А. Koblyakov — Academic of NIKA ,Professor, Dean of the Composition Department at the Moscow Conservatory Scientific Director of the Scientific Center of Interdisciplinary Research of Musical Creativity, Moscow Conservatory.

Introduction

What is art as a whole, and specifically creative art? What is the difference between an art form and a craft? And what is the purpose of a work of art? What is the process through which, by the laws of creativity, the Complex emerges from the Simple, the Whole from its parts, or a new quality emerges in the end product (result)? To this day there is no one specific answer to these questions. As a result, there is an absence of a general theory of creativity in modern scientific knowledge, as well as an absence of such fundamental categories as Purpose, Goal, a Whole. One thing is obvious: our existing knowledge about the world does not include some very important type of relation, which determines the very heart, and the essence of the creative process.

So, what is this essence? One of the most general definitions of creativity is as follows: «Creativity is the process of solving a problem which gives rise to a new quality, and a new result» [1, 164-192], [2, 254], [5, 162]. Let us be reminded, that the problem in this instance is created by the inherent contradiction (juxtaposition). What does it mean to «creatively solve a problem» (to remove the contradiction)?

As studies have shown [1, 164], the strategy of computer in a problematic situation (in this case, the presence of mutually contradictory statements) comes down to a simple rejection of one of the alternatives, in other words, reduction. In turn, the strategy of synthesis, which combines alternatives is characteristic of a creative mind, which, at its core, differentiates it from computer [1, 165-169]. The ability to synthesize as a characteristic quality of a creative style of thinking has been pointed out by many scientists. So, for example, Alfred Whitehead, in his book «Process and reality» (1929) writes that the purpose of art is «to move from disjunction to conjunction, and the formation of a new essence, different from the one given by disjunction» (15, 162). Let us be reminded, that in logic, the transition from disjunction to conjunction also means a transition from a disjoining statement to a conjoining one, from a juxtaposition (separation) of qualities, processes, and phenomena, to there synthesis (connection, see ex. 1). In light of this idea, we can add to the previous definition of creativity, to clarify it using Whitehead’s logical component: Creation is a process of problem solving through the transition from disjunction to conjunction, with the birth of a new quality, and a new result (the synthesis of antitheses, forming a «new essence»). And in this transition from disjunction to conjunction, specifically, lies the very essence of creativity. So, how does this transition from disjunction to conjunction, from opposition to unity, occur? How do antagonists become partners, competitors become allies? How, SPECIFICALLY, does the synthesis of antitheses occur? How does a new quality appear in the Whole, which is absent in its parts? The answer can be found in creative art, and first and foremost — in music, which is the field of our professional interests. It is precisely music, which became the bases of a specific and detailed study of these new types of relations.

Transdimensional relations in music

How does the synthesis of antitheses occur in the simplest musical statement possible — a single voice melody (monody). Lets look at pre-Baroque European monody (ex. 2a). It’s well known that this and similar chants have two types of foundations — the lower and upper stable tones («finalis» and «repercussa», see ex. 2b) . The Finalis is the beginning and ending note of the chant, or the «harmonic tonic», while the Repercussa is the «melodic tonic». The horizontal distance between them is a separately sounding melodic fifth (fourth). In the initial case, the finalis and repercussa are juxtaposed as a disjunction. With the appearance of hypo modes, the functions of the finalis and repercussa are expanded in such a way that each of them takes on the quality of an antagonist: the finalis is presented by the middle tone (by the repercussa), while the repercussa is presented by the lower tone (by the finalis, see ex. 2c). The next stage is there combination: the same chant now sounds both from the finalis tone and repercussa tone SIMULTANIOUSLY . Which is more important- the finalis or the repercussa? BOTH have gained equal status, since they have coalesced into a simultaneously sounding harmonic fourth (fifth). The melody has been doubled: from each note of the fifth (fourth), now sounds the same melody (see 2d)! Disjunction has become conjunction, and the so called «Organum style” (X cen.) has come into being. What has occurred is a transition from single voice melodies to two voice melodies, from a single dimensional musical space (the building block of which is the tone) to a dual dimensional space (the building block is the interval). The new dimension gives us a new parameter: consonance (the vertical coalescing of sounds); a new whole with a different quality has come into being, one which was absent in its initial parts. [The dimesion of a space (system, element), refers to the degree of spatial freedom (system, element), which allows for both a geometric as well as a parametric interpretation]. And so, the fundamental contradiction in a single voice melodic line is resolved through the synthesis of antitheses in a two voice melody. We then find that the entire history of music is a chain of such transitions. So, the next historical stage is the transition into a three dimensional musical space, which occurred in the XVII cen. — it occured in the same manner. The fundamental problem of a two dimensional space was the contradiction between the fifth and the third. In the “Organum style”, the fifth is the dominant interval (vertically, it is based on the parallel motion of fifths). Whereas the third is on the periphery of musical space («dissonance»). After the change in tuning systems (from Pythagorean tuning to tuning by perfect fifths) the third gradually becomes a consonance and «takes over» the role of leadership from the fifth (this is the «strict style» era, XV cen., vertically it is based on parallel thirds while parallel fifths are forbidden). Finally, the «battle» between the fifth and the third is resolved in there synthesis — a new building block of the now three dimensional musical space has come into being — the chord, namely the triad (example 2e, beginning of the XVII cen.). This marks the beginning of the era of homophonic music, which follows the previous (polyphonic) era of modal polyphony. The contradiction between two dimensional intervals of thirds and fifths has been resolved through the synthesis within a three dimensional chord consisting of thirds and fifths (a triad) (ex.2 e). A transition from disjunction to conjunction has occurred, opposing sides now complement one another within a new building block of a multi-dimensional musical space. Finally, the next transition from disjunction to conjunction occurs at the beginning of the XX century: it involves the merging of competing (sub)-chords and (sub)-tonal systems into polychords and polyharmonies; again the synthesis of antitheses has brought about atransition into a new dimension of musical space(see different «well-known» polychords in ex.3).

