This book offers an in-depth analysis of musical variation through a systematic approach, heavily influenced by the principles of Grundgestalt and developed variations, both created by the Austrian composer Arnold Schoenberg (1874-1951). The author introduces a new transformational-derivative model and the theory that supports it, specifically crafted for the examination of tonal music.
The idea for this book emerged during a sabbatical at Columbia University, while the content is the product of extensive research conducted at the Federal University of Rio de Janeiro, resulting in the development of the Model of Derivative Analysis. This model places emphasis on the connections between musical entities rather than viewing them as separate entities. As a case study, the Intermezzo in A Major Op.118/2 by Brahms is selected for analysis.
The author's goal is to provide a formal and structured approach while maintaining the text's readability and appeal for both musicians and mathematicians in the field of music theory. The book concludes with the author's recommendations for further research.
Author(s): Carlos de Lemos Almada
Series: Computational Music Science
Publisher: Springer
Year: 2023
Language: English
Pages: 328
City: Cham
Preface
Acknowledgements
References
Contents
List of Figures
List of Tables
Part I Decontextualized Variation
1 Basic Concepts
1.1 Derivative Work
1.2 Derivative Impact and Relations of Similarity
1.3 Derivative Space
1.4 Levels of Derivative Work
References
2 Decomposable Variation
2.1 Phases of Decomposable Variation
2.2 Classifications of Decomposable Variation
2.3 Attributes
2.3.1 Pitch
2.3.2 Time
2.3.3 Harmony
2.4 Matrix of Attributes
References
3 Measurement of Similarity (I)
3.1 General Information
3.2 Measurement of Similarity in the Pitch Domain
3.3 Measurement of Similarity in the Temporal Domain
3.4 Measurement of Similarity in the Harmonic Domain
3.5 Global Similarity Between Musical Ideas
3.6 Spatial Representation of Similarity Relations
References
4 Transformational Operations
4.1 A Set-Functional Conception
4.2 Classifications of Operations
4.2.1 Types
4.2.2 Domains of Application
4.2.3 Scope of the Argument
4.2.4 Complementary Symbology
4.2.5 Alteration of Cardinality
4.2.6 Collateral Effects
4.3 Descriptions
4.3.1 Addition
4.3.2 Augmentation
4.3.3 Change of Mode
4.3.4 Change of Register
4.3.5 Chordal Inversion
4.3.6 Chromatic Alteration
4.3.7 Chromatic Inversion
4.3.8 Chromatic Transposition
4.3.9 Contextual Dyadic Transformation
4.3.10 Deletion
4.3.11 Diatonic Inversion
4.3.12 Diatonic Transposition
4.3.13 Diminution
4.3.14 Extension
4.3.15 Interpolation
4.3.16 Merging of Durations
4.3.17 Metric Displacement
4.3.18 Permutation
4.3.19 Re-harmonization
4.3.20 Re-partition
4.3.21 Replication
4.3.22 Rest Substitution
4.3.23 Retrogradation
4.3.24 Rotation
4.3.25 Split Duration
4.3.26 Subtraction
4.3.27 Suppression
4.4 Composition of Operators
4.5 Summary
4.6 Transformational-Derivative Analysis
4.6.1 Case 1: Simple/Pure Decomposable Variation
4.6.2 Case 2: Compound/Pure Decomposable Variation
4.6.3 Case 3: Simple/Hybrid Decomposable Variation
4.6.4 Case 4: Compound/Hybrid Decomposable Variation
References
5 Measurement of Similarity (II)
5.1 Similarity Between Non-compatible Cardinalities
5.1.1 Example 1
5.1.2 Example 2
5.1.3 Example 3
5.2 Some Concluding Remarks
Part II Variation on Time
Reference
6 Grundgestalt
6.1 Grundgestalt-Components
6.2 Variables
6.3 Orders of Grundgestalten
6.4 Identification of a Grundgestalt
References
7 Developing Variation
7.1 A Transformational Typology of Developing Variation
7.2 Relative and Absolute Developing Variation
7.3 Genealogical Notation
7.4 Derivation of Variables
7.5 Quotient
7.6 Crossover
7.7 Inter-Attribute Equivalence
7.8 Thematic Transformation, Linkage, and Metric Displacement
7.9 Speciation
7.10 Teleology
7.11 Involution
References
Part III Analysis: Brahms—Intermezzo in A Major Op. 118/2
References
8 Formal, Harmonic, and Metric Structure
8.1 Form
8.2 High-Level Harmony (Tonal Relations)
8.3 Low-Level Harmony (Chord Progressions)
8.4 Metric Structure
8.4.1 Segmentation
8.4.2 Metric Organization of Passages I–IX
References
9 Derivative Analysis
9.1 Methodology
9.2 Grundgestalt, Variables, and Segmentation
9.3 Derivative Segment 1
9.3.1 Derivation of uA 02
9.3.2 Derivation of uA 03
9.3.3 Derivation of uA 05
9.3.4 Derivation of uA 06
9.3.5 Derivation of uA 07
9.3.6 Derivation of uA 08
9.3.7 Derivation of uA 10
9.3.8 Derivation of uA 13
9.3.9 Derivation of uA 14
9.3.10 Derivation of uA 15
9.3.11 Derivation of uA 16
9.3.12 Overview of Segment 1
9.4 Derivative Segment 2
9.4.1 Derivation of uA 17
9.4.2 Derivation of uA 18
9.4.3 Derivation of uA 19
9.4.4 Derivation of uA 21
9.4.5 Derivation of uA 22
9.4.6 Derivation of uA 23
9.4.7 Derivation of uA 24
9.4.8 Overview of Segment 2
9.5 Derivative Segment 3
9.5.1 Derivation of uA 25
9.5.2 Derivation of uA 26
9.5.3 Derivation of uA 28
9.5.4 Derivation of uA 29
9.5.5 Derivation of uAs 31 and 32
9.5.6 Derivation of uAs 33 and 34
9.5.7 Derivation of uAs 36
9.5.8 Overview of Segment 3
9.6 Derivative Segment 4
9.6.1 Derivation of uA 37
9.6.2 Derivation of uA 41
9.6.3 Derivation of uA 42
9.7 Derivative Segment 5
9.7.1 Derivation of uA 45
9.7.2 Derivation of uAs 46 and 48
9.7.3 Derivation of uAs 47, 49, and 50
9.7.4 Derivation of uA 51
9.7.5 Derivation of uAs 52 and 54
9.7.6 Derivation of uAs 53 and 55
9.7.7 Overview of Segment 5
9.8 Derivative Segment 6
9.8.1 Derivation of uA 56
9.8.2 Derivation of uAs 57, 58, and 59
9.8.3 Derivation of uA 60
9.8.4 Derivation of uAs 61 and 62
9.8.5 Overview of Segment 6
9.9 Derivative Segment 7
9.9.1 Derivation of uA 63
9.9.2 Derivation of uA 64
9.9.3 Derivation of uA 65
9.9.4 Derivation of uA 67
9.9.5 Derivation of uA 68
9.9.6 Derivation of uA 69
9.9.7 Derivation of uA 71
9.9.8 Overview of Segment 7
9.10 Segment 8
9.11 Discussion
9.11.1 A Qualitative Perspective of the Variants Production
9.11.2 High-Level Variants
9.11.3 Developing Variation
9.11.4 Low-Level Variants
9.11.5 The Derivative Role of the Kopfnote
References
Afterword
Afterword
A Variation in Non-tonal Contexts
B MDA*
Pitch Filtering
Rhythmic Filtering
Variation in MDA*
C Algorithms
