Talk:Schillinger System
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it seems the author of this article has problems with the markup - especially images are missing. If you don't feel comfortable with the syntax just point me to the location of the images and I'll do the links for you. Just answer me on my discussion page. It would be a shame for this great article to stay in this half-finished condition.
- Edit of 31 December 2005 seems to have introduced something from [1], a Ph.D. dissertation (see Ch.2 Overview, matching closely the Overview section). Possible copyvio issue, therefore. On the whole the article needs much work, to cut out diffuse and POV parts. Charles Matthews 11:54, 1 February 2006 (UTC)
Copying here the long central section of the page, which is a likely copyvio/paste in by author (see comment above). Charles Matthews 19:07, 7 February 2006 (UTC)
[edit] Overview
The Schillinger System of Musical Composition is an ambitious attempt to provide a complete theory of musical creation. The entire work is contained in two volumes divided into twelve sections each of which occupies a separate Book.
Book I. Theory of Rhythm. Book II. Theory of Pitch Scales. Book III. Variations of Music By Means Of Geometrical Projection. Book IV. Theory of Melody. Book V. Special Theory of Harmony. Book VI. The correlation of Harmony and Melody. Book VII. Theory of Counterpoint. Book VIII. Instrumental Forms. Book IX. General theory of Harmony. Book X. Evolution of Pitch Families (Style). Book XI. Theory of Composition. Book XII. Theory of Orchestration.
In order to communicate the essence of Schillinger's work I offer this brief summary of each Book.
[edit] Book I the Theory of Rhythm
The Theory of Rhythm is the foundation of Schillinger's work and its ideas and techniques apply throughout his writing on music. Every dimension of music must occur in a time-period: without measurement of time, either intuitively through memory or objectively in beats and seconds, there can be no musical experience. The temporal flow of music is primarily a matter of rhythm. Schillinger’s ideas about rhythm come from basic wave theory: waves collide and their energy is either cancelled or reinforced, the patterns formed by these encounters are rhythms. Nature may produce irrational, complex or chaotic rhythms, but music is largely dependent regular periodic motion. Schillinger founds his theory on the phenomenon of pulse interference, in which the colliding pulses have different frequencies but begin at exactly the same moment. Pulse interference contributes to almost every aspect of music, from the animation of a phrase to the timbre of a sound; ultimately orchestration becomes a branch of rhythm.
[edit] Meter
In figure 1, each column represents a unit of time. Pulse A recurs every three units of time and pulse B recurs every one unit of time. (A = 3, B = 1). The pulses are represented by down arrows. The double arrows show the effect of two pulses combining to create an especially strong pulse.
A ↓ ↓ B ↓ ↓ ↓ ↓ ↓ ↓ Result ⇓ ↓ ↓ ⇓ ↓ ↓ Figure 1. Pulse interference creates metrical accents
The strong pulse can be interpreted as a downbeat or bar line and in this way, Schillinger explains the phenomenon of meter. Meter occurs when B = 1 and A is an integer multiple of B, for instance, 2, 3, 4, 5 etc… A meter such as 3/4, three beats in the bar, can be described as a pulse ratio, 3:1, and a ratio of 4:1, produces the effect of a bar with four beats, for example, 4/16, 4/8 or 4/4.
[edit] Rhythm
In figure 2, interference between the two pulses A and B produces the effect of rhythm. The requirements for this are as follows: 1) B ≠ 1 2) A and B have no common divisor other than one. Pulse ratios with common divisors, such as 4:2 or 9:3, simply breakdown into forms of metrical regularity, whereas, 3:2, 4:3 or 5:2, produce various alternative types of rhythm.
A=3 ↓ ↓ ↓ ↓ B=2 ↓ ↓ ↓ ↓ ↓ ↓ Result (A+B) ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ Result displayed numerically 2 → 1 1 2 → 2 → 1 1 2 → Result in music notation q e e q q e e q Figure 2. Two cycles of the rhythm created by pulse interference at the ratio of 3:2.
In figure 2, two complete cycles of interference are shown; the right-angled bracket above the diagram indicates the beginning of the second cycle. Pulse A recurs every three units of time and pulse B recurs every two units of time, (A=3, B=2). The third row shows the result of interference, that is, the combination of the first and second rows. In this example the moments when A and B combine, potentially creating a downbeat or bar line, are not shown (as they are in figure one) in the result row, (A+B), since the rhythm can be barred in several different ways.
[edit] Number Patterns and Notation
As can be seen from figure 2, rhythms can be expressed as numbers representing duration between the onsets of pulses. Number patterns are extremely versatile and alongside traditional notation allow for the easy comparison of rhythmic structures and proportions as well as the possibility of applying mathematical techniques to the problem of rhythmic composition and development. Traditional notation has proved extremely efficient in recording musical ideas but it does have defects. For example, in figure 3, the rhythmic pattern (1,2,3) remains unchanged from one bar to the next but each notated example looks completely different. The only audible change is the tempo of the rhythm; the second example is faster than the first.
Figure 3. Music notation can obscure the fact that rhythmic proportions remain unchanged.
On the other hand, numbers represent duration between the onsets of pulse but do not tell us anything about their final musical presentation. For example, the number 2, might represent a note held for two beats but could equally represent a staccato attack followed by a measured silence.
Figure 3. Numbers do not prescribe the final musical presentation of duration.