Now we understand the specific qualities of the transition from disjunction to conjunction: it is a special kind of transition — from a state of smaller dimension to one of larger dimensions. Such a nontrivial quality of this transition demands that it be given its own term. We will call this transition a «transdimensional transition» (from the Latin root “trans”- meaning “through”, abbreviated TDt [13, 20-42].

From everything that has been said, it follows that precisely in this transition, from our point of view, lies the heart of the creative process as a transition from disjunction to conjunction, from opposition to unity, or, in general, a transition to greater degrees of freedom. (It should be clarified, that one must distinguish between two types of transdimensional transitions: the transition from lower dimensions to higher ones and vice versa. The former is naturally seen as a «positive transdimentional transition (+TDt), while the second is a negative one (-TDt). From this point forth, we will use the term «transdimensional transition» to mean, (with certain exceptions), only a transition from lower to higher dimensions, i.e. +TDt. Let us point out once more, that the resolution of the inherent contradiction through the synthesis of antitheses is possible only in a musical (semantic) meta-space relative to the initial one, i.e. only through a transdimensional transition into a multi-dimensional space (this assertion is logically proven in [11, 33-35]).

Now we can see the reason for our previous unsuccessful attempts at rationally describing the synthesis of antitheses: the disregarding of different dimensional states of disjunction and conjunction. Conjunction is located at a higher dimension, than disjunction!

When seen as a process, the musical space of the previous millennium is a «transdimensional horizontal axis» consisting of a chain of transdimensional transitions: a single tone (one dimensional space) — interval (two dimensional) — chord (three dimensional) — polychord (four dimensional). From a structural point of view, the given musical space is a multi-dimensional whole, containing one, two, three, and four dimensional spaces, which are linked by a mutual problem — the contradiction at the foundational single dimension. In this way, a special link occurs between the spaces (similar to stratification): a «transdimensional vertical axis». The transdimensional vertical and horizontal axes, as well as the transdimensional transition are all examples of new types of relations. Using our terminology — a «transdimensional relation» or «transdimensionalism» (from the Latin root “trans” — meaning «through», abbreviated TD). Transdimensionalism refers to the entire set of relationships between spaces of differing dimensions within a multidimensional whole, i.e.: insertion, mutual reflection, folding-unfolding, connections between dimensions, transition from one dimension to another etc. (a few types of transdimensional relations can be seen in ex.4). It is precisely transdimensionalism, which allows one to rationally define such complex categories as Purpose (the solution to a problem), Goal (the transition to greater degrees of freedom), Creativity (the attaining of a new quality though a transdimensional transition), a Whole (a structure with a transdimensional vertical and horizontal axis, or a «transdimensional cross»). We will show how the presence of this «transdimensional cross» distinguishes a true Whole (a masterpiece) from a Conglomerate (an imitation of the Whole).

The new methodology and transdimensionalism

What is the purpose of creative work in general and of an artistic work in particular? What is the reliable value criteria, which makes a masterpiece different from a hack counterfeit? There are no answers, yet, to these «eternal questions of art criticism». The problem is especially acute in the domain of music theory, as the non-verbal and abstract nature of music makes both its meaning and criteria of value rather indefinite, causing a great deal of confusion and paradoxes.

