PARTITIONS GRATUITES :
- MATHEMATICS 1

SKU: HL.49046544

ISBN 9781705122655. UPC: 842819108726. 9.0x12.0x0.224 inches.

I composed the Piano Concerto in two stages: the first three movements during the years 1985-86, the next two in 1987, the final autograph of the last movement was ready by January, 1988. The concerto is dedicated to the American conductor Mario di Bonaventura. The markings of the movements are the following: 1. Vivace molto ritmico e preciso 2. Lento e deserto 3. Vivace cantabile 4. Allegro risoluto 5. Presto luminoso.The first performance of the three-movement Concerto was on October 23rd, 1986 in Graz. Mario di Bonaventura conducted while his brother, Anthony di Bonaventura, was the soloist. Two days later the performance was repeated in the Vienna Konzerthaus. After hearing the work twice, I came to the conclusion that the third movement is not an adequate finale; my feeling of form demanded continuation, a supplement. That led to the composing of the next two movements. The premiere of the whole cycle took place on February 29th, 1988, in the Vienna Konzerthaus with the same conductor and the same pianist. The orchestra consisted of the following: flute, oboe, clarinet, bassoon, horn, trumpet, tenor trombone, percussion and strings. The flautist also plays the piccoIo, the clarinetist, the alto ocarina. The percussion is made up of diverse instruments, which one musician-virtuoso can play. It is more practical, however, if two or three musicians share the instruments. Besides traditional instruments the percussion part calls also for two simple wind instruments: the swanee whistle and the harmonica. The string instrument parts (two violins, viola, cello and doubles bass) can be performed soloistic since they do not contain divisi. For balance, however, the ensemble playing is recommended, for example 6-8 first violins, 6-8 second, 4-6 violas, 4-6 cellos, 3-4 double basses. In the Piano Concerto I realized new concepts of harmony and rhythm. The first movement is entirely written in bimetry: simultaneously 12/8 and 4/4 (8/8). This relates to the known triplet on a doule relation and in itself is nothing new. Because, however, I articulate 12 triola and 8 duola pulses, an entangled, up till now unheard kind of polymetry is created. The rhythm is additionally complicated because of asymmetric groupings inside two speed layers, which means accents are asymmetrically distributed. These groups, as in the talea technique, have a fixed, continuously repeating rhythmic structures of varying lengths in speed layers of 12/8 and 4/4. This means that the repeating pattern in the 12/8 level and the pattern in the 4/4 level do not coincide and continuously give a kaleidoscope of renewing combinations. In our perception we quickly resign from following particular rhythmical successions and that what is going on in time appears for us as something static, resting. This music, if it is played properly, in the right tempo and with the right accents inside particular layers, after a certain time 'rises, as it were, as a plane after taking off: the rhythmic action, too complex to be able to follow in detail, begins flying. This diffusion of individual structures into a different global structure is one of my basic compositional concepts: from the end of the fifties, from the orchestral works Apparitions and Atmospheres I continuously have been looking for new ways of resolving this basic question. The harmony of the first movement is based on mixtures, hence on the parallel leading of voices. This technique is used here in a rather simple form; later in the fourth movement it will be considerably developed. The second movement (the only slow one amongst five movements) also has a talea type of structure, it is however much simpler rhythmically, because it contains only one speed layer. The melody is consisted in the development of a rigorous interval mode in which two minor seconds and one major second alternate therefore nine notes inside an octave. This mode is transposed into different degrees and it also determines the harmony of the movement; however, in closing episode in the piano part there is a combination of diatonics (white keys) and pentatonics (black keys) led in brilliant, sparkling quasimixtures, while the orchestra continues to play in the nine tone mode. In this movement I used isolated sounds and extreme registers (piccolo in a very low register, bassoon in a very high register, canons played by the swanee whistle, the alto ocarina and brass with a harmon-mute' damper, cutting sound combinations of the piccolo, clarinet and oboe in an extremely high register, also alternating of a whistle-siren and xylophone). The third movement also has one speed layer and because of this it appears as simpler than the first, but actually the rhythm is very complicated in a different way here. Above the uninterrupted, fast and regular basic pulse, thanks to the asymmetric distribution of accents, different types of hemiolas and inherent melodical patterns appear (the term was coined by Gerhard Kubik in relation to central African music). If this movement is played with the adequate speed and with very clear accentuation, illusory rhythmic-melodical figures appear. These figures are not played directly; they do not appear in the score, but exist only in our perception as a result of co-operation of different voices. Already earlier I had experimented with illusory rhythmics, namely in Poeme symphonique for 100 metronomes (1962), in Continuum for harpsichord (1968), in Monument for two pianos (1976), and especially in the first and sixth piano etude Desordre and Automne a Varsovie (1985). The third movement of the Piano Concerto is up to now the clearest example of illusory rhythmics and illusory melody. In intervallic and chordal structure this movement is based on alternation, and also inter-relation of various modal and quasi-equidistant harmony spaces. The tempered twelve-part division of the octave allows for diatonical and other modal interval successions, which are not equidistant, but are based on the alternation of major and minor seconds in different groups. The tempered system also allows for the