes a major seventh relationship is established (A-G#). Historically, Verdi’ s colour chord developed from the ’ dominant’chord.

It is interesting to note that also Debussy favoured this type, for example in La Mer (we quote Movement III, No 44): the ’ acoustic’scale of the melody appears above a # B-D -A seventh chord (A=seventh note) - the top-note of the theme is G # (=major sixth of the root): Fig. 196

We grasp the importance of this colour chord if we call in mind occurrences like the ’diatonic’form of the fugue-theme in Bartók’ s Music for Strings, Percussion and Celesta (Movement IV, b. 204). The C ’ acoustic’scale is accompanied by two organ points: the major sixth above the root (A) and the seventh note below the root (Bb): Fig. 197

* The most powerful contrast is created by the complementary keys (See: pp. 31-33). After a dominant seventh (G7) we expect a C major or a C minor tonic. C major can be substituted by E minor, and C minor by Abmajor. The ’ command’motif in Tristan 7 ("Befehlen liess") is introduced by a dominant G and followed by the E minor and Ab major substitute chords - in place of C major or minor. The E minor and A b major triads, as we know, merge in a 1:3 model. Fig. 198

The opening theme of Bartók’ s Fourth Quartet is a true ’ challenge’to the aesthetes. Fig. 199

Bars 1-2 and 3-4 constitute a question-and-answer relationship. Therefore the end of b. 2 corresponds to the traditional dominant half-close cadence. (We note that in bs. 1-2 the viola part is silent and thus a contrast - a ’ split’- is produced between the violins and the cello.) At the end of b. 2, the last chord is a B minor triad in the cello (= the substitute chord of the dominant G major), and an Eb major triad in the violins (= the substitute chord of the dominant G minor). - As a result, the two substitute chords unite in a 1:3 model. The "Klang" character is determined by the fact that in this instance the 1:3 model consists of two separate augmented triads. Fig. 200

The same combination (Eb major + B minor) occurs in Movement II of the ’ Dance Suite’- at the appearance of the Ritornello! In addition, the theme itself (taking on a modal character) is conceived in G. Fig. 201

In Romantic music, the individual harmonies appear not as ’ atoms’but as elements of a ’ giant molecule’(a compound arrangement of tones resulting in an organic structure and unity) - where all particles are associated and closely interlinked; and as a consequence, every chord has an organic connection with every chord. Starting from a major melody, e.g. from Eb major, the substitute chord of the tonic is G minor - and its relative key is C minor:

Fig. 202

The question arises: which is the chord that could establish a contact between these two triads (i.e. G minor and C minor)? The ’ complementary’key of C minor is F b major - while the polar key of G minor is likewise F b major (= a difference of 6 keysignatures): Fig. 203

The melody quoted from Verdi’ s Aida consists of the four chords outlined above. Even the closing cadence - at the end of b. 4 - contains a G minor chord (and in addition, the polar Fb-Ab-D seventh chord also blends into the Bb dominant seventh). Fig. 204

Let us choose now a minor theme - e.g. C minor. In this case, the substitute chord is Ab major (= a negative substitute chord), and the relative key of Ab major is F minor: Fig. 205

This time, the connecting link between F minor and C minor is created by A major because the A major triad constitutes the complementary chord of F minor and the polar chord of C minor (= a difference of 6 key-signatures): Fig. 206

On p. 73 we quoted the Death-motif from Wagner ’ s Tristan (Fig. 123). The basic tonality is C minor. The structure of the theme is determined by the chords described above. In our first example the positive substitute chord, in our second example the negative substitute chord prevails; what is more, both examples start with the substitute chord of the tonic. In the Verdi example we find tonic and dominant, whereas in the Wagner example tonic and subdominant chords only. As a matter of course, this duality predestines the content, the character, the colour and even the ’ lifestyle’of the two melodies. Judging from the analysis of Romantic music, one might speak of the surprising frequency - one might even say preponderance - of substitute chords, which (taking into account the ambiguity inherent in Romantic harmony) reveals something characteristic of Verdi’ s or Wagner’ s ’ double spirited’ harmonies.

TONALITY: MODALITY: ATONALITY I am inclined to think that it was two linguistic elements which made possible the development of the dialectic system of Western music: the discovery of the ’ tensionresolution’ system of atonality and tonality - and the discovery of polymodal tensions. Bach is credited with sowing the seeds which determined the evolution of music for two subsequent centuries. The very first sentence of St. John Passion can give proof of this. When the ’ divine’element is set against the ’ devilish’ , what does Bach do but oppose the idea of ’ tonality’to that of ’ atonality’ : Jesus is represented by the nearest natural overtone, i.e. the perfect fifth, whereas Judas’ s symbol is the diminished fifth - an interval expressing the distance principle in the most direct way, because it symmetrically halves the tonal system (as well as the diminished seventh, which quarters the fifth-circle). Fig. 207

The characteristic ’ tension-resolution’system of Western music could scarcely have evolved without the antagonism of the tonal and the distance principles; moreover, this opposition - an age-old struggle between tonal asymmetrical and atonal symmetrical elements - became a seminal impulse in the development of European music. The characteristic ’ tension’chord of Baroque music Fig. 208

is represented precisely by that diminished seventh harmony which divides symmetrically the fifth-circle round the RE symmetry center (and RE-SI symmetry axis) of our diatonic system. (See. Fig. 214, first diagram on p. 110). Every key has

accordingly a ’ tonal’center and a ’ symmetry’center (e.g. in the E major key, the tonal centre is E - while the symmetry-axis of the E major scale is F# or C). However, the idea of consonance and dissonance is only one structural element of this tonal world. The other element is that tension-principle which is made possible by polymodal thinking. In St. John Passion, at the very first words of Jesus, we feel as though the very air has changed since Bach renders the miracle tangible by replacing the F major harmony with the 3 sharps higher D major: the same elevation arises here that caused the sound to ’ brighten’in our first Verdi example (Fig. 109 on p. 67). Fig. 209

* To give a summary, the main idea of Otello: the ’ kiss-theme’ includes in a concentrated way everything that has been said above. Fig. 210

It emerges literally from the ’ sea’ : the C#-E-G-A# tones. The modal structure of the theme would serve as a model example. In order to step from E major into the relative C# minor, or the relative A# subminor, we have to

move the fifth of E major - the note SO - to LA or FI (the note B to C # or A#). The opening bar of the kiss-theme is almost an embodiment of this principle. Fig. 211

The real driving force of the melody, however, lies in the polymodal changes namely in the potential differences between E major-E minor and C# major-C# minorC# subminor. Fig. 212

From b. 4 on (Fig. 210), the major and minor characters alternate periodically: the MI and MA in the bass change from bar to bar.

In b. 7

E minor is substituted by ’ C major’ , and by this means, it turns grave, In bs. 8-9 E major close - which on the other hand is rendered immaterial by the substitute ’ G# minor’colour (b. 8). These two substitute chords (G # minor and C major) constitute a 1:3 model - and, as such, annihilate each other’ s tonality due to the atonal character of the 1:3 model. The symbol of Otello’ s excitement: the DI rise (E#) also plays a constructive role in the theme: middle of the melody (b. 4)! Note the contrast between the FI (A #) in b. 2 and the FA (A) in b. 3. The FI-FA change is one of the most peculiar phenomena in Romantic harmony - of which more see: pp. 144-145.***) The middle of b. 4 contains a LA-DI-MI (C # major) triad, while b. 5 involves a DOMA-SO (E minor) triad: a difference of 6 key signatures has been reached. It has been frequently observed in Verdi or Wagner that the ’ fulfilment’is realized in the major sixth of the tonic (we have termed it the ’ pastoral’ sixth); this is accomplished in the culminating sixth bar of the kiss-theme, too! Madách ’ s

psychology applies to the gloomy ’ C major’following the fulfilment: ’ The price of the kiss’ s honey is in the dejection which comes after it ’ . NB: the MI-RE-DO motif rounding off the theme which (after the inserted C major) continues the pentatonic scale of b. 6, is the symbol of repose and smoothing throughout the work. The pentatonic scale in question (Fig. 213) creates a counterpole relationship with the pentatonic opening chord of the work (cf. Fig. 93 on p. 54): Fig. 213

Thus, the beginning and end of the opera reflect a pole-counterpole relation.

SUMMARY We have established: asymmetry is coupled with tonal, symmetry with atonal relations. The symmetry-center of our tonal system is constituted by the RE. If we divide the fifth-circle symmetrically around the RE central note - by way of halving we will obtain RE-SI and FA-TI tritones (see the first sketch below). - It is not by chance that the role of the dissonant ’ sensitive notes’is filled by the tritone TI-FA in the major key, and by the tritone SI-RE in the minor key. This is how the LEADING NOTES came into existence. PENTATONY in this system is represented by the five upper notes (second sketch in the Figure). From the 5-note system we arrive at the 7-note system by expanding pentatony by one degree upwards and one downwards: the extreme points (FA and TI) thus enter into a tritonic relationship with each other (third sketch), originating the ’ tensest’points of the scale. Therefore, the major scale (compared to the REcenter) fills in the upper arc of the circle of fifths. The missing notes result in pentatony. Thus the pentatonic and seven-note scales mutually complement each other. The ACOUSTIC scale (harmony) became a static ’ colour’chord because it lacks the two sensitive notes that characterize the major key: TI and FA (fourth sketch in the Figure). NB: in this sense, the major scale can be considered a ’ tensional’scale. If we omit the ’ atonal’degrees of our first sketch from the 12-degree system, model 1:2 is created: the basic scale of chromaticism (DO-DI-MA-MI-FI-SO-LA-TA eightnote scale). The two most static points of our tonal system: DO and MI coincide with the ’asymmetry’points of the system (fifth sketch in the Figure). DO and MI reflect a major third relationship. Fig. 214

