Copyright and use of this thesis · PDF fileconstructive approach using relative chord tones,...

Copyright and use of this thesis

This thesis must be used in accordance with the provisions of the Copyright Act 1968.

Reproduction of material protected by copyright may be an infringement of copyright and copyright owners may be entitled to take legal action against persons who infringe their copyright.

Section 51 (2) of the Copyright Act permits an authorized officer of a university library or archives to provide a copy (by communication or otherwise) of an unpublished thesis kept in the library or archives, to a person who satisfies the authorized officer that he or she requires the reproduction for the purposes of research or study.

The Copyright Act grants the creator of a work a number of moral rights, specifically the right of attribution, the right against false attribution and the right of integrity.

You may infringe the author’s moral rights if you:

- fail to acknowledge the author of this thesis if you quote sections from the work

- attribute this thesis to another author

- subject this thesis to derogatory treatment which may prejudice the author’s reputation

For further information contact the University’s Director of Copyright Services

sydney.edu.au/copyright

Formal methods for the design of imitative

polyphonic structures

Jurjen Lippold van Geenen

A thesis submitted in partial fulfilment ofrequirements for the degree of

Doctor of Philosophy

Sydney Conservatorium of MusicUniversity of Sydney

2013

Abstract

This thesis defines novel and efficient methods for the designof stacked canonsand their use in imitative

polyphonic structures. Chapter 1 discusses the development of canon- and fugue-techniques and their con-

nection in stretto-fugues such as found in Bach’s ‘Kunst derFugue’. Several examples show that larger

polyphonic structures are sustainable by a main theme whichcan appear in many different canons, called

stretti. Hence, techniques to effectively design such themes require the availability of efficient techniques

for the creation of several types of canons. In search of suchtechniques, chapter 2 provides a theoretical ba-

sis for the remainder of the thesis. An analysis of the establishedcounterpointingandintervallic approaches

to the construction of stacked canons shows that these provide limited harmonic control and are computa-

tionally complex. While efficient and in complete control ofharmony, Morris’ Tonnetz approach targets

serial stacked canons ad minimum and does not encompass voice-leading constraints. A style-independent,

constructive approach usingrelative chord tones, chord sequencesandchord sequence modulationsis pre-

sented along with its connections to graph-theory in address of these issues. An analysis of Rameau’sCanon

at the Fifthfrom hisTraite de l’harmonieintroduces the concept of relative chord tones and incorporates

two constraints discussed in the Traite in the newly proposed relative chord tone model, namely, obtaining

complete chords, and, preparation and resolution of sevenths. The analysis explains Rameau’s choice of

dux and chord sequence in terms of the conjunction of these constraints. Using my definition ofrestless

dux graphs, the problem of obtaining complete chords is reduced to the Hamiltonian cycle problem in a

dux graph. The problem of finding adux according to the conjunction of the aforementioned constraints

is reduced to the definition of a generating function. An approach to the incorporation of voice leading

constraints is sketched, by the detailed discussion of prohibitions of firstly parallel octaves, and secondly,

parallel fifths. After deriving a least upper bound on the maximum number of voices in a stacked canon

without parallel octaves in terms of the canon’schord sizes, the problem of finding adux for such a stacked

canon with a maximum number of voices is reduced to the Eulerian cycle problem in a restless dux graph.

It is also proved that the conjunction of either constraint reduces the aforementioned least upper bound by

a single voice. My definition of dux graphs with rests allows the definition of two near lineardux gen-

eration algorithms which respectively satisfy the first andeither constraint. The problem of constraining

the inversion of chords is discussed in terms of the incorporation of a constraint which prohibits64 chords.

The remainder of the chapter discusses two methods which allow variations in a stacked canon’s melody

based on a predetermined chord sequence, followed by a discussion which counts the number of distinct

sub-canons of a stacked canon. It is established that the latter number shows double exponential growth

in the chord sizes of a chosen chord sequence, in demonstration of the applicability of my methods to the

design of thematic material for use in larger scale imitative polyphonic structures.

Chapter 3 discusses my compositionsSpiral which is based on several themes, the main two of which

iii

iv

were derived as relative chord tone sequences using resultsfrom chapter 2. The chapter also discusses

several techniques related to the organization of polythematic stacked canonic structures in larger scale

composition. The first movement was designed in terms of several harmoniolas, in which neither of the

aforementioned themes appears in a stretto. The second movement is designed as a passacaglia structure

based on the aforementioned second theme, onto which many distinct combinations of either theme are

superimposed. The final movement mirrors the structure of the first movement, testing my conjecture that

the use of a chromatic mirror axis can mirror the affects experienced by an audience.

Chapter 4 discusses my compositionFugue in G, the main theme of which was derived as a Hamiltonian

cycle in the underlying dux graph. The fugue contains many stretti whence the chapter provides further a

discussion on the organization of polythematic canonic structures in larger scale composition. While the

theory presented in chapter 2 enables one to derive efficientalgorithms for the generation of stacked canons

adhering to personal stylistic preferences, their derivation may be tedious. This is especially so when

experimenting with different rules.

Chapter 5 discusses a method which usesconstraint logic programswith relative chord tone domains

to efficiently search for the thematic material of polythematic stacked canonic structures. The composition

process for my ‘Missa ad Fugam’ is discussed in demonstration of this technique. The Mass uses three

main themes, each of which allows a four-voice stacked canon, and each pair of which allows a four-voice

stacked double canon. Finally, chapter 6 summarizes the main results of this thesis and provides an outlook

on future research.

Acknowledgments

In none-decreasing order of gratitude, I wish to give thanksto:

• Ivan Zavada from the Composition and Music Technology department for his continued support and

supervision of my candidature;

• My parents, brothers and sister, for their continued support of my efforts, wise or unwise;

• My grandparents, for still being there and always having been there in a lovely way, and for the

beautiful cover my grandfather designed;

• The Sydney Conservatorium of Music, University of Sydney,and Commonwealth of Australia, for

the University of Sydney Postgraduate Award which allowed me to pursue a doctorate;

• Kees van den Bergh, for the many hours we talked over the phone during my eventful time in Sydney;

• Ralf Pisters, for the enthusiastic discussions involvingcomputer science and music, and for proof-

reading parts of my thesis. You were one of very few whom I could share my findings with;

• Jacqui Ingram and Linda Knowles, for providing a pleasant micro-climate in often stormy weather;

• The department of mathematics and computer science at the Eindhoven University of Technology,

for teaching the ‘Eindhoven style of programming’[Dij96] during my computer science studies;

• My teachers Henk de Croon, Marinus Kasbergen, Pieter van Moergastel and Kees Schoonenbeek at

the ‘Brabant’s Conservatorium’ (now Fontys Conservatorium) during my music theory studies for the

enthusiasm, experience and knowledge you transferred;

• My beloved Caroline, for a pleasant and worthwhile return to the Netherlands.

v

List of Figures

1.1 Condensed representation of ‘Sumer is icumen in’ . . . . . .. . . . . . . . . . . . . . . . . . 5

1.2 Bars[171..177] of ‘Fantasia Chromatica’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

1.3 Bars[33..45] of ContrapunctusV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Beginning of ContrapunctusV I (in stilo francese) . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Beginning of ContrapunctusV II (per augmentationem et diminutionem) . . . . . . . . . . . 6

2.1 Rameau’s Canon at the fifth with reduction . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8

2.2 Dux graph of Rameau’s Canon at the fifth . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 16

2.3 Realization ofSC with parameters(V = 3, L = 4, cs = [3], dux = [0,1,0,1]) . . . . . . . . . 19

2.4 Dux graph ofSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 unembellished stretto and chord sequence of ‘Spiral’:theme0 . . . . . . . . . . . . . . . . . 20

2.6 dux graph of stretto of ‘Spiral’:theme0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 D before (a) and after (b) Eulerization . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 22

2.8 Unconstrained dux graph patterns considered in proposition 19 . . . . . . . . . . . . . . . . . 28

2.9 Stacked canonsSC andSC′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 a spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 37

3.2 Spiral:theme1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Spiral part 1: harmoniola . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 40

3.4 Spiral part 2 section B: stretti withtheme0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1 Design of theme and countersubject for ‘Fugue in G’ . . . . .. . . . . . . . . . . . . . . . . 46

5.1 Composition assisted composition process followed in composition of ‘Missa ad Fugam’ . . 54

5.2 Chord sequence of Missa ad Fugam . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 56

5.3 An embellished form of themeA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 An embellished form of themeB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 An embellished form of themeC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 Two repetitions of the mirrored chord sequence . . . . . . . .. . . . . . . . . . . . . . . . . . 70

vii

viii LIST OF FIGURES

List of Tables

2.1 Commonly used parameters of stacked canons . . . . . . . . . . .. . . . . . . . . . . . . . . 7

2.2 type I non-duplicating rotational array . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 11

2.3 classical Tonnetz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 12

2.4 chord roots in Rameau’s canon at the fifth . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13

2.5 relative chord tone structure of Rameau’s canon at the fifth . . . . . . . . . . . . . . . . . . . 14

2.6 relative chord tone structure ofSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 relative chord tone structure of adapted stretto of ‘Spiral’: theme0 . . . . . . . . . . . . . . . 20

2.8 Occurrences of64 chords in ascending (a) and descending (b) stacked canons . .. . . . . . . 29

2.9 number of distinct sub-canons of a stacked canon withoutparallel octaves . . . . . . . . . . . 36

3.1 Spiral: two-voice stretti withtheme1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 canonic structure of part 1 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 41

3.3 canonic structure of part 3 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 44

3.4 Contrasts between part 1 and 3 . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 44

4.1 relation betweenthemeu and[1,0,3,2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 (sub-)sections of ‘Fugue in G’ . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 47

5.1 Conceptual relative chord tone structures ofC1(top) andC2(bottom) . . . . . . . . . . . . . . 57

5.2 Conceptual relative chord tone structure of configurationC0 = [A,A,A,A] . . . . . . . . . . 62

5.3 Relative chord tone structure of configurationC0 = [A,A,A,A] . . . . . . . . . . . . . . . . 63

5.4 Relative chord tone structures ofC1(top) andC2(bottom) . . . . . . . . . . . . . . . . . . . . 64

5.5 Conceptual relative chord tone structures ofC3(top),C4(2nd), C5(3rd) andC6(bottom) . . . . 65

5.6 Conceptual structure of the Gloria . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 67

5.7 Conceptual structure of the Credo . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 69

5.8 Conceptual structure of the Sanctus . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 70

5.9 Conceptual structure of the Agnus Dei . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 72

ix

x LIST OF TABLES

Contents

List of Figures vii

List of Tables ix

Contents xi

1 Introduction 1

1.1 From canon to fugue (and back) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 1

1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 4

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 4

2 On designing stacked canons with relative chord tones 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 7

2.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 9

2.2.1 Counterpointing approach . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 9

2.2.2 Intervallic approach . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 9

2.2.3 Harmoniola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 11

2.2.4 Serial approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 11

2.2.5 Summary and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 12

2.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 13

2.4 Analysis of Rameau’s canon at the fifth . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 13

2.4.1 Obtaining complete chords . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 14

2.4.2 Preparation and resolution of sevenths . . . . . . . . . . . .. . . . . . . . . . . . . . 15

2.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 17

2.5 Voice leading constraints . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 18

2.5.1 preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 18

2.5.2 Avoiding parallels of the class0 (mod 12) . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.3 Avoiding parallels of the class7 (mod 12) . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Avoiding six-four chords in triadic chord sequences . . .. . . . . . . . . . . . . . . . . . . . 27

2.7 The construction of a chord sequence . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 29

2.7.1 Chord sequence modulations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 30

2.7.2 Choosing chord sizes greater thanV . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Counting distinct sub-canons of a stacked canon . . . . . . .. . . . . . . . . . . . . . . . . . 35

xi

xii CONTENTS

3 Spiral 37

3.1 Thematic elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 37

3.1.1 theme0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2 theme1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 39

3.3 Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 39

3.3.1 Section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 41

3.3.2 Section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 41

3.3.3 Section A’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 43

3.3.4 Section B’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 43

3.4 Part 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 43

4 Fugue in G 45

4.1 Thematic elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 45

4.1.1 Main theme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 45

4.1.2 Countersubject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 46

4.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 47

4.2.1 section A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 47

4.2.2 section B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 49

4.2.3 section A’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 51

5 Constraint logic programming stacked canonic structures 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 53

5.2 Esthetic considerations . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 55

5.3 Constraint specifications . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 56

5.3.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 57

5.3.2 Auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 58

5.3.3 Preventing similarity . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 58

5.3.4 Preventing chord tone doubling . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 59

5.3.5 Regulation of dissonance . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 59

5.3.6 Regulation of parallel perfect intervals . . . . . . . . . .. . . . . . . . . . . . . . . . 60

5.3.7 ThemeA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3.8 ThemeB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3.9 ThemeC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Analytical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 66

5.4.1 Kyrie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 66

5.4.2 Gloria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 66

5.4.3 Credo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 68

5.4.4 Sanctus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 68

5.4.5 Agnus Dei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 70

CONTENTS xiii

6 Summary, conclusion and outlook 73

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 73

6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 75

6.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 76

A B-Prolog sources 77

A.1 themeA.pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 77

A.2 themeB.pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 79

A.3 themeC.pl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 82

Bibliography 85

Glossary 89

xiv CONTENTS

Chapter 1

Introduction

This thesis represents an attempt at the provision of efficient methods for the creation of larger scale imita-

tive polyphonic structures. After a brief discussion of thedevelopment of canon and fugue-techniques and

their relation in section 1.1, the problem statement is given in section 1.2, followed by an outline of this

thesis’ structure in section 1.3.

1.1 From canon to fugue (and back)

Within a musical context, the term canon originally referred to an inscribed formula or instruction which the

performer would implement in order to realize one or more parts from the given notation[SG01]. Nowadays

canons are commonly associated with a compositional technique in which derivations of a single melody

are overlappingly imitated. Derivations can be obtained from the original melody through (a combina-

tion of) techniques such as transposition, mirroring, retrograding and variation of the original melody’s

temporal proportions. The original melody is referred to asthe dux (leader), an imitation as acomes

(companion)[Cal92].

Example 1 The English13th century composition ‘Sumer is icumen in’ shown in figure 1.1 on page 5 is

perhaps one of the oldest canons passed down through historyin written form. Its upper four voices form

a rotawhich can be condensed into a structure of twenty-four68 bars in modern notation. The dux appears

in the top staff, followed by its three comites at successivetime intervals of one measure in the three staffs

below it. The initial entry points in figure 1.1 are marked by ‘❉’ in the score. The term rota, nowadays

commonly called round or circle canon, refers to the repetitive structure such a canon allows. Its two bass

voices form apes: the Latin equivalent of a ground or ostinato. Note that the pes is itself a short round

which can be condensed into four68 bars.

The ‘Sumer canon’ is remarkable in several ways[Tar12, ch.9]. First, it is a clear testament to the gradual

acceptance of thirds as (semi-)consonants besides octavesand fifths, a crucial factor in the development

of European polyphony. Indeed, the Sumer canon features a repeated succession of the triadsI, V II (or

I, II, V II). Second, its structure is rather complex for such an early composition. In modern terms the

composition would be called adouble canon, in reference to the simultaneous occurrence of the two afore-

mentioned canons in the upper four and lower two voices.