The traditional research methodology, and the system-functional analysis in particular, long established in music theory, do not provide answers to these eternal challenging questions. To illustrate the point, let us briefly review the analysis of the B flat-major fugue by Bach, from the first volume of Well-Tempered Clavier, which travels from one student manual to another (ex.5).

The fugue in B flat-major is a three-voice fugue. It is a triple counterpoint (with two retained countersubjects). The subject is made up of two contrasting parts: nucleus and micro-development, which is typical for Bach. The nucleus is of a scherzo-like nature, with leaps within a seventh; micro-development — a conjunct rotary motion. The subject reveals the mode functions rather extensively, which is again typical for Bach and finishes with an imperfect melodic cadence on the third tone. The answer is tonal, as the subject includes the sound of a fifth which is the primary tone of the dominant. The first retained countersubject is «anticipatory» (Chougaev,[6]), and is melodically related to the subject. The subject and the countersubject have the same amount of melodic cells (the interval of a seventh). They are also connected in terms of rhythm (which is complementary).

The second retained countersubject does not reveal any obvious connection with the subject and has an abundant number of rests. There are two episodes in the fugue (measures 17–21, 30–34). In the episodes the two parts of the subject are connected by polyphonic means. Measures 17–21 are a composite episode: the final figure in the subject is sequenced which is followed by its inverted nucleus. The second episode (measures 30–34) comes from the second part of the first episode (double counterpoint) and one more voice appears. The fugue is a standard three-part fugue («normative») [6, 201] with exposition, development and reprise. In terms of harmony it is again a standard («normative») tonal fugue, whose ternary structure reflects 1) the tonic domain,

2) deviation from this domain (modulation), 3) return to the original tonic domain [7, 30]. Going beyond the original tonic-dominant frame of the exposition — towards parallel keys (in development) — is typical of a Bach’s fugues of this type in general [7, 31]:

The episodes frame the central structure (statement of subject in minor keys) making a symmetrical concentric form: exposition — episode I — development — episode II — reprise. The tonal structure is also typical for a fugue by Bach: the tonic-dominant domain of exposition is replied to by subdominant area in the development section and beginning of reprise. The question is: why is the fugue in B flat-major so remarkable? What is its meaning, purpose, integrity and value? What was Bach’s motivation when choosing certain artistic means, why has he chosen them and not something else?

These questions and similar ones have been a matter of concern for music theorists. However, they still remain unanswered. The brief analysis of the fugue above again confirms that the traditional analytical approach is unable to answer the «accursed questions» of music theory, as it describes (at its best!) the fugue in terms of its normative-technical aspect in general. However, compliance with “grammatical” rules is not a guarantee of a good artistic result, in the same way that knowledge of a language is not enough to create a literary masterpiece. The analysis of bonds, relations, functions, elements within this approach provides only a logical frame of the piece — logical bonds, however, can be also found in a student’s work in the harmony class. In actuality, neither form(normative three-part), nor harmony (the normative tonal plan, standard TSD support), or polyphony (a standard triple counterpoint), melody or rhythm seemingly have anything which would be the evidence of a unique masterpiece, which the fugue undoubtedly is.

In this connection we offer a new metasystemic methodology, which we refer to as «the problem-meaning approach». It consists in the following: The musical whole is not only an interrelation of various functions of a single system, but also a connection of different principles of organization, cooperation («synergy») of various systems [14]. A piece is interpreted as a problem statement (initial antithesis-opposition as systemic contradiction) and its resolution (unification of systems into a meta-system by means of a trans-dimensional transition, converting disjunction into conjunction. Resolution of the problem is the purpose and aim of the pitch-forming process. The result produced by a composer is estimated in terms of interrelation of all elements of the music outline and in relation to the main problem of the piece [8,317]. The new model of the piece gives rise to a new research methodology accordingly. Its trend (from analysis to synthesis) reflects the development of the piece itself (moving from opposition to complementarity, to the synthesis of antitheses). So, compared to the systemic approach, our method is metasystemic, the piece is approached not only on the functional level but also on a higher level, in terms of various organizational principles (various systems). What can be found in this fugue by Bach if it is approached in terms of problem and meaning?

The systemic-functional approach investigates the meanings of elements of the piece [8]. Let us now try and find their implication. If studied carefully, the fugue evokes a number of questions which are beyond systemic analysis. If the fugue has two retained countersubjects, why then, is the second countersubject changed (see ex.6)? Why is the dissonance in the tonic seventh chord is dropped? Why are these changes present (chromatics instead of diatonicism)? Why is the strange dissonant harmony repeated several times, which sounds especially striking in measures 39 and 43? Why the non-trivial for a polyphonic work concluding part (measures 43–46) with repeated cadences, typical of homophony? As is clear from what has been said above, traditional methodology does not provide answers to these questions.

