Schillinger – Part 2

THE SHILLINGER SYSTEM OF MUSICAL COMPOSITION

2.1 Overview

The origins of Schillinger’s ’system’ and his concept for it are unknown at this stage however; similarities are evident between the foundation of his system (the Theory of Rhythm) and Nikola Tesla’s theoretical work on the invention of alternating current electricity. As is widely known today, alternating current operates on the same principle as sound waves where the periodic motion moves through a 360º ‘phase’ and generates energy. That energy can be increased when the phases of two or more alternating currents are combined. While Schillinger doesn’t speak directly of a connection, he does write extensively of this type of periodic motion as being the scientific basis of all the arts. Furthermore, where phase interference between two cyclical currents generates electrical energy, Schillinger asserts that the ‘interference of periodicities’ (his term for phases or cycles) generates aesthetics.

Understanding Schillinger’s Theory of Rhythm is central to understanding his whole approach. For him, rhythm is not simply a matter of time-rhythm. Schillinger begins by applying rhythm to time durations and then extends it to all other stages of composition – the way in which block harmonies change, intervals in scales and melodies, entrances of counter-themes in counterpoint, distribution of parts through a score, and other processes of composition. These processes are broken down into twelve ‘books’ within the Schillinger System of Composition (hereafter referred to as the SSMC).

2.2 Book I – Theory of Rhythm

Schillinger states at the outset of chapter one of this book that the ‘customary method of musical notation…is inadequate for the analysis and study of rhythmic patterns.’ He then introduces three parallel systems of notation that he will use, initially for the notation of time-durations but ultimately for other components of music and composition in every other book within the SSMC. These are: (1) numbers, (2) graphs and (3) musical notes.

The use of musical notes is self-explanatory and numbers are used in their normal mathematical operations, such as the four actions of addition, subtraction, multiplication and division as well as raising to powers, extracting roots, and for the ease with which they can be used to create permutations of patterns.

However, it is the way in which Schillinger employs graphs that is the key to his system. The graph method he uses is the same used to record variations of components during a time period (stocks in finance; diseases in medicine, etc.). In Theory of Rhythm, Schillinger only deals with time (pitch, intensity and so on are dealt with later). The horizontal coordinate (the abscissa) read from left to right expresses time. The vertical coordinate (known as the ordinate) is used to express the recurrence of a phase, i.e., the moment of attack.

He then goes on with an explanation of basic wave motion and illustrates the way a sine wave is represented in his graph system as a square wave.2 This periodic motion is measured in phase units Schillinger calls periodicities of phases or simply, periodicity. The coincidence of two phases of two difference periodicities intensifies the attack. The recurrence of intensified attacks (“accents”) constitutes musical measures (“bars”). In Schillinger’s words, “The reality of bars depends actually on the placement of attacks, not on the placement of bar lines on music paper.”

Schillinger then illustrates various forms of continuous recurrence of attack groups – i.e., periodicity. The most basic form of periodicity in which all attack groups are identical is called uniform periodicity or monomial periodicity. Using numbers, these could be represented as:

1 + 1 + 1 + 1 + …
2 + 2 + 2 + 2 + …
3 + 3 + 3 + 3 + …
n + n + n + n + …

where each number represents a unit of graph measurement on the abscissa.

From here, Schillinger explains the way the numbers and graph representation can then be converted into musical notation simply by using those units to represent note durations. For example, “1″ could be said to represent a quarter note in which case “2″ represents a half note, “3″ a dotted-half note, “4″ a whole note and so on.

What follows after this is what Schillinger calls “generation of resultant rhythmic groups as produced by the interference of two monomial periodicities – that is to say, the way in which one monomial periodicity (say, 3, 3, 3, 3) may be combined with another (say, 4, 4, 4, 4) so as to produce another rhythm.” In its simplest form, the resultant of this type of interference is syncopation. The remainder of Book I is then given over to a thorough investigation of every possible type of interference and the way these resultant patterns (non-uniform periodicities) can themselves be combined to generate an infinite number of rhythmic patterns.

Ex. 1 below shows both rhythm and pitch relationships of Beethoven’s Minuet in G in graph form. The perceptual similarity to a city skyline should also be noted as Schillinger asserted his graph method worked both as an analytical tool and as one that could generate any musical component from anything with a pattern that could be represented in the two dimensions of his graph system. In other words, composers didn’t need to be limited to musical patterns when composing. He also asserted that the aesthetics of any pattern that was visually appealing could be directly translated into something of comparable aural appeal.

2.3 Book II – Theory of Scales

 The same universality of approach characterizes Schillinger’s Theory of Scales. He classifies any number of tonal units, beginning with a one-unit scale that represents a single note repeated. He also introduces the concept of symmetric scales based on the division of the octave into equal parts (2, 3, 4, 6, 12). Still more new groups are obtained by division of two octaves into three parts, three octaves into four parts, and so on. Interpolating passing notes to fill in gaps between pivotal tones can also create more new scales.

Familiar and unfamiliar scales can also be created by simple permutation of half-step combinations (2 + 2 + 1 + 2 etc.). Any scale can be created this way.

Schillinger then relates the resulting progressions to chordal structures derived from these scales. In other words, he explores modulation as a melodic event as well as a harmonic event. Example 4 shows how an octave divided symmetrically into three equal parts can generate a chordal root movement, as was done to great effect by John Coltrane in “Giant Steps”.

2.4 Book III – Variation by Means of Geometric Projection

This is the slimmest of all the books within the SSMC and perhaps for the non-mathematically minded, the most difficult to penetrate for meaning. The book is divided into two parts: (1) Geometrical Inversions and (2) Geometric Expansions.

