Dissertation Index

Author: Scotto, Ciro, G.

Title: Can Non-Tonal Systems Support Music as Richly as Tonal Systems?

Institution: University of Washington

Begun: September 1991

Completed: December 1995


Can non-tonal systems support music as richly as the tonal system? The question has many possible interpretations, and each interpretation requires a different type of answer. Moreover and more important, each interpretation of the proposition and its corresponding answer directs the investigation down a different path. A proposition's context often determines its sense, and determining a proposition's sense is one method of specifying its interpretation.

Chapter one begins by examining the role context plays in establishing the proposition's sense and the role context plays in determining the sense of a musical expression in music. The sense of a musical expression in music leads to a discussion about expressed and unexpressed music theories which lays the groundwork for establishing the context and sense for proposition P . This essay examines proposition P in the comparative context. The goal of the investigation in this context is to build bridges between domains by integrating tonal structures, such as prolongation and structural levels, and non-tonal structures, such as trichords and uniform trichordal arrays, into a single system called the hybrid system. However, since a recent argument has suggested that the design of compositional systems may be limited by certain cognitive constraints, the author addresses this question in depth in chapter one before constructing the hybrid system.

Since uniform trichordal arrays form the hybrid system's rear end, and since uniform trichordal arrays are a subclass of the class of self-deriving rows, chapter two constructs a detailed model of all self-deriving rows. Using the model of self-deriving rows as a foundation, chapter three constructs a model of the subclass of uniform trichordal arrays, while chapter four compares the subclass of self-deriving rows capable of generating uniform trichordal arrays to other self-deriving rows capable of generating other array types and twelve-tone arrays in general. Chapter five begins the transformation of a uniform trichordal array's structure into the foundations of the hybrid system. Chapter six integrates uniform trichordal arrays and tonal theory to produce the engine that drives the hybrid system. Chapter seven demonstrates how intra-set-type relations translate into inter-set-type relations, and finally, chapter eight briefly reexamines proposition P.

Keywords: non-tonal system, tonal system, cognitive constraints, self-deriving rows, structural levels, hybrid system


Chapter 1: Philosopheme 1
Review of Meaning and Truth 2
Meaning in Music 14
Context, Meaning, and Theory 23
Expressed and Unexpressed Theories 28
Context 38
Lerdahl's Cognitive Constraints 41
Universals 62
Representations 70
Linguistics, Psychology, and Generative Music Theory 81
The Terms of Proposition P 121
Outline of the Hybrid System 139
Chapter 2: Ampliation--A Model of Self-Deriving Arrays
2.1 Towards a Model of Self-Deriving Arrays 145
Set Properties and Folding Sub-Category 3 Type 2 Arrays 164
Partitioning Type 1 and Type 2 Arrays into Subclasses 171
Pruning Table 2.4 by Means of Algorithms for Type 1 and
Type 2 Self-Deriving Arrays 193
2.2 Type 1 Combinations 195
Subclass 1a--General 195
Algorithm for Type 1 Subclass 1a Arrays 201
Preliminaries 201
Procedure 208
Procedural Changes 216
Multiple Orderings, Cycle Length, and Hidden Cycles 221
Merging Schemata 234
Subclass 1c--General 240
Algorithms for Type 1 Subclass 1c Arrays
T0-TnIR/TnI-TnR Special Case 243
T0-TnIR/Tn-TnIR 248
Procedural Changes 253
T0-TnIR/TnM-TnIMR 268
Subclass 1b--General 277
Algorithm for Type Subclass 1b Arrays
T0-TnR/TnM-TnMR 280
T0-TnR/TnIM-TnIMR 286
Subclasses 1d and 1e--General 292
Algorithms for Type 1 Subclasses 1d and 1e Arrays
T0-TnMR/Tn-TnMR, T0-TnMR/TnIM-TnIR, and T0-TnIMR/TnM-
TnIR 294
T0-TnMR/TnM-TnR, T0-TnIMR/Tn-TnIMR, T0-TnIMR/TnIM-TnR 300
2.3 Type 2 Combinations
Preliminaries 306
Subclass 2a: Hexachords 325
Algorithm: Partition Identity Arrays 326
Commentary 328
Subclass 2a: Tetrachords 329
T0-Tn/TnR 334
T0-Tn/TnM 335
T0-Tn/TnMR 337
Commentary and Summary 338
Subclass 2b: Hexachords 339
Algorithm: Partition Identity Arrays 340
Commentary 346
Algorithm: Intersecting Partition Arrays, Subclass 2b,
Schema 3-3, T0-TnI/TnR Combination 347
Algorithm: Intersecting Partition Arrays, Subclass 2b,
Schema 3-3, T0-TnI/TnIR Combination 352
Commentary 355
Algorithm: Intersecting Partition Arrays, Subclass 2b,
Schema 4-2 358
Subclass 2b: Tetrachords 359
Algorithm: Partition Identity Arrays 360
Summary 362
Subclass 2c: Hexachords 363
Algorithm: Partition Identity Arrays 363
Commentary 366
Subclass 2c: Tetrachords 367
Algorithm: Partition Identity Arrays 367
Summary 370
Subclass 2d: Hexachords 371
Algorithm: Partition Identity Arrays 371
Commentary 374
Algorithm: Intersecting Partition Arrays, Subclass 2d,
Schema 3-3, T0-TnIM/TnMR Combination 375
Subclass 2d: Tetrachords 381
Algorithm: Partition Identity Arrays 381
2.4 Summary 383
Chapter 3: Cosmogony--Uniform Trichordal Arrays
Preliminaries 385
Algorithm for Generating a Uniform Trichord Array from
a Seed Array 399
Structural Model of Uniform Trichordal Arrays 419
Chapter 4: Alternative Worlds--Semi-Uniform Trichordal
Arrays and Non-Self-Deriving Arrays
Prelininaries 473
Generating a Semi-Uniform Trichordal Seed-Array from an
Alternative-Merging Schemata 481
Three-Lyne Self-Deriving Arrays 487
Non-Self-Deriving Arrays 494
Chapter 5: The Hybrid System--from Array to Schema
Uniform Trichordal Arrays: Definition of Function 500
Array as Schema: From Pitch-Class to Pitch 512
2'Schematic Structure as Compositional Determinant
From Order to Content 519
Chapter 6: The Schema of Causality 537
Exposition and Translation 538
Trichord Progressions: Part I 580
Trichord Progressions: Part II 614
Translevel Connections 653
Chapter 7: Second Plane Tertiary Structures 673
Chapter 8: Epilogue 703
Bibliography 716
Appendix i: Common-Tone Vectors for Hexachords Under M5
and M7 728
Appendix ii: Hexachordal Information for the
Generation of Uniform Trichordal Arrays and the Arrays
They Generate 733
Appendix iii: Trichord Partitioning Patterns for
Uniform Trichordal Arrays 845
Appendix iv: Detailed Trichord Vectors for Tn/TnI Type
Hexachords 858
Appendix v: Episodes for Guitar 871
Appendix vi: Tetralogy 886


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