1

2 CHAPTER 1. INTRODUCTION

Thirteenth and fourteenth century France saw the related development of thechace(‘hunt’) in which one

voice ‘hunts’ the other. In comparison to the Sumer canon, a chace typically pays less attention to the

creation of consonant harmony but more to contrasting rhythms between voices. Indeed, thehoquetus

(‘hocket’) technique is commonly used in a chace: the dovetailing of sounds and silences by means of the

staggered arrangement of rests[SG01]. The Italian versionof the chace,caccia, became a popular form in

the fourteenth century. Besides a two-voice canon at the unison, a tenor voice was often created as a coun-

terpoint against the two-voice canon in support of the harmonic structure. Besides rather frivolous use of

canons as in the chace or caccia, the composition technique also appeared in sacred music such as Ciconia’s

‘O felix templum jubila’. In the Renaissance era, imitativepolyphony became common. Josquin’s ‘Ave

Maria’ for example is an almost complete chain of imitationsbetween (pairs of) voices. This era produced

several compositions of exceptional quality and contrapuntal complexity in which varying canon techniques

are extensively used. Examples are Ockeghem’s ‘Missa Prolationem’, Josquin des Prez’ ‘Missa L’homme

arme super voces musicales’ and Palestrina’s ‘Missa ad Fugam’. Melson provides interesting insights into

the construction of mensuration and proportion canons of this era[Mel], in which a melody’s temporal pro-

portions are changed upon imitation.

The aforementioned compositions are predominantly vocal.Transcription and ornamentation of vocal mu-

sic for instruments started to gain popularity in the sixteenth century. Initially, an existing motet would be

transcribed for instruments such as the lute, vihuela, guitar, or even viola da gamba. This practice led to

the establishment of the fantasia and ricercar as an independent genre. The introduction of keyboard instru-

ments increased the popularity of the ricercar, as those instruments are better suited to playing polyphony

than the aforementioned plucked instruments. Initially, polythematic ricercars such as Adriaan Willaert’s in-

herited the sectional structure of the motet with often different themes in different sections. The seventeenth

century saw the introduction of the monothematic ricercar such as Sweelinck’s ‘Fantasia Chromatica’, in

which a single theme is predominantly present. It also became common practice to have voices enter the

ricercar such that the theme entries do not overlap as was common in the earlier ricercars. This turned the

canonic overlap of the main theme with itself in astretto into a polyphonic climax. Indeed, having heard

multiple none overlapping imitations of the main theme, a stretto can be perceived as a compression of time

which naturally generates tension.

Example 2 A stretto in Sweelinck’s ‘Fantasia Chromatica’ is shown in figure 1.2 on page 6. The main

theme is basically a downward chromatic scale filling the interval of a perfect fourth. Successive entries

shown in figure 1.2 appear, all with their durations halved, in the alto (bar 171), soprano (bar 172), tenor

(upbeat to bar 174) and bass (upbeat to bar 177).

The monothematic ricercar gradually developed into the fugue, through composers such as Scheidt, Schei-

demann, Reinken, Buxtehude, Frescobaldi, Pachelbell and J.C. Bach, in the Northern- and Southern Ger-

man organ schools. The music of J.S. Bach (brother and pupil of J.C. Bach) probably represents the summit

of polyphonic development and achievement in this regard. E.g. his ‘Kunst der Fuge’ features fourteen

fugues and four canons, all based on a single theme which appears in many overlapping (and/or mirrored

and/or temporally altered) stretti in e.g. ContrapunctiV , V I andV II.

Example 3 Figure 1.3 on page 6 shows bars[33..45] of ContrapunctusV which contains two stretti:

• On the first beat of bar33, the main theme is stated in the bass (rectus), the soprano enters one beat

later with the inverted main theme (inversus);

1.1. FROM CANON TO FUGUE (AND BACK) 3

• On the first beat of bar41, the inversus is stated in the tenor, the alto enters one beatlater with the

rectus.

Example 4 The beginning of ContrapunctusV I is shown in figure 1.4 on page 6. The bass opens with the

rectus, followed by overlapping entries in the soprano in bar 2 (inversus, durations halved) and alto in bar

3 (rectus, durations halved).

Example 5 The beginning of ContrapunctusV II is shown in figure 1.5 on page 6. The tenor starts opens

with the rectus (durations halved), followed by overlapping entries in the soprano in bar 2(inversus), alto

in bar 3(inversus, duration halved) and bass in bar 5 (inversus, doubled durations).

The predominantly homophonicclassical and romantic styles provide increasing competition against polyphony

after Bach’s death. Yet from Haydn and Mozart to Brahms, Bruckner, and Reger, Faure and Vaughan

Williams, Verdi and Dvorak, and then again, up to the present, they have all been permeated with the ef-

fects of the Bach legacy[BW64]. Shostakovich’ opus 87 and Hindemith’s ‘Ludis Tonalis’ are20th century

attempts at contemporary versions of Bach’s ‘wohltemperiertes Klavier’ and several other composers such

as Bartok, Ligeti, Lewis, Messiaen and Stravinsky used thefugue-technique, often within a larger compo-

sition.

Although the fugue lost popularity among serial or dodecaphonic composers, perhaps due to its obvious

connection to tonality, canon techniques can be found in both tonal and atonal20th century music. Webern

extensively used canon techniques both before and during his dodecaphonic period [Per71, Bai88, Mea93].

Schonberg created a number of canons throughout his life, most of which were published by Rufer as

a collection after his death[Sch63]. The common factor in all of these canons however is that none is

dodecaphonic. Schonberg’s own comments indicate that these canons were meant for working out ideas

[Nei64]. Use of canon techniques can however also be observed in dodecaphonic works such as his fourth

string quartet[Sto08]. Charles Ives combines an early use of near dodecaphony and canon in his ‘Steeples

and the Mountains’, e.g. between trumpet and trombone in bar13. Schnittke’s use of canon techniques

not always results in a polyphonic texture in which independent layers are comprehensible. Instead, the

short durations between the entrances and closely-spaced imitations create a dense texture which is not

easily divisible to its components by ear [Hon12], alike Ligeti’s use of ‘Micropolyphony’[Sea89]. A re-

lated technique called ‘beat shifting’ is used in Reich’s phase-shifting compositions [Roe03]. Mediating

between these techniques is the ‘tempo-proportion canon’ of Nancarrow, who displayed a rare and persis-

tent devotion to the technique of canon which appears in somethree quarters of his 51 studies for player

piano[Tho00]. Finally, Vuza and Tangian provide extensivecontributions to the theory on rhythmic canons

[Vuz91, Vuz92b, Vuz92a, Vuz93, Tan03].


1.2 Problem statement

The previous section briefly discussed the development fromcanon to fugue and connections between the

two techniques in stretto-fugues. The repeated use of the main theme in ‘Die Kunst der Fuge’ in various

combinations such as those shown in examples 3 through 5 demonstrates that larger imitative polyphonic

structures are sustainable by a contrapuntally capable (main) theme. According to Bach specialist Christoph

Wolff, ‘Die Kunst der Fuge’ “is an exploration in depth of thecontrapuntal possibilities inherent in a single

musical subject”[Wol02, p.433]. One might infer from this statement that the discovery of the theme’s

inherentcapability of appearing in so many combinations with itselfwas made after its conception. While

such a view seems accurate in relation to ‘Das Musikalische Opfer’1, the theme of which is rather handi-

capped in terms of allowing stretti, this puts the cart before the horse in case of ‘Die Kunst der Fuge’. Bach

probably designed this theme as a very capable one in terms ofstretti, yet the knowledge which enabled him

to do so was not passed on to us. Indeed, the literature on canon techniques abounds with formulations of

what is to be accomplished in various types of canons, yet offers relatively few algorithms for their creation

with often exponential running time in terms of their input (I return to this point in section 2.2). One must

more often than not resort to the proverbial search of a dux ina haystack. Being especially fond of compos-

ing and listening to distinctively polyphonic music, the lack of efficient methods which would enable me to

create larger compositions based on a contrapuntally capable theme inspired me to invent them.

1.3 Outline

Chapter 2 provides a style independent methodology for the derivation of algorithms for the creation of

stacked canons. Chapters 3 and 4 respectively discuss my compositionsSpiralandFugue in G, for which I

used some of the results presented in chapter 2. While the theory presented in chapter 2 enables us to derive

efficient algorithms for the generation of stacked canons adhering to our own stylistic preferences (i.e. our

own rules or similarly, constraints), their derivation may be tedious or difficult for composersunskilled

in this art. Chapter 5 sketches a method which allows computers to efficiently search for canons using

constraint logic programming. The composition process formy ‘Missa ad Fugam’ is used in demonstration

of this technique. Finally, chapter 6 summarizes the main results of this thesis and provides a conclusion

and outlook on future research.

1King Friedrich II of Prussia dictated to Bach the theme for the composition which would become ‘Das Musikalisches Opfer’.Hence it is probable that Bach had to investigate the theme’scapabilities after it was construed. Interestingly, Sassoon argues thatBach may have been familiar with a similar theme used by Handel[Sas03].

1.3. OUTLINE 5

V

V

V

V

?

?

b

b

b

b

b

b

86

86

86

86

86

86

..

..

..

..

..

..

œ* Jœ œ JœSum er is i

œ Jœ œ jœswik thu na ver

œ jœ œ jœWel sin ges thu

.œ .œCu cu,

.œ .œsing cu.œ .œsing cu

œ Jœ œ œ œcum en in

.œ Œ .nu.

.œ œ Jœcu cu! Ne

.œ œ jœcu cu!

.œ œ Jœcu nu.œ Œ .cu

œ jœ œ jœLhu de sing cucœ* Jœ œ JœSum er is i



.œ .œsing cu

.œ .œsing cu

.œ Œ .cuœ Jœ œ œ œ

cum en in

.œ Œ .nu.

.œ œ Jœcu cu! Ne

.œ Œ .cu

.œ œ Jœcu nu

œ jœ œ JœGrow eth sed and




œ jœ œ jœblow eth med and


cum en in

.œ Œ .nu.


œ Jœ œ Jœspring'th the wu de



.œ .œsing cu

.œ .œsing cu

.œ Œ .nu.



cum en in

.œ Œ .cu

.œ œ Jœcu nu

-

-

- - - - - - - -

- - - - - - --

- - - - - -- -

- - - - - -- -

-

-

V

V

V

V

?

?

b

b

b

b

b

b

9 .œ .œSing cuc



œ jœ œ jœLhu de sing cuc

9 .œ .œsing cu.œ .œsing cu

.œ Œ .cu!

.œ Œ .nu.


.œ Œ .cu


œ jœ œ jœA we ble teth.œ .œ

Sing cuc



.œ .œsing cu

.œ .œsing cu

œ Jœ œ jœaft er lomb, lhough.œ Œ .cu!

.œ Œ .nu.


.œ Œ .cu

.œ œ Jœcu nu

œ jœ œ jœaft er cal ve


Sing cuc



.œ œ jœcu. Bul luc


.œ Œ .nu.


œ Jœ œ Jœster teth, bu cke



Sing cuc

.œ .œsing cu

.œ .œsing cu

œ Jœ œ Jœver teth Mu rie

.œ Œ .cu.


.œ Œ .cu

.œ œ Jœcu nu

- - - -

- - - - - - -

- - - - - - -

- - - - - -

-

-

-

-

- - - - - - -

V

V

V

V

?

?

b

b

b

b

b

b

..

..

..

..

..

..

17 œ Jœ .œsing cu cu!

œ jœ œ JœBul luc ster teth,


œ jœ œ jœA we ble teth

17 .œ .œsing cu.œ .œsing cu

.œ .œCu cu,

œ Jœ œ Jœbu cke ver teth

.œ Œ .cu.

œ Jœ œ jœaft er lomb, lhough


.œ œ jœcu cu!œ Jœ œ JœMu rie sing cu



.œ .œsing cu

.œ .œsing cu

œ jœ œ jœWel sin ges thu.œ Œ .cu!


.œ Œ .cu.

.œ Œ .cu

.œ œ Jœcu nu

.œ œ Jœcu cu! Ne

.œ .œCu cu,œ Jœ œ JœMu rie sing cu




.œ œ jœcu cu!.œ Œ .cu!



.œ Œ .nu.


.œ .œCu cu,œ Jœ œ JœMu rie sing cu

.œ .œsing cu

.œ .œsing cu

∑

.œ œ Jœcu cu! Ne

.œ œ jœcu cu!.œ Œ .cu!

.œ Œ .cu

.œ œ Jœcu nu

- - - - - - - - -

- - - - - - - - - -

- - - - - - - - - - -

-

-

-

-

- - - - - -

Figure 1.1: Condensed representation of ‘Sumer is icumen in’


&?

c

c

171 w# Œ œ œ œ.œ œ œ ˙w

Œ œ œ œœ# œn œ# œn∑

œ# œn œ œb˙ Œ œÓ Œ œ

˙ Œ œœ œ œ œœ œ œ# œn

œ œ œ# œn˙ Œ œœ œb ˙

œ œb .œ jœnœ œ œb Œ˙ Œ œ œÓ Œ œ

.œ jœ ˙Œ œ œ œ˙ ˙œ œ œ# œn

Figure 1.2: Bars[171..177] of ‘Fantasia Chromatica’

&?

b

b

c

c

33 Ó ˙.œ Jœ œ œ œ

œ œ ˙ œ˙ .œ Jœ

.œ jœ .œ jœœ Œ ‰ œ œ œœ œ œ œ œ Œ

.œ Jœ ˙

˙ ˙œ œ œ œ œ œŒ ‰ jœ .œ jœ˙ .œ Jœ

.œ jœ ˙œ œ ˙œ œ œ ˙˙ œ œ œ œ

œ œ œ œ ˙œ œ œ œ ˙œ œ œ œ ˙˙ œ œ œ œ

œ œ œ œ œ œœ œ œ œ œ œœ œ œ œ œ Œ˙ œ œ œ œ

&?

b

b

39 œ ˙ œ˙ ˙

œn œ œ œ œ œ œ œ

œ ˙ œ˙ ˙œ œ œ œ œ œ œ œ

.œ jœ œ œn œœ Œ ˙˙ .œ jœœ œ ˙ œb

œ# œ œ œ œ œ.œ Jœ .œ Jœ.œ jœ ˙w

.œ jœ œ Œ˙ ˙#˙b .œ jœœ œ œ œ œ œ œ

.œ Jœ ˙˙ œ œ œ œœ œ œ œ# ˙

œ œ œ œ Jœ ‰ Œ˙ Óœ œ ˙b

Figure 1.3: Bars[33..45] of ContrapunctusV

&?

b

b

c

c

∑

˙ .œ Jœ

œ .œ œ .œ œ œ

.œ Jœ ˙

œ .œ œ œ œ œ œ œÓ œ .œ œ

˙# .œ Jœ

.œ œ œ œ œ œ œ œ.œ œ œ œ# .œ œ˙ .œ œ .œ œ

.œ œn œ .œ œ .œ# œœ œ œ œ œ œ .œ œ

.œ œ .œ œ .œ œ .œ œ

œœ

.œFigure 1.4: Beginning of ContrapunctusV I (in stilo francese)

&?

b

b

c

c

∑∑

œ .œ œ .œ œ œ

˙ .œ jœ∑

œ# .œ œ œ œ œ œ œ.œ jœ ˙œ .œ œn .œ œ œ

œ œ# œ œ œ œn œn œ œ# œ œ œ œ œ#

˙ .œ jœœ .œ œ œ# œ œ œn œœ œn œb œ œ œ# œ œ œ œ œ .œn œ

&?

b

b

5 ˙ œ œ œ œœ œb œ œ œ œ œ œn œ# œ

œ œ œ œn œ# œ œ œ œ œw

œ œ œ œ œ œ œ œ œ œœ œ œn œn œ œ œ œ œ œ#œ .œ œ .œ œ œ.˙ œn

œ œ œ œn jœ ‰ Óœ Œ Óœ œ œ œ œ œ# œ# œ Œ.˙ œ

∑

w

Figure 1.5: Beginning of ContrapunctusV II (per augmentationem et diminutionem)

Chapter 2

On designing stacked canons with

relative chord tones

This chapter was adapted for publication into theJournal of Mathematics and Music[vG12].

2.1 Introduction

Gosman [Gos97] introduced the term “stacked canon” to identify a special kind of canon in which each

comes is a transposition by a fixedtime intervalTI andscale intervalSI of its immediate predecessor.

Such a canon is completely determined by its number ofvoicesV , scaleS, TI, SI and melodydux with

lengthL Table 2.1 lists the aforementioned parameters of a stacked canons for easy reference.

