A.A. Koblyakov – Professor, Composer's faculty Dean, Moscow State Conservatory named by P.I. Tchaikovsky's, a member of the Composers' Union of Russia, member of the Cinematographers' Union, the Board of the Moscow branch of the Composers' Union of Russian Federation, the Admissions Committee of the Composers' Union of Russia, the Academy of Сinema Arts. E-mail: email@example.com
A new approach to the analysis of a musical piece has revealed new typesof relationships, which we refer to as «transdimensional relations», or «transdimensionalism» (abbreviated TD).
Transdimensionalism refers to the entire set of relationships between spaces of differing dimensions within a multidimensional whole. («Dimension» – is the number of degrees of freedom, which allow both geometric and parametric interpretations). Let us look at a special case of transdimensional relations- «polydimensions» (abbreviated PD), meaning an object, which belongs to spaces of different dimensions. We will show that polydimensionalism lies at the base of music. We can look at the basic cell in modal music – the relationship between a stable tone and an unstable tone.
The stable tone is independent on the unstable one, it is defined unambiguously, and it has one degree of freedom. Whereas the unstable tone is dependent on the stable one and it has two degrees of freedom: 1) it can resolve into stable tones, or 2) it can move away from it. From the point of view of musical grammar, the unstable tone is unambiguous like as the stable tone (music as Language). Whereas from a contextual meaning the unstable tone is bimodal (music as Speech), it is 2-dimensional.
And so, the pair «stable tone-unstable tone» is polydimensional, one of its parts is 1- dimensional, while the other is 2-dimensional (PD=1D:2D). The polydimension 1D:2D is the most basic cell of transdimensionalism, which allows one to separate contextual meanings from non-contextual meanings.Let us look at the same pair, «stable sound-unstable sound» in the space of language. Here it is also polydimensional. From one side, both units in the pair are equal and symmetrical, differentiated only by the plus and minus signs (or by «color»). From the other side, the word «un» (negation) creates a new quality (or parameter): not only the negation of a statement («stable sound»), but also an unbreakable bond with it (since we are talking about negation not in the abstract, but of a specific statement). As a result, we find that negation is a complex unit of the pair, as opposed to the simple statement. From this point of view, the pair is polydimensional (PD=1D:2D) and asymmetrical. Out of this, all of the pairs, which are associated with negation, are polydimensional and asymmetrical («truth-untruth», «to be»-«not to be», etc.). The second units (context dependent) of these dyads form a pair that is "linkage" with the first units (context independent). The pair «symmetry-asymmetry» is also polydimensional!
As we all know, our thinking is based on binary oppositions, which are considered as symmetrical. As it turns out, oppositions can be simple and complex (context dependent), and complex oppositions are asymmetrical! This includes not only pairs with negative, but also with reflexive units (object-subject, action-observation), a secondary action (stimulus-reaction, vibrator-resonator), synthesis inside one of its parts (horse-rider), etc. All of these are examples of PD of the type 1D:2D or «transdimensional asymmetry». From this we get several important logical consequences. Firstly, context-independent terms are separated from context dependent ones; in particular, we find the polydimensionality (which was not so obviously till now) of logical pairs such as «yes-no», «true-false», etc. Secondly, when one takes into account transdimensional relations and polydimensionality, classical paradoxes cease to be paradoxical. For example, «the barber paradox» (Bertrand Russell): all inhabitants of the village are 1-dimensional, while the barber is polydimensional, PD=1D:2D. The king's order does not take this into account, thereby creating a paradox! (We can also consider other paradoxes such as «I am lying» or «I am asleep», etc.).
Here arises the possibility of creating a new type of logic – the «logic of creativity», which integrates classical and non classical logic into a single whole through the use of PD=1D:2D
Let's move on to information theory, the basic cell of which is a «bit», that is the same pair as «yes-no», or 0 and 1. It was previously thoughtthat the membersof the pairare equal andsymmetrical. However, this is only partly true. The negation «no» forms a «linkage state» with the affirmation, which justifies the transition to Q-bits. Let us remember, that the idea of Q-bits came from the physical sciences (quantum mechanics). We arrived at the idea from studying the humanities, which underscores the universality of these new relations.