Schillinger System Table of Contents

The Schillinger System of Musical Composition

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BOOK 1
Theory of RhythmChapter 1
Notation System
A. Graphing Music
B. Forms of Periodicity

Chapter 2
Interferences of Periodicities
A. Binary Synchronization
B. Grouping

Chapter 3
The Techniques of Grouping

Chapter 4
The Techniques of Fractioning

Chapter 5
Composition of Groups by Pairs

Chapter 6
Utilization of Three or More Generators
A. The Technique of Synchronization

Chapter 7
Resultants Applied to Instrumental Forms
A. Instrumental Rhythm
B. Applying the Principles of Interference to Harmony

Chapter 8
Coordination of Time Structures
A. Distribution of a Duration-Group
B. Synchronization of an Attack-Group
C. Distribution of a Synchronized Duration-Group
D. Synchronization of an Instrumental Group

Chapter 9
Homogeneous Simultaneity and Continuity (Variations)
A. General and Circular Permutations

Chapter 10
Generalization of Variation Techniques
A. Permutations of the Higher Order

Chapter 11
Composition of Homogeneous Rhythmic Continuity

Chapter 12
Distributive Powers
A. Continuity of Harmonic Contrasts
B. Composition of Rhythmic Counterthemes

Chapter 13
Evolution of Rhythm Styles (Families)
A. Swing Music

Chapter 14
Rhythms of Variable Velocities
A. Acceleration in Uniform Groups
B. Acceleration in Non-uniform Groups
C. Rubato
D. Fermata

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BOOK TWO
Theory of Pitch ScalesChapter 1
Pitch Scales and Equal Temperament

Chapter 2
First Group of Pitch Scales: Diatonic and Related Scales
A. One-unit Scales. Zero Intervals
B. Two-unit Scales. One Interval
C. Three-unit Scales. Two Intervals
D. Four-unit Scales. Three Intervals
E. Scales of Seven Units.

Chapter 3
Evolution of Pitch-scale Styles
A. Relating Pitch-Scales through the Identity of Intervals.
B. Relating Pitch-Scales through the Identity of Pitch-Units
C. Evolving Pitch-Scales through the Selection of Intervals.
D. Evolving Pitch-Scales through the Selection of Intervals.
E. Historical Development of Scales.

Chapter 4
Melodic Modulation and Variable Pitch Axes
A. Primary Axis
B. Key Axis
C. Four Forms of Axis-Relations
D. Modulating through Common Units
E. Modulating through Chromatic Alteration
F. Modulating through Identical Motifs

Chapter 5
Pitch-Scales: The Second Group: Scales in Expansion
A. Methods of Tonal Expansion
B. Translation of Melody into Various Expansions
C. Variable Pitch Axes (Modulation)
D. Technique of Modulation in Scales of the Second Group

Chapter 6
Symmetric Distribution of Pitch-Units

Chapter 7
Pitch-Scales: The Third Group: Symmetrical Scales
A. Table of Symmetric Systems Within 12/2
B. Table of Arithmetical Values
C. Composition of Melodic Continuity in the Third Group

Chapter 8
Pitch-Scales: The Fourth Group: Symmetrical Scales of More Than One Octave in Range
A. Melodic Continuity
B. Directional Units

Chapter 9
Melody-Harmony Relationship in Symmetric Systems

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BOOK THREE
Variations of Music By Means of Geometrical ProjectionChapter 1
Geometrical Inversions

Chapter 2
Geometrical Expansions

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BOOK FOUR
Theory of MelodyChapter 1
Introduction
A. Semantics
B. Semantics of Melody
C. Intentional Biomechanical Processes
D. Definition of Melody

Chapter 2
Preliminary Discussion of Notation
A. History of Musical Notation
B. Mathematical Notation, General Component
1. Notation of Time
C. Special Components
1. Notation of Pitch
2. Notation of Intensity
3. Notation and Quality
D. Relative and Absolute Standards
E. Geometrical (Graph) Notation

Chapter 3.
The Axes of Melody
A. Primary Axis of Melody
B. Analysis of Three Examples
C. Secondary Axes
D. Examples of Axial Combinations
E. Selective Continuity of the Axial Combinations
F. Time Ratios of the Secondary Axes
G. Pitch Ratios of the Secondary Axes
H. Correlation of Time and Pitch Ratios of the Secondary Axes