Example 6 The first sixteen bars of Rameau’s Canon at the fifth [RG71, ch.44] are shown in the upper

four staves of figure 2.1 on page 8. The bottom four staves are discussed in section 2.4, these can be ignored

for now. Rameau’s Canon at the fifth is a stacked canon withV = 4, TI = 2 whole notes,SI = 7 semitones

of the chromatic scaleS, with thedux appearing in the bass. Note that it is also a spiral canon: thedux

is a succession of transpositions of its first eight bars. These bars are henceforth referred to as the unique

part of the dux. ThelengthLU of theunique part of the dux equals8.

parameter domain descriptionV N

+ number ofvoicesSI Z scaleintervalTI N

+ time intervalL N

+ length of thedux, equals∣dux∣LU N

+ length of theunique part of theduxS Scale scale

Table 2.1: Commonly used parameters of stacked canons

Besides forming a category of its own in the musical literature, stacked canons are also useful in the design

of thematic material for imitative polyphonic structures.By designing thematic material such that it allows

a stacked canon for many voices, it is guaranteed that many mutually different imitations are possible.

This chapter offers a novel approach to the systematic design of stacked canons. Section 2.2 reviews the

7

8 CHAPTER 2. ON DESIGNING STACKED CANONS WITH RELATIVE CHORD TONES

&

&

V?

&

&

V?

###

###

C

C

C

C

C

C

C

C

S

A

T

B

S

A

T

B

∑

∑

∑

Ó ˙Ah!

∑

∑

∑

w2

1 ∑

∑

∑

˙ œ œLoin de

1 ∑

∑

∑

w1

2 ∑

∑

Ó ˙Ah!

.˙b œ œri

2 ∑

∑

w2

wb3

3 ∑

∑

˙ œ œLoin de

˙ ˙3 ∑

∑

w#1

w2

4 ∑

ÓAh!

.˙n œ œri

˙ Óre,

4 ∑

w2

wn3

w0

5 ∑

˙ œ œLoin de

˙ ˙w

Pleu

5 ∑

w#1

w2

w3

6 Ó ˙Ah!

.˙n œ œri

˙ Óre,

Jœ w#rons,

6

w2

wn3

w0

w#1

7

˙ œ œLoin de

˙ ˙w

Pleu

˙n ˙Pleu rons.

7

w#1

w2

w3

w0

- - - - - - - - - -

- - - - - - - - -

- - - - - -

&

&

V?

&

&

V?

###

###

S

A

T

B

S

A

T

B

8

.˙n œ œri

˙ Óre,

Jœ w#rons,

Ó ˙Ah!

8

wn3

w0

w#1

w2

9

˙ ˙

wPleu

˙n ˙Pleu rons.

˙ œ# œ#Loin de

9

w#2

w3

w0

w#1

10

˙ Óre,

Jœ w#rons,

Ó ˙Ah!

.˙n œ œri

10

w0

w#1

w2

wn3

11 wPleu

˙n ˙Pleu rons.

˙ œ# œ#Loin de

˙# ˙11 w

3

w#0

w#1

w#2

12

Jœ w#rons,

Ó ˙Ah!

.˙n œ œri

˙ Óre,

12 w#1

w#2

wn3

w0

13 ˙n ˙Pleu rons.

˙ œ# œ#Loin de

˙# ˙w

Pleu

13 w#0

w#1

w#2w3

14 Ó ˙Ah!

.˙n œ œri

˙ Óre,

Jœ w#rons,

14 w#2

wn3

w#0w#1

15 ˙ œ# œ#Loin de

˙# ˙w

Pleu˙n ˙#Pleu rons.

15 w#1

w#2w#3

w#0

- - - - - - - - - -

- - - - - - - -

- - - -

- - - - - - - - - -

Figure 2.1: Rameau’s Canon at the fifth with reduction

literature concerning known concepts and approaches to thedesign of stacked canons. Section 2.3 provides

an outline of the structure of this chapter.

2.2. LITERATURE REVIEW 9

2.2 Literature review

This section provides a review of the literature regarding the creation of stacked canons. The earliest

sources, which describe what I call the counterpointing approach, are discussed in section 2.2.1. Several

authors contributed to what I call the intervallic approach, discussed in section 2.2.2, largely in response to

the shortcomings of the counterpointing approach. Section2.2.3 discusses Burmeister’s harmoniola, which

is an efficient technique for the creation of rounds, but moreuseful as an analytical device in relation to

stacked canons at an interval different than the unison. Section 2.2.4 discusses approaches used for the

creation of rounds in serial music, after which section 2.2.5 summarizes the aforementioned sections.

2.2.1 Counterpointing approach

Slightly varied accounts of the counterpointing approach are given by Simpson [Sim67, vol. 5], Bathe[BK05,

p.36] and Morley[Mor97, pp.110-130]. Simpson’s approach,which is taught even today [Gos97, p. viii], is

summarized in algorithm 7 below.

Algorithm 7 Repeat, until the desired length is obtained:

1. extend thedux melody byTI time units, such that no style dependent constraint is violated;

2. for all v, 1 ≤ v < V : paste the transposition byv ∗ TI time units,v ∗ SI scale-tones of a copy of the

extension created in the previous step into voicev.

Exemplary style dependent constraints algorithm 7 refers to include the restriction of harmonic intervals to

consonants, or voice-leading constraints such as the prohibition of parallel perfect fifths or octaves between

voices. While the above approach is simple and intuitive, several drawbacks are associated with it. Thedux

is repeatedly extended byTI time units. The result is a “fragmented process of composingcounterpoint

upon counterpoint”[Gos97, p. viii]. The above algorithm indeed provides no means for establishing larger

scale goals at a certain point in time. Consider for examplesthe problem of reaching a cadence in a particular

mode att = 160. Having written some part of thedux, the composer may need to backtrack from some

point in thedux while trying all possibilities until the cadence is reached, if it is reachable at all. In a worst

case scenario, one may need to try all possibilities. Let us think of S as a record in which the fieldtones

represents its set of scale tones. Then such an algorithm would have a running time ofO (∣S.tones∣V ∗∣dux∣),where∣S.tones∣ equals the size of the scale and∣dux∣ equals the length of thedux.1

2.2.2 Intervallic approach

The intervallic approach approach discussed in this subsection was developed and refined in reaction to

the shortcomings of the counterpointing approach [Mor95, Gau96, Gos97, Gos00b]. An important charac-

teristic of this approach is its connection to Fuxian species counterpoint[Fux42, pp.109-115]. Thedux is

designed as a series of notes of equal length, after which theresulting structure is embellished into florid

counterpoint.

Morris[Mor95] considers stacked canons without rests andTI = 1. GivenV = 2, TI = 1, and someSI,

he creates a table listing the melodic intervals which generate consonants. Based on the table, a directed

1Somewhat abusing notation, the∣ ⋅ ∣ operator is used both in reference to the cardinality of a set, or the length of a list or array.


graph (digraph) is created in which each vertex represents a valid melodic interval (called acase), and each

arc represents a succession of two of such intervals. Morrisprunes “contours of lines or polyphony that are

usually forbidden by the rules of counterpoint and, of course, parallel or similar octaves and fifths”[Mor95,

p.42]. To generate canons longer than two notes, he suggeststo “concatenate two linear intervals”. To

deal with the case1 < TI he suggests a few different approaches. First, two canons with TI = 1 could be

composed separately and then interleaved. Morris correctly remarks that interleaving two canons which are

independently derived according to the aforementioned graphs neither guarantees that the result satisfies all

stylistic constraints. Nor does this method allow the generation of all possible canons with a greaterTI.

To see this letSC0, SC1 be two stacked canons withTI = 1, which are interleaved in a resultant stacked

canonSCr. Parallel perfect fifths occurring inSC0 or SC1 do not necessarily occur inSCr due to the

interleaving method. Hence constrainingSC0 orSC1 as such actually over-constrains the resultant solution

space, leaving out suitable candidates which would satisfysuch constraints, besides introducing candidates

which may violate those.

Second, he notes that a “better and wholly general method” ... “is to generate charts that sequence two

or more intervals and list the concatenations of these sequences in the body of the chart. Once again, cases

that deviate from stylistic conventions are pruned. While this will generate graphs, the number of cases to

consider grows very quickly so that resulting graphs becometoo cumbersome for humans to use.”[Mor95,

pp.47-48] Indeed, it is not difficult to show that such a graphgeneration algorithm would have an (exponen-

tial) running time ofO (∣S.tones∣TI+1). Finally, he proposes a transformation process in which a canon

with TI = 1 is embellished with passing tones, suspensions, et cetera,by listing the ornamentation patterns

which can replace a melodic interval without introducing violations of counterpoint rules. It is however

questionable whether or not this approach actually generates a canon with1 < TI, as the underlying har-

monic structure is again determined atTI = 1. One could argue that this method leavesTI invariant as the

embellishments are perceived as local ornamentations of the original chord structure (although withV = 2

the term ‘interval structure’ seems more appropriate).

Gauldin’s research of Renaissance stretto canons[Gau96] considers canons which are not necessarily stacked

and offers a similar approach to Morris’. He considers canons for two or three voices, with a time interval of

one or two, and modal scale intervals between two consecutive voices of a fourth, fifth or octave. In terms

of large scale design, Gauldin suggests to reach a cadence inthe canon by writing it in reverse order. This

however moves the problem of reaching a cadence, to creatingthe link from the beginning of the canon to

the cadence thus created. He provides no algorithm for this but notes that since “this link will vary with

each system and previous melodic ideas, it is best left to theingenuity of the composer”[Gau96, p.49].

Gosman’s contribution to this topic[Gos97] follows a similar approach to Morris’ and Gauldin’s though

his focus is on analysis rather than construction of stackedcanons. In terms of designing a larger scale

structure he identifies repeated three note dux patterns (concatenations of two of Morris’ cases) and also

suggests palindromic duces to be of use by identifying thesein several stacked canons. It is however not

difficult to show that the condition of a dux to be palindromicis neither sufficient nor necessary a condition

for the dux to adhere to the associated stylistic constraints. Also, as Gosman studies stacked canons from

the Renaissance era, the limited number of choices available upon incremental extension of the dux may

unintentionally introduce palindromic structures.

2.2. LITERATURE REVIEW 11

When considering a chord as a stack of melodic intervals, as does the intervallic approach, all(V2) oc-

curring intervals must be taken into consideration in deciding whether or not a chord is consonant because

the stacking of several consonant intervals does not necessarily result in a consonant chord. None of the

contributors provide a method for3 < V which would be useful if thedux is used as a theme in an im-

itative polyphonic form, in which case a largerV provides the possibility of a larger number of mutually

distinct imitations with the theme. Analogous to Morris’ argument regarding an increase ofTI, it could be

argued that an increase ofV also renders this approach less suitable for humans. Also, the accepted use of

dissonant (seventh) chords as in Baroque music, with associated voice-leading constraints, seems unwieldy

to implement in the intervallic approach. Indeed, no obvious control of the resulting chord sequence of the

resulting canon is provided. Reconsider the problem of constructing a stacked canon in which a cadence

must be reached att = 160. Using the intervallic approach, the way to try and achieve this is again to try all

possibilities which would again give rise to an exponentialalgorithm.

2.2.3 Harmoniola

Another approach to the design of canons is the creation of a harmoniola. An early description of the har-

moniola technique is given in Burmeister’sMusical Poetics[BR93, p.193]. The basic idea is to create a

short harmonic passage (the harmoniola) forV voices with lengthTI. Thedux of the resulting stacked

canon is the concatenation of transpositions of theV melodies of the harmoniola. The order in which the

harmoniola’s melodies are concatenated and its individualtranspositions respectively determine the order

in which the voices enter the canon and their scale interval(s). E.g. the top four staves of bars 6 and 7 in

figure 2.1 are a harmoniola for Rameau’s Canon at the fifth. When labeling the voices top down with an

integerv starting at0, thedux can be reconstructed by concatenating the voices in ascending order, each

voicev transposed upward by7v semitones. While Burmeister provides guidelines for the construction of

rounds (canons at the unison), for which the method works well, other intervals are not discussed. Hence,

Gosman mainly regards the harmoniola of a stacked canon as ananalytical device[Gos00b, ch. 4].

2.2.4 Serial approaches

A concept related to stacked canons are rotational arrays and their applications in mainly serialism. Rogers

describes three types of such arrays[Rog68]. Table 2.2 shows a type I non-duplicating rotational array. All

entries are pitch classes from some generating pitch-classsetgs. The ∣gs∣2-sized array shown in table 2.2

is based on the pitch-class set{0,2,5,9,3} which appears in every column and row. Let us label the rows

bottom up, with an integerv, 0 ≤ v < V = ∣gs∣ = 5. Each rowv + 1 (mod V ) equals rowv, rotated

one position to the right. Hence the structure naturally connects to a round forV = ∣gs∣ voices, with

TI = 1 andSI = 0 semitones of the chromatic scale[Mor88]. Beyond the definition of the type I non-

0 2 5 9 32 5 9 3 05 9 3 0 29 3 0 2 53 0 2 5 9

Table 2.2: type I non-duplicating rotational array


duplicating rotational array, Rogers’ paper mostly studies two other types of rotational arrays (types II and

III) in some depth, both of which have an obvious connection to non-stacked canons. Morris describes a

method which transforms an existing atonal stacked canon with TI = 1 andSI = 0 to a resultant canon at a

different scale interval [Mor05]. The original canon’sdux is constructed by flipping triangles on a Tonnetz,

a lattice representing tonal space. In the classical Tonnetz shown in table 2.3, vertical neighbours are at

a pitch-distance of4 (mod 12), whereas horizontal neighbours are at a pitch-distance of3 (mod 12).Note how minor triads can be formed from a starting vertex, bygoing east(E), south(S), and returning

North-West(NW), back to the initial vertex (E-S-NW), e.g. the triangle formed by pitch classes{0,3,7}.

Similarly, major triads can be formed by going S-E-NW, e.g. the triangle formed by pitch classes{0,4,7}.

Also note how either triangle can be flipped over its diagonaledge into the other, which corresponds to

the traversal of related pitch class sets. By repeatedly choosing pitch classes from a triangle, and flipping

0 Ð 3 6 8∣ Ó ∣4 Ð 7 10 1

8 11 2 5

Table 2.3: classical Tonnetz

it over one of its edges to a next triangle, one can obtain thedux for a canon at the unison ad minimum,

the harmonic structure of which is derived from the choice oftriangles. The method is efficient but neither

considers time intervals larger than one, nor voice leadingconstraints, the latter perhaps due to Morris’

focus on atonal music.

2.2.5 Summary and remarks

While intuitive and easy to learn as an algorithm, the counterpointing approach provides a fragmented pro-

cess of composition with hardly any structural control. Dueto its reliance on composing counterpoint upon

counterpoint a composer may face the dread of trying all possibilities in order to obtain a desired harmonic

structure at some point in the canon, perhaps to no avail. Theintervallic approach developed in response to

this encodes harmonic and some voice leading constraints interms of constraints on the melodic intervals

of thedux. While the latter approach is efficient forV = 2, TI = 1, an increase of either parameter makes

the method too cumbersome for practical use by humans. Also,none of the aforementioned contributors to

the intervallic approach deal with rests in a dux even if these can be instrumental in avoiding the violation

of several types of constraints. Associated with this is a possible definition problem: what is the interval

between a rest and a dux tone, especially for1 < TI? Also, the intervallic approach requires, for each

musical style, time- or scale-interval, different tables for differentSI, which hardly contributes to a better

understanding of the relation betweendux and harmony. Morris’ Tonnetz approach marks an important

step forward in reasoning about stacked canons in this regard, in the sense that the harmony is chosen be-

forehand in terms of a pitch-class set. Yet, his approach targets serial stacked canons withTI = 1 and

provides no directions for dealing with voice-leading constraints. Another important observation in this

regard is Gosman’s recognition of a pattern of occurrences of the root, third, fifth and seventh of a dominant

seventh chord in thedux of Rameau’s canon at the fifth[Gos00a, p.53]. We return to this point in section

2.4.