Let us summarize the fundamental phenomena of polymodal chromaticism (this is how Bartók himself called his own style) - on the basis of the position the 12 degrees occupy in the system. If we group the 12 notes of the chromatic scale into symmetrical pairs (in relation to the RE center-note), the meaning of the individual degrees will be as follows: dynamic active elements (’ major’and ’ minor’ DI and MA — tension). FI and TA — static colour elements (cf. acoustic scale). In turn: DI and FI rouses a feeling of ascent, MA and TA rouses a feeling of descent. RE and SI — the symmetry centers of the system: ’ atonal’poles. TI and FA — the tension-points of the scale: sensitive leading notes. DO and MI — the most tonal pillars of our tonal system: the roots of the ’ DO’and ’ MI’system. As a result of the mirror relation, the DO-MI third is impressionistic in character, the MI-DO sixth is expressionistic in character. SO and LA — their meaning is determined by the fact that the ’DO’system, LA ® MI step downwards is the basic cadence of the ’MI’system. Among the modal keys, the

SO-scale represents the most elevated mode (hymnic), while the emotional depth characteristic of the minor key is reflected by the LA scale. Once more we may add that compared to DO: LA is an impressionistic ’ pastoral’sixth, compared to MI: SO is an expressive ’ pentatonic’third (the SO-MI motif is the most elementary manifestation of pentatonic tension). The four polymodal colouring elements - DI, MA, FI, TA - enable the following combinational possibilities. Jointly, MA+FI produce a passionate (=MA), but at the same time, an elevated (=FI) atmosphere. On the other hand, TA+DI embody a specifically Hungarian (oriental) attitude (TA = bent for ’melancholy’ , and DI = bent for temper and impulsiveness). The DI+FI together create a cheerful, even transcendental character. The MA+TA, however, bring about a sorrowful or languid character. Finally, the DI+MA bring about a polar tension (’ choleric’temperament) - while FI+TA represents balance: see acoustic (overtone) harmony. This study has attempted at defining the place of ’ relative solmization ’ in musicology. As far as I am concerned, I believe this method to be the missing link which might connect abstract theoretical research with living music. According to Eddington, the theoretical mathematician working with symbols of his own creation never knows what he is doing. It is the duty of the practical physicist to recognize that playing around with letters might reflect or describe such structure systems that could yield him knowledge about the existing physical world: armed with the abstract plan of structure he penetrates external reality once deemed unfathomable - the function and nature of which would, with traditional means, be beyond his grasp and comprehension. With the seemingly primitive symbols of relative solmization, I have attempted to draw the map of a musical world that manifests itself elementarily in these symbols provided we know what the operations concern. I consider Kodály’ s concept (based on our capacity to identify any pitch in relation to a tonal center) to be a clue which enables us to pull down that boundary which separates theory from everyday experience and leads us directly to the structure and contents of music; and at the same time, it is also suited to the historical analysis of the musical material. *) Concerning the reversed turn - minor second upwards - see Fig. 188 on p. 99 (ed.) **) Concerning FA-MI step, see p. 144 (ed.) ***) The fourth polymodal alteration (TA) is also present (D - b.4) (ed.)

THE ETHOS OF THE AXIS SYSTEM (tonal serialism) Comparisons are, by their very nature, delusive. Still, I would like to venture the following — comparative — statement: classical harmony is to modal harmony what a geocentric world concept is to a heliocentric one. Let us arrange the tone set of the Jupiter Symphony in a circle of fifths. In a traditional representation, the tonic –C – will be placed at the top of the fifth-circle, in the center of symmetry of the system. However, earlier we came to the conclusion and one glance at a piano keyboard (with the white and black keys) will make us understand that the axis of symmetry of the C major scale is not C... but D (or Ab), in relation to which every note of the C major scale has a symmetrical counterpart both upward and down. Of course, C major, with its zero key signature, occupies a special place in our system of musical notation. Moreover, as we can see on the second diagram below, F major with 1 flat and E minor with 1 sharp (or B major with 5 sharps and B b minor with 5 flats ) are symmetrical opposites in relation to a D center note. Wouldn’ t it be more appropriate, in a depiction of the circle of fifths, to place the D at the top of the system and the Ab at the bottom? We would thus illustrate the actual relationship between the notes: Fig. 215

Now let us replace the preceding fifths with their corresponding keys. In the upper half of the circle, the chords F major and D minor, as well as G major and E minor belong to the scale of C major (or A minor). Similarly, the chords Db major and Bb minor, as well as B major and A b (G#) minor all belong to the Gb (F#) major and Eb minor scales in the lower half of the circle. (See on next page.) The relative C major and A minor keys rest on the same notes and (as it appears from

Fig. 216 on p. 114) have identical weight! The symmetry thus obtained remains flawless even when the major triads are replaced by their parallel minor triads, and vice versa (for example, D minor and G major by D major and G minor respectively; D major and G minor will still occupy symmetrical positions in relation to the D center). The above figure will henceforth be referred to as the basic formula of our system. Fig. 216

the basic formula of our chapter The subject of our study will be Verdi’ s Don Carlos (we must be satisfied with a sketchy summary only, as a detailed analysis would require a separate volume). According to our basic formula the symmetrical counterpart of Bb major is B minor. Is it not striking that the opera begins in B b major — with 2 flats, and ends in B minor — with 2 sharps? Or, why is it that in the nocturnal garden scene the settling effect of the F major terzett is followed by Eboli’ s E minor revenge-aria? And why is the effect so convincing? As has been seen, F major’ s mirror is E minor. The dramatic turning-point of the dialogue between Posa and Filippo: Filippo ’ s confession in F minor, is likewise preceded by an E major play with ’ colours’ . The thought of Carlos’’ salvation’in Ab major (Eboli’ s aria) and Carlos’’ fall’in C# minor display a similar relationship (see our basic formula). Following Posa’ s Db major farewell, the revolution breaks out — strikingly enough — in Ab minor: in this system the counterpart of Db major is Ab minor. The clue to this tonal riddle comes from Verdi himself (he places, so to speak, the clue into our hands). The KEY-SENTENCE is sung by the ’ Monk’(whose disguise

of anonymity conceals Charles V bound for the monastery). The first half of this sentence is about ’ world cares’ , the second half about ’ heavenly consolation’ . The Cb major tonality representing earthly concerns is contrasted with the Bb minor tonality denoting heavenly affairs. The mirror image of C b major is (as can be seen in our basic formula) Bb minor: Fig. 217

Incidentally, it was Verdi in his late works who brought this system to the highest perfection. If we were to classify (to ’ catalogue’ ) the scales and themes of Don Carlos, we would discover not only the special meaning of each key, but also the multidimensional relationships between the keys. Let us take as an example a D minor triad, which has the following characteristics: symmetrical counterpart (in our basic formula): counterpole: relative major: substitute key:

G major Ab minor F major Bb major

complementary (annihilating) key: F# major parallel major (with the same name): D major polar major key (with a difference of 6 accidentals): B major functional meaning: subdominant it is the modal dominant of E, etc. Therefore, the meaning of a chord changes according to its relation with another harmony. In the case of counterpoles, a difference of plus 6 = minus 6 key signatures equalize, compensate one another — thus, these necessarily imply the existence of common traits. The upper and lower halves (or left and right halves) of our basic formula also conceal surprising symbols. Each of the tables below represents a specific tonal relation and, more important still, one of ’ content’ . Any one of the 24 keys can form 23 direct relations with the rest of the possible keys. Moreover, one is also led to realize that second- and third-degree relations perforce produce identical results — indeed, they even reinforce each other! In other terms, should we only know the ’ meaning’of 23 keys, we could deduct the meaning of the unknown 24th key from these relations.

Hence it follows that the system holds true only if every dimension is verified and confirmed by the work’ s dramaturgical content or poetical meaning. Our ’ serial ’ study — our musical Rubik ’ cube’ — as it appears in the tabulation below, is based on the five-act version of Don Carlos, 1886 (Ricordi Edition, 1982). The tonic is C major.

POLE-COUNTERPOLE RELATIONS C the center of the ’ physical’world, firm ground, the image of tangible major: reality, natural light (its condition of existence is musical ’ space’ ). the center of the supernatural world (the Church), the basic key of F# major: spiritual existence: the temple of Religion, stability and immovability.

the basic formula for ’ expression’ , it has an emotional charge (its A dynamics and tension is most often created by the course of music in minor: ’time’ ). the conveyor of mystical experiences, frequently expressing the loneliness caused by remoteness (religiosity cherishing feelings and E minor: illusions): yearning for eternity, longing to get away to another world (as Carlos’appearance in the Court Scene in Act II). b

C minor:

dark passion, despair, emotional revolt, opposition to the existing world order.

F# minor:

the forlorn hope of redemption, unrelievable and incurable heart-ache (e.g. the beginning of Act II).

serenity - caused by the ’ senses’(good taste, love of beauty, acceptance of all that is good and noble), enthusiasm: the delusion of the Veil Song A (’ the eye winks through the veil on the face of beautiful ladies’ ), just as major: Posa’s sensual manoeuvre (his A major romance in the Court Scene of Act II). humanity, benign sympathy and understanding (the reflection not of Eb active life but of contemplation), intellectual and human dignity, spiritual major: wisdom, friendship.

G major:

living force, vital impulse; desire for action, direct effect, often success.

Db major:

’fulfilment’through love - or through redeeming death (without any connotation of direct action or militant deed).

E visible greatness and strength, predominance of will, heavy Royal major: splendour, external festive pomp (Auto-da-fé Scene). if E major is the image of strength, Bb major is that of beauty Bb (occasionally, glitter of mundane finery which adheres to external beauty major: and stems from the senses), attractiveness caused by direct ’ impression’ .

