Warning: The following essay is not intended for children as it has content suitable only for the adult mind. Worse than that, it assumes a fair degree of musical sophistication, thought, vocabulary, education, experience and knowledge. Someone else might be able to expand this brief essay, wherein I say everything that I care to say, into a full scale textbook - following the title with "- For Dummies". My intention is quite the opposite. Since what I say I think should be intuitively obvious to the intended audience, the brief exposition is very short on argumentation to a viewpoint and shorter yet on didactic examples that specialize the abstractions. If this makes no sense to you whatever, know that it makes perfect sense to me, and that I figure you should be reading something else.
The Origins of Musical Forms
The musical forms of the era of "classical harmony" are derived from those very harmonic laws. In this sense the forms have a natural integrity intertwined with the language in which they are expressed. Should one expect less from an atonal idiom? I would hope not.
The existence of the problem of musical form can be seen in the aphoristic works of both Schoenberg and Webern, e.g., already in Schoenberg's Sechs Kleine Klavierstuecke Op. 19.
The problem is, if your language is strictly atonal, you have discarded the harmonic and tonal underpinnings of form; your compositions must, per force, return to small, "durchkomponiert" forms. This, of course, is exactly what both Schoenberg and Webern did. After that, with two pieces of clearly similar material and contrasting treatment, binary AA, AA' or ABA forms become possible. In extending such patterns to a ABACADABA (rondo) forms it becomes increasingly difficult to justify these architectures, unless perhaps the extended form is itself brief in which case one has a structure for a unary form. One might tend to more palindromatic forms. Large scale forms of several distinct movements are indeed difficult to justify without tonal or quasipalindromatic underpinnings.
Once Schoenberg introduced his method of composing with twelve tones, he seems to have thought the problems of form had been solved, but then proceded to stuff a dodecaphonic language back into the set of classcial forms. This idea truly lacks artistic integrity, and I believe that this was not a good idea.
Schoenberg's dodecaphonic method, in solving the problem of pitch coherence in an atonal idiom actually introduced an equally pernicious problem in that dodecaphony sought to destroy systematically all tonality. It seems that at that time only someone with the rarified mind of Anton v. Webern could inhabit such a severely restricted space. He refined further the dodecaphonic concept by by imposing certain symmetries on the row; these symmetries then suggested the symmetries of increasingly larger structures, starting with thoroughly minimal structures, like his piece of solo cello that lasts all of 11 seconds. In this way, Webern created forms that were integral to the musical material.
The Origins of Musical Gesture
In classical harmonic language, the language is conceived in terms of scales (sequences of pitches ordered by ascent or descent) and triads built on each degree of a scale. The natural gestures or language fragments are determined by combinations of scale like and chord like motions of lines and events. The fundamentals of the language go further in determining the next level of fractal like structure in determining how the various gestures are combined and related. The successive fractal selfsimilarities of structure ascend all the way up the the structural hierarchy to the global architecture, so that ultimately the entire lingustic structure is determined by an assumed set of logical fundamentals. Exactly how this becomes instantiated depends on the composer and the work; obviously, even with such meager beginings, there is much room for creativity and inventiveness. The point here is that the nature of the set of logical fundamentals matters a great deal to the gestural, expressive and architectural structures that flow from it.
Similarly, if dodecaphony is assumed, new fractal like types are evolved that will not be the same as those generated within the langauge of classical harmony. A significant artistic problem is to understand all over again, with dodecaphony, how gestures, their combining and the successive combining of combinations conspire to evolve architectural forms possessed of a musicolinguistic integrity.
More on Webern's Solution
Webern realized that this new seemingly sufficient constraint of Schoenberg's method was not enough of a constraint because it did not provide for natural higher levels of organization, and was not enough of a constraint again because, with a twelve tone row, too many possible transformations of material were possible within a necessarily small and simple architecture.
His novel solution was to steal from the simple transformations of fugal context, inversion, retrograde and retrograde inversion in order to create twelve tone rows with a high degree of symmetry, thus reducing the effective musical material within a row, creating an inherantly higher automatic degree of cross relations, and at the same time, give even a greater latitude in dramatic expression. However, given that in his Konzert Op. 24, for orchestra, the core material is a simple three note motive, the gestural expressiveness becomes naturally laconic in the extreme, where contextually, a phrase can be reduced to a single note.
The works of Webern will be seen to have a profound influence on the works of practically all composers who followed him. While he took his own language to its logical extreme, the ideas and techniques implicit in them have still left a remarkably fertile and insightful legacy from which composers have drawn.
Klangfarben and its Elucidation of Structure
In his later orchestral works, Webern sought further tools of cohesion for yet larger architectures and found one in the structured use of tone colors (Klangfarben). These works are very sparsely orchestrated so that instrumental color is always almost of a single color; contrasting colors are frequently used. The repetition of a color can sensibly bind and group pitches that would otherwise not be so bound, thus overlaying an additional, possibly interpretational (semantic), structure on the underlying pitches.
In his orchestration of the Ricercare from Bach's Musical Offering, using this technique, Webern creates a new perspective of the material, an elucidation of the material that exposes an existing fine grained structure of the material that would otherwise be hidden both by the sustained density of the counterpoint, and by a historical aculturation of the ear unaccustomed to the density of relationships existing in Bach's own music. Whether or not one likes or appreciates this revisting of Bach's music, it is nonetheless pointedly didactic.
Music of all artistic forms is the one most related to mathematics; indeed, it has been intertwined with mathematics since before Pythagoras, and remained so conceptually and theoretically. The connection is very simple (not that all such connections are necessarily simple); all musical events, structures and relationships can be specified in principle by numbers or sequences of numbers. That given, a great body of mathematics can be applied to transform and manipulate the substance of music. See, e.g., my rather longer essay on Musical Patterns and transformations described through combinatorics and group theory, or another rather more mathematical essay Lie Algebraic Analysis of Motivic Structure (and then ultimately of all structure). In this context, it is a great temptation to reduce the composition of music to mathematical calculation.