2.3. OUTLINE 13

2.3 Outline

The remainder of this chapter attempts to remove the schismsof harmony and counterpoint on the one hand,

and tonal and atonal music on the other hand, in the known approaches to the creation of stacked canons,

by providing an efficient yet style-independent method for their derivation in terms of an abstraction of

the chord tones of a predetermined chord sequence and variations thereof. The definition ofrelative chord

toneswhich enables this was in part inspired by Gosman’s analysisof Rameau’s canon at the fifth. My

relative chord tone approach to the design of stacked canonsis introduced in section 2.4 through further

analysis of this canon, along with the study of both a harmonic- and voice-leading constraint within the

domain of relative chord tones instead of pitch classes. A general approach to dealing with voice leading

constraints is sketched in section 2.5, by studying some which are common in music theories. Section

2.6 provides an example of the incorporation of a constrainton the inversion of chords. The creation and

variation of chord sequences for stacked canons is discussed in section 2.7, after which section 2.8 counts

the number of distinct sub-canons of a stacked canon in demonstration of the applicability of my method

to the design of larger imitative polyphonic structures. Note that no attempt is made to fully describe or

analyze any particular style of stacked canons in the provided model. Interested composers or theorists

could attempt this by inclusion of the associated constraints and extension of the model.

2.4 Analysis of Rameau’s canon at the fifth

Example 6 briefly introduced Rameau’s canon at the fifth. Thissection develops several concepts related

to the design of stacked canons with relative chord tones through further analysis of the canon. It also

demonstrates the encoding of several constraints on stacked canons into generative algorithms.

Rameau considered the dominant seventh chord “the most perfect of all dissonant chords”[RG71, p.42],

usable in all inversions. Gosman argues that Rameau wrote the canons in his treatise partly in response to

Zarlino[Gos00a], critiquing Zarlino for not writing chords with four different tones. The harmonic reduc-

tion given in the four bottom rows of figure 2.1 reflects this: the canon is a repeated succession of complete

dominant seventh chords in all inversions. Bars[8..16)2 are the first in which the entire unique part of the

dux sounds in all four voices. The pitch classes of the roots of the seventh chords starting from bar8 have

the structure of table 2.4. The bottom row of table 2.4 contains the roots of the chords in bars[8..16). A

+SI +SI +SI+2 +2 +2 +2

9 11 4 6 11 1 6 8

Table 2.4: chord roots in Rameau’s canon at the fifth

chord sequence comprising two dominant seventh chords, their roots being2 semitones apart as indicated

in the second row, is continuously transposed byTI bars andSI semitones(mod 12) as indicated in the

top row. Note that his choice ofSI results in the “most perfect progression”[RG71, p.95] between the roots

of the chords in progressions from odd to even bars: the descending fifth. According to his “Rule for the

2The interval notation is also used in reference to bar numbers. Lower- and upper bounds are separated by two dots, round andsquare brackets have their usual meaning. I.e.[8..16) refers to the numbers8,9, . . . ,15.


progression of Dissonances, derived from the progression of fundamental chords”[RG71, p.95], this is one

out of three progressions in which the seventh can be prepared. Another one of such progressions is the

ascending second (or descending seventh) between roots, which corresponds to the progression from odd to

even bars. Also note that the combination of these progressions as in the chord sequence chosen by Rameau

allows for the stepwise downward resolution of each seventhaccording to his rule. We return to these points

in section 2.4.2.

In his analysis of Rameau’s canon at the fifth [Gos00a, p.53],Gosman notes that the canon’sdux in-

troduces each degree of the seventh chord, the root, third, fifth, and seventh in order to obtain complete

chords. His analysis implies a time-interval of1 where every odd bar is a (Schenkerian) prolongation of the

preceding even bar. While understandable from an analytical viewpoint, recall that the chords at odd bars

are also complete, which impliesTI = 2 whole notes. Nevertheless, his remarks demonstrate that when

studying harmonic- or voice-leading constraints between chords in a stacked canon, the pitch(-class) of a

melody-tone is often of lesser importance than its relativeposition within its chord. To simplify the study

of this relation, I define the concept of relative chord tones: the relative position of a chord tone within a

chord. While in triadic harmonies, the ‘root’, ‘third’, ‘fifth’ and ‘seventh’ are established relative chord tone

labels, we adopt the approach here of labeling relative chord tones with integers starting from0. To deter-

mine the relative chord tone labeling of a chord in general, first transpose the chord such that its (possibly

subjective) root is at pitch class0. Then label the chord tones in ascending pitch order, starting by0. Hence

the aforementioned relative chord tone labeling in triadicharmonies correspond to0,1,2 and3 respectively.

The latter labeling is easier in calculations and better serves the treatment of non-triadic harmonies.

2.4.1 Obtaining complete chords

Having explained the notion of relative chord tones, let us investigate how to obtain complete chords in

a stacked canon such as Rameau’s. The number below each note in the four bottom staffs of figure 2.1

equals the relative chord tone the note corresponds in the canon. Table 2.5 gives the relative chord tones

in bars[8..16). The rows correspond to the four voices in their natural order with the bass, voice0, on the

bottom row. Each column corresponds to one bar. Note the connection with rotational arrays in the sense

3 2 0 3 1 0 2 10 3 1 0 2 1 3 21 0 2 1 3 2 0 32 1 3 2 0 3 1 0

Table 2.5: relative chord tone structure of Rameau’s canon at the fifth

that the relative chord tones at botht = 8 andt = 9 are each cyclicly shifted upward everyTI bars. Next

I explain how this cyclic shift resulted from the choice ofV , the chord sequence, and desire for complete

chords, and why this naturally led to a spiral canon. Letrct[v, t] denote therelativechord tone of voicev

at timet. I.e. rct is a two-dimensional array, e.g.rct[0,8] = 2 andrct[0,9] = 1. Thedux of the canon

in relative chord tone notation corresponds torct[0], the one-dimensional array[2,1,3,2,0,3,1,0, . . . ]corresponding to a repetition of table 2.5’s bottom row. Because we are dealing with a stacked canon with

a fixed chord sequence, we have thatrct[v + 1, t + TI] = rct[v, t] for 0 ≤ v < V − 1 ∧ 0 ≤ t < L − TI.

Hence, the chord att = 14, the seventh column of table 2.5 (in bold font), is completely determined by

rct[0, t], t ∈ {14 − v ∗ TI ∣ 0 ≤ v < V } = {8,10,12,14}. In relative chord tone notation, this is the same

2.4. ANALYSIS OF RAMEAU’S CANON AT THE FIFTH 15

chord as the one att = 6, the first chord in which four voices sound. The latter chord is completely

determined byrct[0, t], t ∈ {6 − v ∗ TI ∣ 0 ≤ v < V } = {0,2,4,6}. Hence, if a complete chord is to sound

at t = 6, then mutually different relative chord tones must be chosen for rct[0, t], t ∈ {0,2,4,6}. After

picking mutually differentrct[0, t] for t ∈ {0,2,4}, there is just a single choice left forrct[0,6] because

the dominant seventh chord has only four different (relative) chord tones. Hence we have no choice but to

chooserct[0,8] = rct[0,0] if a complete chord is to sound att = 8. By repeating this argument, it follows

that the relative chord tones in the stacked canon in the evenbars are cyclic shifts of some permutation of

the numbers[0..V ). Obviously, the same applies to the odd bars. Hence, becauseeach chord in the chord

sequence has sizeV , the maximum and actuallength of theunique part of the dux in Rameau’s canon,LU ,

equalsV ∗TI = 8. Further extensions always result in a repetition ofrct[0,0..LU): a spiral canon, as is the

case with Rameau’s. The predicate enforcingcompletechords,Pcc, is given below. Note that the Dijkstra

notation for quantifications[Dij02] is used in this thesis,in which a quantification’s dummies, range and

term are separated by colons.

Pcc ∶ (∀v, t ∶ 0 ≤ t ∧ 1 ≤ v < V ∧ v ∗ TI < t ∶ rct[0, t] ≠ rct[0, t − v ∗ TI]) .

Algorithm 8 offers an efficient choice ofrct[0,0..T I∗V ) satisfyingPcc, if each chord in our chord sequence

has sizeV .

Algorithm 8 Precondition:each chord in the chord sequence has sizeV .

1. for eachtime classtc (mod TI): pick a permutationπtc of [0..V ), corresponding to the desired

cyclic shift of relative chord tones at time classtc.

2. for eacht, 0 ≤ t < V ∗ TI: put rct[0, t] = πt (mod TI) [t/TI (mod V )], where ‘/’ denotes integer

division.

Example 9 Algorithm 8 can generate Rameau’sdux as follows:

1. Choose permutationsπ0 = (2,3,0,1), π1 = (1,2,3,0) respectively for time classes0,1 (mod TI);

2. The interleaving ofπ0 andπ1 yieldsrct[0] = [2,1,3,2,0,3,1,0].Algorithm 8 yields adux, rct[0], with lengthLU = V ∗ TI, in justO(V ∗ TI) time. SinceV ∗ TI

equals the number of assignments to be made in choosing adux with this unique length, the problem’s time

complexity is inΘ (V ∗ TI) whence algorithm 8 is optimal. Thedux can always be repeated in a stacked

spiral canon. We summarize this section’s main result in proposition 10.

Proposition 10 There exists a stacked spiral canon in which only complete chords sound withV voices,

time intervalTI, chord sequencechord, and length of the unique part of the duxLU if and only if each

chord in the chord sequence has sizeV andLU = V ∗ TI.

2.4.2 Preparation and resolution of sevenths

Having explained how one may achieve complete chords, let usattempt to explain Rameau’s choice for

[2,1,3,2,0,3,1,0] out of all possible solutions. As mentioned earlier in this section, a constraint assert-

ing that the minorsevenths in each chord areprepared (if possible) andresolve in stepwise downward


motion,Pspr, is clearly formulated within his “Rule for the progressionof Dissonances, derived from the

progression of fundamental chords”[RG71, p.95]. Also, hisrule states that major sevenths must resolve

in ascending stepwise motion. The latter aspect, in combination with his preference for ‘the most perfect’

dominant seventh chords, may explain the chromatic descension from the third at odd bars to the seventh

at even bars, as is common in thorough bass practice. Indeed,without this chromatic alteration, a major

seventh chord would result in all inversions, which is why itwould have to resolve upward according to his

rule. GivenTI, Pcc, and his choice of chord sequence, the preceding observations result in the following

rules:

• any root in an even bar must be followed by a seventh and third(which corresponds to the relative

chord tone sequence[0,3,1] starting at from an even bar),and,

• any third in an odd bar must be followed by a seventh and fifth (which corresponds to the relative

chord tone sequence[1,3,2] starting at from an odd bar).

A closer look at[0,3,1], [1,3,2], and the entiredux reveals that all three conform to the functionf below,

defined for1 ≤ t.

f(t + 1) =⎧⎪⎪⎨⎪⎪⎩

f(t) − 1 (mod V ), if t is even,

f(t) − 2 (mod V ), if t is odd

A weakest predicate expressing conformance tof would be stronger thanPspr , which indicates that

Rameau’s choice ofdux (and chord sequence) may well have been formed byPspr. Note thatf(t + 2) =f(t) − 3 (mod V ) = f(t) + 1 (mod V ), which offers another explanation for its generation of complete

chords in the presence of chords withV voices. Indeed, the right summand ‘1’ is a generator of the cyclic

group(Z/V Z,+), the members which represents the relative chord tones available at both even and odd

bars in the presence of a chord sequence in which all chords have sizeV .

In order for constraints likePcc orPspr to be expressed intuitively, I introduce the notion of a dux graph.

Vertices in a dux graph always represent the relative chord tones of the chord sequence. An arc in a dux

graph represents an actual or possible voice leading, from one relative chord tone to the next. A dux graph

of Rameau’s canon is shown in figure 2.2. Each vertex in a dux graph is a tuple of the form(tc, rct): tc

represents thetime class moduloTI (even or odd bars in this case),rct the relativechord tone. The arcs

(0,2)

(0,3)

(0,0)

(0,1)

(1,0)

(1,3)

(1,2)

(1,1)

Figure 2.2: Dux graph of Rameau’s Canon at the fifth

2.4. ANALYSIS OF RAMEAU’S CANON AT THE FIFTH 17

in figure 2.2 correspond to the actual dux chosen by Rameau. The respective initial and terminal vertices

(0,2), (1,0) are drawn in bold. The arcs which ensure the proper preparation and resolution of sevenths

are drawn in bold. This section is only concerned withrestless dux graphs. As their name suggests, restless

dux graphs contain no rests. Section 2.5 introduces dux graphs with rests. Definition 11 formally defines

restless dux graphs. The arraycs contains thechord-sizes of the chord sequence’s chords. E.g. within the

context of Rameau’s canon at the fifth we havecs = [4,4] andTI = ∣cs∣ = 2. In our model of stacked

canons, the two constantsTI and∣cs∣ are always equal. Hence we may write∣cs∣ as the time interval of the

canon from now on.

Definition 11 The unconstrained, restless dux graphD (cs) = (V (D (cs)),A (D (cs))) for a stacked

canon with chord sizescs, is a digraph defined by:

• the set of verticesV (D (cs)) = {(tc, rct) ∣ 0 ≤ tc < ∣cs∣ ∧ 0 ≤ rct < cs[tc]}.

• the set of arcsA(D(cs)) =⎧⎪⎪⎨⎪⎪⎩⟨(tc, rct) , (tc + 1 (mod ∣cs∣), rct′)⟩

RRRRRRRRRRR0 ≤ tc < ∣cs∣ ∧ 0 ≤ rct < cs[tc] ∧0 ≤ rct′ < cs [tc + 1 (mod ∣cs∣)]

⎫⎪⎪⎬⎪⎪⎭Having introduced the notion of dux graphs, let us return to figure 2.2. Starting in(0,2), the path corre-

sponding to Rameau’sdux visits each vertex exactly once before returning to(0,2). I.e. the problem of

finding a dux for a stacked canon in which only complete chordssound, corresponds to the problem of

finding a Hamiltonian cycle in the corresponding unconstrained restless dux graph. The general problem of

finding a Hamiltonian cycle is NP-hard[GJ90, p.56-59]. However, ifPcc is our only constraint, Hamiltonian

cycles in a dux graph in which all chord sizes are equal are intuitively found as follows. Starting from some

initial vertexv with time class0 in the unconstrained dux graph, we walk through the graph while avoiding

vertices already used. This walk always ends in some vertexv′ with time classTI − 1 after which the arc

⟨v′, v⟩ closes the cycle. Note that such a graph algorithm basicallycorresponds to algorithm 8, although

strictly speaking, the setup of the unconstrained restlessdux graph in the graph algorithm makes it more

expensive.

2.4.3 Summary

The previous subsections introduced the modeling of stacked canons by means of relative chord tones,

chord sequences and dux graphs. Several algorithms were presented: one for designing stacked canons

with complete chords inO (∣cs∣ ∗ V ) time (the graph version of which runs inO (∣A (D(cs))∣) time), and

one for designing stacked canons with complete chords in which sevenths present in the chord sequence

are prepared and resolve by deriving a generating function.The efficiency of either algorithm objectively

shows that the design of a stacked canon such as Rameau’s canon at the fifth is not a computationaltour

de force3. The remainder of this chapter further develops the relative chord tone approach to the design of

stacked canons, which comprises the following activities:

1. create a chord sequence;

3Gosman concludes that music printers in the Renaissance eraseemed to have concurred with his belief that the creation ofstacked canons is a compositional tour de force. He infers this from their prominent placement as opening or closing works incollections[Gos00b, p.3]. My complexity analyses do not contest that they believed so, they do contest their belief.


2. design adux as a list of relative chord tones adhering to all stylistic constraints in force;

3. embellish the harmonic reduction of the stacked canon.

