Mathematics and Computation in Music: 8th International Conference, MCM 2022, Atlanta, GA, USA, June 21–24, 2022, Proceedings

Preface
Organization
An Afternoon of Math+Music@MODA (Abstract of Invited Talk)
Contents
Mathematical Scale and Rhythm Theory: Combinatorial, Graph Theoretic, Group Theoretic, and Transformational Approaches
A Set-Theoretic Model of Meter and Metric Dissonance
	1 Introduction
	2 A Set Theoretic Model of Meter
		2.1 Time point and span
		2.2 Pulse
		2.3 Meter
		2.4 Metric Relations
	3 Kinds of Metric Relations
		3.1 Pulse-Association Classes
		3.2 Metric-Relation Classes
		3.3 Three Genera of Metric Relations
	4 Extensions
	References
New Insights on Diatonicity and Majorness
	1 Progressive Transposition Scales
		1.1 Properties of PT Sets
		1.2 The Case N Prime
	2 Generalized Diatonic Scales
	3 Generalized Major and Minor Scales
		3.1 Case N Even
	References
Parsimonious Graphs for Selected Heptatonic and Pentatonic Scales
	1 Introduction
	2 Selection of Heptatonic Scales
		2.1 Combinations of Two Tetrachords
		2.2 Combinations of the Altered Notes from the Nearest Key Signatures
		2.3 Harmonic Characteristics of the Selected Heptatonic Scales
	3 Parsimonious Graphs for the Selected Heptatonic Scales
	4 Pentatonic Scales
		4.1 Selection of Pentatonic Scales
		4.2 Parsimonious Graphs for the Selected Pentatonic Scales
		4.3 Relation Between the Selected Heptatonic and Pentatonic Scales
	5 Example of Musical Analysis
	6 Conclusions
	References
An Interactive Tool for Composing (with) Automorphisms in the Colored Cube Dance
	1 An Algebraic Introduction to the Colored Cube Dance
	2 The Automorphism Group of the Monoid Action of MU,P, L
	3 An Interactive Interface for Composing (with) Automorphisms
	References
Combinatorial Spaces
	1 Introduction
	2 Music-Theoretical and Mathematical Background
	3 Results and Applications
		3.1 I Combinatoriality
		3.2 P Combinatoriality
		3.3 R Combinatoriality
		3.4 RI Combinatoriality
	4 Conclusions
	References
Euler's ``Tentamen'': Historical and Mathematical Aspects on the Consonance Theory
	1 Introduction
	2 Some Historical Aspects on Euler's Musical Interests and the Birth of the ``Tentamen''
	3 Sound and Hearing
	4 Pleasure and Consonance
		4.1 Study of Chords with Two Sounds
		4.2 Study of Chords with More Than Two Sounds
	5 Comparison Between Tentamen and Other Mathematical Consonance Theories
	References
Categorical and Algebraic Approaches to Music
A Projection-Oriented Mathematical Model for Second-Species Counterpoint
	1 Introduction
	2 General Overview of Mazzola's Counterpoint Model
	3 Dichotomies of 2-Intervals
	4 Species Projections
	5 Algorithm for the Calculation of Projections
	6 Comparison with Fux's Approach
	References
When Virtual Reality Helps Fathom Mathemusical Hyperdimensional Models
	1 Context and Definitions
		1.1 Mathemusical Models
		1.2 Virtual Reality
		1.3 Immersive World Generation
	2 Methodology
		2.1 Tools
		2.2 VR Techniques
	3 Applications
		3.1 2D Space: Virtual Swan Lake
		3.2 4D Space: Entangled Hypersphere
	4 Feedback and Future
		4.1 Feedback
		4.2 Future Developments
	5 Conclusion
	References
SUM Classes and Quotient Generalized Interval Systems
	1 Introduction
	2 SUM-Class Transformation Groups and Quotient Generalized Interval Systems
	3 The Dual Quotient Generalized Interval System
	4 Other Set-Classes
	References
Extended Vuza Canons
	1 Prelude
	2 Aperiodic Tiling Canons
	3 Extended Vuza Canons
	References
Some Mathematical and Computational Relations Between Timbre and Color
	1 Introduction
	2 Spaces and Mappings: An Overview
		2.1 The CIE 1931 Color Space
		2.2 Timbre Space
		2.3 Maps Between Timbre and Color
	3 Categorical Enrichment
		3.1 Induced Infinity-Groupoids
		3.2 Induced Functors
		3.3 A Computation of Colors from Timbres
	4 Gestural Considerations
		4.1 From Paths to Gestures and Gestural Similarity
		4.2 Induced Maps Between Spaces of Gestures
	5 Conclusion
	6 Glossary
	References
Transformations for Pairwise Well-Formed Modes
	1 Motivation
	2 Letter Frequencies and Periods in Pairwise Well-Formed Words
	3 Syntonic and Diatonic Morphisms Revisited
	4 Specific PWWF Superpositions of Specific Syntonic and Diatonic Modes
	5 Supersymmetries: Exploring a Refined Graham Construction
	References
Algorithms and Modeling for Music and Music-Related Phenomena
Spline Modeling of Audio Signals with Cycle Interpolation
	1 Introduction
		1.1 Motivation and Background
	2 The Basic Model
	3 Cycle Interpolation
	4 Problems with Cycle Interpolation
	5 Future Work
	References
Transposition and Time-Scaling Invariant Algorithm for Detecting Repeated Patterns in Polyphonic Music
	1 Introduction
	2 Problem Definition
		2.1 Example
		2.2 Lower Bound
		2.3 Simple Algorithm
	3 Algorithm Description
		3.1 Example
		3.2 Filtering Repetitions
	4 Experiments
		4.1 Efficiency
		4.2 Pattern Discovery
	5 Conclusions
	References
On the Memory Usage of the SIA Algorithm Family for Symbolic Music Pattern Discovery
	1 Introduction
	2 Background
	3 Sweepline Algorithm
		3.1 Correctness and Analysis of the Algorithm
		3.2 Example
	4 Classifying Patterns
		4.1 Finding All TECs
		4.2 Finding Distinct TECs
	5 Experiments
		5.1 Efficiency
		5.2 Filtering Patterns
	6 Conclusions
	References
A Proposal to Compare the Similarity Between Musical Products. One More Step for Automated Plagiarism Detection?
