Let’s Calculate Bach: Applying Information Theory and Statistics to Numbers in Music

Foreword
Preface
Acknowledgements
Contents
Abbreviations
List of Figures
List of Tables
1 Introduction
	1.1 The Science of Musicology
	1.2 Numerology and Bach
	1.3 About This Book
2 An Information Theory Approach
	2.1 Information and Communication
	2.2 Measuring Information—The Bit
	2.3 The Bit as Binary Digit
	2.4 Signal, Noise, Redundancy and Encoding
	2.5 Messages and Symbols
	2.6 Throughput and Protocols
	2.7 Gematria as Hash Coding
	2.8 An Unambiguous Coding
	2.9 Codings and References
3 Some Possible Codings in Music
	3.1 Preamble
	3.2 Number of Bars
	3.3 Notes
	3.4 Intervals
	3.5 Note Lengths
	3.6 Number of Notes
	3.7 Number of Pieces, Movements or Sections
	3.8 Sum of the G-Values of Notes
	3.9 Key Signature
	3.10 Accidentals
	3.11 Occurrences of Words
	3.12 Rests
	3.13 Time Signature
	3.14 Figured Bass
	3.15 Entries of a Theme
	3.16 Other Possibilities
		3.16.1 Acrostics
		3.16.2 More Subtle Ways
	3.17 Beyond Bach
		3.17.1 BWV Numbers
		3.17.2 Frequencies
		3.17.3 Morse Code
		3.17.4 Colours and Shapes
		3.17.5 Other Puzzles
	3.18 Combined Codings
	3.19 A Cryptographic Example
	3.20 Summary
	3.21 The Real Coding
	3.22 Notes for Researchers
4 Ambiguity in Decoding
	4.1 Preamble
	4.2 Sources
	4.3 Modern Dictionary
		4.3.1 Method
		4.3.2 Modern Dictionary with Latin Natural Coding
		4.3.3 Modern Dictionary with Latin Milesian and Trigonal Coding
	4.4 Historic Sources
		4.4.1 Luther Bible
		4.4.2 Cantata Texts
		4.4.3 Combining Historic Sources
	4.5 Summary
	4.6 Notes for Researchers
5 Multiple Words and Partitioning
	5.1 Partitioning and Permutations
	5.2 Partitioning G-Values
	5.3 Composers’ Names
	5.4 Notes for Researchers
6 Score Analysis
	6.1 The Method
	6.2 Counting Bars
	6.3 Statistics
	6.4 Further Applications
	6.5 Summary
	6.6 Other Representations and Tools
	6.7 Notes for Researchers
7 Statistical Methods
	7.1 Preamble
	7.2 Probability and Distributions
	7.3 Hypothesis Testing and Significance
	7.4 Confidence Interval
	7.5 Monte Carlo Simulation
	7.6 Bayes Theorem
	7.7 Notes for Researchers
8 Exploring Proportions
	8.1 Preamble
	8.2 Simple Proportions and Terminology
		8.2.1 Sets and Pieces
		8.2.2 Proportion
		8.2.3 Combinations
		8.2.4 Solutions, Targets, Opposites and Complements
		8.2.5 Symmetries, Signatures and Patterns
		8.2.6 Binary Signatures
	8.3 Layers of Proportion
	8.4 Summary of Terms
	8.5 The Proportional Parallelism Explorer Program
		8.5.1 Solution Search
		8.5.2 Solutions Search Through Layers
		8.5.3 Pattern Matching
		8.5.4 Pattern Matching in Layers
		8.5.5 Colour Coding for Visual Pattern Recognition
		8.5.6 Monte Carlo Simulation
9 Applying the Methods to the Well Tempered Clavier Book 1 BWV 846–869
	9.1 Preamble
	9.2 Solutions
	9.3 Probability
	9.4 Monte Carlo Simulation
	9.5 Hypothesis Testing and Significance
	9.6 Bayes Theorem
	9.7 Patterns
	9.8 Preludes and Fugues Separately
	9.9 Ariadne Musica
10 Consolidated Observations
	10.1 Preamble
	10.2 The Effect of the Number of Pieces
	10.3 Works Which Could Have More Than One Layer
	10.4 Probability
	10.5 Types of Distribution
	10.6 Real Works Versus Single-Layer Simulations
	10.7 Accuracy
	10.8 Proportions and Other Structures
	10.9 Proportions in Durations
	10.10 Works with No Proportions
