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THE FORMAL STRUCTURE AND THEORETICAL FOUNDATIONS OF HEXAGRAM NOTATION





A. The formal structure of Hexagram notation derives from that of Pentagram notation, it is founded upon traditional concepts, which it simultaneously transcends. Consequently, it seems to us unprecedented and at the same time familiar; it is both varied and uniform. The whole applied system is more complex and at the same time more simple than that of Pentagram notation.

B. Hexagram notation is based upon three essential elements: the theoretical and artistic achievements of Pentagram notation; the rational nucleus of the formal logic and of the mathematical logic; the system of philosophical thought and the deductive modality of the Yi Jing and the "Ba Gua"*. The staff system thus devised is characterized by its rational structure, concise concept, complete function, and simplified rules.

   The formative modalities of the new system are the following: 1) Each set of three horizontal parallel lines forms a half staff — precisely, based on the traditional Pentagram, with the three lower lines of the "Treble staff" or the three upper lines of the "Bass staff" — which, combined with the transitory natural ledger line, constitutes a mutable scale group, i.e. precisely an octave, a series of twelve tones or a series of twenty-four tones; each set of six horizontal parallel lines, made up of the combination of these two adjacent and opposite groups of mutable scales, forms a staff that, by means of the principle of reciprocal complement, becomes an autonomous expressive musical mode. 2) Each staff is a symmetrical combination of the 2:1 ratio both formally and mathematically; each staff can be evolved and duplicated logically according to the modalities of the binary system and of the graduality, so as to produce the "Complete staff" and the "Inferential staff". The "Complete staff", that is to say, a combination of the staves of 4 different levels with the clefs of the seven applicative programs made up of 27 lines and 8 octaves, comprises all the ranges of executable tones; the "Inferential staff", i.e. a series of variable combinations structured according to the principle of equivocality, can instead express specific categories of frequency and of the imaginary, beyond a practicable use. They can be concrete and abstract, realistic or fantastic, all at the same time. The application of this system, having just one unified rule and one logical formula, which can be unfolded from the microcosm to the macrocosm, from the interior towards the exterior, always beginning again in an endless cycle, capable of infinite variations without detaching from its principle.

C. In the Hexagram notation, the "Middle-c" is the centre of the staff structure and of the scale system. The staves and scales are formed by expansion from "Middle-c"; the "Middle-c" is the centre of unification for every combination of the staves and scales. They are at the same time complete and divided, similar and different. The rules are invariable and can be applied to different ranges.

D. The formal structure and theoretical basis of Hexagram notation, and especially its formal and mathematical logic, can also form the theoretical framework for the "Demonstrative staff" in the field of contemporary and avant-garde music, offering practicable and standardized rules for its application and future development.



* "Ba Gua": The system of 64 different "hexa-grams" in the Yi Jing, obtained from the combination, in pairs, of 8 different fundamental "tri-grams", used ever since antiquity both as a means of divination and as a philosophical deductive modality. Its formal and mathematical logic is that of the 2:1 ratio (i.e. a unity formed by doubling the half), which is characteristic of the unity of opposites, and possesses also the capacity for always beginning the circle again in an endless cycle; the alternation of the Yin and the Yang corresponds to the principle of the binary system.

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