G failure in one’ s vocation; setback, fiasco, defeat suffered in social life. minor: ill fate caused by outside violence (e.g. violent death): Eboli’ s fall, C# s resistance (Act V). minor: Posa’s funeral music, the breaking of Elisabeth’

E elevation: through physical weightlessness, bodilessness, the cessation minor: of constraints (e.g. social constraints). active elevation: through a transcendental (religious) experience; someBb times sacred anger: a sense of calling (with signs of aggressiveness in minor: the case of the Grand Inquisitor).

shadow effect produced by an active force (the dark colour appears in F the form of an effective and shaping force), it may also be the image of major: hard-won calm. human worries, renunciation of the vanity of this world or resignation; B major: the burden of bodily existence (e.g. the Cb major farewell duet).

a result achieved by volitional factors (in spite of something), or an D advantage obtained through rank (birth, privilege, cleverness, skilmajor: fulness); courtly manners belong here as well. Ab heroism at the cost of self-sacrifice, hymnic unfolding. major:

F

inward brooding, self-doubts, inclination for self-torture: the

minor: predominance of psychical aspects. events determined not by human intervention but by ’ destiny’and law; B unavoidable and unimpressionable - fateful - events (in a milder form: minor: the fixed order of courtly etiquette).

D annihilation - dead point: stemming from the absence of driving forces; minor: deadly and unconscious dream (Filippo’ s aria). annihilation through ’ burning’- turbulence, riot and destructive Ab minor: instincts (cf. revolution scene).

THE SYMMETRICAL CORRESPONDENCES OF THE SYSTEM C major:

the primary experience of existence, the visible world (perceivable with the senses).

A minor: the elemental manifestation of passion.

longing for the unattainable, yearning for mystical experiences: ancient Eb minor: and heroic, distant and mysterious - like the ancient Tasso-songs. F# faith as certainty: belief in God and spiritual existence. major:

G major:

positive and immediate effect, the spontaneous (active) manifestation of will and consciousness.

D minor:

passive quietude, inclination to melancholy, the silence of annihilation and destruction.

Ab minor:

the overthrow of the existing world order: upheaval, revolution, destructive forces.

Db major:

happiness (or even happy death), the ’ art’of ultimate love: emotional nobility and superiority.

F natural gravitation, intellectual depth or deep calm; often a shadow effect major: that reinforces a subsequent light-effect. physical weightlessness - or flight from reality (the latter will be found E in the form of spiritual drunkenness as well, as in Eboli’ s vow for minor: revenge in the garden scene).

Bb minor:

transcendental elevation, the ecstasy of the soul.

B major:

the giving up of worldly thoughts - through renunciation.

C blind passion, a feverish state: fanatic vehemence, challenge and burst of minor: fury (see "L’ ora fatale" preparing the finale of the Fontainebleau scene). A elation, rapture, devotedness, intellectual elevation. major:

Eb major:

spiritual harmony, goodwill and warmth, sympathy (sharing the worries of others), service for mankind.

F# minor:

spiritual sorrow and anguish: heart-sore; an emotional state of suffering (even the weight of an anathema).

G failure (ill-success), frustration, unfulfilled desires, defeat (conflict with minor: the law or with social conventions). meeting with success (e.g. a successful appearance at court), advanceD ment and good fortune in social life; triumph over one’ s self or over major: others.

an unselfish act, sacrifice raised to a ritual degree: the peremptory call of Ab humanism - which one must even be willing to die for (see the Peace major: Song in the Auto-da-fé Scene). violence exerted upon others, falling victim to arbitrariness, tragedy C# minor: inflicted by tyranny - deep mourning.

F self-torment or qualms of conscience, introversion; pensiveness, endless minor: worry, self-reproach. E

the external signs of Power: artificial light, pomp; energy and vitality

major: (also in the form of an excessive test of strength).

the magic of beauty (’ external’beauty), experience radiating happiness, Bb major: festive cheerfulness, love of live. a climate of tragedy: the constraining force of a coercive external power, B predestination: as ’ written in the stars’ ; inescapable destiny or strictly minor: regulated order (e.g. the strict rules of etiquette). Actually, our traditional method of notation also reflects the above principles (see the basic formula). Those keys in which the number of sharps and flats is identical occupy a symmetrical position in the system.

RELATIONS BETWEEN PARALLEL KEYS In the case of parallel major and minor keys one of the two keys always belongs to a family of ’ natural’keys, and the other to one of ’ modified’keys. C major, for example, differs from C minor in that the natural DO-MI-SO chord is altered to DOMA-SO. C static force; self-evident, popular naturalness. major: C rebellious defiance, furious temper, obsessed and vehement, minor: uncontrollable and capable of anything.

G major:

hope, life-force, self-consciousness; desire for action, success (the art of getting on).

G minor:

humiliation, miscarriage, social disgrace (loss of honour, abasement, branding).

D victory over somebody: the rising above the commonplace, or the rising major: above the instincts. D passivity, the lack or total exhaustion of life forces, oblivion, the longing minor: for a dream; fatigue caused by excessive burden.

A major: elevated spirits - where the voice has an ’ impressionistic’colouration. A minor:

passion or expressiveness - where the voice has an ’ emotional’ quality.

E size and weightiness: the power of the masses, luxury, heavy pomp, major: spectacular ceremony, authority (massive, weighty forces). E incorporeal hovering, weightlessness, aerial quality (but also a ’ raised’ minor: and edged voice).

B the sphere of worldly concerns, self-surrender, passive acceptance of the major: unavoidable, reconciliation with destiny. B inappellable and severe power (higher authority) that cannot be shunned; minor: in another form: the shackles of convention and court-manners.

F# major:

the temple of everlasting life.

F# minor:

the misery of earthly existence.

Db major:

fulfilled desire - salvation.

C# minor:

a broken existence (physical death and mourning).

Ab major:

redemption - through self-denial: absolution.

Ab minor:

perdition, burning out - by way of self-destruction; turbulence: rebellion against order.

Eb major:

exalted thoughts: love and compassion with belief in mankind, friendly devotedness.

Eb minor:

mystery, foreignness, longing to be elsewhere, loneliness.

alluring power, force of attraction (as a vehicle of a refined outward Bb appearance); vain beauty or loveliness that arouses desire; the joy of major: feast and celebration (cf. the Royal Hunt). Bb the appeal of the afterlife: irrational desires. minor:

F well-deserved repose after a time of worries and troubles, profoundity major: of ideas and thoughts. F spiritual conflict and pensiveness, inner motivation. minor:

RELATIONS IN CONTENT BETWEEN SUBSTITUTE KEYS

C major and E minor:

materiality and immateriality (often in the form of ’ spiritual drunkenness’as well).

the Beginning and the End: openness and restriction - the latter G major and exposed to fate or courtly etiquette. (Awakening to consciousness B minor: and tragic end.)

D major and F# minor:

A major and C# minor:

cheerfulness and sorrow (the glitter and the bitterness of courtly life).

sensual attraction and its breaking down (e.g. Posa’ s personal charms and his fall).

E major and Ab minor:

greatness and decay; domination and revolution.

B major and Eb earthly and heavenly philosophy; the renunciation of mundane desires and the magic of faith. minor:

F# major and static tranquility and high aspirations (in an extreme case a militant sense of calling, as in the scene of the Grand Inquisitor). Bb minor:

Db major and F minor:

redemption and remorse; sensual satisfaction and masochistic tendencies.

Ab major and C minor:

the hymnic and the angry word; constructive and destructive instincts.

Eb major and G minor:

Bb major and D minor:

dignity and indignity, human warmth and a state of being outcast.

impression and expression; physical beauty and spiritual agony; love of life and depression.

profoundity of ideas and spontaneous passion; the dark and light F major and side of the soul; gravitation caused by seriousness and the rising of A minor: emotions.

RELATIONS IN CONTENT BETWEEN COMPLEMENTARY (ANNIHILATING) KEYS

C major and Ab minor:

G major and Eb minor:

D major and Bb minor:

existence and non-existence.

sense of reality and imagination; self-consciousness and intuition; physical well-being and the longing for the hereafter.

persistent attachment to life and sacred sense of calling; thisworldly and other-worldly language (may figure as the contrast between courtly lustre and the ’ sacred wrath’ ).

confidence and doubt; external serenity and inner conflict.

A major and F minor: E major and C minor:

power and the challenge of power.

B major and G minor: F# major and D minor: Db major and A minor:

resignation and excitement.

the eternal order of theology and the longing for nonexistence. passive release (looseness) and active reaction; emotional elevation and passion.

awareness of one’ s vocation and intoxication: controlled and Ab major and uncontrolled action (may also be the contrast between devotion E minor: and revenge). Eb major and man-centered and fate-centered world; the opposition between the personal and the impersonal (humanity and inhumanity). B minor: Bb major and F# minor:

poetry of glaring light and of the night; happiness and death-wish.

F major and C# minor:

profoundity of thoughts and senseless violence.

POLAR MAJOR AND MINOR CHORDS The major and minor keys displaying a difference of 6 accidentals reflect the same contrast, in terms of content, as the pole– counterpole relations. These correspondences may be found in our first tabulation — if the connections below are considered:

FUNCTION TONIC chords reflect static motionlessness. This applies not only to the tonal pillars of C major (and A minor), or F# major (and Eb minor), but to the variations of these as well: the positive A major and Eb major (of LA- DI- MI principle) and the negative C minor and F# minor (of DO-MA-SO principle) are also characterized by the fact that they cannot be further developed — there is no way out of them. DOMINANT keys differ from tonic ones by the ’ active’force they exert (complying with the principles of the axis system). This is how E major became the symbol of Power, Bb major the symbol of Beauty, G major the symbol of Life force, Db major the symbol of Happiness. The dominant minor chords can engender a rise, like the E minor and B b minor keys, but a violent and tragic turn as well, like C# minor and G minor. SUBDOMINANT

chords

function

in

a

Ab major represents self-sacrifice, D major represents nobleness and dignity, F major represents calm and seriousness,

similar

manner:

B major represents spiritual peace. The minor subdominant chords are distinguished by their passivity: D minor is the key of sleep and nonexistence, Ab minor is the key of annihilating turbulence, B minor is the key of defenselessness, F minor is the key of doubt.