	1 Introduction
	2 Theoretical Background
		2.1 Clustering Methods
		2.2 Hard and Soft Partitions
		2.3 K-Means Clustering
		2.4 Fuzzy C-Means Clustering (FCM)
		2.5 Fuzzy Ordered C-Means Clustering (FOCM)
		2.6 Definition of a Dissimilarity Based on FOCM Clustering
	3 A Comparison of Musical Products Based on FOCM
		3.1 FFT Process
		3.2 Peak Detection Algorithm
		3.3 Constellation Map
		3.4 Calculation of the Dissimilarity Based on FOCM
	4 Experiments
	5 Conclusions
	References
A New Fitness Function for Evolutionary Music Composition
	1 Introduction
	2 Related Work
	3 Theoretical Background
		3.1 Mathematical Definition of a Melody
		3.2 Neighbourhood Functions
		3.3 Fitness Function
	4 Genetic Algorithm
		4.1 Genotype Representation
		4.2 Restrictions on the Evolution Strategy
		4.3 Mutation
		4.4 Initialization of the Population, Parents Selection and Crossover
		4.5 Performance Indicators
	5 Experiments
		5.1 First Experiment
		5.2 Second Experiment
		5.3 Results
	6 Discussion
	7 Conclusions
	References
A Mathematical Model of Tonal Function (I): Voice Leadings
	1 What Is Tonal Function?
	2 Basic Concepts
		2.1 Pitch Classes and Chords
		2.2 The Nabla Distance
	3 A Mathematical Model of Tonal Function
		3.1 Voice Optimization
		3.2 Cadence Endomorphisms and Chord Classification
	4 Conclusions
	References
A Mathematical Model of Tonal Function (II): Modulation
	1 Introduction
	2 Tonal Function in Chords with Different Number of Voices
	3 Dual Tonal Function
	4 Chord Classification
	5 Modulation
	6 Conclusions
	References
Hypercube + Rubik's Cube + Music = HyperCubeHarmonic
	1 Introduction
	2 Mathematical Concepts
		2.1 Ars Combinatoria, Permutations, and the Rubik's Cube
		2.2 The Hypercube
		2.3 The Hyper-Rubik's Cube
		2.4 The Hyper-Rubik's Cube with Music: The HyperCubeHarmonic
	3 First Implementations
		3.1 Implementation 1
		3.2 Implementation 2
	4 Discussion and Conclusions
	References
Applications of Mathematics to Musical Analysis
Mathematical Morphology Operators for Harmonic Analysis
	1 Introduction
	2 Algebraic Framework
		2.1 Time Group
		2.2 Frequency Groups
		2.3 Time-Frequency Groups
	3 Mathematical Morphology
		3.1 Erosion and Dilation
		3.2 Opening and Closing
		3.3 Hit-or-miss Transform
	4 Applications
		4.1 Mathematical Morphology on Z12
		4.2 Mathematical Morphology on Chroma Rolls
	5 Conclusions
	References
Computational Analysis of Musical Structures Based on Morphological Filters
	1 Introduction
	2 Creating a Self-distance Matrix from Symbolic Music
		2.1 Converting Symbolic Music to Sequence Using Chord Contour
		2.2 Distance Matrix of a Chord Contour Sequence
	3 Analysis of the Self-distance Matrix Using Morphological Operations and Filters
		3.1 A Short Introduction to Morphological Filters
		3.2 Application of Mathematical Morphology to the Self-distance Matrix
	4 Changing the Shape of the Morphological Filter to Detect Different Musical Structures
	5 Conclusions
	References
Non-spectral Transposition-Invariant Information in Pitch-Class Sets and Distributions
	1 Pitch-Class Set Theory and Homometry
	2 Products of DFT Coefficients
		2.1 Definitions
		2.2 Features
	3 Example: Tonal Pitch-Class Distributions
	4 Example: All-Interval Tetrachords and Takemitsu
	5 Conclusion
	References
Tetrachordal Folding Operations
	1 Trichordal Folding
	2 Tetrachordal Folding
	3 Application to Morton Feldman's ``For Stephan Wolpe''
	4 Conclusion
	References
Mathematical Techniques and Microtonality
Continuous Chromagrams and Pseudometric Spaces of Sound Spectra