Let us go back to the analysis of the fugue. Let us now consider deviations from the major pitch “grammars” (mode, intervals and chords). The nucleus of the subject is remarkable in the following way: the tone c is dropped, it does not move towards the b flat of the tonic (there is no direct resolution II–I), and the b flat¹ is followed by the II scale tone — c² (measure 2). Moreover, the II scale tone leads into the III scale note (measure 3) so that the lower horizon of the theme rests not upon the I scale note (b flat), but the III (d, melodic support). In the development of the subject (measures 3–4) d is again the melodic support (second octave). Here we see a distinctive tritone contour a—es with a unilateral resolution: es—d, whereas the VII scale tone does not directly resolve into the tonic b flat (on the strong beat).

Thus, the supporting fourth-and-fifth frame of the d¹—a¹—d² theme (instead of the normative b flat—f¹—b flat¹). We see a typical example of “discord” between linear-melodic and mode-functional bases [14]. The II and VII scale tones of the mode behave “abnormally” in terms of typical (harmonic) tensions in major, which is, however, perfectly normative in terms of modal relations! The countersubject reinforces the contradiction between modal and tonal systems. Measures 6 and 7 (answer) feature a dropped dissonance on the strong beat (seventh), whose upper tone is g² (again the II scale tone —in F-dur). The strong beat dissonance in the critical point of the answer (the border between the nucleus and micro-development) is not allowed even at a distance (which is a unique case for Bach, see ex.5).

The problem becomes even more acute with the advent of the second countersubject. In measures 10–11 instead of a dropped dissonance of a seventh we see a non-tertiary harmony (fourth-structure), which is also not allowed even at a distance. Thus, contours of one and the same problem are outlined (the modal is opposed to the tonal, linear-melodic — to the functional harmonic) on different structural levels of the piece: the mode, the intervals, the chords. It should be noted that the second, so-called «retained» countersubject is constantly changing (compare measures 9–11, 22–24, 35–36, 41–43; see example 5,6).

If studied carefully, these changes give the key to the meaning and purpose of the whole intonational process of the fugue. The fact is that gradual transformation of the second countersubject provides new chords of a tertiary type, which in the last development, the climax of the fugue (measures 41–43) make for a full functional sequence of S–D–T in the main key (ex.7, a,b)! The conclusion (measures 43–48) with a repeated cadence consolidates the transition from a quartial system to tertiary system, from polyphony to homophony, from modal to tonal rules (see examples 5,6,7). A more powerful unit of the sound-pitch system is created. And this unit is the chord. As a result we get an illusion of the initial contradiction: the I scale note c resolves within the newly-formed functional-chord sequence D₇–T (measures 42–43).

As for the theme, is hasn’t changed, whereas relations of the counterpointing voices have: by slight changes in the second countersubject, Bach makes the following: the fourth-structure concords are transformed into the third-structures first without (measures 22–24), then with a root note (measures 35–37), making a full functional sequence in the climax (measures 41–43, see ex.7). Bach solves the problem of the subject by means transforming accompanying voices. A strikingly witty and only possible way, as complying with the rules of a classical fugue the subject is unchangeable!

Let us sum up what has been said above. The subject of the fugue bears a contradiction, as the actual (modal) and the supposed (tonal) bases are mismatched here, which leads to «anomalies» — violations of “grammar”. By means of a wide range of special techniques (first and foremost, elaborately reconstructing the counterpoint) Bach consciously transforms the pitch system into a more complex condition: from modality, with the interval as the unit of harmony, into tonality, with the chord, synthetic unification of intervals, as the basic unit. The chord resolves the initial contradiction in the subject between competing pivots of different types [8,312–316].

We see a transition from a dyad-interval (unit of the two-dimensional music space of the modal harmony) to the triad chord, which is a unit of the three-dimensional tonal-harmonic space. Such important transition of the system into another dimensional condition requires special terminology. We refer to it as a «transdimensional transition», and relations between spaces of different dimensions are called «transdimensionalism». (We remind: dimension of the (system) element is the number of degrees of freedom of the (system) element, which allows for both geometrical and parametrical interpretations).Transdimension and transdimensional transition are the key notions of our approach [12,13].