NATURAL AND MODIFIED KEYS The upper half of our basic formula (p. 114) makes up the diatonic scale (7-degree scale). The diatonic scale contains six perfect triads: C major and A minor have a zero key signature, G major and E minor have one sharp, F major and D minor have one flat. This is to say: the ’ highest’minor : E minor and the ’ lowest’major: F major are in fact the positive substitute chord of the C major tonic, the negative substitute chord of the A minor tonic. All of the above applies to the lower half of our basic formula as well (in which case F#=DO). If we change the ’ gender’of the keys, that is, if we substitute a minor key for its parallel major and a major key for its parallel minor, we may note that all ’ natural ’ thoughts correspond to unmodified keys and ’ stimulated’ emotional states to modified ones. Even in Eboli’ s revenge melody, E minor is associated with ’ natural ’ (i.e. instinctive) emotions — while E major radiates an artificial, ’ ’ intensified light (Auto-dafé Scene). In the same way, Ab minor denotes self-destruction, but Ab major suggest a heroic sacrifice at the cost of one’ s own life (as in the finale of Eboli ’ s aria or in the Peace Song of the Auto-da-fé Scene).

UPPER AND LOWER SPHERES The system is ’ closed’ . The upper and lower halves of our basic formula are mirror reflections of one another. This means that, for instance, F# minor is dialectically

related not only to F# major — but to the C major tonic as well. In the first case we perceive it as a DO-MA-SO chord, expressing ’ despair’—as in Elisabeth ’ s heartrending lament ("Ben lo sapete...") after the chest has been forced open. But in relation to the C major tonic, ’ F # minor’represents the most immaterial harmony to be found in Verdi’ s music: that is why we sense the FI-LA-DI formula —Elisabeth ’ s F# minor melody — as being so "immacolata" (as indicated in the libretto). Symbolically speaking, the upper and the lower halves of our basic formula relate to each other as the empirical Aristotelian world to the Platonic one — to the world of Ideas.

THE POTENTIALITIES OF THE AXIS SYSTEM A similar ’ dictionary’may be compiled from the relations between relative major and minor keys — or what is even more fascinating, those between axis-related major or minor keys. Let us cite only one example here. The cheerful Bb major of the Royal Hunt in Act I is transformed into D b major: into a love-scene. (If Bb major is a DO-MI-SO tonic, then the D b major must be interpreted as MA major!) — In Act II, too, the finest moments of the CarlosElisabeth — duet are b b marked by a B major— D major turn. In the nocturnal garden scene it is just the opposite what happens: Carlos and Eboli –hidden behind masks –declare their ’love’to each other in Db major; but at the moment of unmasking, we immediately return to Bb major!

OVERTONE RELATIONS According to the classical definition, the major triad consists of the 3 closest overtones of a basic note — while the minor triad is made up of the 3 closest basic notes of a common overtone: Fig. 218

This alone helps us understand why C major represents the external (visible) world and F minor the internal (invisible) one. Using the above principle as a new basis, we may discover a new dimension of our tonal system (which also corresponds to the ’Western’ and ’ Eastern’ ways of thinking):

C major and F minor:

the empirical and the psychical world (Ccenter).

G major and C minor: affirmation and denial (G-center). D major and G minor: bright cheerfulness and dark passion (D-center). A major and D minor: wakefulness and sleep. E major and A minor: impersonal greatness and personal feeling. B major and E minor: immobility and desire to act. F# major and B minor: spiritual haven of refuge and blind fate. Db major and F# minor: fulfilment and longing (Db = C# center). Ab major and C# minor: unfolding and failure (Ab = G# center). Eb major and Ab minor: constructive (humane) and destructive forces. Bb major and Eb minor: sense of beauty and sadness. F major and Bb minor: low and high

WIE LENKT’ICH SICHER DEN KIEL (modal serialism) The difference between ’ tonal serialism’and ’ modal serialism’lies in the fact that in the former the variety and diversity of the 24 major and minor chords are exhausted, while the latter exploits the possibilities of the 12 chromatic degrees. The ’ exposition’of Act I in Wagner’ s Tristan brings into focus the following sentence:

Wherever I stand, faithfully I serve her, the glory of all women; were I to leave the helm just now, how could I safely steer the ship to King Marke’ s country? (Act I, Scene 2) Fig. 219

Our analysis has been centered around five moments:

(1) "Auf jeder Stelle wo ich steh ... " Where is Tristan standing? Above the maelstrom: in the gate of hell, so to speak (bs. 1-4). (2) And where is Isolde standing? High above Tristan in social rank. Her name is "Frauen höchster Ehr'" (bs. 5-7). She is to be King Marke’ s bride. (3) Tristan’ s spontaneous reflex: he has to pull down Isolde’ s pride. That’ s exactly what b. 8 implies with its slap-like MA major chord (Eb). (4) Tristan strains every nerve (b. 10) –’ the colour of his face changes’(b. 12). (5) Where can this road lead to –and where is Tristan guiding the boat? Over a mystical "Styx" –to the land of the dead. Let us consider these points one by one. (1) The MI-FA step has the pressing force of a steam boiler. Wagner begins the melody with A minor and its negative substitute chord: F major. The precipitous fall at the beginning of the Death motif was also effected by the motif bursting forth in the negative substitute chord, in place of the tonic. The ’ balance’tips over at the meeting point of the two worlds –in b. 3: the transition is marked by the two ’ atonality points’of our tonal system, RE and SI = D and G # (and FA-TI = F-B). In bars 2-3 it is already the C minor that we sense to be the tonic –which means that the third bar (dominant diminished seventh) should be continued like this: Fig. 220

Nevertheless, it resolves in a polar way to A major (six accidentals away), entering thereby into a new sphere (b. 4). The C minor triad manifests itself openly as well in b. 9! (2) And conversely, the musical analogy of "Frauen höchster Ehr'" is the upshooting FI-SO (the sweeping dash of notes F#-G in bs. 5-8). Wagner further heightens the light effect by raising bs. 5-7 to the dominant (G). Stripping all the fringes and frills off the melody, we find that the two surprises in the first line are produced by: the F minorA —major complementary keys (the ’ colour of Tristans’ face changes); ’ the A major and Eb major + C minor (bs. 8-9) polar chords (the symbolic ’ slap’ ); the two areas are hallmarked by DI and MA! All this is extended by another two elements. The strophe sets out from A minor and its negative substitute chord, F major. This itself represents a big charge of tension. But Wagner does not stop here: he submits the two chords to another ’ load test ’by converting A minor into A major (b. 4) and F major into F minor (b. 2). And with this

he creates the possibility for a ’ metamorphosis’ , since the F minor + A major triads are complementary keys, annihilating each other (producing jointly a 1:3 model), as indicated in Fig. 221: Fig. 221

These chords act like litmus paper, which changes colour according to the acidity or alkalinity of solutions: the change of colour in b. 4 is eloquent proof of this. Thus, if we combine three principles: the substitute relationship: A minor and F major the parallel major and minor relation: A minor- A major, and F major- F minor the complementary relation: F minor and A major (resulting from the above connection) then the tonal structure of the theme can be clearly seen. In addition, the end of line 1 (C minor) and that of line 2 (E major) also reflects a complementary (annihilating) relationship. (See: Fig. 219 on p. 127) We note that also the F minor chord (b. 2) rhymes with the D major chord (bs. 5-7) in a polar way. (3) "Frauen höchster Ehr'": b. 8 gives away that the praise actually conceals ’contempt’ . As if Tristan wanted to push Isolde off the throne! We hear a two-step negative cadence: G minor - in place of G major, indeed, its deceptive cadence! This is how the word "Ehr'" acquires a rough and brusque tone (E b major = MA major), instead of reverence. It has repeatedly been pointed out that the deceptive cadence of the minor dominant is tantamount to MA major (Eb major), which is suggestive of sinister passions. And (typically of Wagner) even the deceptive cadence of the deceptive cadence is employed: C minor (b. 9). (4) The second line of the melody (from b. 10) repeats the first line a major second degree higher; this turn is known to represent the most powerful form of intensification: Tristan raises his voice! [NB: A major second rise = a change based on the modal dominant-tonic principle.] Bar 4 emerges as if the colour of the characters’faces had all of a sudden changed. The litmus effect is elicited again: G minor (b. 10) and B major (b. 12) are complementary (annihilating) keys. The four analogous chords (bs. 2-4 and 10-12) encompass all twelve degrees of chromaticism: Fig. 222

The divergence of the first melodic line rests on the two-directional pull of the FA and FI. Similarly to the opposite attractions of the FA and FI, TA tends downward – while TI lures upward. This TA-TI (Bb-B) turn speaks for itself in bs. 11-12. The character of the E subminor (b. 10) is determined by the TA, and that of the E seventh chord (b. 13) by the TI degree. (Moreover, E subminor and E major are a polar distance apart: the monologue is propelled by an expansion of 6 accidentals.) The formula preparing the dénouement is one of the oddest achievements of Romantic music. The ’ TI major’chord (B-D #-F# in b. 12) borrows its peculiar quality from the MA and FI notes: D# (=Eb) gains a MA colouration, while the note F# a FI character. The chord couples the mortal ’ human’element (MA) with the uplifting ’ spiritual’element (FI). It is small wonder that the chord ushering the melody to its ’ destination’is the B major chord at issue. We have arrived at a point where Tristan’ s famous sentence "Wie lenkt’ich sicher den Kiel zu König Markes Land?" is uttered. (5) I can’ t be far from the truth claiming that bs. 13-16 condense the gist of the Tristanian "Lebensgefühl" (’ experience of being’ ). Wieland Wagner appropriately remarks that Tristan’ s boat is headed towards the realm of the night –towards Nirvana –over the "Styx". The Oriental effect of the two final chords is due partly to the Phrygian mode (bs. 1516), partly to the six-four tonality of b. 13, but first of all to b. 15, which simultaneously condenses the A minor + F major triads (known from the starting bar): it unites the tonic minor chord with its negative substitute chord (F major), as seen more clearly below: Fig. 223

This is the very moment when stage action metamorphoses into mythology: the ’ sea ’ becomes perceivable in its full depth. With A minor we have retraced the circle to the starting point. The first half-sentence ends with a deceptive cadence (dramaturgically, too) as it brings the process to a halt; the second half-sentence remains open on the dominant. Consequently, the C minor (b. 9) and the E major (b. 16) enter into a complementary relationship again. The main conclusion for us to be drawn from this analysis is that Wagner uses the 12

degrees of the chromatic scale the way the dodecaphonists used the 12 notes of the "Reihe". Each degree has its exact, logically defined place in the row: besides the tonic notes LA-DO, the degrees FA-MI and FI-SO (bs. 1-8), the atonal RE-SI tritone on the borderline (b. 3), the notes DI (b. 4) and MA (b. 8) in polar opposition, then the TA and TI rise (bs. 10-12). The modal melody exploits the potentiality of the set of tones to the full. This is what accounts for the indivisible unity of the theme, and first and foremost for the organic cohesion between the tones. I am not the only one to have been struck by the ortographic ’ beauty’and tidiness of Wagner’ s score (so much in contrast with the emotional turmoil and ’ debaucheries’of the work). The analyst tends to conclude that the score of Tristan –a coliseum of notes –could never have been erected without the help of an infallible ’ absolute pitch’ . It must be one of the many contradictions that Wagner did not have absolute pitch. This again (i.e. the identification of any pitch in relation to a tonal center) underpins the argument: it is the modal content of the 12 degrees in which the vital principle of Wagner ’ s music inheres.

