]]>

Once this broad goal is accomplished, or perhaps as I go, I will consider some potential programming implementations of the techniques within the software musicians frequently use today.

]]>**BOOK 1
Theory of Rhythm**

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

**BOOK TWO
Theory of Pitch Scales**

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**

**BOOK THREE
Variations of Music By Means of Geometrical Projection**

Geometrical Inversions

**Chapter 2
Geometrical Expansions
**

**BOOK FOUR
Theory of Melody**

Introduction

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**

**BOOK FIVE
Special Theory of Harmony**

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**

**BOOK SIX
The Correlation of Harmony and Melody**

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

**BOOK VII
Theory of Counterpoint**

The Theory of Harmonic Intervals

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

**BOOK EIGHT
Instrumental Forms**

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

**BOOK NINE
General Theory of Harmony: Strata Harmony**

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→

**BOOK TEN
Evolution of Pitch-Families (Style)**

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

**BOOK ELEVEN
Theory of Composition**Introduction

**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

**BOOK TWELVE
Theory of Orchestration**

The vast strides in musical software development during the last thirty years have yet to integrate the benefits of a system which had already systematized many of the musical processes that composers spend much of their time with.

The Schillinger system uses a graphical notation nearly identical with that of MIDI sequences, yet Schillinger didn’t have the advantage of a computer to implement his system, and most sequencer programmers don’t know what benefits it could offer them.

Theorists who discuss the system often are so intellectually entranced by analyses of classical works that they fail to consider its potential implementation in a contemporary setting.

I hope to begin to consider some of the potential uses of the system, and to see how the mathematical language could be made transparent to a musician.

]]>What input and outputs relate to the materials of each chapter in his books?

How can the mathematical jargon be hidden from the musician, making it accessible to someone with traditional musical training? Are his uses of mathematical terms misleading or even incorrect?

Can his transformational processes be integrated into existing sequencers or music notation programs like Digital Performer, Cubase, Logic, Finale, and Sibelius?

Can they be applied to digital sound file DSP transformations, or only MIDI streams?

Can they be offered as Open Source software? Or will proprietary implementations be necessary?

What elements of a computer or musical interface would best serve the implementation of each process?

What aspects of computer interface design are most important in the development of such a system?

]]>