The thing is that such a resolution of the initial contradiction by means of the antitheses- synthesis is only possible in a semantic meta-space relative to the initial, that is by means of a trans-dimensional transition into a space of greater dimension (this statement is proven in [11,33–35]). As for Bach’s fugue, all its shape-generating means (mode, chords, functional sequence, sequences, the tonal plan e. t. c.) are highly thematic as they refer to the initial problem, the actual theme of the piece. The continuous interconnection of all elements of the fugue, in particular in terms of their relation to the main problem, and the validity of their choice in an actual context makes the piece integral and intellectual, an indisputable masterpiece. The small piece, lasting one minute and a half, has been proven to show a crucial historical transition from modality to tonality, the transition lasting several centuries which followed the same standard scheme [8].

So, the whole history of music appears to be is a chain of transdimensional transitions. The key transition was from monody («one-dimensional music space) to two-voice texture, or music «two-dimension», the X century, the “Organum style”. Thus, the new methodology (from analysis to synthesis via trans-dimensional transition) reveals in musical pieces some dynamical standard scheme, an ideal model (including the Fugue by Bach) which we believe to be universal [13, 32–36]. It should be pointed out, that all the pitch spaces of the given fugue, which make up the whole (one dimensional- mode, two dimensional-intervals, three dimensional- chords) are linked by the initial problem, which is given in the first statement of the theme. Such a transdimensional link is a “stratification” (on the vertical axis) in conjunction with a transdimensional transition (a horizontal link), forms a «transdimensional cross» (see ex.4), on which the whole is divided. The presence of such transdimensional links is the factor which distinguishes an organic Whole from a mechical Conglomerate.

Transdimensional transitions in other art forms

And how does one explain the issue of transdimensional transitions in other art forms?

A careful study of this question reveals that the transition from disjunction to conjunction and the transdimensional relations that come with it is characteristic of the majority of masterpieces from all different styles and eras in literature (Vygotsky), painting (Florensky), cinema (Eisenstein), theater (Stanislavski), as well as for tales (Propp), myths (Levi-Strauss) etc. [9,10,13].

Let’s look into the transdimensional transitions (or how disjunction proceeds into conjunction) in the works of literature, for example. We should address to the superb analyzes that L. Vygotsky makes in his book “Psychology of art” [2]. The works are constructed upon the same scheme: a narrative layer is given with some “anomalies” — contrasting elements of the antithetic second layer (disjunction) — appearing. In the process of development of the plot these two layers are starting to intercross each other with a bigger intensity. The climax (or, “the catastrophe”, as Vygotsky says) is based upon the combination of two opposite layers (conjunction) and creation of qualitatively new unity in which two different layers become a projection, one dimension of the whole. For example, this (universal!) scheme is found by Vygotsky in a novel of I. Bunin called “Easy breathing”.

The opposition of “light” and “heavy” is synthesized in the climax of the novel into a “light-heavy” two-dimensional unity [2, 193-195]. That’s how two opposing processes in the narration of “Hamlet”, that are connected with the action and its procrastination or reflection (“To be or not to be”), find an bright and unusual solution in a climax of the whole piece – in the scene of a duel and Hamlet’s death. Here the processes that are opposite to each other and are developing separately are finally united: Hamlet is active (“To Be”) when he is already poisoned (“Not to Be”). In connection with this a thought comes to mind, that the mysterious “catharsis” actually is a transdimensional transition into a bigger panoramic view, into a meta-space which isn’t reflected in our minds consciously, but is deeply felt.

Let’s make a short excursus into mythology. It is known that K. Levi-Strauss has found a universal scheme of a myth, in which an oppositional synthesis (“mediation”, as he called it) is the main semantic kernel. “The synthesis of opposites is a universal teleological principle of the myths of all time” [3,47]. A “successful” mediation leads the hero into a new dimension, a new stage of freedom (quite often a literal, not a metaphorical one). It seems that the transdimensional transition is the next step. Even Levi-Strauss himself had a thought about a new quality of a resulting conjunction in reference to the initial disjunction, which had found its reflection in the so-called “canonic formula of Levi-Strauss”[3, 45]. But even in such a formula the main idea isn’t represented. The idea is that the disjunction and conjunction are placed in different dimensions!

To summarize: the meaning of a myth, of music, creative work and evolution in general is the intention to reach a state of freedom or, to be more accurate, augmentation of the degrees of freedom via trans-dimensional transitions! This approach made in clear the «ideal essence», and the «universal plot» of all masterpieces ever created.

Polydimensionalism and symmetry-asymmetry

Let us look at a special case of transdimensional relations- «polydimensions» (abbreviated PD), meaning an object, which belongs to spaces of different dimensions (see ex.4). 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 [14]. 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.