COMPUTER LANGUAGE OF MUSIC It was the introduction of the computer that brought about the most unexpected turn in these analyses. In 1983 I compiled a program, making use of the simplest combinations and permutations found in Bartók’ s music. After running the program, the computer ’ dictated’—to our no small surprise —well-known melodic and harmonic passages from Tristan, Parsifal, Otello, and Boris Godunov. We must be content with some basic operations. In my programs no more than 3 numbers and 3 letters are employed. Each number or letter tells us something profoundly interesting and new about music and its perception. Number 1 indicates a perfect fifth. We mark the symmetry center of our tonal system (i.e., RE) with letter Z –while the root of the DO system is indicated by X, and that of the MI system by Y. If Z=0, then as shown in Fig. 224: Fig. 224

X = Z - 2 and Y = Z + 2 The difference between X and Y is a major third: of all the equidistant scales, the augmented triad (major third + major third) is the only one in which the number of notes (three) cannot be divided by 2! Oddly enough, the symmetry center (Z) marks the ’ point of atonality’ . In the axis system, besides degree RE (=Z), there is to be found one more symmetry center –and this is the tritone of RE: the SI; in C major, this is the G # (=Ab) note. In the language of geometry, we have: Z + 6 = Z –6 Naturally, in the case of modulation (or the choice of a new key) the value of Z changes. The three notes of the major and relative minor triads show an inverted relationship: Major: X X + 1 Y Minor: Y Y –1 X Number 1 is an important element here, because it determines the tonal character of the chord (being a perfect fifth). Both X and Y are included in the tabulation above. Number 3 expresses a modal change –according to the fact that in the axis system a modal change implies a difference of three key signatures. Let us take the simplest case: LA-DI-MI and DO-MA-SO: DI = Y + 3 and MA = X –3

Logically, if number 3 is related to the Z center (RE), it signifies a ’ dissonant ’note (=sensitive note): Z + 3 = TI Z –3 = FA The tritonic relationship between the two notes is expressed in the difference of six perfect fifths. Note the ’ outward’and ’ inward’acting force functioning in TI and FA, respectively. It is no accident that in the subdominant D minor and dominant G major the D note (=Z) plays the role of the common note. In the subdominant chords we find a FA note (Z –3) and in the dominant a TI note (Z + 3). Moreover, the subdominant chord involves a LA note (Z + 1), while the dominant chord a SO note (Z –1). And because as indicated in Fig. 225 Fig. 225

LA: Z + 1 = Y - 1, and SO: Z - 1 = X + 1 we see that these two notes provide for the ’ connecting link’between Z and X on the one hand and between Z and Y on the other. That is, these two notes make the connection possible between T– S and between D– T, respectively. The formula of the positive and negative substitute chords TI - MI - SO: Y + 1 Y X + 1 FA - DO - LA: X –1 X Y –1 coincides with the psychological observation that we compare the former with point Y (see the role of Y and Y + 1 in the first chord) and the latter with point X (the second chord includes X and X - 1). Complementary keys express their ’ annihilating’quality in figures as well. As we know, the complementary key of C major is Ab minor — while that of A minor is C# (Db) major. In both we find the Ab (G#) note, that is, the symmetry center of our tonal system (Z + 6 = Z –6). In the Ab minor complementary key the sensitive note TI (Z + 3) manifests itself, whereas in the C# major complementary key we find the sensitive note FA (Z –3). The most interesting is, however, the role of the third element: in the ’ negative ’ b complementary key (A minor) the MA figures (i.e., X –3), whereas in the positive complementary key the DI plays the same role (Y + 3).

* We give one single example. Let us harmonize an A minor melody with its relative major harmony, C majorand — these, in turn, with their substitute chords: F major and E minor, respectively (See: Fig. 183 on p. 96). If these triads are interchanged by their parallel triads (i.e., E minor by E major, C major by C minor, A minor by A major, and F major by F minor), the symmetry remains untouched, as shown in Fig. 226. Fig. 226

The difference between C minor (with three flats) and A major (with three sharps) is six accidentals and reflects a polar opposition. On the other hand, F minor and A major are complementary keysannihilating — each other. Similarly: E major and C minor reveal the same relationship (the two triads result in an 1:3 model.*)

*) F minor and E major are chords with a common third - see: p.140 (ed.)

THE SPRECHER SCENE OF THE MAGIC FLUTE When I was to give a lecture on the computer analysis of music, my colleagues, having been asked to give me a ’ subject for examination’ , chose The Magic Flute’s Sprecher scene. The question was whether the ’ artificial brain’could add anything new or astonishing to the known interpretations of the work. This analysis is no more than the result of a comparison between the computer-made ’ map’and the libretto. What can a computer do? Its inventor, the Hungarian-born John Neumann said: ’ the computer is the most stupid animal — but it works terribly fast.’To put it more precisely, the computer is capable of what the human eye is useless for: it can find a needle in a hay-stack. For some time it has been recognised that the sound dramaturgy of music can only be partly understood through the rules of classical harmony. Therefore we know that instead of conforming to the accustomed rules of music, Verdi ’ s or Wagner’ s operas are rather like organic cells where every component is organically related to every other element. Earlier, we sought the logic of how one element is born from another one: what logic governs the succession of chords. At the historic moment of abandoning the heptatonic system, the tonal system immediately became multi-dimensional: so complex and indecipherable for the human mind that without computer programs it would be hard to give an answer to such seemingly simple questions as to what the true meaning of the dialogues in The Magic Flute is. Let us start our investigation with a few ’ test-bars’ . Mozart constructed each section of the memorable dialogue around a dramatic ’ breaking point’— which also determines the tonal structure of the scene. The major and harmonic minor scales differ from each other in two tones: the minor scale contains the minor third and minor sixth (instead of the major third and major sixth). For example, modifying the D major by ordering a minor third and minor sixth to the root D (i.e. the notes F and B b), we obtain a Bb major triad which (compared to D major) gives the impression of the most violent ’ major-minor ’contrast. In the scene, the sharpest contrast comes with the command "Zurück!" sounding from behind the temple gates: it all but gives the feeling of a physical blow: the D major harmony is knocked down, shattered by a Bb major chord.

It is worth noting: a similarly sharp major-minor contrast is generated by the Ab major chord within the C major tonality. A scenic and a tonal surprise appear in each section of the dialogue. The first genuine astonishment accompanies the sighting of the temple columns. Earlier, the ’ effect mechanism’ of modal thinking was explained in the following way: with each coming chord a comparison is made between the chord which would be expected, according to the natural logic of music, and the chord which actually occurs. The tension difference between the two determines the message, the meaning (modal quality) of the music. What happened now was the A major dominant being followed by a D minor tonic: "Wo bin ich? was wird mit mir?". Now the A major dominant is unexpectedly followed by F# major (instead of the D minor): Fig. 1

That implies at least three consequences: (a) F# major — in the context of the C major scene — performs the role of the ’counterpole’ . (b) More important still, after the dominant A major the F# major a minor third lower registers a 3-sharp rise; to put it in another way, an ’ axis-rise’takes place in # the positive direction (F major also appearing as a dominant). As was said, the inner elevation captures the moment when Tamino catches sight of the columns of Sarastro’ s temple.1) (c) It must not be forgotten, however, that the F # major appeared instead of D minor, and significantly, D minor and F# major are complementary keys: they ’destroy’each other. As is indicated by the Allegro theme, our hero gets into a state of excitement. That is how we arrive (via the resolution of the F # dominant) into the B minor key which depicts Tamino’ s character in vivid colours (Allegro). What do this B minor and the following E minor key mean (forte: "mutig zur Pforte hinein?") As we know, in Romantic music the greatest contrast is created by complementary keys that

neutralize each other. The Sprecher’ s Ab major utterance followed by the Eb major reply (it is still the Sprecher who strikes the tone of "Heiligtum") constitutes such a contrast compared to B minor and E minor that it has no match in expressiveness even in Romanticism. E minor and Ab major, as well as B minor and Eb major are complementary keys: they extinguish each other. Has it ever struck anyone why Bartók gave E minor to the right hand and Ab major to the left in Burlesque No.2 (or in the piano piece See-saw)? Or why he put the movement II. of the String Quartet No.4 in ’ E minor’and its b variation — movement IV — in A ’ major’ ? (Later it will also come to light why the former assumes a ’ Phrygian’ character, and the latter, necessarily, a ’ Lydian’ 2) character.) Let us play Tristan’s love duet in C major and we shall immediately get a hint of the electric charge in the contrast between B minor and E b major. The major tension of G major can be hightened by B minor (=positive substitute chord), the minor tension of G minor by Eb major (=negative substitute chord).3) Fig. 2