[edit] Symmetry
All rhythms generated by pulse interference are periodic and each cycle is symmetrical around its centre. For example,
Ratio of pulse interference Numeric result 3:2 2,1↔2, 1 4:3 3,1, 2↔2, 1, 3 5:2 2, 2, 1↔1, 2, 2 Figure 4. Pulse interference patterns are always symmetrical. Double arrows indicate the point of symmetry.
Schillinger suggests that symmetrical rhythms have important musical qualities: economy, since one half generates the other, balance, due to the mirror symmetry and a quality Schillinger refers to as contrast, the difference between successive numbers. For example, taking the pattern generated by 3:2, in figure 4, the contrast between the numbers two, and one, is two minus one equals one. The greater the difference between numbers the greater the contrast.
[edit] Applying rhythm
A rhythmic pattern is a set of proportions that can be used in numerous ways throughout the process of composition, providing the basic rhythm of a phrase, or, in a structural capacity, as the key to flow and form. Numbers are flexible: they can be sub-divided, added together, scaled, re-ordered and subtly transformed from one value to another using nothing more than simple arithmetic and yet the musical results can be dramatic. For example, coefficients of recurrence are frequently used throughout the System as a means of controlling flow and proportion. Here the elements of a rhythmic pattern determine the frequency with which any musical parameter is heard. For instance, the rhythm (2, 1, 1, 2), might be used as coefficients for two themes, A and B, resulting in the scheme 2A, 1B, 1A, 2B, where theme A gives way to theme B. Rhythms, as number patterns are not just temporal schemes and can work equally well in the field of melody and harmony to generate scales and chords and progressions.
[edit] Organic forms of rhythm
The ancient Egyptians and Greeks succeeded in incorporating the quality of growth into their architecture and design. The mathematics of Dynamic Symmetry, in which each number of a series is the sum of the preceding two numbers, was formalised in the early 13th Century by Leonardo Pisano Fibonacci, after whom the series is named. The Fibonacci series is simply represented by the following series of numbers:
1, 2, 3, 5, 8, 13, 21…
During the 19th and early 20th century, it was shown that the Fibonacci series, described the angles and patterns of growth in many biological forms, for instance, the horns of wild animals, the shells of crustaceans, and the petals of flowers . In music growth series effect rhythmic acceleration and retardation. The connection between biology and systems of proportions in art, led Schillinger to develop a theory of psychological response.
" Thus we see that forms of organic growth associated with life, well-being, self preservation and evolution appeal to us as forms of beauty when expressed through the art medium. Intuitive artists of great merit are usually endowed with great sensitiveness and intuitive knowledge of the underlying scheme of things. This is why a composer like Wagner is capable of projecting spiral formations.... without any analytical knowledge of the process involved. "
Building on the idea that art imitates nature, Schillinger says,
" Musical patterns, viewed in the universe of physical, biological, and aesthetic objects, are only special cases in the general scheme of pattern making."
[edit] Distributive powers
One of Schillinger's most original ideas was to apply power series to musical proportions. He observed a common tendency in Western music towards bars of four beats and bar groups of four bars. The classical bar group consists of 16 beats, the square of the number of beats in one bar of 4/4. Other meters, for instance, bars of three or six beats also tend towards grouping in units of four bars, a fact which lead Schillinger to suggest that the number two, its multiples and powers were a defining feature of Western music.
Figure 5. Mozart, Eine Klieine Nachtmusik, a typical classical bar group: four bars of four beats.
Schillinger’s knowledge of folk music from various cultures and his strong mathematical instinct, lead him to believe that metrical structures of bar groups should be considered as belonging, at least ideally, to a power series. Existing music could be explained as aspiring to a pure form given by the power series. Bar groups need not be subject to grouping by two and its multiples, but instead in accordance with the meter itself: a bar of three beats forms part of a group of three bars and a bar of five beats forms part of a group of five bars. This idea leads to the development of a series of extremely potent routines and formulae, which generate related but independent rhythms. All rhythms made in this way are perfectly in accordance with the bar group and consequently a polyphonic score of great rhythmic clarity and sophistication may be constructed.
[edit] Variation
One of the most common experiences encountered by students of composition is of having a good idea but finding it impossible to develop it into a more extended form. Frequently, the composer requires more-of-the-same but to avoid monotony, not exactly-the-same. Musical themes are explored and developed through varied repetition. The degree of variation can be subtle, where an idea changes its features only slightly, or extreme, leading to a complete break with the original idea, not so much a variation but a contrast. Schillinger's primary technique of creating variation is by the re-ordering of elements of a group whether they be attacks of a rhythm, the notes of a scale or the sections of a composition. Two basic methods are presented: general permutation and circular permutation. General permutation reveals all possible combinations of the elements of a group. Circular permutation generates fewer variants than general permutation but these tend to be more closely related to the original pattern. Two elements, (A, B), have only one permutation, (B, A). With three or more elements, the direction of permutation (clockwise or counter clockwise) becomes important.
Figure 6. Circular permutation: the rotation of a pattern in either clockwise or anti-clockwise direction.
In a clockwise direction, rotation produces the following variants: ABC, BCA, CAB. In a counter clockwise direction, rotation produces the following variants: ACB, CBA, BAC Permutation techniques play a highly influential role in Schillinger’s work and appear in some surprising places, for instance, as a means of controlling voice leading and registration in chord progressions.