The first two of the above activities do not necessarily take place in any particular order. Indeed, our

reasoning within the relative chord tone domain induces theequivalence relation ‘equality modulo chord

sequence instantiation’ which expresses whether or not twostacked canons are equal in relative chord tone

notation. We have already seen that the design of adux is a style dependent activity in the sense that

it depends on the constraints which define a composer’s style. Instead of attempting to fully discuss one

particular style, section 2.5 discusses the enforcement oftwo well known voice-leading constraints and their

encoding in dux graphs, section 2.6 sketches an approach to dealing with six-four chords. These are meant

to serve as examples: the approach followed can be used by thereadership for encodingtheir constraints in

a dux graph. The creation and variation of chord sequences isdiscussed in section 2.7. The embellishment

of the harmonic reduction is left to the discretion of the reader as it is an entirely style dependent activity.

2.5 Voice leading constraints

From the high Renaissance on, theorists generally forbade the use of parallel fifths and octaves in strict

counterpoint, and their occurrence in music up to the late19th century was incidental[SG01]. This section

discusses the encoding of the associated constraints in ourrelative chord tone model. Section 2.5.2 discusses

parallel octaves, section 2.5.3 discusses parallel fifths.These sections also serve as examples on encoding

voice leading constraints in general, in our relative chordtone model. Section 2.5.1 provides a few required

definitions.

2.5.1 preliminaries

We use the record typeScale defined below to denote a scale as a transposed subset of the twelve pitch

classes. The fieldroot is used to denote its transposition. The whole tone scale ‘rooted’ at pitch class1 for

example, corresponds to⟨{0,2,4,6,8,10},1⟩.

type Scale =⎧⎪⎪⎨⎪⎪⎩

tones ∶ 2[0..12) {0 ∈ tones}root ∶ [0..12)

We denote chords by instances of the record typeChord given below.

type Chord =⎧⎪⎪⎨⎪⎪⎩

tones ∶ 2[0..∣S.tones∣) {0 ∈ tones}root ∶ [0..∣S.tones∣)

The setc.tones of a chordc contains scale tone classes, the fieldroot denotes a transposition in scale

tones. Within the context of the scale⟨{0,2,4,5,7,9,11},7⟩ for example, the chord⟨{0,2,4},0⟩ denotes

the G-major triad. For a chordc, we refer to∣c.tones∣ as the chord size ofc. The chord sequence of a

stacked canon is represented by the arraychord : [0..∣cs∣) → Chord. This implies that in our model

of stacked canons, the chord sequence is transposed every∣cs∣ time units at intervalSI. This may seem

a restriction, we return to this when discussing chord sequencing. The notion of a scale is required when

studying constraints on e.g. parallel fifths in a diatonic context. Due to my own compositional needs, my

discussion focusses mostly on a chromatic context in which scale tones correspond to pitch classes.

2.5. VOICE LEADING CONSTRAINTS 19

2.5.2 Avoiding parallels of the class0 (mod 12)

This subsection discusses the avoidance of parallel intervals of the class0 (mod 12), ‘parallel octaves’,

considered undesirable in many styles. Before developing the theory on this subject, we study a few exam-

ples.

Example 12 LetSC be a stacked canon with parametersV = 3, L = 4, cs = [3], dux = [0,1,0,1]), with

dux in relative chord tone notation. Figure 2.3 gives an examplerealization ofSC in G-major withSI = 3

andchord = [⟨{0,2,4} ,0⟩]. A parallel interval of the class0 (mod 12) occurs in the chord change from

t = 2 to t = 3 between voices0 and2.The harmonic reduction in relative chord tone notation is given in

table 2.6, again with the voices in their natural order.

&?

#

#22

22

∑Ó ˙˙ ˙

˙ ˙˙ ˙˙ ˙

˙ ˙˙ Ó

∑Figure 2.3: Realization ofSC with parameters(V = 3, L = 4, cs = [3], dux = [0,1,0,1])

0 1 0 10 1 0 1

0 1 0 1

Table 2.6: relative chord tone structure ofSC

The parallel octave in example 12 corresponds to the consecutive occurrence, of two equal relative chord

tones in some pair of voices. Indeed: we have two occurrencesof relative chord tone0 at t = 2 in the outer

voices, and two occurrences of1 at t + 1 = 3 in the same outer voices. Figure 2.4 shows the derivation of

this dux in a dux graph. An arc label again corresponds to the order in which it is traversed in thedux.

Note that arc⟨(0,1), (0,1)⟩ carries a double label: it is used twice.

1 0,2

(0,2)

(0,0)

(0,1)

Figure 2.4: Dux graph ofSC

Example 13 My composition ‘Spiral’ for string orchestra, fully discussed in chapter 3, features several

themes, one of which istheme0. This theme allows a consonant six-voice stretto ad minimumand many

sub-stretti, all without parallel octaves, while adheringto a simple chord sequence. The six-voice stretto

is a stacked canon with parametersV = 6, cs = [3], SI = 11, L = 10, chord = [⟨{0,4,8} ,0⟩], dux =


[0,1,1,2,0,0,2,1,1,0]. The unembellished six-voice stretto is shown in the top three staves of figure 2.5.

The harmonic reduction is given in the bottom6 rows of table 2.7. The top row represents a seventh voice,

&

V?

?

22

22

22

22

∑∑∑∑

Ó ˙˙ ˙b˙# ˙n#

∑∑

Ó ˙˙b ˙b˙ ˙#˙ ˙˙## ˙nn#

Ó ˙˙b ˙b˙ ˙˙ ˙b˙ ˙˙b ˙˙# ˙#n

˙b ˙˙b ˙b˙ ˙˙b ˙˙# ˙#˙ ˙˙## ˙#nn

˙ ˙b˙b ˙b˙ ˙˙ ˙˙# ˙n˙b ˙b˙# ˙n#n

˙ ˙˙b ˙˙# ˙n˙b ˙b˙ Ó∑˙#n# ˙n#n

˙ ˙b˙b ˙b˙ Ó∑∑∑

˙# ˙n#

˙b Ó∑∑∑∑∑

˙## ˙nn#Figure 2.5: unembellished stretto and chord sequence of ‘Spiral’: theme0

not present in figure 2.5. The corresponding dux graph is given in figure 2.6. The addition of a seventh voice

0 1 1 2 0 0 2 1 1 00 1 1 2 0 0 2 1 1 0

0 1 1 2 0 0 2 1 1 00 1 1 2 0 0 2 1 1 0

0 1 1 2 0 0 2 1 1 00 1 1 2 0 0 2 1 1 0

0 1 1 2 0 0 2 1 1 0

Table 2.7: relative chord tone structure of adapted strettoof ‘Spiral’: theme0

85

6

3

1,72

0

4

(0,1)

(0,0)

(0,2)

Figure 2.6: dux graph of stretto of ‘Spiral’:theme0

in the stacked canon would introduce a parallel octave highlighted in bold font, between the outer voices

from t = 7 to t = 8. The reason for this is the dual use of arc⟨(0,1), (0,1)⟩ in thedux at t = 1→ 2,7→ 8.

Let us summarize the former two examples and think of adux as the arc-list of its path in an unconstrained

dux graph. GivenV voices, we eventually haveV voice leadings from one chord to the next if and only if

V ∗ ∣cs∣ < L. Because a voice-leading corresponds to a unique arc, we have that a parallel octave occursif

and only if within a sublist ofV ∗ ∣cs∣ arcs, some arc appears twice. The question of how to create such

adux is answered as follows. First we derive an upper bound on the number of voices in a stacked canon


without parallel octaves, in terms of the canon’s chord sizes. Next we show that choosingV according to

this bound is also a sufficient condition for such a canon to exist by providing an algorithm. This section

concludes with an algorithm for the generation of a stacked canon without parallel octaves which also con-

siders rests.

As explained, we start by deriving an upper bound onV . To this end, consider any chord progression

in the canon fromt → t + 1. Let A−(D(cs), t) denote the set of arcs leaving time classt (mod ∣cs∣),which, by definition 11, equals the set of arcs entering time classt + 1 (mod ∣cs∣), A+(D(cs), t + 1). In

the unconstrained dux graph, we have:

∣A− (D (cs) , t)∣ = cs [t (mod ∣cs∣)] ∗ cs [t + 1 (mod ∣cs∣)] . (2.1)

I.e. the number of mutually different possible voice leadings fromt to t + 1 equals∣A− (D (cs) , t) ∣. Obvi-

ously, if we have more voices than mutually different voice leadings between two consecutive chords, then

a parallel octave occurs. This observation leads to the following upper bound on the number of voicesV

for a stacked spiral canon without parallel octaves or rests:

V ≤ (↓ tc ∶ 0 ≤ tc < ∣cs∣ ∶ ∣A− (D (cs) , tc)∣) . (2.2)

This bound is useful when designing themes for stretto-fugues, in which a large number of different

stretti is desirable. The question however is of course: is this bound sufficient for adux to exist, such

that it results in a stacked canon without parallel octaves?We answer this question by first studying

the extreme case in whichV equals the right-hand side of (2.2). Let, for an unconstrained dux graph

D(cs) = (V(D(cs)),A(D(cs))):

edg (D (cs))) ≡ (∀tc ∶ 0 ≤ tc < ∣cs∣ ∶ ∣A− (D (cs) , tc)∣ = ∣A− (D (cs) , tc + 1)∣) . (2.3)

We consider the following two cases:

edg(D(cs)): from definition 11 and (2.3), it follows that every vertex has the same in- and out

degrees, and the graph is strongly connected. I.e. it is Eulerian [Eul36]. A Eulerian cycle exists

which uses every arc exactly once, before returning to the initial vertex. The arc-list corresponding

to such a cycle has length(∑ t ∶ 0 ≤ t < ∣cs∣ ∶A−(D(cs), t)), which, byedg(D(cs)), equals∣cs∣ ∗A− (D (cs) ,0). Hence, by repeating this cycle, using (2.2) it follows thatno arc is used twice within

any sublist of∣cs∣ ∗ V = ∣cs∣ ∗A−(D(cs),0) arcs whence the resulting stacked canon indeed has no

parallel octaves. The Eulerian cycle can be constructed using Fleury’s algorithm [Fle83].

¬edg(D(cs)): the basic idea in this case is to delete vertices and their adjacent arcs fromD, thus

obtainingD(cs′), such thatedg(D(cs′)) and the out-degree of each time-class equals the minimum

out-degree of all time-classes inD(cs). We call this process the Eulerization ofD(cs).

Example 14 Figure 2.7 shows a dux graphD(cs) before (a) and after (b) Eulerization. By re-

moving vertex(2,1) fromD([1,2,2,2]) and all adjacent arcs, we obtain the Eulerian dux graph

D ([1,2,1,2])


0 1 2 3

(1,1)

(1,0)(0,0) (2,0)

(2,1) (3,1)

(3,0)

(a) before Eulerization

0 1 2 3

(1,1)

(1,0)(0,0) (2,0)

(3,1)

(3,0)

(b) after Eulerization

Figure 2.7:D before (a) and after (b) Eulerization

Having established that there exists a stacked canon without parallel octaves for the extreme cases where

(2.2) is an equality, it is easy to show that one also exists for smallerV . Indeed, removing voices from the

canon obtained for the extreme case of (2.2) obviously cannot introduce parallel octaves. We summarize

this section’s main result in proposition 15.

Proposition 15 There exists a restlessdux, ∣cs∣ ∗ V < ∣dux∣, for a stacked canon without parallel intervals

of the class0 (mod 12) and chord sizescs if and only if (2.2).

The preceding discussion on avoiding parallel octaves did not involve the use of rests, even if they can

be useful in avoiding parallel octaves. Also, the algorithms described leave out many potential solutions.

Indeed, the Eulerization process reduced a non Eulerian duxgraph by removing vertices which are never

used if the Eulerized graph is used as an input to Fleury’s algorithm. Also, if (2.2) is a strict inequality,

then creating adux with Fleury’s algorithm only considers Eulerian cycles, which is overly restrictive. We

conclude this section by introducing an algorithm which successfully deals with rests and also generates

many more solutions. To this end we need the following definition of the unconstrained dux graph with

rests.

Definition 16 The unconstrained, dux graphD<(cs) = (V (D<(cs)) ,A (D<(cs))) for a stacked canon

with chord sizescs, is a digraph defined by:


• its set of verticesV (D<(cs)) =

{(tc, rtc) ∣ 0 ≤ tc < ∣cs∣ ∧ rct ∈ [0..cs[tc]) +<} .

• its set of arcsA(D<(cs)) =⎧⎪⎪⎨⎪⎪⎩⟨(tc, rct) , (tc′, rct′)⟩

RRRRRRRRRRR0 ≤ tc < ∣cs∣ ∧ tc′ = tc + 1 (mod ∣cs∣) ∧tc ∈ [0..cs[tc]) +<∧ tc′ ∈ [0..cs[tc′]) +<

⎫⎪⎪⎬⎪⎪⎭

In comparison to the restless dux graph, the dux graph with rests has an extra vertex for every time-class,

with relative chord tone ‘<’: a rest. This extra vertex has one incoming arc from each vertex of the previous

time class, and one outgoing arc to each of the vertices of thenext time class. If an arc is neither entering nor

leaving a rest, we call it restless. We create adux with rests for a stacked canon without parallel octaves by

walking through its dux graphD<(cs). During this walk, we must once more avoid the reuse of any restless

arc within a subsequence of∣cs∣ ∗V arcs. We do this by removing the restless arc corresponding to dux[l −1..l] immediately after choosingdux[l]. This arc is then restored immediately after choosingdux[l − 1 +V ∗ ∣cs∣]. Reusing an arc containing a rest obviously cannot result inparallel octaves. Hence, if we allow

rests, the number of voices in a stacked canon is no longer constrained by (2.2). Algorithm 17 uses these

observations to find such adux. It is given in a slightly adapted version of the Guarded Command Language

(GCL) [Dij76][Kal90] and explained further below. Three reasons motivated this choice of formalism. The

GCL is a formal language with well defined semantics, borrowed from logic and mathematics. Second,

the GCL uses language constructs quite similar to imperative programming languages. Hence it is easy

to understand or translate to an actual computer program. Third, the derivational style of programming

associated with the GCL was used in deriving most results andalgorithms in the chapter, although their

formal derivation is probably out of scope. The program and all of its GCL-constructs are explained below.


Algorithm 17

∣[const V,L ∶ N

+;

const cs ∶ [0..∣cs∣) → [1..12] ;const CS = (↑ t ∶ 0 ≤ t < ∣cs∣ ∶ cs[t]) ;

var as ∶ [0.. ∣cs∣) × [0..CS) +<→ 2[0..CS)+<;

var dux ∶ [0..L)→ [0..CS) + <;

var l ∶ [0..L];

for all tc, rct ∶ tc ∈ [0..∣cs∣) ∧ rct ∈ [0..CS) +<∶as[tc, rct] ∶= [0..cs [tc + 1 (mod ∣cs∣)]) +<;

dux[0] ∶ dux[0] ∈ [0..cs[0]) ∨ dux[0] =<;l ∶= 1;

do l ≠ L →

dux[l] ∶ dux[l] ∈ as [l − 1 (mod ∣cs∣), dux[l − 1]] ;if <∈ dux[l − 1..l] → skip;

[] </∈ dux[l − 1..l] → as [l − 1 (mod ∣cs∣), dux[l − 1]] ∶=as [l − 1 (mod ∣cs∣), dux[l − 1]] − dux[l];

f i

const l′ = l − ∣cs∣ ∗ (V − 1) ;if 0 < l′ → as [l′ − 1 (mod ∣cs∣), dux[l′ − 1]] ∶=

as [l′ − 1 (mod ∣cs∣), dux[l′ − 1]] + dux[l′];[] l′ ≤ 0 → skip;

f i

l ∶= l + 1;

od

]∣Let us examine the global variables and constants used in algorithm 17. The constantsV , L andcs have

their usual meaning, as does the variabledux. The constantCS equals the maximum of all chord sizes.

