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The Haskell School of Music -- From Signals to Symphonies.pdf
Table of Contents
List of Figures
List of Tables
Preface
Computer Music, Euterpea, and Haskell
The Note vs. Signal Dichotomy
Basic Principles of Programming
Computation by Calculation
Expressions and Values
Types
Function Types and Type Signatures
Abstraction, Abstraction, Abstraction
Haskell Equality vs. Euterpean Equality
Code Reuse and Modularity
[Advanced] Programming with Numbers
Simple Music
Preliminaries
Notes, Music, and Polymorphism
Convenient Auxiliary Functions
Absolute Pitches
Polymorphic & Higher-Order Functions
Polymorphic Types
Abstraction Over Recursive Definitions
Append
Fold
[Advanced] A Final Example: Reverse
Currying
Errors
A Musical Interlude
Modules
Transcribing an Existing Score
Simple Algorithmic Composition
Syntactic Magic
Sections
Anonymous Functions
List Comprehensions
Function Composition
Higher-Order Thinking
Infix Function Application
More Music
Delay and Repeat
Inversion and Retrograde
Polyrhythms
Symbolic Meter Changes
Computing Duration
Super-retrograde
takeM and dropM
Removing Zeros
Truncating Parallel Composition
Trills
Grace Notes
Percussion
A Map for Music
A Fold for Music
Crazy Recursion
Qualified Types and Type Classes
Motivation
Equality
Defining Our Own Type Classes
Haskell's Standard Type Classes
Other Derived Instances
The type of play
Reasoning With Type Classes
Interpretation and Performance
Abstract Performance
Players
Putting it all Together
Self-Similar Music
Self-Similar Melody
Self-Similar Harmony
Other Self-Similar Structures
Proof by Induction
Induction and Recursion
Examples of List Induction
Proving Function Equivalences
Useful Properties on Lists
Induction on the Music Data Type
[Advanced] Induction on Other Data Types
An Algebra of Music
Musical Equivalance
Some Simple Axioms
The Fundamental Axiom Set
An Algebraic Semantics
Other Musical Properties
L-Systems and Generative Grammars
Generative Grammars
An L-System Grammar for Music
Random Numbers ... and Markov Chains
Random Numbers
Probability Distributions
Markov Chains
From Performance to Midi
An Introduction to Midi
Converting a Performance into Midi
Putting It All Together
Basic Input/Output
IO in Haskell
do Syntax
Actions are Just Values
Reading and Writing MIDI Files
Higher-Order Types and Monads
The Functor Class
The Monad Class
The MonadPlus Class
State Monads
Type Class Type Errors
Musical User Interface
Introduction
Basic Concepts
The UISF Arrow
Non-Widget Signal Functions
Musical Examples
Special Purpose and Custom Widgets
Advanced Topics
Sound and Signals
The Nature of Sound
Digital Audio
Euterpea's Signal Functions
Signals and Signal Functions
Generating Sound
Instruments
Spectrum Analysis
Fourier's Theorem
The Discrete Fourier Transform
The Fast Fourier Transform
Further Pragmatics
References
Additive and Subtractive Synthesis
Additive Synthesis
Subtractive Synthesis
Amplitude and Frequency Modulation
Amplitude Modulation
Frequency Modulation
Physical Modelling
Introduction
Delay Lines
Karplus-Strong Algorithm
Waveguide Synthesis
Sound Effects
Appendix
The PreludeList Module
The PreludeList Module
Simple List Selector Functions
Index-Based Selector Functions
Predicate-Based Selector Functions
Fold-like Functions
List Generators
String-Based Functions
Boolean List Functions
List Membership Functions
Arithmetic on Lists
List Combining Functions
Haskell's Standard Type Classes
The Ordered Class
The Enumeration Class
The Bounded Class
The Show Class
The Read Class
The Index Class
The Numeric Classes
Built-in Types Are Not Special
Pattern-Matching Details
Bibliography
The Haskell School of Music
— From Signals to Symphonies —
Paul Hudak
Yale University
Department of Computer Science
Version 2.6 (January 2014)
i
The Haskell School of Music
— From Signals to Symphonies —
Paul Hudak
Yale University
Department of Computer Science
New Haven, CT, USA
Version 2.6 (January 2014)
Copyright c Paul Hudak
January 2011, 2012, 2013, 2014
All rights reserved. No part of this publication may be reproduced or
distributed in any form or by any means, or stored in a data base or
retrieval system, without the prior written permission of the author.
Cover image: Euterpe, the Greek Muse of Music
(attribution unknown)
Contents
Table of Contents
List of Figures
List of Tables
Preface
1 Computer Music, Euterpea, and Haskell
1.1 The Note vs. Signal Dichotomy . . . . . . . . . . . . . . . . .
1.2 Basic Principles of Programming . . . . . . . . . . . . . . . .
1.3 Computation by Calculation . . . . . . . . . . . . . . . . . . .
1.4 Expressions and Values
. . . . . . . . . . . . . . . . . . . . .