In many respects, the processes of inversions that Schillinger explores have origins that date back at least as far as Bach. His graph method provides an easy to interpret visual guide to inversion. The second part with its mathematical permutations used to generate new scale and melodic material has all the hallmarks of pitch class set theory. However, Schillinger uses visual art references to illustrate the way ‘geometric rotation’ can be used to distort an image without losing the connection to the aesthetics of the original. This type of graphic manipulation we take for granted these days in computer applications such as Photoshop.

2.5 Book IV – Theory of Melody

In Schillinger’s conception, melody is the musical projection of a curve that thus can firstly be plotted using basic trigonometric operations and ultimately synthesized. Much of the introduction he writes to this book speaks of his most basic aesthetic that composition, particularly insofar as the composition of melodies is no different to the engineering involved in building a house. He is openly critical at this point about the lingering romantic-era notion of the composer as ‘divinely inspired genius’. It is perhaps this outspoken opinion, more than anything else Schillinger did, that led to music institutions of his time dismissing him and his theories out-of-hand.

Book IV then applies the graph method as a means to represent ‘axes’ of balance. The dynamic quality of a melodic line is determined by alternation of balancing and unbalancing axes, which are melodic fragments directed to and away from the axis of perfect balance. This latter represents the tone of maximum duration in a given melodic fragment. Schillinger further relates the arithmetical summation series with melodies in which each successive interval is the sum of two or more preceding intervals counted in semitone units. He also demonstrates how intervals can my permutated using multiplication so that a Bach fugue would sound like a piece by Debussy in whole tones if the intervals were multiplied by a factor of two, since all the semitones would vanish in the process of intervallic doubling.

2.6 Book V – Theory of Harmony

Schillinger includes an account here of orthodox procedures but its methodological value lies in the greatly generalized idea of stratified harmony in which single tones are formed into two-note, three-note and four-note combinations. These then serve in turn to form larger harmonic structures or, as Schillinger calls them, “pitch assemblages”. He also includes the concept of the voice leading ‘circle’ that produces motion between chords in a systematic manner (see Example 5). Ascending and descending root motion patterns are also described in mathematical (and therefore permutable) terms. Not only is the familiar cycle of fourths explored, but cycles of 2, 3, 5, 6 and 7 are also detailed.

2.7 Book VI – The Correlation of Harmony and Melody

Additional harmonic considerations are shown and exhaustively examined in numerous musical exercises both in this book and the one that follows. The material points out that the familiar Western European system of diatonic harmony is just the tip of the iceberg since, according to Schillinger, “any scale can form the basis of a diatonic or symmetric system from which characteristic intervals and chords may be derived.”

Other things explored here cover 7th, 9th, 11th and 13th chords and techniques for voice-leading upper structures (see Example 6) – hybrids and upper structure triads.

2.8 Book VII – Theory of Counterpoint

This book is sub-titled ‘The Technology of Correlated Melodies’ and it delves deeply into things such as correlation between two melodies, attack groups in two-part counterpoint, counterpoint with symmetric scales, canons and imitations, and so on. Harmonization of two-part counterpoint is also explored, as are melodic, harmonic and contrapuntal ostinato.

2.9 Book VIII – Instrumental Forms

Instrumental form for Schillinger means “a modification of the original melody and/or harmony which renders them fit for execution on an instrument.” Discussion of attack rhythms first explained in Book I is further explored here in instrumental forms. Further explanation is also given to the concepts of stratified harmony and counterpoint from preceding books.

2.10 Book IX – General Theory of Harmony

The focus on stratified harmony continues here. Schillinger begins with ‘one part harmony’ and ‘one stratum of one-part harmony’. From here he works through two-part harmony, starting with one stratum of two-part harmony, then two parts, then hybrids of those as well as diatonic and symmetric ‘limits’ of compound structures. Similarly, three-part and then four-part harmony is explored before he moves into the area of quartal harmony.

The second half of this book looks deeply into applications of the preceding. Schillinger addresses the issue of melodic and harmonic continuity, accompaniment, composition of canons from strata harmony as well as ‘density’ and its applications to strata.

2.11 Book X – Evolution of Pitch Families

This slim book is a summary of sorts of pitch scales, harmony and melodization of harmony.

2.12 Book XI – Theory of Composition

This book is perhaps the crowning achievement of Schillinger’s ’system’. Here, he methodically explores the components of a composition, beginning with thematic units, moving into thematic continuity and strategies for planning a composition, and finally into ‘Semantic (Connotative) Composition’.

It is in this section that Schillinger introduces what he calls the “psychological dial” – a representation of ‘quadrants’ analogous with the Myers-Briggs Personality Type indicator. A dozen or so chapters are devoted to the semantic basis of music, composition of sonic symbols and finally, the composition of semantic continuity.

2.13 Book XII – Theory of Orchestration

This final book in the SSMC addresses timbral considerations for all orchestral instruments as well as electronic instruments that Schillinger divides into two groups: (1) Varying Electro-Magnetic Field instruments such as the Theremin and (2) Conventional Sources of Sound such as an electrified piano and the Hammond Organ.

The second half of Book XII is devoted to ‘instrumental techniques’ (orchestral forms and so on) and then intensities, attack forms and pitch ranges as they relate to instrumental combinations.

In the final chapter of Book XII, is titled ‘Acoustical Basis of Orchestration’. It’s only a page long and the editors note the original manuscript didn’t finish at this point. However, the publisher decided the material that followed was incomplete and that much of the material on orchestration had already been presented in earlier books.

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2 It’s important not to confuse this square wave as representing a sound. It merely represents attack and intensity of a recurring phase cycle.