The variableas stores theadjacencyset of each vertex. I.e.as[tc, rct] is initialized such that it contains

all relative chord tones at time classtc+ 1 (mod ∣cs∣), including the rest. These are all relative chord tones

of the vertices accessible from vertex(tc, rct) in D<(cs). The variablel is a counter ranging from[1..L].It keeps track of the progress made so far in thedo-od repetition. Having explained the declarations in the

first five lines, let us analyze the algorithm.

The first three statements of algorithm 17 form the initialization. Thefor all statement initializes the

adjacency setsas. Formally, a statement of the formfor all v0, v1,⋯ ∶ P (v0, v1,⋯) ∶ S executesS with

all values for the dummy variablesv0, v1,⋯ for which P (v0, v1,⋯) holds. It is a convenience construct

which avoids writing down a trivial repetition. The next statement chooses a value fordux[0]: any relative

chord tone or rest available at time class0. Formally, in a statement of the formv ∶ P , the binary choice

operator ‘∶’ assigns a value to the variablev such that the predicateP evaluates totrue: P is a valid

postcondition to this statement. In this chapter, the choice operator models the composer’s choice. The last


initialization initializesl to 1, reflecting that thedux created thus far has length1.

The repetition which follows the initialization expandsdux[0..l) by one relative chord tone during each

stroke, untill = L. The first statement within the repetition chooses some available relative chord tone from

the adjacency setas [l − 1 (mod ∣cs∣), dux[l − 1]], using the choice operator ‘∶’. Note that the choice of

a rest is always available, as those are never removed in thisprogram. Theif -fi selection following this

choice considers two cases: the first case corresponds to theharmless choice of an arc containing a rest, the

second to a choice of an arc without a rest. If a restless arc ischosen, we must take care not to use it again

within a subsequence of(V − 1) ∗ ∣cs∣ arcs or parallel octaves will be the result. Hence, the removal of the

end vertexdux[l] from the adjacency set ofas [l − 1 (mod ∣cs∣), dux[l − 1]]. The next two statements

deal with adding the arc⟨(l′ − 1 (mod ∣cs), dux[l′ − 1]) , (l′ (mod ∣cs), dux[l′])⟩ back to the graph, for

l′ = l − (V − 1) ∗ ∣cs∣, as it can now be safely reused. Initially, this arc may be undefined, since we haven’t

proceeded far enough in the generation of ourdux: hence the selection.

Algorithm 17 is a straightforward derivation using the technique of replacing the constantL by the

program variablel[Kal90, pp.57-62] in the invariants expressing thatdux[0..L) results in a stacked canon

without parallel octaves. Let us briefly analyze its complexity. The most costly initialization step is the gen-

eration of the adjacency setsas. LetCS = (↑ tc ∶ 0 ≤ tc < ∣cs∣ ∶ cs[tc]). Then this step takesO(∣cs∣ ∗CS2)steps. The repetition takesO (L) time. The result,O (∣cs∣ ∗CS2 +L), is near linear for largerL.

2.5.3 Avoiding parallels of the class7 (mod 12)

Some styles consider parallels of the interval classI (mod 12) unacceptable, whereas parallels of the

interval−I (mod 12) are not. A well known example are ‘parallel fifths’ (I = 7), the inverse of which,

‘parallel fourths’ (I = 5) are considered less problematic. We sketch an approach to avoiding such parallels

in stacked canons in this subsection.

Whether or not two relative chord tones form an undesirable parallel interval classI (mod 12), ulti-

mately depends on the scale and chords used, the type of stacking (ascending or descending), and whether

or not voices may cross. In this section we assume thatI results from a particular relative chord tone inter-

val such as 2, which corresponds to a fifth in triadic chord sequences. For the sake of brevity we assume

that no voice crossings occur. We use the boolean constantASC ∶ ASC ≡ 0 < SI to denote whether or

not a canon is stacked inascending fashion. Forb ∈ B, we defineJbK to equal 1 ifb and0 if ¬b. The boolean

functioncr ∶ 2[0..L) → B , cr(T ) ≡ (∃t ∶ t ∈ T ∶ dux[t] =<), expresses whether or notdux contains a

rest for somet ∈ T .

The predicatePnpar(rcti) given below expresses that no parallel relative chord tone intervalsrcti

occurs.

Pnpar (rcti) ∶⎛⎜⎜⎜⎜⎜⎝

∀v, t ∶ 1 ≤ t < L ∧ 1 ≤ v < V ∧ 1 ≤ t − v ∗ ∣cs∣ ∶cr ({t − 1 − v ∗ ∣cs∣, t − v ∗ ∣cs∣, t − 1, t}) ∨dux [t − 1..t] + JASCK ∗ rcti ≠dux [t − 1 − v ∗ ∣cs∣ ..t − v ∗ ∣cs∣] + J¬ASCK ∗ rcti

⎞⎟⎟⎟⎟⎟⎠.

Note thatPnpar(0), with L replaced byl, was used as a loop invariant in the derivation of algorithm 17.

The summands containingASC reduce to0 in that case. For parallel fifths we putrcti = 2. Using the

aforementioned technique of replacing the constantL by the program variablel, we obtainPnpar(rcti, l)


given below, as a precondition to an increase ofl by one.

Pnpar(rcti, l) ∶

⎛⎜⎜⎜⎜⎜⎝

∀v ∶ 1 ≤ v < V ∧ 1 ≤ l − v ∗ ∣cs∣ ∶cr ({l − 1 − v ∗ ∣cs∣, l − v ∗ ∣cs∣, l − 1, l}) ∨dux [l − 1..l] + JASCK ∗ rcti ≠dux [l − 1 − v ∗ ∣cs∣ ..l − v ∗ ∣cs∣] + J¬ASCK ∗ rcti

⎞⎟⎟⎟⎟⎟⎠.

Algorithm 18 Adapt algorithm 17 as follows to prevent parallel relative chord tone intervalsrcti, using

the auxiliary constant

δrcti = (J¬ASCK − JASCK) ∗ rcti ∶

1. The second alternative of the first if-statement must be catenated with the below fragment :

∣[ var b ∶ B;

b ∶= 0 ≤ dux[l − 1] + δrcti < cs [l − 1 (mod ∣cs∣)] ∧0 ≤ dux[l] + δrcti < cs [l (mod ∣cs∣)] ;

if b → as [l − 1 (mod ∣cs∣), dux [l − 1] + δrcti] ∶=as [l − 1, dux [l − 1] + δrcti] − (dux[l] + δrcti);

[] ¬b → skip;

f i ]∣.

The local variableb expresses whether or not the arc corresponding todux[l − 1, l] can intro-

duce the undesirable parallels corresponding to the relative chord tone intervalrcti. If so, the

arc ⟨(l − 1 (mod ∣cs∣), dux[l − 1] + δrcti) , (l (mod ∣cs∣), dux[l] + δrcti)⟩ is removed fromA(cs).Again, an arc containing a rest is never removed whence deadlock cannot occur.

2. The first alternative of the second if-statement must be catenated with the below fragment, which

allows an arc deleted by the first modification to be used againit no longer poses a danger.

∣[ var b ∶ B;

b ∶= 0 ≤ dux [l − 1 − ∣cs∣ ∗ (V − 1)] + δrcti < cs [l − 1 (mod ∣cs∣)] ∧0 ≤ dux [l − ∣cs∣ ∗ (V − 1)] + δrcti < cs [l (mod ∣cs∣)] ;

if b → as [l − 1 (mod ∣cs∣), dux [l − 1 − ∣cs∣ ∗ (V − 1)] + δrcti] ∶=as [l − 1 (mod ∣cs∣), dux [l − 1 − ∣cs∣ ∗ (V − 1)] + δrcti] +(dux [l − ∣cs∣ ∗ (V − 1)] + δrcti) ;

[] ¬b → skip;

f i ]∣.

Note that neither adaptation changes the computational complexity of the resulting algorithm from our pre-

vious analysis.

Proposition 15 provided us a bound on the number of voices in terms of the chord size of a long stacked

2.6. AVOIDING SIX-FOUR CHORDS IN TRIADIC CHORD SEQUENCES 27

canon without parallel octaves. Here we provide a similar bound for a stacked spiral canon without parallel

fifths or octaves: proposition 19. Because of the connectionbetweenPnpar(2) and traditional harmonies

and music, there is not much use in investigating chord sizeslarger than4.

Proposition 19 Let cs be of the pattern(3)+, (3,4)+, (4,3)+, or (4)+ ∧ ∣cs∣ (mod 2) = 1. There exists a

restlessdux with LU = ∣cs∣ ∗A− (D (cs) ,0), for a stacked spiral canon for whichPnpar(0) ∧ Pnpar(2)holdsif and only if V ≤ (↓ tc ∶ 0 ≤ tc < ∣cs∣ ∶ A− (D (cs) , tc)).Proof. Note that for each chord size pattern under consideration, the corresponding dux graph is Eulerian.

Hence the constructive proof of proposition 15 in section 2.5.2 provides adux for a stacked spiral canon

with V = A− (D (cs) ,0) voices without parallel octaves. Such adux corresponds to a Eulerian cycle

in a Eulerian dux graph withLU = ∣cs∣ ∗ A− (D (cs) ,0) which repetition results in a spiral canon. Let

σn,∣cs∣(dux) denote the cyclic right-shift ofdux by n ∗ ∣cs∣ places. The occurrence of a parallel fifth

between voices0 andv corresponds to the existence of a cyclic shiftσv,∣cs∣ which maps two consecutive

relative chord tonesi + 2, j + 2 onto i, j. Let the respective time classes betci, tcj , i.e. tcj = 1 + tci

(mod ∣cs∣). Such a cyclic shift exists if the corresponding chords havesizes at least 3. Indeed, the system

of equations0 ≤ i < cs[tci] − 2, 0 ≤ j < cs[tcj] − 2 has(cs [tcj] − 2) ∗ (cs [tcj] − 2) solutions, which

indicates that multiple shifts may exist for larger chord sizes. By choosing adux in which for all solutions

i, j, the segmenti + 2, j + 2 is mapped ontoi, j by σv,∣cs∣(dux), for somev ∈ [0.. ∣cs∣ ∗ cs [tci] ∗ cs [tcj]),we only lose one of the maximum ofcs[tci] ∗ cs[tcj] voices provided by the aforementioned constructive

proof of proposition 15. It remains to show that such duces indeed exists for a stacked spiral canon based on

either of the given chord sequence patterns. Consider figure2.8. Its subfigures represent the unconstrained

restless dux graphs for the patterns ofcs under consideration. Each of those graphs have the same structure.

The black arrows are the ones inD(cs) potentially partaking in parallel fifths. The gray arrows are their

complement inA(D (cs)). All vertices are labeled only with relative chord tones. For each of the given

patterns ofcs, note that the black arrows, in each sub-graph of figure 2.8 form two disjoint cyclesc1, c2,

such thatc2 is the upward transposition by two relative chord tones ofc1. The gray arrows span a single

strongly connected subgraph containing a Eulerian cyclec0, adjacent to all vertices ofc1 andc2. Hence if

we insertc1 into c0 at some vertex(tc, rct), we can always insertc2 at (tc, rct + 2). I.e. this procedure

generates a Eulerian cycle in the unconstrained dux graph, in which only a single shiftσv,∣cs∣ results in all

possible parallel fifths for somev.

2.6 Avoiding six-four chords in triadic chord sequences

Since the Renaissance the fourth has been considered a consonance only when it is understood as the inver-

sion of the fifth[SG01]. Hence the six-four chord is banned inFux’ ‘note against note’ counterpoint [Fux42,

part 2]. This section sketches an approach to the incorporation of the associated constraint in our relative

chord tone model.

Table 2.8 shows the relative chord tone structures of ascending (a) and descending (b) stacked canons with

cs = [3] andV = 3, in which a perfect fourth occurs between the bass and another voice in column three

(in bold font). We assume the use of a triadic chord sequence within this section, which seems appropriate

in light of the constraint being discussed. In the ascendingstacked canon, the six-four chord results from

relative chord tone 0 being followed by relative chord tone 2, within a dux-segment of lengthV . In the


0

2

0

1

2

1

(a) cs = (3)+

1

2

0

1

2

0

3

2

1

0

3

(b) cs = (3,4)+ ∨ cs = (4,3)+

0

1

2

0

3

2

1

33

2

0

1

(c) cs = (4)+ ∧ ∣cs∣ (mod 2) = 1

Figure 2.8: Unconstrained dux graph patterns considered inproposition 19

descending stacked canon, the six-four chord results from relative chord tone 2 being followed by relative

chord tone 2, again within adux-segment of lengthV . The predicatePnf defined below generalizes the

preceding observations in forbidding the undesirable fourths in stacked spiral canons with an arbitrary time

distance based on a triadic chord sequence. The boolean constantASC again indicates whether or not a

stacked canon isascending. IfASC, then the right summand in the consequent ofPnf becomes negative,

otherwise positive.

Pnf ∶⎛⎝∀ t, v ∶ 0 ≤ t < L ∧ 0 < v < V ∧ dux[t] = 0

∶ dux [ t + (−1)J¬ASCK∗ v ∗ ∣cs∣ (mod L) ] ≠ 2

⎞⎠

2.7. THE CONSTRUCTION OF A CHORD SEQUENCE 29

(a) ascending stacked canon:

0 1 2 00 1 2 0

0 1 2 0

(b) descending stacked canon:

2 0 1 22 0 1 2

2 0 1 2

Table 2.8: Occurrences of64 chords in ascending (a) and descending (b) stacked canons

As for choosing a value fordux[l] for incremental values ofl as in algorithms 17 or 18, the following rules

can be inferred, with0 < v < V :

1. If ASC, then, upon choosingdux[l] = 0, a choice of 2 must be avoided fordux [l + v ∗ ∣cs∣];2. If ¬ASC, then, upon choosingdux[l] = 2, a choice of 0 must be avoided fordux [l + v ∗ ∣cs∣].

The incorporation of these rules into dux-generation algorithms 17 or 18 cannot introduce a deadlock, since

one can always resort to choosing rests in avoiding relativechord tones 2 or 0. The adjacency setsas must

be manipulated in a manner similar to the definition of algorithm 18, which again hardly changes the com-

putational complexity of the resulting algorithms.

Later styles adopt what is described by theorists as cadential six-four chords as in I64 V53, among other

accepted functions of six-four chords such as passing-, neighbouring- or arpeggiated six-four chords. Note

that both examples in table 2.8 can be understood as a relative chord tone structure which contains the

progression I64 V53. Indeed, with an appropriate choice of chord sequence, the chords I64, V5

3 respectively

correspond to columns 3 and 4 of either sub-table. Prototypical voiceleadings in I64, V53 corresponds to the

following permutations in relative chord tone notation:

• (0,1): the root of I64 steps down to the third of V53;

• (1,2): the third of I64 steps down to the fifth of V53;

• (2,0): the fifth of I64 is tied to the root of V53.

Composing the above cycles we obtain(0,1,2), which, if repeated ad libitum, corresponds to a rotationalarray,

corresponding to a chain of transpositions of the aforementioned progression. While an exhaustive treat-

ment of six-four chords in stacked canons according to the rules of classical harmony is beyond the scope

of this chapter, the examples in this chapter should enable an interested reader to provide it. However, note

that other techniques can also be used to avoid six-four chords. By Rameau’s argument that seventh chords

are acceptable in all inversions, any chord sequence of seventh chords avoids the undesirable use of six-four

chords once all voices have entered the canon, if only complete chords result. Recall from section 2.4.1

that it is not difficult to obtain complete seventh chords in astacked canon. The only potential occurrence

of dissonant fourths remains during the setup phase of the canon, when voices enter one by one. Hence it

suffices to design the initial part of thedux accordingly in such cases.