1.5 Types
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Function Types and Type Signatures . . . . . . . . . . . . . .
1.7 Abstraction, Abstraction, Abstraction . . . . . . . . . . . . .
1.8 Haskell Equality vs. Euterpean Equality . . . . . . . . . . . .
1.9 Code Reuse and Modularity . . . . . . . . . . . . . . . . . . .
1.10 [Advanced] Programming with Numbers . . . . . . . . . . . .
2 Simple Music
2.1 Preliminaries
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Notes, Music, and Polymorphism . . . . . . . . . . . . . . . .
2.3 Convenient Auxiliary Functions . . . . . . . . . . . . . . . . .
ii
viii
xii
xiii
xiv
1
2
3
4
8
10
11
13
22
23
24
28
28
30
34
CONTENTS
iii
2.4 Absolute Pitches . . . . . . . . . . . . . . . . . . . . . . . . .
41
3 Polymorphic & Higher-Order Functions
3.1 Polymorphic Types . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Abstraction Over Recursive Definitions . . . . . . . . . . . . .
3.3 Append . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Fold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
[Advanced] A Final Example: Reverse . . . . . . . . . . . . .
3.6 Currying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 A Musical Interlude
4.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Transcribing an Existing Score . . . . . . . . . . . . . . . . .
4.3 Simple Algorithmic Composition . . . . . . . . . . . . . . . .
5 Syntactic Magic
5.1 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Anonymous Functions . . . . . . . . . . . . . . . . . . . . . .
5.3 List Comprehensions . . . . . . . . . . . . . . . . . . . . . . .
5.4 Function Composition . . . . . . . . . . . . . . . . . . . . . .
5.5 Higher-Order Thinking . . . . . . . . . . . . . . . . . . . . . .
5.6
Infix Function Application . . . . . . . . . . . . . . . . . . . .
6 More Music
6.1 Delay and Repeat . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Inversion and Retrograde
. . . . . . . . . . . . . . . . . . . .
6.3 Polyrhythms
. . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Symbolic Meter Changes . . . . . . . . . . . . . . . . . . . . .
6.5 Computing Duration . . . . . . . . . . . . . . . . . . . . . . .
6.6 Super-retrograde . . . . . . . . . . . . . . . . . . . . . . . . .
6.7
takeM and dropM . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Removing Zeros . . . . . . . . . . . . . . . . . . . . . . . . . .
45
46
47
51
53
58
60
64
67
67
69
74
77
77
79
80
83
84
85
87
87
88
90
92
93
93
94
95
CONTENTS
6.9 Truncating Parallel Composition . . . . . . . . . . . . . . . .
6.10 Trills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
97
99
6.11 Grace Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.12 Percussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.13 A Map for Music . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.14 A Fold for Music . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.15 Crazy Recursion . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Qualified Types and Type Classes
111
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3 Defining Our Own Type Classes
. . . . . . . . . . . . . . . . 115
7.4 Haskell’s Standard Type Classes
. . . . . . . . . . . . . . . . 121
7.5 Other Derived Instances . . . . . . . . . . . . . . . . . . . . . 126
7.6 The type of play . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.7 Reasoning With Type Classes . . . . . . . . . . . . . . . . . . 130
8 Interpretation and Performance
134
8.1 Abstract Performance . . . . . . . . . . . . . . . . . . . . . . 135
8.2 Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.3 Putting it all Together . . . . . . . . . . . . . . . . . . . . . . 149
9 Self-Similar Music
153
9.1 Self-Similar Melody . . . . . . . . . . . . . . . . . . . . . . . . 153
9.2 Self-Similar Harmony . . . . . . . . . . . . . . . . . . . . . . . 157
9.3 Other Self-Similar Structures
. . . . . . . . . . . . . . . . . . 158
10 Proof by Induction
161
10.1 Induction and Recursion . . . . . . . . . . . . . . . . . . . . . 161
10.2 Examples of List Induction . . . . . . . . . . . . . . . . . . . 162
10.3 Proving Function Equivalences
. . . . . . . . . . . . . . . . . 164
10.4 Useful Properties on Lists . . . . . . . . . . . . . . . . . . . . 168
10.5 Induction on the Music Data Type . . . . . . . . . . . . . . . 172
CONTENTS
v
10.6 [Advanced] Induction on Other Data Types . . . . . . . . . . 176
11 An Algebra of Music
182
11.1 Musical Equivalance . . . . . . . . . . . . . . . . . . . . . . . 182
11.2 Some Simple Axioms . . . . . . . . . . . . . . . . . . . . . . . 184
11.3 The Fundamental Axiom Set
. . . . . . . . . . . . . . . . . . 187
11.4 An Algebraic Semantics . . . . . . . . . . . . . . . . . . . . . 189