Chapter 4
Melody: Climax and Resistance
A. Forms of Resistance Applied to Melodic Trajectories
B. Distribution of Climaxes in Melodic Continuity

Chapter 5
Superimposition of Pitch and Time on the Axes
A. Secondary Axes
B. Forms of Trajectorial Motion

Chapter 6
Composition of Melodic Continuity

Chapter 7
Additional Melodic Techniques
A. Use of Symmetric Scales
B. Technique of Plotting Modulations

Chapter 8
Use of Organic Forms in Melody

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BOOK FIVE
Special Theory of HarmonyChapter 1
Introduction

Chapter 2
The Diatonic System of Harmony
A. Diatonic Progressions (Positive Form)
B. Historical Development of Cycle Styles
C. Transformations of S(5)
D. Voice-Leading
E. How Cycles and Transformations are Related
F. The Negative Form

Chapter 3
The Symmetric System of Harmony
A. Structures of S(5)
B. Symmetric Progressions. Symmetric Zero Cycle (C0)

Chapter 4
The Diatonic-Symmetric System of Harmony (Type II)

Chapter 5
The Symmetric System of Harmony (Type III)
A. Two Tonics
B. Three Tonics
C. Four Tonics
D. Six Tonics
E. Twelve Tonics

Chapter 6
Variable Doublings in Harmony

Chapter 7
Inversions of the S(5) Chord
A. Doublings of S(6)
B. Continuity of S(5) and S(6)

Chapter 8
Groups With Passing Chords
A. Passing Sixth Chords
B. Continuity of G6
C. Generalization of G6
D. Continuity of the Generalized G6
E. Generalization of the Passing Third
F. Applications of G6 to Diatonic-Symmetric (Type II) and Symmetric (Type III) Progressions
G. Passing Fourth-sixth Chords: S(6/4)
H. Cycles and Groups Mixed

Chapter 9
The Seventh Chord
A. Diatonic System
B. The Resolution of S(7)
C. With Negative Cycles
D. S(7) in the Symmetric Zero Cycle (C0)
E. Hybrid Five-Part Harmony

Chapter 10
The Ninth Chord
A. S(9) in the Diatonic System
B. S(9) in the Symmetric System

Chapter 11
The Eleventh Chord
A. S(11) in the Diatonic System
B. Preparation of S(11)
C. S(11) in the Symmetric System
D. In Hybrid Four-Part Harmony

Chapter 12
Generalization of Symmetric Progressions
A. Generalized Symmetric Progressions as Applied to Modulation Problems

Chapter 13
The Chromatic System of Harmony
A. Operations from S3(5) and S4(5) bases
B. Chromatic Alterations of the Seventh
C. Parallel Double Chromatics
D. Triple and Quadruple Parallel Chromatics
E. Enharmonic Treatment of the Chromatic System
F. Overlapping Chromatic Groups
G. Coinciding Chromiatic Groups

Chapter 14
Modulations in the Chromatic System
A. Indirect Modulations

Chapter 15
The Passing Seventh Generalized
A. Generalized Passing Seventh in Progressions of Type III
B. Generalization of Passing Chromatic Tones
C. Altered Chords

Chapter 16
Automatic Chromatic Continuities
A. In Four Part Harmony

Chapter 17
Hybrid Harmonic Continuities

Chapter 18
Linking Harmonic Continuities

Chapter 19
A Discussion of Pedal Points
A. Classical Pedal Point
B. Diatonic Pedal Point
C. Chromatic (Modulating) Pedal Point
D. Symmetric Pedal Point

Chapter 20
Melodic Figuration; Preliminary Survey of the Techniques
A. Four Types of Melodic Figuration

Chapter 21
Suspensions, Passing Tones and Anticipations
A. Types of Suspensions
B. Passing Tones
C. Anticipations

Chapter 22
Auxiliary Tones

Chapter 23
Neutral and Thematic Melodic Figuration

Chapter 24
Contrapuntal Variations of Harmony

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BOOK SIX
The Correlation of Harmony and MelodyChapter 1
The Melodization of Harmony
A. Diatonic Melodization
B. More than one Attack in Melody per H