[edit] Instrumental Forms
Every time we collaborate with one or more people, we create patterns and forms. Imagine a group of musicians sitting in a circle, each makes a sound in turn. The sound is an attack, and the position of the instrument making the sound is the place of attack. Patterns of musical sound emanating from different points in space are instrumental forms. The term Instrumental form refers to the relationship between rhythm and timbre. Beyond simple attacks, complete or partial rhythms and melodic phrases can be distributed between the various instruments and instrumental groups creating new expressions of rhythm; a rhythm of timbre, an instrumental form. The idea of instrumental form can be taken to great lengths: elements of a rhythm or phrase can be woven between instruments and instrumental groups of the orchestra creating the effect of a complex co-ordinated musical whole. Beyond this, lies the idea of a rhythm of texture and density.
[edit] Book II, the Theory of Pitch Scales
If the attacks of a rhythm are plotted along the horizontal axis of a graph, then scales can be understood as rhythmic patterns projected onto the vertical axis, the frequency dimension, of the same graph. In theory, an infinite number of frequencies might participate in a scale, and so a system of selection is required. Fortunately, nature has provided such a system in the form of the octave, the founding interval of the harmonic series , which, in turn, underpins the structure of our own nervous system . The octave, then, provides the frequency limits of a scale but what determines the mood and character of a particular scale? The answer lies in the proportions used to sub-divide the octave, thereby determining the intervals of the scale. In figure 7, each square on the vertical axis equals a semitone rise in pitch. Each square on the time axis represents the duration of a crotchet. The interval pattern of the major scale (2, 2, 1, 2, 2, 2, 1) is plotted along both the pitch and time axes as arrows.
c ← b ← a# a ← g# G ← f# f ← e ← d# d ← c# c ←↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ q q q q q q q q q q q q q q Figure 7. The rhythm (2, 2, 1, 2, 2, 2, 1) projected onto the frequency and time axes of a graph.
The patterns and types of intervals that fill the octave define the essential character of a scale. Tradition has provided a relatively small number of models, such as the family of five-note pentatonic scales or seven-note diatonic scales. Taking a mathematical approach and simply sub-dividing the number twelve (the octave) yields a vast number of scales. Rhythmic techniques, first described in Book I, are involved in this process. For instance, networks of closely related scales are revealed when two parent intervals are synchronised, or for example, the traditional church modes emerge when a major scale undergoes circular permutation. These methods empower the composer by providing an abundant source of relevant and related melodic and harmonic forms.
[edit] Symmetric scales
There is a special class of symmetrical scales in which the octave is divided into equal portions. For example, the chromatic scale,
12 = (1+1+1+1+1+1+1+1+1+1+1+1)
Various symmetrical scales have established themselves in musical tradition, for instance, the Whole Tone scale or the Octatonic scale. Others are superficially similar to the diminished seventh and augmented arpeggios.
12 = (3+3+3+3) or 12 = (4+4+4)
Figure 8. Scales created through symmetrical divisions of the octave.
The concept of symmetric choice is of great importance in the System, and later the idea is extended to include any pattern that has been pre-selected from the range of available intervals. A pattern originally conceived as a rhythm may be equally effective as a scale. For example, rhythms generated by pulse interference provide excellent source material for pitch scales and themes. Figure 9, illustrates a scale based on the rhythm 4:3 (3,1,2,2,1,3). This particular scale has no name and neither does it adhere to any conventional key signature.
Figure 9. A scale created from the interference rhythm 4:3.
[edit] The Primary Axis of Melody
Permutation is the basic method used to explore the melodic potential of a scale. In addition, Schillinger introduces a new and potent idea concerning the mechanics of melody. For a sequence of pitches to be truly melodic, it must have a primary axis (PA): a pitch that appears more frequently and for a longer duration than any other. The primary axis need not be the root of a harmony and in theory may be any note belonging to the prevailing key. The primary axis is the pivot around which the melody revolves - it is not necessarily fixed throughout but may change from one phrase to the next. This particular discussion of melody and the important concept of the primary axis is a preliminary to the extended study of the subject presented in Book IV.
[edit] Modal modulation
An established scale fixes itself in the listener’s memory producing the effect of key or tonality. The process whereby one scale is displaced in favour of another is called modulation. In the absence of harmony, modulation between scales is a matter of identifying common features, pitches, intervals or melodic contours, and after a suitable period of ambiguity, produced by dwelling on those common features, substituting the original scale for its equivalent in the new key. Other modulation techniques involve imposing patterns of accidentals belonging to other scales, on the fixed pitches of the melody. This is a most useful technique as it reveals highly contrasting variations in key and mode without effecting the pitch names or contour of the original melody.
[edit] Scale expansions
Traditional music theory views a scale as having a single tonic but Schillinger identifies four types of scale of which two have multiple tonics. The different categories are as follows:
1) Those with one tonic contained within the range of an octave (group one). 2) Those with one tonic arranged over two or more octaves (group two). 3) Symmetric scales consisting of two or more tonics contained within the range of an octave (group three). 4) Symmetric scales consisting of two or more tonics extended over a range of many octaves (group four).
Scales that extend across multiple octaves are the products of Expansion, a method involving nothing more complicated than re-ordering the scale omitting adjacent pitch units. For example,
Original C D E F G A B First expansion C E G B D F A
Figure 9. The C major scale and its first expansion.