On the one side there is Tamino’ s youthful passion, spontaneous, almost militant bravado: the challenging activity — a combination of physical strength and a keen mental interest, fresh receptivity. We must feel from our hero's enraptured and incalculable outbursts, his enthusiasm, his brave resolution, that he would not balk at the obstacles in his path. — On the other side there is the world of the Sprechers’ mature age, the attitude of thoughtful and profound wisdom. The always predictable spiritual power draws on a deep experience of life — while bridled passion hides ascethic perspicacity. At a superficial glance, B minor and E minor might suggest a ’minor’world, and Ab major and Eb major a ’major’world —but actually, it is the other way round. Just like in the above example, B minor appears in the C major scene as the substitute chord of G major of the 5th degree (cf. "das Laster nicht leicht") — positive substitute chord, we must add. Similarly, E minor occurs as the (positive) substitute key of the tonic C major (the resolution of the former G major chord is not C major; instead, the monologue shifts toward E minor). On the other hand, the Ab major chord indicating the Sprecher’ s entry upon the stage is introduced as a ’ deceptive-cadence’ , taking the place of C minor. As a result, Ab

major appears as the negative substitute key of C minor. What, however, calls for some more words is the perception of E b major (first of all) as the substitute chord of the minor dominant — G minor — and not as the relative key of C minor. An example from Romanticism comes to mind. In the Prelude to Parsifal (at the second entry of the theme, see Fig. 42 on p. 30) Ab major appears as the suspension of C minor while the E minor answer motif is enveloped by C major. It can clearly be seen that Fig. 3

the substitute chord of C minor is Ab major, while the substitute chord of C major is E minor. To close the circle: what gives the pith and marrow to A b major and E minor — the negative and positive substitute chords — is that the two neutralize each other because they enter into a complementary relationship. As for the relationship between B minor and Eb major, B minor replaces the dominant G major (positive substitute chord), while Eb major substitutes for the minor dominant G minor (negative substitute chord; earlier we said that Eb major = the deceptive cadence of G minor). Returning to the Tristan love duet it should not pass unmentioned that although the second step of the sequence is identical with the first, it is heard like this: Fig. 4

Here, E minor and Ab major are none other than the substitute chords of the tonic C major and C minor, respectively. I often used to ask why Bartók insisted on the E minor and Ab major keys? The central turn of Cantata Profana — "A fáklyák már égnek..." [The candles are all lit...] — was clearly conceived in C major, yet the melody is accompanied by E minor and Ab major chords alternating from bar to bar. One colours the motif as the positive substitute chord of C major, the other as the negative substitute chord of C minor. And what weighs most is that the two are united in a 1:3 model, that is, their interrelation is complementary.

Fig. 5

In the second movement of Bartók’ s Dance Suite, the Ritornell melody in G is similarly buttressed by B minor and E b major chords as the substitute chords for G major and G minor. (See Fig. 201 on p. 104) What marked the most significant turn in the development of classical heptatonic diatony was the birth of the SI degree as the leading note of the minor scale (upon the model of the major scale). The ’ fermentation’of diatony, the Romantic slackening of the tonal system was indeed caused by the difference between SO and SI. A palpable example of this process is the slow movement of Beethoven’ s 7th Symphony (in A minor) in wich the C major triad appears as the substitute chord of the minor (E minor) dominant, and not as the relative key of the tonic A minor. This is aptly shown by the refrain of the theme: Fig. 6

In the first four bars the G # note means SI, in the next four bars G means SO. The SO note gets a separate leading note thanks to the FI-SO (F #-G) melodic step. The difference between SI and SO gives the impression that the character of SO is far more ’ clouded’ , blunter (darker), more sensitive. The same takes place when the refrain melody is repeated (in bs. 15-18): Fig. 7

in bs. 15-16 we hear SI, but in b. 17 the more ’ cloudy’ SO appears again. It can be regarded as a rule that moving from a minor to a relative major means progressing in the ’ negative’unnatural , direction, because the leading note (SI) does not resolve upwards, but unnaturally downwards (towards SO). In Tristan’s love motif, too: the B leading note, instead of guiding us to the expected C, goes downwards towards the Bb note. Fig. 8

Let us return to The Magic Flute. In the dialogue the first rhetoric contradiction — also in the libretto —is the emphasized word " allein", which might be translated as ’only,...’. The Sprecher ’ raises’ his voice: but no matter whether the key is interpreted as Eb major or C minor, the Bb note has the meaning of SO and the B note means SI: Fig. 9

Mozart puts his finger on the main feature of the mentioned ’ fermentation ’ , realizing that the birth of SI entails the possibility of the augmented triad: the Sprecher sings an augmented triad melody, Fig. 10

and this possibility is exploited by Mozart at other points as well: think of "Zurück!" Fig. 11

Reference must be made to an essential connection between chords with a common third, of which the classical study of harmony forgets to mention the most important fact: namely, that (in the overwhelming majority of cases) chords with a common third appear as the VI. degree — as the VI. degree of the major and minor scale. The VI. degree of C major is A minor and the VI. degree of C minor is Ab major. The common third (the note C) is identical with the key note. It is not the most effective way of pitting a major against a minor to put C major opposite C minor, but to have their VI. degrees contrasted: A minor and Ab major, just as in the following Brangäne-Tristan dialogue. When Tristan (and with him the A b major) appears, he almost ’ loses his mind, his consciousness’(though in an ironical sense here): Wagner presents a

genuine ’minor effect’to us: Fig. 12

As for the Sprecher’ s scene, the minor VI. degree and the major VI. degree determine the formal outline of the entire dialogue: the Sprecher enters the stage on the minor VI. degree (with an Ab major harmony) and when he exits, Tamino arrives at the major VI. degree (A minor key). This is what carries the liberating thought: VI. degree of are chords with VI. degree of C major: A minor

C a

minor: Ab common

major third.

Mozart undoubtedly reserved the most intriguing harmonic event to the development. It cannot be accidental that the greatest performers like Toscanini and Bruno Walter placed this moment — "Man opferte vielleicht sie schon?" — into the focus of the plot dramaturgically as well. Already the polar turning-point indicates that Mozart is to ’ try out’his most daring harmonic effect on us: D minor and B major are separated by six accidentals, just like the keys a tritone apart (e.g. F major and B major). Fig. 13

But let us go back to an earlier point. (Formerly, I devoted a detailed study to this problem.) In a major tonality the tonal centre is DO. That causes the predominance of the TI-DO leading-note steps. In minor keys, however, it is not necessarily LA that constitutes the tonal centre, but much rather the note MI: the core of a melody mostly comprises FA-MI steps (think of the main idea in Mozart’ s great G minor Symphony, or the main theme in Beethoven’ s Appassionata sonata: Db-C turns in bs. 1-24). In our tonal system, the symmetry-pair of the TI-DO leading note step is the FA-MI step; Mi is the mirror image of DO. 4) The note MI (note E in A minor) can most effectively be approached from two sides: from the directions of FA and RI (F and D #). What is the ’ augmented six-five‘

chord? It is a special chord typical of the minor tonality obtained when the V. (MI) degree is prepared not simply by the IV. degree, Fig. 14

but the MI centre is approached from both sides with chromatic steps: with FA-MI and at the same time RI-MI ’ direction notes’ , (that is, with F-E + D #-E turns). In such a case this tight harmonic relation will almost produce a ’ physiological ’effect, as it were! Fig. 15

Between each statement of the three-times repeated motto-theme ("Sobald dich führt..."), which in terms of form expresses the attainment of the goal, the augmented six-five chord occurs, lending special weight to the MI centre. The chord first accents the word "Licht", then underscores the sentence "saget mir: lebt denn Pamina noch?" Fig. 16

The nucleus of this harmony — F major — is the VI. degree of the tonic A minor. B major preparing the development is, on the other hand, the counterpole — the tritone — of F major. The model of the polar turn must be traced back to the Neapolitan chord. The Neapolitan chord and the V-I step following it serve to circumscribe a tonal centre note, with semi-tone steps, to boot; in A minor key, the notes B b and G# chromatically head towards the note A: 5) Fig. 17

By the way, we know from earlier experience that the B b major Neapolitan chord is nothing else but the substitute chord of the subdominant D minor. The above quoted polar turn ("Man opferte... sie schon?") is based on a similar

attraction — proved unquestionably by the renderings of Toscanini, Bruno Walter or Karajan — : the melody falls over the central E note. Fig. 18

This time, too, FA and RI play the role of the ’ leading notes’from two sides (I prefer to use the term ’ direction notes’ : the meaning of both FA and RI is expressed by the sphere of attraction of the MI centre). However, the circumscription of the note LA cannot be missing, either. Before the outburst of the ’motto-theme’repeated ritually three times ("Die Zunge bindet Eid und Pflicht... schwinden") the notes Bb and G# perform a similar task: 6) Fig. 19

In terms of taxonomy, the ’ symmetry-pair’of the G major dominant seventh is the BD-F-A subminor chord. The character of the latter is far more lyrical and melodious (’ singing’ ): thanks to the LA note. Instinctively, that is how it became the backbone of the repeated motto-theme: Fig. 20

The dialogue has a recurring ’ leitmotif’which owes its existence to the directionnotes — precisely to an element that is the opposite of the leading note step. Let us do ’violence’upon the leading note of the V. degree (e.g. the note B of G major) by modifying the note B to Bb (instead of the tonic resolution): that is, let us have the major third (B) replaced by a minor third (B b). This charges the note Bb with tension, turning it into a dissonant element, a ’ direction note’requiring further leading — towards D major. From that point onwards the function of the dominant is not performed by the G major but by D major (= the secondary dominant: the dominant of the dominant). The task of the tonic belonging to D major (i.e. G minor) is simply to restore the equilibrium — as it was expressed by a great conductor in connection with "Ja, ja, Sarastro herrschet hier". Fig. 21