Chapter 2
Composing Melodic Attack-Groups

A. How the Durations for Attack-Groups of Melody Are Composed
B. Direct Composition of Durations Correlating Melody and Harmony
C. Chromatic Variation of Diatonic Melodization
D. Symmetric Melodization: The Σ Families
E. Chromatic Variation of a Symmetric Melodization
F. Chromatic Melodization of Harmony
G. Statistical Melodization of Chromatic Progressions

Chapter 3
The Harmonization of Melody
A. Diatonic Harmonization of a Diatonic Melody
B. Chromatic Harmonization of a Diatonic Melody
C. Symmetric Harmonization of a Diatonic Melody
D. Symmetric Harmonization of a Symmetric Melody
E. Chromatic Harmonization of a Symmetric Melody
F. Diatonic Harmonization of a Symmetric Melody
G. Chromatic Harmonization of a Chromatic Melody
H. Diatonic Harmonization of a Chromatic Melody
I. Symmetric Harmonization of a Chromatic Melody

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BOOK VII
Theory of CounterpointChapter 1
The Theory of Harmonic Intervals
A. Some Acoustical Fallacies
B. Classification of Harmonic Intervals Within the Equal Temperament of Twelve
C. Resolution of Harmonic Intervals
D. Resolution of Chromatic Intervals

Chapter 2
The Correlation of Two Melodies
A. Two-Part Counterpoint
B. CP/CF = a
C. Forms of Harmonic Correlation
D. CP/CF = 2a
E. CP/CF = 3a
F. CP/CF = 4a
G. CP/CF = 5a
H. CP/CF = 6a
I. CP/CF = 7a
J. CP/CF = 8a

Chapter 4
The Composition of Contrapuntal Continuity

Chapter 5
Correlation of Melodic Forms in Two-Part Counterpoint
A. Use of Monomial Axes
B. Binomial Axes Groups
C. Trinomial Axial Combinations
D. Polynomial Axial Combinations
E. Developing Axial Relations Through Attack-Groups
F. Interference of Axis-Groups
G. Correlation of Pitch-Time Ratios of the Axes
H. Composition of a Counterpoint to a Given Melody by Means of Axial Correlation

Chapter 6
Two-Part Counterpoint With Symmetric Scales

Chapter 7
Canons and Canonic Imitations
A. Temporal Structure of Continuous Imitation
1. Temporal structures composed from the parts of resultants
2. Temporal structures composed from complete resultants
3. Temporal strucres evolved by means of permutations
4. Temporal structures composed from synchronized involution-groups
5. Temporal structures composed from acceleration-groups and their inversions

B. Canons in All Four Types of Harmonic Correlation

C. Composition of Canonic Continuity by means of Geometrical Inversions

Chapter 8
The Art of the Fugue

A. The Form of the Fugue
B. Forms of Imitation Evolved Through Four Quadrants
C. Steps in teh Composition of a Fugue
D. Composition of the Theme
E. Preparation of the Exposition
F. Composition of the Exposition
G. Preparation of the Interludes
H. Non-Modulating Interludes
I. Modulating Interludes
J. Assembly of the Fugue

Chapter 9
Two-Part Contrapuntal Melodization of a Given Harmonic Continuum

A. Melodization of Diatonic Harmony by means of Two-Part Diatonic Counterpoint
B. Chromatization of Two-Part Diatonic Melodization
C. Melodization of Symmetric Harmony
D. Chromatization of a Symmetric Harmony
E. Melodization of Chromatic Harmony by means of Two-Part Counterpoint

Chapter 10
Attack-Groups For Two-Part Melodization

A. Composition of Durations
B. Direct Composition of Durations
C. Composition of Continuity

Chapter 11
Harmonization of Two-Part Counterpoint

A. Diatonic Harmonization
B. Chromatization of Harmony accompanying Two-Part Diatonic Counterpoint (Types I and II)
C. Diatonic Harmonization of Chromatic Counterpoint whose origin is Diatonic (Types I and II)
D. Symmetric Harmonization of Diatonic Two-Part Counterpoint (Types I, II, III, and IV)
E. Symmetric Harmonization of Chromatic Two-Part Counterpoint
F. Symmetric Harmonization of Symmetric Two-Part Counterpoint

Chapter 12
Melodic, Harmonic, and Contrapuntal Ostinato

A. Melodic Ostinato (Basso)
B. Harmonic Ostinato
C. Contrapuntal Ostinato

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BOOK EIGHT
Instrumental FormsChapter 1
Multiplication of Attacks
A. Nomenclature
B. Sources of Instrumental Forms
C. Definition of Instrumental Forms