use of the anhemitonic pentatonic scale (the black keys of the piano). From equidistant scales, therefore interval formations which are based on the division of an octave in equal distances, the twelve-tone tempered system allows only chromatics (only minor seconds) and the six-tone scale (the whole-tone: only major seconds). Moreover, the division of the octave into four parts only minor thirds) and three parts (three major thirds) is possible. In several music cultures different equidistant divisions of an octave are accepted, for example, in the Javanese slendro into five parts, in Melanesia into seven parts, popular also in southeastern Asia, and apart from this, in southern Africa. This does not mean an exact equidistance: there is a certain tolerance for the inaccurateness of the interval tuning. These exotic for us, Europeans, harmony and melody have attracted me for several years. However I did not want to re-tune the piano (microtone deviations appear in the concerto only in a few places in the horn and trombone parts led in natural tones). After the period of experimenting, I got to pseudo- or quasiequidistant intervals, which is neither whole-tone nor chromatic: in the twelve-tone system, two whole-tone scales are possible, shifted a minor second apart from each other. Therefore, I connect these two scales (or sound resources), and for example, places occur where the melodies and figurations in the piano part are created from both whole tone scales; in one band one six-tone sound resource is utilized, and in the other hand, the complementary. In this way whole-tonality and chromaticism mutually reduce themselves: a type of deformed equidistancism is formed, strangely brilliant and at the same time slanting; illusory harmony, indeed being created inside the tempered twelve-tone system, but in sound quality not belonging to it anymore. The appearance of such slantedequidistant harmony fields alternating with modal fields and based on chords built on fifths (mainly in the piano part), complemented with mixtures built on fifths in the orchestra, gives this movement an individual, soft-metallic colour (a metallic sound resulting from harmonics). The fourth movement was meant to be the central movement of the Concerto. Its melodc-rhythmic elements (embryos or fragments of motives) in themselves are simple. The movement also begins simply, with a succession of overlapping of these elements in the mixture type structures. Also here a kaleidoscope is created, due to a limited number of these elements - of these pebbles in the kaleidoscope - which continuously return in augmentations and diminutions. Step by step, however, so that in the beginning we cannot hear it, a compiled rhythmic organization of the talea type gradually comes into daylight, based on the simultaneity of two mutually shifted to each other speed layers (also triplet and duoles, however, with different asymmetric structures than in the first movement). While longer rests are gradually filled in with motive fragments, we slowly come to the conclusion that we have found ourselves inside a rhythmic-melodical whirl: without change in tempo, only through increasing the density of the musical events, a rotation is created in the stream of successive and compiled, augmented and diminished motive fragments, and increasing the density suggests acceleration. Thanks to the periodical structure of the composition, always new but however of the same (all the motivic cells are similar to earlier ones but none of them are exactly repeated; the general structure is therefore self-similar), an impression is created of a gigantic, indissoluble network. Also, rhythmic structures at first hidden gradually begin to emerge, two independent speed layers with their various internal accentuations. This great, self-similar whirl in a very indirect way relates to musical associations, which came to my mind while watching the graphic projection of the mathematical sets of Julia and of Mandelbrot made with the help of a computer. I saw these wonderful pictures of fractal creations, made by scientists from Brema, Peitgen and Richter, for the first time in 1984. From that time they have played a great role in my musical concepts. This does not mean, however, that composing the fourth movement I used mathematical methods or iterative calculus; indeed, I did use constructions which, however, are not based on mathematical thinking, but are rather craftman's constructions (in this respect, my attitude towards mathematics is similar to that of the graphic artist Maurits Escher). I am concerned rather with intuitional, poetic, synesthetic correspondence, not on the scientific, but on the poetic level of thinking. The fifth, very short Presto movement is harmonically very simple, but all the more complicated in its rhythmic structure: it is based on the further development of ''inherent patterns of the third movement. The quasi-equidistance system dominates harmonically and melodically in this movement, as in the third, alternating with harmonic fields, which are based on the division of the chromatic whole into diatonics and anhemitonic pentatonics. Polyrhythms and harmonic mixtures reach their greatest density, and at the same time this movement is strikingly light, enlightened with very bright colours: at first it seems chaotic, but after listening to it for a few times it is easy to grasp its content: many autonomous but self-similar figures which crossing themselves. I present my artistic credo in the Piano Concerto: I demonstrate my independence from criteria of the traditional avantgarde, as well as the fashionable postmodernism. Musical illusions which I consider to be also so important are not a goal in itself for me, but a foundation for my aesthetical attitude. I prefer musical forms which have a more object-like than processual character. Music as frozen time, as an object in imaginary space evoked by music in our imagination, as a creation which really develops in time, but in imagination it exists simultaneously in all its moments. The spell of time, the enduring its passing by, closing it in a moment of the present is my main intention as a composer. (Gyorgy Ligeti).