Following the onerous admonition "Tod und Rache dich entzünden" (concealing an augmented six-five again), Tamino surfaces with the G major dominant: Fig. 22

The old priest suddenly tones down his voice: instead of G major we hear G minor and the continuation conforms to what was described above. The G minor and D major chords shown in Fig. 21 constitute a symmetry-pair, mirroring each other taxonomically. In this sense, the mirror image of the note Bb (TA) is F# (FI). It would be redundant to mention this, had Mozart not made use of its opposite upon the return of the motif. This time the emphasis is laid not on TA but on FI, which immediately gives the motif a ’ challenging’ , provocative character: it becomes a threat to murder, striking the key of passionate protest, of indignation. The key is F major (or F minor): "das ist mir schon genug". Were this to occur in a classical oratorio, the homophonic melody would be answered in the following way: Fig. 23

What accounts for the sharp ’ challenge’is the replacement of the C major domi-nant by C minor (the passionate minor dominant), and just like earlier, the natural continuation is in G major — where the note B means FI, striking the tone of danger and threat. Fig. 24*)

The scale-degrees FA and FI have a distinguished place even among the ’ directionnotes’ . FA pulls downwards (to MI), while FI pushes upwards (to SO). This is a very modern idea in dramaturgy. The introductory part of the dialogue is tied to C major, its middle part to C minor and its related keys, while the closing section returns to the

keys without accidentals: first to A minor and finally to C major. The question arises what scenic moment elicits the reprise. In Tamino, the image of the "unglückliches Weib" (disconsolate mother) evokes the tragic G minor. For the old priest, however, the "Weib" means something quite different. At first one might think he is only envious (that is why the sarcastic intonation) but it shows through the music more and more apparently that she is the opponent, or even the ’ enemy’ . Isn’ t it peculiar that "Ein Weib" as well as the image of the enemy is associated with the C major chord? Let us take a closer look at what is happening here. In G minor the Eb–D steps give the impression of emotional FA-MI steps; the guiding thread of the harmonies is made exclusively of E b–D steps (earlier I pointed out that FA– MI is the ’emotional’ , introverted element in music): Fig. 25

As against that, when the Sprecher begins to speak, — what a turn! — FA is replaced (ousted, to be precise) by FI, which ushers the dialogue towards SO: Fig. 26 (Play before it the former example!)

In connection with the mentioned G minor, Bb7and Eb major turns let me refer to another idiomatic turn also behaving like a leitmotif. In my analyses of Verdi and Wagner I termed this element the turn-motif. In Mozart’ s music the role assigned to it is to give emphasis (emotionally charged emphasis) to a word. When, for instance, the root of the A minor chord is raised a semitone higher, that is, modified to B b (NB: the modification giving accent to the chord), a major seventh (C7) is gained which automatically leads into the substitute key (F major). Fig. 27

That’ s how we arrive from the first "Zurück!" to the second gate, from G minor to E b major: Fig. 28

Later we move from the G minor of "Sarastro herrschet hier" to E b major in the same way: raising the basic note of G minor gives edge to Tamino’ s violent "nicht". Fig. 29

The sentence "Er ist ein Unmensch" is stressed by a similar motif (G minor, B b7, Eb major). Fig. 30

The same takes place after "Erklär’dies Räthsel" (E minor, G 7, C major; the raising of the root note falling on the sentence "täusch mich nicht!"). 7) Before sketching the tonal structure of the scene, let us remember an analogy: Tristan’ s ’ dream chords’ (Brangäne’ s first monologue in Act 2: "Einsam wachend..."). Each group of chords springs from C# major — and the nadir is reached # when C major is followed by A minor and E minor triads: Fig. 31

In the opera C# major signifies the ’ mother’ s lap’(the womb), the ’ dream’ , the ’night’—the Nirvana. The A minor and E minor nadir presupposes the renunciation of this, too: after C# major A minor establishes an annihilating relation (not to speak

of the fact that the leading note within the C # major chord [E#] is resolved unnaturally downwards), while E minor represents a polarly distant relationship with C# major. If in the cited dialogue of the Magic Flute D major symbolizes the ’ gate’of Sarastro’ s temple, the concept of the ’ non-gate’—the moment when Tamino gets ready to leave disappointed and ’ renounces’the gates —will be represented by Bb minor and F minor, according to the above logic. D major and Bb minor are complementary keys: renouncing each other, while F minor is removed to the other pole from D major: in our tonal system (e.g. the circle of fifths) the largest possible distance is expressed by a difference of 6 accidentals. As if it were the model for the Wagnerian technique: the very point where Mozart noted in the score "er will gehen" (and the Sprecher asks Tamino: "Willst du schon wieder geh ’ n?"), we find ourselves in the Bb minor key, followed four bars later by F minor. The contrast becomes even sharper when it is considered that D major is prepared by a salient, conspicuously emphatic A major dominant (almost 9 bars in length); attraction and repulsion are made even more apparent: Fig. 32

A similar contrast was noted earlier: Tamino’ s youthful, spontaneous utterances were expressed in B minor and E minor, while the old priest opened up the tonal scope of Ab major and Eb major: Fig. 33

As a result of the above-described connections, a dual, intertwining ’ spiral system’is created in which each element is balanced off by its counterpole (its tritone-pair): Fig. 34

Another two arguments are elicited by the transformation in the wake of "Zurück!". In C major tonality, the symmetry centre of our tonal system is marked by the D or the Ab pole. The gate is evoked by D major, the Sprecher appears on the counterpole: in Ab major. For Bartók, the inversion of the major pentatony — DO pentatony — became incarnated in MI pentatony, (projecting the DO pentatonic scale built on the note D downwards of D, a MI pentatonic scale is produced). The basic idea of Cantata Profana cannot be separated from the fact that our tonal system (notation, the stringing of the keyboard instruments, and in many respects the string instruments) is based on the ’’ d symmetry axis. Bartók contrasts the DO pentatony based on D with MI pentatony also based on D. The starting scale of the work rests on the D=MI pentatonic frame, while the acoustic scale closing the work unfolds from the D=DO pentatonic scale: (see Fig. 85 on p. 49) 8) Thus the above two scales are the exact mirror images of each other. The D=MI pentatonic scale incorporates the G minor and B b major triads. The D=DO pentatonic skeleton of the finale contains first of all the D major triad, but also the B minor triad. In The Magic Flute, D major symbolizes the ’ gate’(cf. also Tamino’ s rapture in B minor upon sighting Sarastro’ s temple). The D=MI pentatonic scale, on the other hand, contains the Bb major and G minor triads: at the turning point, as a consequence of the word "Zurück!", these very ’ reversed’triads get legitimation: first B b major, followed by G minor (the former after "Zurück", the latter after "Glück"). One more thing should be touched on in connection with the outcry "Zurück!". This is what triggers off the radical change that occurs through the entry from the ’ major ’ world into the ’ minor’world. On the one hand, C major and its V. degree (G major) ensure the tonal aura that can be schematized in the following way (every second step in the figure rhyme in perfect fifths). The formula also implies the possibility of B minor and E minor (as was mentioned earlier, Tamino’ s temper is governed by the positive substitute chords of G major and C major: B minor and E minor). Fig. 35

To produce the above net of fifths (Fig. 35a), a single C major (or A minor) triad would suffice. Should we replace C major by C minor, the fifth steps would take the shape above (Fig. 35bthe — relative key of C minor is Eb major, etc.). Looking at the two schemata side by side, one realizes that the relationship between the alternating major and minor third is reversed. The second row of thirds goes along the same path as is covered by the music after the shock of "Zurück!": Zurück! Bb major

Glück. G minor

Zurück! Eb major

...hier. C minor

Adagio. Ab major.

Fig. 36

In the scheme above C major and C minor, G major and G minor indicate a ’ majorminor’(modal) contrast. A more complicated case of the major-minor relationship is the contrast of the chords with a common third: Ab major and A minor almost present an allegoric contrast (appearance and dissappearance of the old priest). A similar contrast is created between E minor and E b major, B minor and Bb major. By the same token, the chord with the common third as the enthusiastic B minor emanating from Tamino’s personality is to become the Bb major, which expresses the stout resistance of the gate, the shock ("Zurück!"). Tamino’ s flaring up in E minor will also have a chord with a common third: Eb major, which projects to us the state of mind of the old priest together with the spiritual world behind him. And if you add to that, that E minor and Ab major, as well as B minor and E b major are spheres that negate one another (complementary keys), while B minor and A b major are polar spheres, then the computer may help you orientate yourself in this multidimensional network of relations. The broad cadences illuminate the D major episode like spotlights: Tamino is standing in front of the temple gate. The ’ map’of the scene gives similar salience to the importance and frequence of the G minor episodes: "Ja, ja, Sarastro herrschet hier": a G minor arrival, right in the foreground of an explosion. And later: "So gieb mir deine Gründe an!": G minor, before another explosion. And yet another G minor:

"das Gram und Jammer niederdrückt." In C major, the simplest subdominant and dominant are represented by D minor and G major. In taxonomic terms: the symmetry-mirror of D minor is G major. The symmetry remains intact even if we replace D minor and G major by D major and G minor, respectively. In this case, the notes FA and TI are replaced by FI and TA: in our system of notation, the nearest (first) sharp and the nearest flat appears: Fig. 37

(The frequency of D major and G minor not only illustrates how the mentioned principle became one of the most obvious instruments of ’ expanding’the diatonic system, but also suggests how the ’ acoustic’scale: —DO-scale with FI and TA —was 9) born in Bartók’ s style.) Speaking of the 19th century, let it be mentioned that the dominant of the A minor key: the E major chord also has a symmetry pair, the F minor. In the C major key the symmetry axis of the system is constituted — besides D — by the G # = Ab note (in the notation, it is marked by the ’ middle’ , the third sharp or flat, resp.). One points 10) upwards, the other downwards. The nadir of disillusionment is scored by Mozart in the F minor key ("... nie eu’ ren Tempel sehn!"), while the development (mottotheme) is born out of E major. F minor and E major not only satisfy the requirements of the ’mirror relationship’ , but are also chords with a common third.11) The Mozartian chromaticism is omnipresent —ensuring the organic, unbroken connection between the chords. After the statement "Die Absicht ist nur allzu Klar!" the note B leads to C, then (extended into a major third) gets emphasis from the C # note so that the latter could proceed to D as the leading note. The role of the neckbreaking polar change (D minorB —major) seems to imply that D should turn into # D and resolve in E. Via the ’ turn-motif’ , however, this E also rises to F, which eventually becomes reconciled in the central E note. Fig. 38