Scales with multiple tonics are symmetric scales in which every pitch is considered a tonic; literally a series of key centres on which further Sectional scales might be founded. Sectional scales are made by sub-dividing the interval between each consecutive tonic, a technique originally described in the Theory of Rhythm. Expanded scales clearly have harmonic potential the extra octave allows more space between adjacent pitches revealing a pattern of harmonic intervals. A comparison of the two scales in figure 9 reveals clearly how melody and harmony can come from one source. The interrelatedness of melody and harmony is discussed in a preliminary way in the Theory of Pitch Scales and then at great length in books V and IX.
[edit] Book III, the Variation of Music By Means of Geometrical Projection.
In this portion of the text, Schillinger describes methods of creating variations in pitch using geometry. Two fundamental processes are identified: geometrical inversion and geometrical expansion. Inversion requires the rotation of pitches around the axes of a graph. An ordinary graph with two axes representing, for example, pitch (y) and time (x), is really just part of a larger system comprising the four quadrants of a circle. Music can be plotted as co-ordinates on a graph and may be rotated from one quadrant to another producing variants of the original.
Figure 10. Variations produced by rotation of a scale around the quadrants of a graph.
Geometrical expansion involves the scaling of the intervals of a melody or harmony using a coefficient of expansion. This process is quite different to the technique of scale expansion described in Book II, here every interval is multiplied by the same coefficient producing a distantly related form.
Figure 11. Geometrical expansion: 2 × (3,1,2,2,1,3) = (6,2,4,4,2,6)
[edit] Book IV, the Theory of Melody
In The Theory of Melody, Schillinger reveals some of his most interesting ideas concerning the nature of music alongside his most demanding techniques.
He proposed that melody has a distant evolutionary origin: the information flowing through our sense organs stimulates electrochemical and biomechanical responses, fear invokes muscular contraction; joy, lust or desire produces expansion. Our primitive spontaneous vocal responses to such stimuli eventually crystallised into formal melodic utterances. In between the extreme forms of response, such as fear and joy, there is the resting state characterised by simple periodic motion, for instance, regular breathing or the heart beat. Schillinger attempts to translate these ideas into the co-ordinates of a graph. The contours and direction of melody are exhaustively analysed and several important principles are established. Melody must have a Primary Axis (PA) representing balance or rest. Melody develops by revolving around the PA, via Secondary Axes: when the melody moves away from the PA it expresses the process of expansion and growth, movement towards the PA, represents contraction and ultimately death. Figure 12, shows secondary axes, above and below the primary axis, represented by the pitch B.
Figure12. The primary and secondary axes of melody.
The secondary axes represent the direction of the melodic contour and not its detailed surface motion. To properly render the melody an additional ingredient is required: different kinds of oscillatory motion are superimposed on the secondary axes in order to create a more typically melodic outline. Once a characteristic pattern of contours is decided, rhythmic techniques can be used to structure the melody in relationship to the bar group. A pitch scale and rhythm of duration may be imposed on the axes giving the melody its final form. Applying methods of rhythmic and geometrical variation may further develop a melody.
Figure 13. Oscillatory motion in a theme from Carnival (Reconnaissance) by Schumann.
In figure 13, the primary axis of the melody is the pitch C, the third degree of the scale of A flat major, which dominates the opening phrase, especially in the fourth bar of the example. During the last four bars, the original PA is abandoned in favour of a new axis on the tonic A flat.
[edit] Climax
A climax is a pitch time maximum: the focussing on a single pitch for the maximum time. The climax itself represents a release of energy after a period of energetic disturbance, a moment of simplification, achieved by focussing on a single pitch for an extended period. The climax can only be considered in relation to the primary axis, the balance line around which the melody evolves. It is only dramatic when approached by melodic forms which suggest increasing resistance to the eventual singularity of the climax: very much the experience of the mountaineer who struggles through foothills and increasingly difficult terrain towards the summit, a moment of pure relief and ecstasy made more intense by the preceding struggle. Schillinger notes that if a climax occurs suddenly without preparation by a suitable period of struggle, a feeling of disappointment, hilarity or disbelief may be engendered in the listener, a response deliberately or intuitively exploited by composers of music for children’s cartoons.
[edit] Organic Forms of Melody
The final chapter of Book IV, Organic Forms in Melody, deals with the practical application of growth series to melody. Various spiral patterns are described which possess unusual tonal characteristics when compared to the potential forms offered by normal scales .
[edit] Book V, Special Theory of Harmony
The Special Theory of Harmony deals specifically with techniques of traditional harmony derived from seven-note diatonic scales. Schillinger makes a strong distinction between root progressions (bass progression) and the chord structures built upwards from those roots.
Figure 14. Scale degree IV, the root F in C major and its diatonic chord structure (∑).
[edit] Diatonic Harmony
In diatonic harmony, both root progressions and chord structures are derived from one scale. The technique extends the process of scale expansion first described in book two: three expansions are required to construct a diatonic progression and these are termed cycles, of the third (Cy3), the fifth (Cy5) and the seventh (Cy7) . The cycles are first used to create a melody in the bass, or a root progression . The actual pitches of the bass line are determined by the sequence of cycles, which can be mixed in various patterns and controlled using techniques described in the Theory of Rhythm. All three cycles may participate in the composition of a root progression but chord structures come exclusively from the cycle of the third.
Figure 15. Diatonic harmony: both melody and harmony come out of a single scale.