	1 Introduction
		1.1 Pitch and Chroma
		1.2 Motivations
	2 Continuous Chromagrams
		2.1 Illustration and Definition
		2.2 Invariance to Interval of Reduction
		2.3 Computational Implementations
	3 Pseudometric Spaces of Sound Spectra
		3.1 Lp Pseudometrics
		3.2 Metrics Induced from Pseudometrics
		3.3 Comparing Time Variant Spectra
	4 Applications
		4.1 Visualizations of Musical Sounds
		4.2 Creative and Theoretical Possibilities
	References
N2D3P9
	1 Background on Sagittal Notation
	2 A Preliminary Pop Culture Reference
	3 Formula
	4 Justification
	5 Development/Discovery
	References
Performing Easley Blackwood’s Twelve Microtonal Etudes:An Open-Source Software Development Project
	1 Introduction
	2 Examining Some Keyboard Mappings for Microtonal Music
	3 An Open-Source Project for Microtonal Music Software
	4 Conclusion
	References
Short Papers
Identifying Metric Types with Optimized DFT and Autocorrelation Models
	1 Procedure
		1.1 DFT
		1.2 Autocorrelation
		1.3 Corpus and Data Preparation
		1.4 Weighting, Windowing, and Training
	2 Results
	References
Persistent Homology on Musical Bars
	1 Introduction
		1.1 Persistent Homology
		1.2 Context and Problematic
	2 Persistent Homology on Musical Bars
		2.1 Filtration: The Vietoris-Rips Method
		2.2 Musical Bars of a Score
		2.3 The Score as a Point Cloud
	3 Application: Analysis of a Musical Piece
	4 Conclusion and Prospect
	References
Formal Structures of a Harmony in the Parabola
	1 Introduction
	2 The Group Law on the Parabola
	3 Harmonic Polygons over a Parabola
	4 The Ring and Field Law on the Parabola
	5 Parabola over a Module and Affine Transformations
	6 Group Actions over Parabolic Music
	7 Parabolic Planet-8D and Metric
	8 The Discrete Fourier Transform in a Parabolic World
	9 Conclusions
	References
midiVERTO: A Web Application to Visualize Tonality in Real Time
	1 Introduction
	2 midiVERTO: Features and Technical Details
	3 A Brief Case Study
	4 Conclusion
	References
Quantum-Musical Explorations on Zn
	1 Motivation
	2 Quantum Theory on Zn
	3 Exploring the Finite Quantum-Harmonic Oscillator
	References
The Mystery of Anatol Vieru's Periodic Sequences Unveiled
	1 Anatol Vieru's Periodic Sequence: A New Formalization
	2 Decomposing  Pm
		2.1 Decomposition with Primes
		2.2 Decomposition in Nilpotent and Idempotent Part
		2.3 Decomposition of Nilpotent Sequences Using Constants
	3 Unveiling the Period and the Proliferation of Values
		3.1 Period of the Primitives of Vieru's Sequence
		3.2 Proliferation of Values
	4 Recursive Formulas for the Number of {4,8} in 8s-3 V
	References
Benford's Law and Music Note Frequencies
	1 Introduction
		1.1 What is Benford's Law
	2 Numeric Proof
	3 Piano Note Frequencies
	4 Application
	5 Conclusion
	References
Altered Chord Alternatives
	1 Introduction
	2 Motivation
	3 Theoretical Framework
	4 Methodology
	5 Results
	6 Future Directions
	References
Information Synthesis of Time-Geometry QCurve for Music Retrieval
	1 Introduction
	2 QCurve Transform
	3 Harmonic Consistency
	4 Harmonic Leading
	5 Consonance
	6 LSQOP
	7 Evaluation and Analysis
	8 Conclusions
	References
Investigating Style with Scale Embeddings
	1 Introduction
	2 Methodology: Word2vec, Encoding Procedure, and Training
		2.1 Word2vec Algorithm
		2.2 Encoding Procedure
		2.3 Model Parameters
	3 Properties of Embeddings
		3.1 The Circle of Fifths According to the Mozart Model
		3.2 Composer Embeddings Correlated with the Circle of Fifths
	4 Conclusion
	References
Author Index