The Limitations of Serialization
That temptation has been yielded to, with interesting though not exactly spectacularly meritorious results. The first yielding, motivated by dodecaphony itself and Webern's systematic exploitation of Klangfarben, was not only a serialization of tone color, but also the additional serialization of musical parameters other than pitch and color. This would create a higher density of cross relations automatically, and also constrain the material, generally increasing its coherence. There is, however, a deletorious effect in that such overserialization severely restricts dramatic expressiveness of structural semantics, thus causing much music structured in this way to lose contact with a human audience, although I'm sure some computer might ajudicate it to be quite grand.
An essential aspect of music is that it is the product of personal neural systems seeking to communicate, using a common language, with other personal neural systems. If this aspect of music is forgotten, it destroys the distinction between music and applied mathematics. Mathematics is not thereby enriched, and music is sorely diminished.
The Problems Anew
With the breakdown of classical harmony, often signaled by musicologists with the famous tonally amorphic "Tristan Chord" in the opening measures of Wagner's Tristan und Isolde, the problem of musical composition, that of reestablishing some new meaningful sense of tonality remains. In the history of dodecaphony and its successor-imitator of total serialization, the ignoration of the problem and an absence of a solution becomes increasingly selfevident.
The very monotonous circulation of the twelve tones actually prohibits a solution to the problems of "tonality without tonality", and "harmony without harmony".  Pitches provide "chroma", and a uniform mixture of them equally weighted gives a uniform gray to the overall sound; in pitch, the ear is never surprised or refreshed. This is impossible in any other than the smallest of pieces; and in the alternative, worse than the very dullest of academic fugues where all is completely and ineluctably predictable. Music to commit suicide by.
Perhaps, intervals and intervallic contrasts provide an even more immediate sense of chroma, and one can consider within a dodecaphonic context, an intervallic Evolution of Tone Rows generated through a shape preserving evolution of a three note motive whose inversion, retrograde and retrograde inversion combine to form a row of Webernian symmetry. In the Konzert Op. 24, the fundamental row has the overall symmetry of being equal to its retrograde inversion.
Although this kind evolution is a perfectly sensible way of extending the the usefulness of dodecaphony in that it provides surcease of intervallic boredom, and provides also a reasonable force in the dynamic change of material, allowing longer discourses, it still does not address the problem of the tragic loss of tonal center. The equidistributed pitches still circulate restlessly inducing a similar reaction in listener's minds.
A First Postwebernian Nonserialistic Solution
Pitch sets of twelve tones do not work to relieve the lack of chroma, since then you have nothing but what you started with, an unordered set of twelve tones. If, however, the set is restricted, dynamic possibilities arise by connecting, expanding and contracting and contrasting distinct pitch sets. One can see the intimations of this direction in the works of Webern, where fragments of horizontal row material are pulled together in vertical expression.
The clear implication of Webern's transformations between horizontal and vertical expositions is the pitch set, the circulations of which, mixing the vertical and horizontal aspects of its reorderings is perfectly audible. This particular way of breaking with dodecaphony which nevertheless maintains a cognizance of a twelve tone background, immediately restores the freedom required for the pitch material to express its own dynamics dramatically. Serendipitously, it also restores senses of tonality and chroma, and even multitonality where the central tonus can be a set of pitches, and not simply a single pitch. The functioning of a pitch set then reaches back historically to connect with the triads and their chordal extensions to, e.g., seventh and ninth chords built on the degree of the scales.
One also comes to a generalized concept of harmony through pitch sets. One of the primal questions of harmony, the one that defines its dramatic function, is what is Consonant and what is Dissonant? The history and literature of music have shown that these concepts of consonance and dissonance are neither fixed nor absolute concepts, and that they must be understood contextually and functionally.
E.g., in the context of any I chord prevailing harmony, in either major (church Ionian) or minor (church Aeolian) mode, a V chord is relatively dissonant; out of that harmonic context the V chord isn't a V chord, it is simply an isolated chord.
Harmonic dissonance then is a function of contextual relationships that, again contextually, creates a sense of tension; this is why one often speaks of "the resolution of a dissonance".
If one analyzes music written in the classical language of scales and chords to observe the functioning consonance and dissonance, a pattern emerges which shows that there is far more consonance than disonance and that it is the sparse set of disonances that convey the most musical information, exhibit the novelties or "radical events" and which are concommittantly the dramatic peaks.
What then is the great function of consonance?
Consonance provides for dissonance since it is the normative background against which the disonance acquires its substance and meaning.
If the background material has the flavor of minor ninths and minor seconds which provide a definition of consonance, the surprise of a major third is a harmonic dissonance; this is true harmonically, irrespective of one's preconceived feelings about the intrinsic consonance or dissonance of these intervals.
These historically reinforced distinctions between intrinsic consonance and intrinsic dissonance for which there is "apparently" no physiological reason, are exactly that, historical and bound to certain specific prior concepts of harmony. One may then also write, knowingly, pitting the context of a composition against the context of musical history to greater and lesser degree.
This understanding of the dramatic dynamics of: pitch sets; the functions of pitch against a twelve tone background; and the derived generalized concepts of harmony and tonality, together with the historical path leading to all this; is exactly the complex understanding that lurks beneath the music of Raoul Pleskow and also that of Stefan Wolpe.
 The problems are here expressed in the form of "koanic Wheelerisms" after John Archibald Wheeler, one of the world's great theoretical physicists, who has used such forms to express some of the deep questions and problems of physical theory.
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