The harmonic progression shown at the bottom of figure 15 is derived entirely from a single scale (E0) and its expansions (E1 and E2). Lines extending from the cycle clocks, in the middle of the diagram, show how they contribute to the harmonic progression. The interval between consecutive roots always corresponds to a cycle movement. For example, the root progression begins on the pitch C and moves to F: an anti-clockwise movement around the cycle of the fifth (Cy5/E2). The journey from the second root F, to the third root G, is a clock-wise movement on the cycle of the seventh (Cy7/E0). Roman numerals indicate how roots correspond to one of the seven notes of the C major scale. The chord structures are all derived from the cycle of the third (Cy3/E1).
[edit] Voice leading
Once harmonies have been assigned to the root progression, the entire group may be shaped and varied through voice leading. Voice leading methods are based on the techniques of rotation and permutation described in Book I. Voice leading does not alter the meaning of the harmony but rather effects the texture and density of the progression and improves the flow of the music. Figure 16, shows the effect of anti-clockwise voice leading on the progression shown in figure 15
Figure 16. Anti-clockwise voice leading.
In figure 16, the progression of figure 15 has undergone a voice leading process. The harmonic meaning is not changed in any way but the flow of the progression is less jagged and more melodic than the original. The table in the middle of figure 16 shows the first three chords and the horizontal progress of their chord functions: root (1), third (3), fifth (5). This linear progression is always in an anti-clockwise direction as indicated by the clock on the right hand side of figure 16.
[edit] Diatonic symmetric
Diatonicism requires that both melody and harmony belong to one source scale. The diatonic system can be can be extended by composing a root progression in the normal way, from a single scale and its cycles, but imposing chord structures derived from any other seven-note scale. Schillinger calculates that out of all the possible seven-note scales, there are only a small number of diatonic chords structures: four types of triad (major, minor, augmented and diminished) and seven types of tetrad . With these resources, it is possible to create numerous types of rich hybrid harmony, often associated with well-known styles of twentieth century music.
[edit] Traditional techniques
The Special Theory of Harmony deals with many of the concepts found in traditional treatises: pedal points, suspensions, anticipations, auxiliary and passing notes, inversions, doubling and various types of progression are all treated with Schillinger’s customary rigour and originality.
[edit] Book VI, the Correlation of Harmony and Melody
This portion of the text is a bridge between the subject of diatonic harmony and counterpoint. It describes techniques for the composition of melody with harmonic accompaniment, a type of music that might be referred to as homophonic. The subject is divided into three chapters:
1) The Melodisation of Harmony. 2) Composing Melodic Attack Groups. 3) The Harmonisation of Melody.
Schillinger states that the most satisfactory relationships between melody and harmony are those in which melody is derived from an existing chord progression, the subject of Chapter 1. The opposite method, deriving harmony from an existing melody, is covered in Chapter 3. Both chapters describe numerous relationships between melody and harmony, using concepts from Book V, Special theory of Harmony, and Book IV, the Theory of Melody. Techniques are largely dependent on the hierarchical arrangement of chord functions (1,3,5,7,9,11,13) and the organisation of the axes of melody. Many ornamentation techniques are described, which require the insertion of chromatic tones between the main functional pitches of a melody. Chapter 2 is extremely important as it concerns the composition of melodic attack groups, which, in this case, refers to a group of melody notes associated with a chord. Rhythmic patterns derived from techniques presented in Book I are used to determine both the quantity of pitches in an attack group as well as the duration of each pitch so creating a co-ordinated rhythm of melody and accompaniment. The techniques presented in this chapter are some of the most powerful as they control the rhythmic flow and degree of animation of the music. Attack groups are involved in many other areas of the System ranging from the study of melodic flow through to orchestration and the composition of textural density.
[edit] Book VII, the Theory of Counterpoint
The Theory of Counterpoint only deals with counterpoint in two parts. It begins with a traditional classification of intervals and their resolutions. There are four alternative relationships between the original voice, Cantus Firmus (CF) and the added voice, Counterpoint (CP). These cover all forms of counterpoint in which both parts belong to the same scale, as well as more exotic polytonal types in which the voices may belong to different modes and key centres. At this point in the text the different relationships between CF and CP are based on principles established in Book II, they are as follows:
CF and CP belong to the same scale (tonic) and the same mode (major/minor). CF and CP belong to different modes. CF and CP belong to different scales but are identical in mode. CF and CP belong to different scales and different modes.
It is assumed that the two parts (CF and CP) have well established primary axes, and that the interval between them is always consonant. The relationship of the contours (secondary axes) of the two voices is discussed through principles first introduced in the Theory of Melody.
[edit] Canon
The techniques described for the composition of canons and fugues are approached as primarily matters of rhythm. Combining a symmetrical rhythm with itself at a time interval equal to half its duration produces an echo, an imitative form, in music known as a canon. Figure 16, shows how the rhythmic resultant 5:4 (4,1,3,2,2,3,1,4) might be arranged as a two-part canon. This is shown first in the form of a table and then as a score.
Announcement Imitation Continuation Voice 1 4,1,3,2 2,3,1,4 4,1,3,2 Voice 2 -------------- 4,1,3,2 2,3,1,4
Figure 17. The rhythm 5:4 arranged as a canon at the unison.
[edit] Book VIII, Instrumental Forms.
The Theory of Instrumental Forms elaborates upon the ideas first presented in Book I. Schillinger sets out the scope of the discussion as follows:
"What we are to discuss here is all forms of arpeggio and their applications in the field of melody, harmony, and correlated melody"
What seems at first a most dry and unpromising discussion ultimately becomes a fascinating study of the mechanics of musical flow.