2.7 The construction of a chord sequence

This section discusses the creation of chord sequences for stacked canons. The point of departure for our

discussion of chord sequencing is the relation between dux and comites in a stacked canon. As discussed


in section 2.1, every comes is the transposition bySI scale steps andTI = ∣cs∣ time units, of its immediate

predecessor. Let the array ofrelativescaletones,rst, be defined analogous to the array of relative chord

tonesrct. The predicatePsc summarizes the relation between the relative scale tones ofthe voices in a

stackedcanon. Similar to relative chord tones, relative scale tonesidentify pitches in a scale with an integer

number for zero up to the number of scale tones. Within the context of a diatonic scale, the below definition

admits diatonic deviations betweendux andcomites whereas a chromatic scale results in strict imitation.

Psc ∶⎛⎝∀t, v ∶ 0 ≤ v < V ∧ v ∗ ∣cs∣ ≤ t < L + v ∗ ∣cs∣

∶ rst[v, t] = rst[0, t − v ∗ ∣cs∣] + v ∗ SI mod ∣S.tones∣⎞⎠

As the reader may check, algorithm 20 defined below is an obvious way of establishingPsc. All of the chord

sequences in the preceding examples can be constructed by it. The algorithm uses the auxiliary function

rct2rst which, given a chord, and a relative chord tone, returns the corresponding relative scale tone.

Algorithm 20 After choosingcs, rct[0]:

1. repeat the relative chord tones of the dux at the time interval of the canon in all voices, such that

(∀v, t ∶ 0 ≤ t < L ∧ 0 ≤ v < V ∶ rct [t + v ∗ ∣cs∣ , v] = rtc [t,0]) ;

2. choosechord[0..∣cs∣) and repeat it every∣cs∣ time units rootedSI scale degrees higher, such that for

all t, ∣cs∣ ≤ t < ∣rct [V − 1]∣:

chord [t] =⎧⎪⎪⎨⎪⎪⎩

chord [t − ∣cs∣] .tonesrct2rst (chord [t − ∣cs∣] ,0) + SI (mod ∣S.tones∣)

A potential drawback of algorithm 20 is repetition: the chord sequencechord[0..∣cs∣) is transposed every

∣cs∣ bars and literally repeats every∣cs∣∗∣S.tones∣gcd(∣S.tones∣,SI)

bars. I introduce two ways of preventing boredom in this

respect. Section 2.7.1 introduceschord sequence modulations, section 2.7.2 discusses exploiting a choice

of chord sizes greater thanV .

2.7.1 Chord sequence modulations

The basic idea of a chord sequence modulation is to reinterpret the relative chord tones of an incomplete

chord in the canon, as relative chord tones of another chord,while holding the pitches of the incomplete

chord invariant. The incomplete chord is called thepivot chordof the chord sequence modulation. Consider

for example, the incomplete chord which contains only relative chord tones0,1 from ⟨{0,4,7} ,0⟩, these

may be reinterpreted as1,2 of ⟨{0,3,7} ,9⟩. Something similar to this idea is called achord shift degreeby

Gosman[Gos00b, p.155]. The main difference with Gosman’s definition is that chord sequence modulations

may occur anywhere in the canon where an incomplete chord occurs, whereas thechord shift degreeonly

occurs at the end of a canonic thread [Gos00b, p.122-210]). The main purpose of the chord shift degree

is to allow a shortened canonic thread to end in the same harmonic function, whereas chord sequence

modulations allow deviations from a chosen chord sequence.This allows us to view the dux of a harmonic

reduction of almost any stacked canon as an array of relativechord tones. A chord sequence modulation

may actually result in different pitches after the pivot chord. This is demonstrated later on in example 22.


Definition 21 A chord sequence modulationM is a triple (c, f, t), wherec is a Chord, f a member of

the symmetric groupS∣c.tones∣, 0 ≤ t < L + (V − 1) ∗ ∣cs∣, which transforms the stacked canonSC =(S,V,SI, ∣cs∣, dux, chord) into the stacked canonSC′ = (S,V,SI, ∣cs∣,CS, dux, chord′) such thatPMdefined below holds.

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∀t′, v′ ∶ 0 ≤ t′ < L + (V − 1) ∗ ∣cs∣ ∧ 0 ≤ v′ < V ∧ v′ ≤ t′ ∶

chord′[t′] =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

chord[t′] if t′ < t ∨ t′ ≠ t (mod ∣cs∣)⟨c.tones, c.root + SI ∗ t′−t

∣cs∣(mod ∣S.tones∣)⟩ if

t ≤ t′ ∧ t′ = t (mod ∣cs∣)∧

rct′[t′, v′] =⎧⎪⎪⎨⎪⎪⎩

rct[t′, v′] if t′ < t ∨ t′ ≠ t (mod ∣cs∣) ∨ rct[t′, v′] =<f[rct[t′, v′]] if t ≤ t′ ∧ t′ = t (mod ∣cs∣) ∧ rct[t′, v′] ≠<

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

The first conjunct of the consequent expresses thatM affects only the chords at timeclasst (mod ∣cs∣),from t onwards. The second conjunct expresses thatf permutes the relative chord tones at timet + k ∗ ∣cs∣.M is said to becanon preservingif it leavesPsc invariant. The chordc is a pivot chord if and only if

(∀v, t ∶ 0 ≤ v < V ∧ v ≤ t ∶ rct2rst (c, f [rct [t, v]]) = rct2rst (chord [t] , rct [t, v])) .

Example 22 Given below are the arraysrct, rct′ of relative chord tones corresponding to the canons

SC,SC′ respectively shown in figure 2.9. The chords of the sequence are given below each system. The

notation ‘e0’ denotes a diminished triad rooted at e, otherwise small letters denote minor triads, capitals

denote major triads.

rct :

2 1 0 0 1 2 0

2 1 0 0 1 2 0

2 1 0 0 1 2 0

rct’ :

2 1 1 1 2 0 1

2 1 0 1 2 0 1

2 1 0 0 2 0 1

SC is transformed intoSC′ by the canon preserving modulationM,

M = (⟨{0,3,7} ,2⟩ , f ∶ f (x) = x + 1 (mod 3),4) , c = ⟨{3,7} ,2⟩ .

The modulation is demarcated by a horizontal line inrct andrct′. Note thatc is a pivot chord.

Note that definition 21 requires the functionf of a chord sequence modulation(c, f, t) to be a permutation

of the set[0..CS). Although non-bijective functions could also produce canon preserving chord sequence

modulations, they may introduce parallel intervals of the class0 (mod 12). The following proposition

shows that bijective functions cannot.

Proposition 23 LetM = (c, f, t) be a chord sequence modulation which transforms the stackedcanon

SC = (S,V,SI, ∣cs∣,CS, rct, chord) into SC′ = (S,V,SI, ∣cs∣,CS, rct′ , chord′). ThenSC′ contains par-

allel intervals of the class0 (mod 12) if and only ifSC does.

Proof. Let the predicateP be defined as follows for0 ≤ v < v′ < V ∧ v′ ∗ ∣cs∣ ≤ t′ < (V − 1) ∗ ∣cs∣ − 1:

P ≡ rct[t′, v] = rct[t′, v′] ∧ rct[t′ + 1, v] = rct[t′ + 1, v′].


&

?

&

?

b

b

b

b

42

42

42

42

SC

SC'

∑

˙a

∑

˙a

∑

˙d

∑

˙d

˙˙G

˙˙G

˙˙C

˙˙C

˙˙F

˙˙

F~d

˙˙bB

˙˙g

˙˙e0

˙˙C

˙˙Óa

˙˙ÓF

˙

Ód

˙

ÓbB

Figure 2.9: Stacked canonsSC andSC′

P is true, if and only if parallel intervals of the class0 (mod 12) occurs between voicesv, v′ upon moving

from time t′ to t′ + 1. As f is a bijection,applyingf to the first, second or both conjuncts ofP leaves it

invariant. A similar argument can be made forP ′ defined below and the inverse function off :

P ′ ≡ rct′[t′, v] = rct′[t′, v′] ∧ rct′[t′ + 1, v] = rct′[t′ + 1, v′].

Algorithm 24 summarizes the preceding observations regarding chord sequencing. It produces a chord

sequencechord and relative chord tone sequencerct from the given duxrtc−1[0] while maintaining a

global permutation functionf[tc] for each time classtc, 0 ≤ tc < ∣cs∣. An advantage of this approach is

that for a modulation(c, f ′, t), we need only applyf ′ to f[tc] and subsequently storef[tc, rct[t, v]] for

0 ≤ v < V . The result is an algorithm with running timeO((L + (V − 1) ∗ ∣cs∣) ∗ (CS + V )) whereas the

alternative of applyingf ′ to rtc[t + k ∗ ∣cs∣] for 0 ≤ k ≤ ⌊L+(V −1)∗∣cs∣−1−t∣cs∣

⌋ would have a running time of

O ((L + (V − 1) ∗ ∣cs∣) ∗ (CS + V ∗L+(V −1)∗∣cs∣

∣cs∣)). What follows is a detailed description of algorithm

24 which comprises four initialization statements and a repetition. The first initialization statement setsf to

the identity ofSCS . The second initialization statement chooses an initial chord sequencechord[0..∣cs∣), to

be followed by the third initialization statement which initializesrtc according to the first step of algorithm

20. The variabletc always refers to the time class oft.

The repetition extends the chord sequence withchord[t] in every stroke. The first alternative of the

if-statement corresponds to a continuation of the existingchord sequence, i.e. no chord modulation. The

second alternative corresponds to a canon preserving chordsequence modulation(c, f ′, t)which affects the

chord at timest + k ∗ ∣cs∣.


Algorithm 24∣[const S ∶ Scale;

const CS ∶ [1.. ∣S.tones∣) ;const SI ∶ Z;

const dux ∶ [0..L)→ [0..CS);var f ∶ [0..L + (V − 1) ∗ ∣cs∣) × [0..CS)→ [0..CS);var f ′ ∶ [0..CS)→ [0..CS);var rct ∶ [0..L + (V − 1) ∗ ∣cs∣) × [0..V )→ [0..CS) ∪ {<};var chord ∶ [0..L + (V − 1) ∗ ∣cs∣) → Chord;

var t ∶ [0..L + (V − 1) ∗ ∣cs∣];var tc ∶ [0..∣cs∣);var c ∶ Chord;

for all i ∶ 0 ≤ i < ∣cs∣ ∧ 0 ≤ j < CS ∶ f[i, j] ∶= j;

for all i ∶ 0 ≤ i < ∣cs∣ ∶ chord[i] ∶ ∣chord[i].tones∣ = CS;

for all v, t ∶ 0 ≤ t < L ∧ 1 ≤ v < V ∶ rtc [t + v ∗ ∣cs∣, v] ∶= rtc [t,0] ;t, tc ∶= ∣cs∣,0;

do t ≠ L + (V − 1) ∗ ∣cs∣ →if true → chord[t] ∶= ⟨chord[t − ∣cs∣].tones, rst (chord[t − ∣cs∣],0, SI)⟩ ;[] true → c, f ∶

∣c.tones∣ = CS ∧Ð→f ∈ SCS ∧

⎛⎜⎜⎜⎝

∀v ∶ 1 ≤ v ≤min (V − 1, ⌊ t∣cs∣⌋) ∧ rct [t, v] ≠<∶

rst (⟨chord[t − ∣cs∣].tones, rst (chord[t − ∣cs∣],0, SI)⟩ , f [tc, rct [t, v]]) =rst (c, f ′ [f [rct [t, v]]])

⎞⎟⎟⎟⎠

for all i ∶ 0 ≤ i < CS ∶ f [tc, i] ∶= f ′ [f [tc, i]]f i;

for all v ∶ 0 ≤ v ≤min (V − 1, ⌊ t∣cs∣⌋) ∧ rct[t, v] ≠<∶

rct[t, v] ∶= f[tc, rct[t, v]];t, tc ∶= t + 1, tc + 1 (mod ∣cs∣)

od ]∣

2.7.2 Choosing chord sizes greater thanV

Melodic and harmonic variety in a stacked (spiral) canon canalso be obtained by choosing a chord sequence

which contains at least one chord with a size greater thanV . Let tc be its corresponding time-class

(mod ∣cs∣). By varying the subset of relative chord tones chosen at timeclasstc, variety in both harmony

and melody is indeed obtained, yet, the underlying chord sequence remains completely pre-determined.

This section sketches an approach towards the generation ofa longdux, which is the concatenation of

several shorter duces, such that:

• each of the shorter duces is based on some choice of subsets of relative chord tones for each time-

class.


• each one of the shorter duces, and their long concatenation, allows a stacked-spiral canon without

parallel octaves or chord tone doublings.

We first study how to obtain one of the shorter duces, after which the conditions for concatenating (several

of) such duces into the longerdux are derived. For the purpose of creating such a shortdux, let us use the

array RS for storing therelative chord tonesubset for each time-classtc (mod ∣cs∣). I.e.RS[tc] represents

our choices for relative chord tones at time-classtc. Also, let lcm(s) for some set of integerss equal the

least common multiple of its members.

Proposition 25 For all tc, RS, such that:

0 ≤ tc < ∣cs∣ ∧ ∣RS∣ = ∣cs∣ ∧ RS[tc] ⊆ [0..cs [tc]) ∧ V ≤ ∣RS[tc]∣,

there exists adux for a stacked spiral canon forV voices without parallel octaves or chord tone doublings,

such that:

∣dux∣ = ∣cs∣ ∗ lcm ( {∣RS [i]∣ ∣ 0 ≤ i < ∣cs∣} ) .

Proof. Algorithm 26, an adaptation of algorithm 8, can construct each of the duces according to the pream-

ble.

Algorithm 26

1. For each time-classtc (mod ∣cs∣):(a) choose a subsetRS[tc] of the relative chord tones[0..cs [tc]), such thatV ≤ ∣RS[tc]∣(b) choose a permutationπtc of RS[tc] corresponding to the desired cyclic shift of relative chord

tones at time classtc.

2. For eacht, tc, 0 ≤ t < ∣cs∣ ∗ lcm ( {∣RS [i]∣ ∣ 0 ≤ i < ∣cs∣} ) ∧ tc = t (mod ∣cs∣):

dux[t] ∶= πtc [ t/∣cs∣ (mod ∣RS [tc]∣) ] ,

where ‘/’ denotes integer division.

While I referred to adux created by algorithm 26 as ‘short’ in comparison to the concatenation of sev-

eral such duces, note that it can in fact be quite long if (several of) the different subsets chosen in step1a)have sizes which are relatively prime. Yet, the algorithm islinear in ∣dux∣, i.e. very efficient.

Let us now study the concatenation of duces derived with algorithm 26 such that no chord tone doublings

result from the longdux. A way of easing the prevention of chord tone doublings in theresulting canon,

is the insertion of rests between two short duces,duxi andduxi+1. Let the number of rests inserted equal

ri ∗ ∣cs∣. Also, let us superscript permutations chosen in step1a) of algorithm 26 by the corresponding dux’

subscript. The projection onto time-classtc of the concatenation (‘⊕’) of duxi, ri ∗ ∣cs∣ rests,duxi+1, has a

sublist of the structure given below, which generates chords in which relative chord tones of both permuta-

tionsπitc andπi+1

tc sound simultaneously. The firstV − 1 − ri entries correspond toπitc’s tail. Similarly, the

2.8. COUNTING DISTINCT SUB-CANONS OF A STACKED CANON 35

lastV − 1 − ri entries correspond toπi+1tc ’s head.

πitc [∣πi

tc∣ − (V − 1) + ri .. ∣πitc∣)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶V −1−ri

⊕<⊕ ⋅ ⋅ ⋅ ⊕ <

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ri

⊕ πi+1tc [0 .. V − 1 − ri)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

V −1−ri

A sliding window of sizeV in the above concatenation, corresponds to the succession of chords at time-

classtc (mod ∣cs∣) formed by both permutations. It follows that the concatenation introduces no chord tone

doublings, and hence no parallel octaves,if and only if for all time classestc, there exists no such window

containing any relative chord tone other than a rest more than once. Hence, successive permutationsπitc,

πi+1tc , . . . must be chosen accordingly in step1b) of algorithm 26 if so desired.