The basic motif of relief, of ’ smoothing out’is the DO-RE-MI motif (also on the

stage of Wagner and Verdi). Significantly, the three-times repeated motto-theme is prepared by this phrase (see Fig. 19 on p. 143). The cathartic moment follows the introduction of the motto-theme: "O ew ’ ge Nacht" — Mozart has the tonic (bass) and the dominant sound at the same time hence — the timeparalyzing, static effect: Fig. 39

The true unfolding — the absolution — is brought along by the flute solo of Andante. The dual — Eb major and C major — stage of The Magic Flute has often been discussed by analysts (Eb major as the tonality of being initiated, being an ’ insider’in the sacred secret); this dual tonal stage prevails in the examined dialogue as well. Compared to C major, the Eb major chord assumes its expressive character from the notes MA and TA (Eb and Bb). When the flute solo taming the beasts is intoned and Tamino’s aria "Wie stark ist nicht dein Zauberton" is begun, the stage lighting also changes: in our tonal system, the mirror image of TA and MA are the notes FI and DI –the aria is dotted with FI and DI colours, which produce the ’ fabulous’natural aura of the scene: Fig. 40

In connection with the ’ sanctified’key of E b major, let me remind you of the Sprecher’ s theme "... Heiligtum. Der Lieb’und Tugend Eigentum" following his introduction. The more so, because it ‘ rhymes’with the closing act of the above Quintet, quoting almost ’ note for note’its Bb major melody "Drei Knäbchen jung, schön, hold und weise, umschweben..." and its smoothingly blissful parallel thirds. Deepening the chords in thirds (a typical feature of the ’ motto’ -theme as well) is well known from the classical literature: it promotes the constant expansion of the ’inner stage’ , it keeps intensifying the radiance of the theme.

1. In a major key, the turn of the V-III. degree elicits a similar effect. (see. e.g. Beethoven’ s Missa Solemnis from bar 345). Here we are already in a minor key, that is, the dominant effect is enhanced by the dominant chord built upon ’ DI’ . 2. Bartók: Beginning of String Quartet No.4. 3. Cf. the Command motif in the Tristan. 4. See the beginning of Beethoven’ s 5th Symphony. 5. Cf. Verdi’ s Don Carlos (Act II), the Monk’ s second half-phrase in A: F-B. 6. End of page 60, too (The Magic Flute, piano score, Universal Edition):

*) ‘ta‘added by the editors 7. This motif has a counterpart, taxonomically speaking a symmetry-pair, which is created by lowering the SO note of the major chord by a semi-tone, e.g. modifying the G of C major to F#. The resulting F# subminor (F#-C-E, or F#-A-C-E) is a ’ blood-kin’of the secondary dominant (that is, it is related in meaning to the dominant of the dominant: the D7 chord), so much so, that in Mozart’s score the two usually appear simultaneously. Such is the Sprecher’ s b b first sentence where the fifth (E ) of the A major is modified to D, and the simultaneously appearing D subminor + Bb7 chords unite in a Bb9chord:

Tamino’s very first sentence at the beginning of the finale is also governed by a similar rule: the fifth of the C major is changed into F#, resulting in an F# subminor + D7 (= D9 chord). The ’ turn ’ -motif and the turn described above are mirror images of each other: both belong to the basic stock of Mozart’ s idiom. One gravitates towards the subdominant, the other towards the dominant. 8. Bartók himself made mention of this MI scale in his Harvard lectures. 9. Lurking behind the gripping - moved and at the same time exalting - B major breaking point: "Man opferte ... sie schon" and the Bb minor expressing disappointment (see the "er will gehen" instruction) one discerns a similar symmetry relation: compared to the D centre, B major is the mirror reflection of Bb minor (the contrast inhering in the reflection prompted Verdi to create one of his favourite dramatic motifs.) 10. That is how the F minor-C major cadence, which is none other than the counterpart, the mirror image, of the E major-A minor cadence (customary in a minor key), struck root in Romanticism. The D subminor-C major cadence is used with a similar meaning (as the mirror image of the E7-A minor cadence). 11. Wagner even composed a theme on the relationship between F minor and E major: the Marke theme is one of the most enigmatic thoughts of the Tristan.

APPENDIX The perception of music is based on our capacity to identify musical sounds by their relationship to a given key (or root) - not immediately by their absolute pitch. (The sense of perfect pitch is a faculty that is in most cases innate and independent of the direct perception of music.) The advantage of relative solmization over absolute tone names is that it also expresses the musical 'meaning' (function) of the notes. According to Kodály’ s musical concept, each major scale has a DO-RE-MI-FA-SO-LA-TI-DO, and each minor scale a LA-TI-DO-RE-MI-FA-SO-LA meaning. By raising the DO, we obtain degree DI, by raising the FA, degree FI, by raising the SO, degree SI, etc. By lowering the MI, we obtain degree MA, by lowering the TI, degree TA, etc. For example, in the C major (or A minor) key: LA-DI-MI stands for the A major triad (A-C#-E), DO-MA-SO stands for the C minor triad (C-Eb-G). The chromatic scale in the E major or C# minor key runs as shown in the next figure:

Relative solmization is more than just a matter of learning how to write and read music. It is virtually the only method which is capable of expressing modalpolymodal relations. In it, for example, the closedness of the system is expressed by the fact that DO-DI and MI-MA are not only derivatives but also mirror images of each other. This method can tell us something profoundly new about the modal structure and semantics of music.

EDITORS POSTSCRIPT In the last one and a half years of his life Ernő Lendvai frequently entertained the thought of giving a succinct summary of his analytic method in a sort of school-book. His unexpected death prevented the realization of his plan. Upon the request of his wife, Erzsébet Tusa, the editors of this volume undertook this rather difficult task. Ernő Lendvai‘ s professional career spanned some four and a half decades. As is known, he began as a Bartók scholar, analyzing Bartók‘ s works in his first essays (from 1947) and books (from 1955). From the early ’ 70s on, he extended his field of research first toward Kodály and later toward romanticism, particularly toward the music of Verdi and Wagner, and in his last paper (1992) he put a scene of Mozart ‘ s Magic Flute under scrutiny. His method thus gradually evolved into an almost overall analytic system. The book is aimed, on however limited space, to present his theoretical statements as fully as possible. In one of our last meetings he said such a summary could aptly be based on his article ’ Symmetries of Music’(in: SYMMETRY Vol. 1, 1990 VCH Publishers, Inc.). Obviously, that was the most appropriate title to be given this book as well. The study, however, had to be substantially enlarged because some problems were completely missing from it and others were only touched on sketchily. We selected passages from two major synthetizing works, ’ The Workshop of Bartók and Kodály’(Editio Musica Budapest, 1983) and ’ Verdi and Wagner’(International House Budapest, 1988). The former, containing articles written chiefly in the ‘ 60s and ‘ 70s, was mainly used in the first half of this book, the latter in the second half. In addition, we borrowed some passages and chapters from essays published elsewhere. With slight omissions, Lendvai‘ s study ’ The Quadrophonic Stage of Bartók ’ s Music for Strings, Percussion and Celesta’was also included in the abridged form he prepared for the New Hungarian Quarterly, because, on the one hand, we wished to illustrate his way of analyzing a large form, and on the other, because it addresses an intriguing and special problem that is not discussed anywhere else. We present in full professor Lendvai‘ s last essay prepared for publication (The ’Sprecher’ -scene of The Magic Flute) which first appears in this volume. The manuscript being untitled, we assigned it the title by which the author referred to it. This volume is not at all a critical edition. Our task was to unify sections of studies published on the same topic at various points of time and space and to avoid repetitions. Always choosing the most compact and complete wording, at times we borrowed a passage or just a sentence from somewhere else. Appendices and notes containing significant theoretical statements were included in the body of text, but the ones deemed dispenseble were omitted. What entitled us to apply this procedure was that the author himself did the same to different versions of his studies. It cannot be stressed enough, however, that there is not a single sentence, half

sentence or even adjective that Ernő Lendvai would not have put down in the same context (the unavoidable editorial notes or references are added in footnotes and marked /ed./). That applies to music examples and figures as well. Not even the bibliographic data were complemented (apart from the correction of obvious misprints). The selection of music examples was governed by the desire to best enlighten the theoretical statement on the one hand, and to represent as many composers as possible on the other. The great temporal distance between the studies and the transformation and extension of the author‘ s research field (e.g., the use of relative solmization) explains the differences (but not contradictions!) in style and partly in outlook between the first chapters of the volume (Axis System, Nature Symbolism, Harmonic Principles, The Quadrophonic Stage), and the later ones. This difference can sometimes be detected within a chapter as well (e.g., Authentic and Plagal Thinking). The same explanation applies to the preponderance of certain composers in some chapters (e.g. Bartók and in part Kodály in the starting chapters, Verdi and Wagner in the second half), and to the restriction of certain statements to one composer or another. That could not be avoided –nor was it our desire to do so. The essays were translated by Mónika Pálos, Judit Pokoly and the author (and possibly by other, unknown translators), but it was impossible to determine who translated what. The Mozart study was translated by Judit Pokoly. Thanks are due to the Kodály Institute of Kecskemét and specifically to deputy director Mihály Ittzés for the publication of this volume. He was the one to organize Ernő Lendvai‘ s first, and regrettably last, seminar in June 1992 where he could personally meet a younger, unbiased generation open toward a new approach. It was a source of delight both for him and for the participants. Miklós Szabó Miklós Mohay