[edit] Attack groups and arpeggios
An attack is the onset of a sound; it is an event. An attack group is a series of events belonging together. A place of attack might be any number of things from the head of a drum to the voices of a chord or the keys on a piano. The pattern of attacks as they fall on the various places of attack is an instrumental form. In the case of a chord, the result is an arpeggio.
Figure 18. Two part harmony, attack groups and arpeggios.
Figure 18, shows how harmony becomes animated through the application of attacks and instrumental form. Two diads (top line) are modified by two attack groups each containing three attacks (middle line). The result of combining the two upper lines is shown on the bottom stave. The duration of each note in the bottom line is not relevant to this discussion. Figure 18, is an extremely simple example of a technique that can be made to produce highly sophisticated results. Over the course of the first five chapters of Instrumental Forms, Schillinger lists all possible arrangements for attack groups ranging from two to twelve attacks distributed through harmony of two, three and four parts. A large number of examples of ornamented harmonic progressions accompany these tables.
[edit] Harmonic Strata
Chapter 6 introduces the idea of harmonic structures called strata. At this point, the discussion is only about doubling identical harmonies at the interval of the octave. When strata are superimposed the resulting assemblage is referred to as a Sigma (Σ). Strata and Sigmas are important to Schillinger’s concept of orchestration and it is suggested that a crisp sound is best achieved when the simultaneous strata are allotted to different instrumental ensembles within the orchestra. A harmonic stratum may be doubled at the octave under certain conditions: the arrangement of the two strata must be identical otherwise harmonics and difference tones will cause a loss of clarity. When combining non-identical strata, for instance different inversions of the same harmony, the function (1, 3, 5, 7, 9, 11, 13) in the uppermost voice of each stratum must be identical. In addition, non-identical strata must be arranged so that the most closely spaced chord is uppermost. By ensuring that the overall spacing of harmony notes in the score is widest in the bass register and narrowest in the soprano, the composer imitates the natural spacing of the harmonic series and ensures maximum acoustical clarity.
Figure 19. Harmonic strata doubled at the interval of the octave.
[edit] Book IX, the General Theory of Harmony
The General Theory of Harmony develops principles for the construction and co-ordination of all types of harmony. In scope, it ranges from the simplest two-part doubling to the co-ordination of very large chords with potentially tens or hundreds of voices. Book IX is particularly illuminating in the realm of orchestration in which harmony becomes indistinguishable from timbre, texture and density. Schillinger clearly distinguishes between the General and the Special Theory of Harmony.
"Contrary to what was the case in my special theory of harmony, this system has not been based on observation and analyses of existing musical facts only; it is entirely inductive…the special harmony is but one case of general harmony"
This portion of the System pertains directly to the field of orchestration by providing techniques by which the various instrumental groups within an ensemble can be controlled and differentiated through the co-ordination of independent, simultaneous blocks (strata) of harmony.
"As the main purpose of the General Theory of Harmony is to satisfy demands for the scoring of all possible combinations of instruments or voices, or both, it should be flexible enough to make any instrumental combination possible"
Schillinger's method of generating harmonic structure is initially the same as that described in the Special Theory of Harmony. This involves superimposing the pitch units of a scale on itself and its various expansions. Unlike the Special Theory of Harmony, which utilises only the first expansion of a diatonic scale as a source of harmony, the General Theory of Harmony allows chord structures to be derived from all scale expansions. A simple case of two-part harmony will give the reader a good idea of how harmonic strata emerge from a scale. Figure 20 shows a pentatonic scale and its harmonies derived from expansion.
Figure 20. A pentatonic scale is expanded to reveal its harmonic derivatives.
Schillinger observes that only scales with seven different pitches produce regular harmony, that is, expansions consisting of one type of interval such as, thirds, fourths or fifths, quite unlike the products of the five-note scale shown in figure 20.
[edit] Symmetric
A stratum does not have to come from a scale but can instead be chosen symmetrically from any of the eleven intervals contained by of the octave . The chord functions 1, 3, 5, 7, 9, 11, 13, are simply irrelevant in the absence of a seven-note scale, and so instead, alphabet letters represent the chord functions. In figure 20, the sequence of root tones in the lower staff progress according to the cycle of fifths. In the upper staff, function a represents the root of the harmony while b represents a second function at the interval of a major second from the root. Voice leading in two-part harmony is limited to only two possibilities: either chord functions alternate between consecutive chords ab →ba or functions remain unchanged (parallel) between consecutive chords. In figure 21 the alternating voice leading causes the chord structures to switch between open and close position: the major second transforms into a minor seventh.
Figure 21. Two part harmony with alternating voice leading.
Figure 21, could be described as hybrid three-part harmony. The roots in the lower stave represent an added stratum of one part. Such an arrangement might be suitable for distribution between two distinct instrumental groups. For example, a violin might be assigned to the upper line while a bassoon to the roots in the lower stave. Despite the simplicity of the example shown in figure 21, it should be observed that while the two strata are co-ordinated harmonically, their independence in voice leading facilitates the clarity of the chosen orchestration. Over the course of several chapters, ever-richer combinations of strata are described. Added parts and doublings increase dramatically the number of voice leading options. Schillinger describes various techniques for converting the strata into musical forms, such as melody with accompaniment or contrapuntal textures, methods that involve techniques from earlier portions of the System.