Chapter 2
Strata of One Part

Chapter 3
Strata of Two Parts
A. General Classification of I (S = 2p)
B. Instrumental Forms of S-2p

Chapter 4
Strata of Three Parts
A. General Classification of I (S=3p)
B. Development of Attack-Groups by Means of Coefficients of Recurrence
C. Instrumental Forms of S-3p

Chapter 5
Strata of Four Parts
A. General Classification of I (S=4p)
B. Development of Attack-Groups by Means of Coefficients of Recurrence
C. Instrumental Forms of S=4p

Chapter 6
Composition of Instrumental Strata
A. Identical Octave Positions
B. Acoustical Conditions for Setting the Bass

Chapter 7
Some Instrumental Forms of Accompanied Melody
A. Melody with Harmonic Accompaniment
B. Instrumental Forms of Duet with Harmonic Accompaniment

Chapter 8
The Use of Directional Units in Instrumental Forms of Harmony

Chapter 9
Instrumental Forms of Two-Part Counterpoint

Chapter 10
Instrumental Forms for Piano Composition
A. Position of Hands with Respect to the Keyboard

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BOOK NINE
General Theory of Harmony: Strata HarmonyChapter 1
One-Part Harmony
A. One Stratum of One-Part Harmony

Chapter 2
Two-Part Harmony
A. One Stratum of Two-Part Harmony
B. One Two-Part Stratum
C. Two Hybrid Strata
D. Table of Hybrid Three-Part Structures
E. Examples of Hybrid Three-Part Structures
F. Two Strata of Two-Part Harmonies
G. Examples of Progressions in Two Strata
H. Three Hybrid Strata
I. Three, Four, and More Strata of Two-Part Harmonies
J. Diatonic and Symmetric Limits and the Compound Sigmae of Two-Part Strata
K. Compound Sigmae

Chapter 3
Three-Part Harmony
A. One Stratum of Three-Part Harmony
B. Transformations of S-3p
C. Two Strata of Three-Part Harmonies
D. Three Strata of Three-Part Harmonies
E. Four and More Strata of Three-Part Harmonies
F. The Limits of Three-Part Harmonies
1. Diatonic Limit
2. Symmetric Limit
3. Compound Symmetric Limit

Chapter 4
Four-Part Harmony
A. One Stratum of Four-Part Harmony
B. Transformations of S-4p
C. Examples of Progressions of S-4p

Chapter 5
The Harmony of Fourths

Chapter 6
Additional Data on Four-Part Harmony
A. Special Cases of Four-Part Harmonies in Two Strata
1. Reciprocating Strata
2. Hybrid Symmetric Strata
B. Generalization of the E-2S; S-4p
C. Three Strata of Four-Part Harmonies
D. Four and More Strata of Four-Part Harmonies
E. The Limits of Four-Part Harmonies
1. Diatonic Limit
2. Symmetric Limit
3. Compound Symmetric Limit

Chapter 7
Variable Number of Parts in the Different Strata of a Sigma
A. Construction of Sigmae Belonging to one Family
1. Σ=S
2. 1. Σ=4S
B. Progressions with Variable Sigma
C. Distribution of Given Harmonic Continuity Through Strata

Chapter 8
General Theory of Directional Units
A. Directional Units of Sp
B. Directional Units of S2p
C. Directional Units of S3p
D. Directional Units of S4p
E. Strata Composition of Assemblages Containing Directional Units
F. Sequent Groups of Directional Units

APPLICATIONS OF GENERAL HARMONY

Chapter 9
Composition of Melodic Continuity from Strata
A. Melody from one individual part of a stratum
B. Melody from 2p, 3p, 4p of an S
C. Melody from S
D. Melody from 2S, 3S
E. Generalization of the Method
F. Mixed forms
G. Distribution of Auxiliary Units through p, S and Σ
H. Variation of original melodic continuity by means of auxiliary tones

Chapter 10
Composition of Harmonic Continuity from Strata
A. Harmony from one stratum
B. Harmony from 2S, 3S
C. Harmony from Σ
D. Patterns of Distribution
E. Application of Auxiliary Units
F. Variation through Auxiliary Units

