A Basic Introduction to Eleven Pitch Class Systems Analysis

Pitch Class Systems Analysis is quite unique in that it can be applied to a wide range of musical genres that have evolved over the course of music history. Indeed, systems analysis is the only theoretical construct that truly offers a unified approach to the many disparate areas of Western art Music ranging from late medieval modality to twentieth century atonality. Therefore, before we begin with an explanation of the theory itself, a cursory explanation of two very broad terms should be discussed as they relate to music theory in general.

What is Tonality?

Tonality is a musical system that arranges pitches or chords to induce a hierarchy of perceived relations, stabilities, and attractions. In this hierarchy, the individual pitch or triadic chord with the greatest stability is called the tonic. The root of the tonic chord is considered to be the key of a piece or song. Thus a piece in which the tonic chord is C major is said to be “in the key of C”. The most common use of the term …”is to designate the arrangement of musical phenomena around a referential tonic in European music from about 1600 to about 1910″[1]. Contemporary classical music from 1910 to the 2000s may practice or avoid any sort of tonality which is commonly referred to as atonality.

In sum, tonality is an organized system of tones (e.g., the tones of a major or minor scale) in which one tone (the tonic) becomes the central point for the remaining tones. The other tones in a tonal piece are all defined in terms of their relationship to the tonic. In tonality, the tonic (tonal center) is the tone of complete relaxation and stability; the target toward which other tones lead. The cadence (coming to rest point) in which the dominant chord or dominant seventh chord resolves to the tonic chord plays an important role in establishing the tonality of a piece.

What is Chromaticism?

Chromaticism is a compositional technique interspersing the primary diatonic pitches and chords with other pitches of the chromatic scale. Chromaticism is in contrast or addition to tonality or diatonicism (the major and minor scales). Chromatic elements are  considered by the majority of music theorists to be “elaborations of or substitutions for diatonic scale members”.[1] In actuality, this elaboration often becomes a vital core element providing the composer a rich palette of ‘color’ that serves as fertile ground for significant development.

Thus we come to the essential difference between this theory of systems analysis and most  theoretical constructs: Systems analysis is not based solely on the diatonic aspect of the various tonal systems exploited by composers; rather, the theory is chromatically based- the chromatically inflected octave being the source not only for a highly ingenious developmental dialectic, but also encompassing the moment-to-moment progression of the musical narrative itself. In sum, we believe that the process of compositional development may be defined by a chromatic background that coexists with a diatonis contrapuntal background.

With all of this in mind, we may now turn to a more detailed systematic discussion of the basic principles of the theory:

The Two Basic Tenets of Eleven Pitch Class Systems Analysis

  1. The Primary Chromatic Array: Structure and Function

The twelve tones of the chromatic scale are present in almost any piece of tonal music whose length is of any consequence. Indeed, in any work admitting to functional triadic tonality, the introduction of all twelve tones of the chromatic scale will be unavoidable. In its most fundamental form, this underlying chromatic construct unfolds in a precise order, by rising half steps, throughout a single-movement composition or a single movement of a larger work. The question is not whether the total aggregate of pitches will be completed – that is a given – but, rather, how the aggregate is partitioned as a future source of developmental material. As such, the identification of the PCA represents a great departure from the current standard practice of musical analysis which attempts to explicate the composer’s developmental process by concentrating to a great degree  on the manipulation of thematic material or the outlining of a work’s overall harmonic structure as it relates to diatonicism.

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2. The “Missing Pitch”

In all but the simplest of compositions whose length is of any consequence, whether modal or tonal, the act of prolonging the final or tonic through harmonic cadences requires the chromatic alterations of diatonic pitches. Thus any given mode or key will naturally comprise both diatonic and chromatic pitch classes which are drawn from the larger systems to which they pertain. However, no mode or key may have more than eleven pitch classes. The twelfth or “missing pitch” of any given mode or key is that which is not included in the respective system of that mode or key. If the twelfth pitch class is included, this signifies a modulation of the system either up or down depending on how the pitch class is spelled enharmonically and whether we are dealing with a mode or a key. The missing pitch, thus, is a “system motivator;” that is, an individual pitch class that may provoke a modulation from one eleven-pitch-class system to another. The fascinating point to be made here is that Henry Burnett realized that in each composition, the note that was missing from the twelve available ones, taking into account the “key” in which it was written and discounting enharmonic variants, was the same. The missing pitch is invariably the minor third or augmented second above either the central hexachord of the modal system or of the tonic system of a key.

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