11.5 Other Musical Properties
. . . . . . . . . . . . . . . . . . . . 189
12 L-Systems and Generative Grammars
191
12.1 Generative Grammars . . . . . . . . . . . . . . . . . . . . . . 191
12.2 An L-System Grammar for Music . . . . . . . . . . . . . . . . 197
13 Random Numbers ... and Markov Chains
201
13.1 Random Numbers
. . . . . . . . . . . . . . . . . . . . . . . . 201
13.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . 204
13.3 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . 210
14 From Performance to Midi
214
14.1 An Introduction to Midi . . . . . . . . . . . . . . . . . . . . . 215
14.2 Converting a Performance into Midi
. . . . . . . . . . . . . . 220
14.3 Putting It All Together
. . . . . . . . . . . . . . . . . . . . . 223
15 Basic Input/Output
224
15.1 IO in Haskell
. . . . . . . . . . . . . . . . . . . . . . . . . . . 224
15.2 do Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
15.3 Actions are Just Values
. . . . . . . . . . . . . . . . . . . . . 227
15.4 Reading and Writing MIDI Files
. . . . . . . . . . . . . . . . 229
16 Higher-Order Types and Monads
230
16.1 The Functor Class
. . . . . . . . . . . . . . . . . . . . . . . . 230
16.2 The Monad Class . . . . . . . . . . . . . . . . . . . . . . . . . 233
16.3 The MonadPlus Class
. . . . . . . . . . . . . . . . . . . . . . 240
CONTENTS
vi
16.4 State Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
16.5 Type Class Type Errors . . . . . . . . . . . . . . . . . . . . . 245
17 Musical User Interface
Daniel Winograd-Cort
247
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
17.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 248
17.3 The UISF Arrow . . . . . . . . . . . . . . . . . . . . . . . . . 254
17.4 Non-Widget Signal Functions . . . . . . . . . . . . . . . . . . 264
17.5 Musical Examples
. . . . . . . . . . . . . . . . . . . . . . . . 269
17.6 Special Purpose and Custom Widgets
. . . . . . . . . . . . . 274
17.7 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . 282
18 Sound and Signals
289
18.1 The Nature of Sound . . . . . . . . . . . . . . . . . . . . . . . 289
18.2 Digital Audio . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
19 Euterpea’s Signal Functions
311
19.1 Signals and Signal Functions
. . . . . . . . . . . . . . . . . . 312
19.2 Generating Sound . . . . . . . . . . . . . . . . . . . . . . . . 322
19.3 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
20 Spectrum Analysis
331
20.1 Fourier’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 331
20.2 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . 335
20.3 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . 348
20.4 Further Pragmatics . . . . . . . . . . . . . . . . . . . . . . . . 349
20.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
21 Additive and Subtractive Synthesis
351
21.1 Additive Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 352
21.2 Subtractive Synthesis . . . . . . . . . . . . . . . . . . . . . . . 360
CONTENTS
22 Amplitude and Frequency Modulation
vii
369
22.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . 369
22.2 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . 372
23 Physical Modelling
377
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
23.2 Delay Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
23.3 Karplus-Strong Algorithm . . . . . . . . . . . . . . . . . . . . 381
23.4 Waveguide Synthesis . . . . . . . . . . . . . . . . . . . . . . . 382
24 Sound Effects
Appendix
A The PreludeList Module
386
387
388
A.1 The PreludeList Module . . . . . . . . . . . . . . . . . . . . . 389
A.2 Simple List Selector Functions
. . . . . . . . . . . . . . . . . 389
A.3 Index-Based Selector Functions . . . . . . . . . . . . . . . . . 390
A.4 Predicate-Based Selector Functions . . . . . . . . . . . . . . . 392
A.5 Fold-like Functions . . . . . . . . . . . . . . . . . . . . . . . . 392
A.6 List Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 394
A.7 String-Based Functions . . . . . . . . . . . . . . . . . . . . . . 394
A.8 Boolean List Functions . . . . . . . . . . . . . . . . . . . . . . 395
A.9 List Membership Functions
. . . . . . . . . . . . . . . . . . . 396
A.10 Arithmetic on Lists . . . . . . . . . . . . . . . . . . . . . . . . 396
A.11 List Combining Functions . . . . . . . . . . . . . . . . . . . . 397
B Haskell’s Standard Type Classes
399
B.1 The Ordered Class . . . . . . . . . . . . . . . . . . . . . . . . 399
B.2 The Enumeration Class
. . . . . . . . . . . . . . . . . . . . . 400
B.3 The Bounded Class . . . . . . . . . . . . . . . . . . . . . . . . 401
B.4 The Show Class . . . . . . . . . . . . . . . . . . . . . . . . . . 402
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