[edit] Harmonic Density
In Chapter 15, Schillinger introduces an idea he refers to as textural density. This is one of the most challenging discussions in the SSMC, largely written in Schillinger's special system of algebraic notation and accompanied by very few musical examples. It is, however, unusual and far-reaching in its concept and deserves clarification. The density of music changes very rapidly: an orchestral work contains numerous instrumental combinations ranging from solo to tutti; this might be described as the density of orchestration. Schillinger suggests that there is another kind of density, one of texture produced by patterns of arpeggios and harmonies, which is also fundamental to musical flow. The General Theory of Harmony is based on the idea that a score can be made up of independent but co-ordinated harmonic layers or strata, these collectively are referred to as a sigma (∑). One can imagine a sigma as being something like a geological diagram showing a cross section of the Earth's crust. Textural density depends on varying the number of strata in a score from one moment to the next. Alternatively, Imagine a sequence of slides in which the same three story building appears at first compete, then with its ground and top floors missing, and finally with the top and bottom floors intact but without the middle story. For the house substitute Sigma, for the floor levels, substitute harmonic strata. A sequence of sigmas such as this would be referred to as a density group. Once a density group has been composed, its variations can be generated by rotation. Figure 22 shows a three-element density group and two variants produced by rotation in a clockwise direction. Inactivity in one or other stratum is shown as white rectangles while black rectangles represent activity. The first group starts with ∑1, which consists of three active strata; it is followed by ∑2 and ∑3, which exhibit varying degrees of activity.
Strata 1 Strata 2 Strata 3 ∑1 ∑2 ∑3 ∑3 ∑1 ∑2 ∑2 ∑3 ∑1
Figure 22. A density group made up of three ∑
A density group is subject to rotational variation in two directions simultaneously: rotation around the vertical and horizontal axes of a graph. When textures are rotated vertically, it is important to ensure that the harmony belonging to any stratum does not change its position, which would radically alter the harmonic structure of the entire score. Three vertical permutations of the original density group, seen first in figure 22, are shown in figure 23. The different textures and forms of arpeggiation rotate vertically, moving upwards by one place at a time as indicated by the arrows. Each stratum is labelled to indicate a hypothetical form of texture: harmony (H) or melody (M). From these basic types, come two alternative melodic textures, M1 and M2 and one harmonic type H1. The two M types might be different patterns of arpeggiation, while H1 might be a type of harmonic accompaniment using block chords. Figure 24 is a partial realisation of the scheme of figure 23 using simplified patterns. Strata 1 − M1− M1 − H1 H1 − M2 M2 Strata 2 − H1− H1 − M2 M2 − M1 M1 Strata 3 − M2− M2 − M1 M1 − H1 H1
Figure 23. Three variations produced by vertical rotation.
Figure 24. The density group shown in figure 22 realised in music in a schematic form.
[edit] Book X, Evolution of Pitch-Families (Style)
Book X consists of a collection of ideas and techniques described first in Book II, The Theory of Pitch Scales, and Book IX, The General Theory of Harmony. Tracing the evolution of a scale from its original (primitive) form to its modernised fully developed hybrid form, Schillinger attempts to show how a unity of style, in terms of melody and harmony, can be achieved.
"Often styles of intonation can be defined geographically and historically. There may be a certain national style which, in due course of time, undergoes various modifications. These modifications...can also be looked upon as modernisation of the source...
Apart from this historical and evolutionary argument, there are no new concepts offered here. Nevertheless, Book X provides a useful overview of scale development techniques because it combines ideas from different parts of the System in a condensed form and in addition provides new examples and illustrations.
[edit] Book XI, Theory of Composition
In the introduction to Book XI, Schillinger outlines three basic approaches to composing.
1) Composition of parts or themes without prior knowledge of the whole form: this may potentially result in the connection of themes or material that does not belong together. 2) Improvisation: which by definition does not anticipate the whole and tends towards loose structures and or excessive repetition. 3) Conception of the whole form prior to creating its various parts.
The Theory of Composition deals with the last of these approaches but it should be understood that the role of intuition and improvisation are not excluded.
"Each approach contains different ratios of the intuitive and the rational elements by which the process of composition is accomplished. Works of different quality may result from each of these three basic approaches. Often these forms of creation are fused with one another"
The Theory of Composition is divided into three parts:
1) Composition of Thematic Units. 2) Composition of Thematic Continuity. 3) Semantic (Connotative) Composition.
[edit] Part I, Composition of Thematic Units
In Part I, Schillinger introduces the idea of the thematic unit: the basic building block of a composition. A thematic unit, otherwise referred to as a theme or a subject, is a structure that will yield variations and ultimately whole sections of a composition. Schillinger lists seven sources from which to develop thematic units : 1) Rhythm, 2) scales, 3) melodies 4) harmonic progressions, 5) arpeggiated, harmony 6) counterpoint, 7) orchestral resources.
These categories represent the base material from which the thematic unit is developed. The last entry in the list above (7) includes the possibility of tone quality, dynamics, density and instrumental forms as potential components for the composition of a thematic unit. A thematic unit is frequently composed from more than one source, in which case it features both a major and a minor component. A thematic unit derived primarily from rhythm (a major component) might well involve pitch as a secondary (minor) component. The Danse Sacral, the final movement of Stravinsky’s Le Sacre Du Printemps , fits exactly this profile, music dominated by rhythm and inflected by harmony.
[edit] Part II, Composition of Thematic Continuity
Part II, Composition of Thematic Continuity is a discussion of musical form and how thematic units are joined to make a thematic sequence. Letters of the alphabet represent thematic units.