Chapter 11
Melody With Harmonic Accompaniment

Chapter 12
Correlated Melodies

Chapter 13
Composition of Canons From Strata Harmony
A. Two-Part Continuous Imitation
B. Three-Part Continuous Imitation
C. Four-Part Continuous Imitation

Chapter 14
Correlated Melodies With Harmonic Accompaniment

Chapter 15
Composition of Density In Its Applications to Strata

A. Technical Premise
B. Composition of Density-groups
C. Permutation of sequent Density-groups
D. Phasic Rotation of Δ and Δ→
E. Practical Applications of Δ→ to Σ→

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BOOK TEN
Evolution of Pitch-Families (Style)Chapter 1
Pitch-Scales as a Source of Melody

Chapter 2
Harmony
A. Diatonic Harmony
B. Diatonic-Symmetric Harmony
C. Symmetric Harmony
D. Strata (General) Harmony
E. Melodic Figuration
F. Transposition of Symmetric Roots of Strata
G. Compound Sigma

Chapter 3
Melodization of Harmony
A. Diatonic Melodization
B. Symmetric Melodization
C. Conclusion

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BOOK ELEVEN
Theory of CompositionIntroduction

Part I
COMPOSITION OF THEMATIC UNITS

Chapter 1
Components of Thematic Units

Chapter 2
Temporal Rhythm as Major Component

Chapter 3
Pitch-Scale As Major Component

Chapter 4
Melody As Major Component

Chapter 5
Harmony As Major Component

Chapter 6
Melodization As Major Component

Chapter 7
Counterpoint As Major Component

Chapter 8
Density As Major Component

Chapter 9
Instrumental Resources As Major Component
A. Dynamics
B. Tone-Quality
C. Forms of Attack

PART II
COMPOSITION OF THEMATIC CONTINUITY

Chapter 10
Musical Form

Chapter 11
Forms of Thematic Sequence

Chapter 12
Temporal Coordination of Thematic Sequence
A. Using the Resultants of Interference
B. Permutation-Groups
C. Involution-Groups
D. Acceleration-Groups

Chapter 13
Integration of Thematic Continuity
A. Transformation of Thematic Units into Thematic Groups
B. Transformation of Subjects into their Modified Variants
1. Temporal Modification of a Subject
2. Intonational Modification of a Subject
C. Axial Synthesis of Thematic Continuity

Chapter 14
Planning a Composition
A. Clock-Time Duration of a Composition
B. Temporal Saturation of a Composition
C. Selection of the Number of Subjects and Thematic Groups
D. Selection of a Thematic Sequence
E. Temporal Distribution of Thematic Groups
F. Realization of Continuity in Terms of t and t’
G. Composition of Thematic Units
H. Composition of Thematic Groups
I. Composition of Key-Axes
J. Instrumental Composition

Chapter 15
Monothematic Composition
A. “Song” from “The First Airphonic Suite”
B. “Mouvement Electrique et Pathetique”
C. “Funeral March” for Piano
D. “Study in Rhythm I” for Piano
E. “Study in Rhythm II” for Piano

Chapter 16
Polythematic Composition

PART III
SEMANTIC (CONNOTATIVE) COMPOSITION

Chapter 17
Semantic Basis of Music
A. Evolution of Sonic Symbols
B. Configurational Orientation and the Psychological Dial
C. Anticipation-Fulfillment Pattern
D. Translating Response Patterns into Geometrical Configurations
E. Complex Forms of Stimulus-Response Configurations
F. Spatio-Temporal Associations

Chapter 18
Composition of Sonic Symbols
A. Normal (Circle with clock hand at 12) Balance and repose
B. Upper Quadrant of the Negative Zone (Circle with 9 to 12 quadrant dark) Dissatisfaction, Depression and Despari
C. Upper Quadrant of the Positive Zone (Circle with 12 to 3 quadrant dark) Satisfaction, Strength, and Success
D. Lower Quadrant of Both Zones (Circle with 3 to 9 half dark) Association by Contrast: The Humorous and Fantastic

Chapter 19
Composition of Semantic Continuity
A. Modulation of Sonic Symbols
1. Temporal Modulation
2. Intonational Modulation
3. Configurational Modulation
B. Coordination of Sonic Symbols
C. Classification of Stimulus-Response Patterns

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BOOK TWELVE
Theory of Orchestration