	10.11 The Impossible Proportions
	10.12 Reverse Engineering and the Art of Fugue BWV 1080
	10.13 Combining Works
	10.14 Summary of Main Statistics
	10.15 Notes for Researchers
11 Magic Squares
	11.1 The Dieben Rectangle
		11.1.1 Deriving Proportions from Dieben’s Rectangle
	11.2 Use by Modern Composers
12 Psychological Fallacies
	12.1 Preamble
	12.2 Story Bias or Narrative Fallacy
	12.3 Confirmation Bias
	12.4 Neglect of Probability
	12.5 Halo Effect
	12.6 Conjunction Fallacy
	12.7 Notes for Researchers
13 Bach, Science and Technology
	13.1 Mathematics and Philosophy in Bach’s Time
	13.2 Models
	13.3 Computers
	13.4 Artificial Intelligence
	13.5 Quantz and Hi-Fi
	13.6 Summary
14 Conclusion
Appendix A More Parallel Proportion Results
A.1 Preamble
A.2 Mass in B Minor BWV 232
A.3 Brandenburg Concertos BWV 1046–1051
A.4 Canonic Variations on “vom Himmel Hoch” BWV 769
A.5 Cello Suites BWV 1007–1012
A.6 Clavierübung I—Partitas BWV 825–830
A.7 Clavierübung II—Italian Concerto BWV 831 and French Overture BWV 971
A.8 Clavierübung III—Organ Mass BWV 669–689, 552, 802–805
A.9 English Suites BWV 806–811
A.10 French Suites BWV 812–817
A.11 Goldberg Variations BWV 988
A.12 Great Fifteen Organ Preludes BWV 651–665
A.13 Inventions and Sinfonias BWV 772–801
A.14 Musical Offering BWV 1079
A.15 Schübler Chorales BWV 645–650
A.16 Sei Soli for Violin BWV 1001–1006
A.17 Transcribed Concertos BWV 972–980, 592, 981–982
A.18 Trio Sonatas for Organ BWV 525–530
A.19 Violin Sonatas BWV 1014–1019
A.20 Well Tempered Clavier Book 1—See Chap. 9
Appendix B Proportional Parallelism Explorer Program User Manual
B.1 Installing the Program
B.1.1 Download
B.1.2 Prerequisite
B.1.3 Caveats
B.1.4 Installing the Program
B.1.5 Removing the Program
B.2 Running the Program
B.2.1 From the Downloaded File
B.2.2 Running the Program from the Command Prompt
B.2.3 Running Multiple Instances
B.2.4 Starter Script
B.2.5 CSV Separator
B.2.6 Mixed Languages
B.2.7 Error Messages
B.3 File Processing
B.3.1 File Processing Data Flow
B.3.2 Input File—General
B.3.3 Input File—Pieces
B.3.4 Input File—Patterns: Overview
B.3.5 Input File—Patterns: BinaryTemplate
B.3.6 Input File—Patterns: Count
B.3.7 Input File—Patterns: Built in Functions
B.3.8 Solutions Output File
B.3.9 Patterns Output File
B.3.10 File Processing Summary
B.4 Monte Carlo Simulation
B.4.1 Monte Carlo Processing Data Flow
B.4.2 Results Output File
B.4.3 Monte Carlo Summary
B.4.4 Lengths Histogram Output File
B.4.5 Results Histogram Output File
B.5 Further Processing with Other Programs
B.5.1 Import to Excel
B.5.2 Formatting Histograms with Excel
B.5.3 Text Processors
B.6 Main Window
B.7 Program Menu
B.8 Preferences
B.5.4 General Tab
B.5.5 Monte Carlo Tab
B.5.6 Output Tab
B.9 Close
B.10 File Menu
B.11 Open File
B.12 Start from File
B.13 Pause File
B.14 Resume File
B.15 Cancel File
B.16 Monte Carlo Menu
B.17 Start Monte Carlo
B.18 Pause Monte Carlo
B.19 Resume Monte Carlo
B.20 Cancel Monte Carlo
B.21 Help Menu
B.22 Use Cases
B.23 Try Various Proportions
B.24 Try Various Inputs
B.25 Performance
B.26 Large Files
Appendix C Tabular History
Appendix D Alphabet Tables
D.1 Numeric Alphabets
D.2 G-Values for Notes with Sharps and Flats
Appendix E Interval Proportions
Appendix F Excel Functions
F.1 Factors of a Number
F.2 G-Value
Literature
General Index
Index of Names