Binary forms: (A+B). Symmetrical forms: (A+B+A). Rotational forms: (A+B+C)→(B+C+A)→(C+A+B).
One of the most interesting types is the so-called progressive-symmetric form. Here a subject gradually looses its dominance to another subject. For example, in the following scheme subject C replaces subject A:
A+(A+B)+(A+B+C)+(B+C)+(C).
Such an arrangement offers possibilities for the gradual transformation of one idea to another.
Chapter 12 Temporal Co-ordination of Thematic Units offers rhythmic methods of regulating the significance of a subject within the composition as a whole, using techniques described in Book I, The Theory of Rhythm. Schillinger is very clear on the matter of the relative importance of the various subjects.
"This theory repudiates the academic point of view, according to which some themes are so unimportant that they function as mere bridges tying the main themes together. If a certain thematic unit is unimportant.... and merely consumes time, it should not participate in the composition."
In Chapter 14, Planning a Composition, Schillinger describes the process of composition in ten stages.
1) Decision as to total length of composition in clock time. 2) Decision as to degree of temporal saturation. 3) Decision as the number of subjects and thematic groups of subjects. 4) Form of thematic sequence. 5) Temporal definition and distribution of thematic groups. 6) Organisation of temporal continuity. 7) Composition of thematic units. 8) Composition of thematic groups. 9) Intonational co-ordination (key structure). 10) Instrumental development (orchestration / instrumentation).
Temporal saturation (point 2) is the degree of density of events (notes, attacks, harmonies etc.) within a given time: the greater the density of events, the longer our perception of time. Temporal definition and distribution of thematic groups (point 5) refers to the different weight or duration applied to each subject, that is, the ratio or balance between subjects and the form of their distribution. Organisation of temporal continuity (point 6) refers to the basic unit of duration (crotchet, quaver, triplet quaver...) for each subject or thematic unit. The remainder of Part II is devoted to working out examples of monothematic and polythematic compositions.
[edit] Part III Semantic (Connotative) Composition
Part III, Book XI, Semantic (Connotative) composition, is based on the idea that musical forms are sonic symbols: sound forms that convey meaning and can motivate forms of behaviour. This is one of the more ambitious and therefore contentious aspects of the System, in that it assumes that human beings have similar responses to any stimulus and that patterns may be constructed to inspire certain predetermined generalized responses. There is certainly plenty of evidence that listeners respond to a stimulus in similar ways. For example, the threshold of hearing can be measured in the laboratory and neurophysiological investigations have demonstrated the mechanism of pitch sensation in the brain. It has also been shown that sound can motivate behaviour, the factory siren for instance, but Schillinger goes further and proposes the following:
"…music is capable of expressing everything which can be translated into form of motion"
Thus, the temporal patterns of our biology can be transposed into music to produce a predictable emotional response in a listener. While intuition suggests that this idea has more than a grain of truth, the matter cannot be judged scientifically (there is still no accounting for taste or proper understanding of perception) and so I suggest it is viewed artistically, as compelling idea from which the composer may gain unsuspected inspiration.
"As the response to sonic forms exists even in so-called inanimate nature in the form of sympathetic vibrations or resonance, it is no wonder that primitive man inherited highly developed mimetic responses. From this we can conclude that a great many of the early sonic symbols probably originated as imitation of sonic patterns, coming as stimuli from the surrounding world"
The notation used to describe the System of sonic symbols comes from the so called psychological dial, shown below, on which the various possible responses to stimuli are represented.
Figure 25. The psychological dial.
Schillinger illustrates the use of the dial through anecdote. For example, a man who enters a bargain basement store expecting to pay no more than ten cents for any item has his expectations confirmed, his response is normal, which is represented on the dial at 180°. Alternatively he is asked to pay $100 for a pencil, his response is astonishment or disbelief, which can perhaps be represented on the dial at 90° (infranormal). The theory by which psychological states are translated into music is developed from ideas first presented in Book IV, the Theory of Melody. The dial is divided vertically into two halves, the left half is negative (loss of energy, contraction, defense) and the right half is positive (gain of energy, expansion, aggression). Any point on the dial can be translated into the motion of a secondary axis of melody. When the secondary axis moves away from the primary axis, it corresponds to the positive zone of the dial. When moving towards the primary axis it corresponds to the negative zone. As stimulus and response increase, so the angle of the axes with respect to the PA must become more acute. Figure 26 shows five dial positions and their corresponding potential axial configurations.
Figure 26. Psychological dials and axial correspondences.
Schillinger gives examples of how such correspondences can be translated into rhythm, melody, harmony and density of timbre. There are musical examples and verbal descriptions. He also describes how sonic symbols may be combined into sequences and suggests that this technique is invaluable for composition based on narrative forms, such as programme music or film and stage music.
[edit] Book XII, Theory of Orchestration
This portion of the text is mainly a very standard description of the tuning, range and basic performance characteristics of orchestral instruments. A chapter dealing with electronic musical instruments is of historical interest as it contains a description of different types of Theremin : Chapter 8, Instrumental Combination, is an attempt to classify and compare instrumental timbres. Chapter 9, Acoustical Basis of Orchestration, is only a few paragraphs long; Schillinger clearly intended to develop a theory of instrumental combination based on acoustics, but after acknowledging the difficulties inherent in this task, the chapter ends. An editorial note suggests that Schillinger left notes on this subject but had not completed them before his death.