2.8 Counting distinct sub-canons of a stacked canon

Requirements to a dux used solely in a stacked canon may differ from requirements to a main-theme used

in more liberal polyphonic forms. A requirement more typical within the context of the latter application,

is the possibility of a large number of polyphonic combinations with itself, i.e.stretti. As pointed out in

section 2.1, one way of attaining this goal is to design a theme as the dux of a stacked canon for a relatively

large number of voices. In this section we count the number ofdistinct, non-trivial sub-canons of a stacked

canon forV voices. To this end we need a few definitions. In any polyphonic structure, we define inversion

analogous to inversion of chords. I.e. the root inversion isthe original structure, the first inversion is the

original structure with its lowest voice transposed up by the minimal number of octaves required for it to

start above the highest voice in the original stacked canon,et cetera.

Example 27 Reconsider Rameau’s canon at the fifth, shown in figure 2.1 on page 8. The canon’s harmoniola,

the polyphonic structure of bars[6,7], appears in third inversion, transposed up by the scale interval of the

stacked canon, in bar[8,9]. Indeed, the set of bars:

{[m,m + 1] ∣ 6 ≤m ∧m (mod 2) = 0} ,

is an equivalence class induced by the relation ‘equivalentmodulo transposition and inversion to the har-

moniola’.

We define any polyphonic (sub-)structure as trivial if it hasless than two voices. Two polyphonic struc-

tures forV voices areequivalent modulo the operations of transposition and inversion if there exists a

transposition and inversion which identifies them.

Proposition 28 The number of distinct, non-trivial,sub-canons modulo the operations of transposition and

inversion of a stacked canon forV voices,subc(V ), equals2V −1 − 1.

Proof. Let us label the entries in a stacked sub-canon by the number of entries preceding it in the original

stacked canon. By definition of equivalence modulo transposition and inversion, it follows that:

• for every sub-canon not involving voice0, there exists an equivalent sub-canon containing voice0.

Let Vmin be the smallest used in the sub-canon. The equivalent stacked canon is found by replacing

each voicev by v − Vmin;

• for every inverted sub-canon, there exists an equivalent sub-canon with its entries in ascending order.


I.e. it suffices to consider only sub-canons with all entriesin ascending order, containing voice0. Setting

the first entry of a sub-canon to voice0 of the original canon leavesV − 1 voices to in- or exclude. Of the

2V −1 different combinations of theV − 1 voices, exactly one results in a trivial sub-canon with justone

voice.

Corollary 29 Let subc(V,Vsubc) equal the number of distinctsub-canons forVsubc voices modulo trans-

position and inversion of a stacked canon forV voices.

subc(V,Vsubc) = ( V − 1Vsubc − 1

).

Proposition 28 shows that the number of distinct sub-canonsof a stacked canon grows exponentially in

V . Proposition 15 on page 22 shows that the maximum number of voices of a restless stacked canon

without parallel octaves is a quadratic function of the chosen chord sizes. Table 2.9 shows, for the chord

size patterns shown in its first column, the maximum number ofvoices of a restless stacked canon without

parallel octaves provided by proposition 15 in its second column, and, the corresponding number of distinct

non-trivial sub-canons in its third column. LetCS equal the repeated chord size in the first column, then the

corresponding value in the third column equals2CS2−1 − 1. I.e. the third column shows double exponential

growth in terms of the chosen chord sizes. Both the efficiencyof the algorithms for generating stacked

cs V = ∣A (D (cs) ,0)−∣ subc(V )(2)+ 4 7

(3)+ 9 255

(4)+ 16 32,767

(5)+ 25 16,777,215

Table 2.9: number of distinct sub-canons of a stacked canon without parallel octaves

canons developed in this chapter and the large number of sub-stretti a theme designed as a stacked canon

allows, underline the connection between stacked canons and the design of larger scale imitative polyphonic

structures.

Chapter 3

Spiral

Spiral is a multi-part composition for string orchestra. Its title refers to a geometrical shape such as shown in

figure 3.11 , which, emanating from a central point, moves progressively further away. The shape is endless

Figure 3.1: a spiral

and seemingly repetitive, yet ever so slightly changing, which concept is intuitively followed throughout

the composition. Spiral’s main thematic elements are discussed in section 3.1. An analysis of its individual

parts is provided in sections 3.2 and 3.3.

3.1 Thematic elements

The relative chord tone approach to the design of stacked canons developed in chapter 2 was applied to the

design of Spiral’s main themes. Their derivation and properties in sections 3.1.1 and 3.1.2.

3.1.1 theme0

Example 13 on page 19 briefly introduced Spiral’s main theme,theme0. An important requirement was

for it to enable many distinct stretti, in order to sustain a larger imitative polyphonic form. As discussed

in section 2.8, the design of a theme as the dux of a stacked canon is an efficient method enabling this.

Several other esthetic principals also played an importantrole in the derivation oftheme0. While I con-

sider parallel perfect fifths unproblematic in sufficientlydissonant contexts, they can make independence

1the 3D coordinates are determined by(1.08t ∗ sin (t),−t,1.08t cos (t)), 0 ≤ t < 15π

37

38 CHAPTER 3. SPIRAL

of voices difficult to achieve[Per61, p.203], especially inconsonant surroundings or two-voice sections.

Hence a design constraint fortheme0 was the absence of parallel fifths in its stretti. Furthermore, I wished

for the stretti based ontheme0 to be consonant for several reasons. Firstly, dissonance iseasily added but

not removed and consonant stretti therefore provide more freedom in the harmonic design of the composi-

tion. Secondly, dissonant chords in stretti are more difficult to manage in polyphonic contexts in which the

invertibility of chords is required when using double counterpoint at the octave. Indeed, dissonances sound

milder when chords are widely spaced across registers, the inversion of which results in narrow spacing and

thus sharper dissonance and possibly voice crossings. Another design constraint that requires little debate

was the absence of parallel octaves. The augmented triad rooted at pitch class0, repeated a semi-tone lower

on each successive repetition, was chosen to easily satisfymost of these constraints. As this chord sequence

excludes the occurrence of parallel perfect fifths or dissonances, the only constraint still to be accounted for

is the avoidance of parallel octaves.

The dux graph corresponding to the chosen chord sequence wasshown in figure 2.6 on page 20. The

set of edges leaving the one and only time-class has size9. Using proposition 15 we infer that adux for

a stacked canon without parallel octaves exists ifV ≤ 9. The fixture ofV to 6 balanced two opposing

constraints, namely, the possibility of many distinct sub-stretti on the one hand, and the artistic freedom of

choosing a melody suitable for a theme on the other hand. Using proposition 28 we find thattheme0 allows

for 31 distinct sub-stretti modulo the operations of transposition and inversionby design. Besides those, it

is probable that other stretti are possible too, which do notviolate any of the aforementioned constraints.

Indeed, the embellishment of a theme may explain dissonances occurring in a stretto as passing tones,

suspensions, et cetera. In combination with the aforementioned operations of transposition and inversion,

this certainly seems enough to base a multi-part composition on. Having chosen all parameters, the theme

was derived using algorithm 17.

3.1.2 theme1

Depending on the chosen embellishments,theme0 may feature a latent binary meter. As I wished for

theme1 to be a contrast totheme0, I constructed it such that its latent meter is ternary. Thiswas achieved

by choosing the relative chord tone sequence[2,2,2,1,1,1,0,0,0,2,2,2] in the chord sequence used for

derivingtheme0. The ternary meter is exploited in part 2 of Spiral where the theme is used in a passacaglia.

The unembellished version oftheme1 is shown in figure 3.2. Its underlying structure is best explained

& 22 ˙b ˙2 2

˙# ˙b2 1

˙ ˙1 1

˙# ˙n0 0

˙ ˙0 2

˙b ˙2 2

Figure 3.2: Spiral:theme1

using the relative chord tone difference pattern(0,0,−1 (mod 3))∗. After picking a relative-chord tone,

we repeat it twice (the zeroes), and then subtract one modulothree. Due to the augmented triad’s equidistant

division of the octave, we have that:

• the same melodic interval structure results from startingthe relative chord tone difference pattern

from either of the three possible relative chord tones;

3.2. PART 1 39

• theme1 is a sequence in classical music-theoretical terminology.By alternating ascending fifths and

descending fourths we obtain the impression of a longer theme.

The ten possible two-voice stretti at a time interval smaller than6, with a length-12 version oftheme1

are given in table 3.1 in relative chord tone notation. The relative chord tones above the horizontal line

corresponds to the first entry in the stretto, any of the relative chord sequences below the horizontal line

correspond to the second entry.

2 2 2 1 1 1 0 0 0 2 2 21 1 1 0 0 0 2 2 2 1 1 10 0 0 2 2 2 1 1 1 0 0 0

2 2 2 1 1 1 0 0 0 2 2 20 0 0 2 2 2 1 1 1 0 0 0

2 2 2 1 1 1 0 0 0 2 2 20 0 0 2 2 2 1 1 1 0 0 0

2 2 2 1 1 1 0 0 0 2 2 20 0 0 2 2 2 1 1 1 0 0 0

2 2 2 1 1 1 0 0 0 2 2 21 1 1 0 0 0 2 2 2 1 1 1

Table 3.1: Spiral: two-voice stretti withtheme1

3.2 Part 1

Part 1 of Spiral is based on two stacked spiral canons, both based on the fragment shown in figure 3.3.

The themes used are labeled as follows:theme0: Cb:[91..95];theme4: Vc:[90..95];theme3: Va:[91..94];

theme2: Vln. II:[91..95]; theme1: Vln. I:[91..95]. Measures[91..95] are the harmoniola of an ascend-

ing stacked spiral canon withTI = 10 beats,V = 5 andSI = 8 semitones. The different lengths of the

various themes were chosen to obfuscate the inherent blockwise structure provided by the harmoniola. An

overview of the canonic structure is shown in table 3.2. The table was split in three in order to fit on one

page. Column labels correspond to measure numbers, row labels to voices. Its entries correspond to the

subscripts of the different themes. The first stacked spiralcanon starts in measure 1 and is based on the first

four themes only. The second stacked spiral canon starts in bar 41 with the entrance of the double basses.

The third stacked spiral canon is a recurrence of the previous four-voice canon and starts in bar91. The

fourth and final stacked spiral canon is a recurrence of the previous five-voice canon and starts in bar111.

In each canon, each occurrence of the dux (Vc:[1..20]) is followed by an instance transposed four semi-

tones lower. The four-voice spiral canon is descending, thefive-voice canon ascends by four semitones.

theme4 is embellished in three different ways for the sake of variety with first respective occurrences in

Vc:[[40..45] , [64..69] , [110..115]]. Part 1 ends with a stretto oftheme0, with SI = 8 semitones and

TI = 4 beats. I.e. a time-compression of the entries oftheme0 in the preceding structure.

3.3 Part 2

Part 2 of Spiral features a passacaglia structure enforced by a seemingly endless sequence oftheme1. Start-

ing in Va:1,theme1 is repeatedly transposed downwards. It is handed over to Vc:13, Cb:25, Vln. I:37, Vln.


&

&

B

?

?

Vln. I

Vln. II

Vla.

Vc.

Cb.

90 œ œn ≤≤≤

˙b ˙b

œ ˙b œn

≥ ˙dolcissimo

˙ Ó

f

f

f

f

f

Œ ˙#≥ œnC

œ œn ˙n ≤

w≥

œ œ œb œ

≤ ˙bsonore

˙# ˙#

˙ ˙b

.˙b œ

œ œb œ œb

œ œ œ œ

œ œn ˙

œ œn ˙

w

˙ œn ≤ œ

.˙b œ

Œ ˙#≥≥ œn

˙ ˙#

.˙# Œ

œ œ œ# œ#

≤ ˙n≥

˙ ˙

œ ˙n œ#

˙#≥ ˙#dolcissimo

œ œ# œ# Œ

œ œ# ˙# ≤

Figure 3.3: Spiral part 1: harmoniola

II:49, Va:65, Vc: 74, Cb:83, Vln. I:103, Vln. II:119, Va:125, Vc:141, Cb:151, Vln. I:155. Superimposed on

this is an A-B-A’-B’ structure with many overlapping combinations of themes 0,1 and 2. The bar numbers

on which the respective sections are 1, 42, 73 and 118. Beforediscussing the individual sections, we require

a few definitions to more easily describe combinations of themes.

A triple such as⟨theme0, t, p⟩ refers to the occurrence oftheme0 with ontimet (bars or beats, depend-

ing on the context), such thatp equals the pitch class of its first note. A set of such triples is referred to as a

combination. Let the first member of a combination be a memberwith the smallest ontimet, note that there

may be a choice in this appointment. Members of a combinationcan bereduced modulo transposition and

inversionby calculating their transposed ontime and pitch-class by the same transposition used in transpos-

ing the first entry to ontime0 and pitch-class0. The reduced members of a combination are given between

normal brackets, e.g.(theme0,∆b,∆p). If no confusion arises, we may refer to a theme by its subscript

instead of its full name, e.g.(0,∆b,∆p).

All combinations occurring in part 2 are without parallel octaves and dissonances are typically followed

by their resolution. Part 2 nearly exhausts all of the combinations oftheme1 derived from table 3.1 on page

39. Also, several combinations oftheme1 andtheme0 occur, with two overlapping entries oftheme1.

This combinatorial abundance not only hides the rigourous structure of the passacaglia, it also provides for

contrasting movement, meter and volatility in comparison to part 1.

3.3. PART 2 41

m. 1 6 11 16 21 26 31 36 41 46 51 56 61Vln I 0 1 2 3 0 1 2 3 4 0Vln II 0 1 2 3 0 1 2 3 4 0 1Va 0 1 2 3 0 1 2 3 4 0 1 2Vc 0 1 2 3 0 1 2 3 4 0 1 2 3Cb 0 1 2 3 4

m. 66 71 76 81 86 91 96 101 106 111 116 121 126Vln I 1 2 3 4 0 1 2 3 0 1 2 3 4Vln II 2 3 4 0 1 2 3 0 1 2 3 4 0Va 3 4 0 1 2 3 0 1 2 3 4 0 1Vc 4 0 1 2 3 0 1 2 3 4 0 1 2Cb 0 1 2 3 4 0 1 2 3

m. 131 136 141 146 151 156 161 166 171 176 181Vln I 0 1 2 3 4 0 1 2 3 4 0Vln II 1 2 3 4 0 1 2 3 4 0 1Va 2 3 4 0 1 2 3 4 0 1 2Vc 3 4 0 1 2 3 4 0 1 2 3Cb 4 0 1 2 3 4 0 1 2 3

Table 3.2: canonic structure of part 1

3.3.1 Section A

Section A exposes several new ideas based on the themes introduced in Part 1. The section opens with yet

unheard combinations of the form{(0,0,0) , (1,0,8) , (0,6,10)}, where theme0 is often accompanied by

theme2 in the same relation as in part 1. Note that the pitches of the second half of(1,0,8) equal those of

the first halve of(1,6,6), the regular counterpoint to(0,6,10) throughout part 1, which explains its con-

formance to the passacaglia structure. Using this relationthe stretti are chained with entrances oftheme0

in measures Cb:1, Vln. II: 3, Vln. I: 5, Vc: 7, Vln. II 9. A divertimento based ontheme0’s tail spans

measures[12..27)mainly based ontheme1’s tail andtheme3.

From measure 27 onwards there is a gradual buildup to a first climax starting from measure 37 onwards.

Entries oftheme0 in Vln I: 27, Va: 30 and Vln. II:33 are chained in the relation{(0,0,0) , (0,9,3)}. Note

again that similar to the previous stretti, the pitches of the fourth quarter of(1,0,0) correspond to the first

quarter of(1,3,0). Themes 2 and 3 accompanytheme0 as throughout part 1. This creates accentuates

the ternary meter and creates a slight compression of the preceding structure which, together with the

expansion of the register and stretti withtheme1 starting from measure 33, creates a drive towards measure

37. The stretti withtheme1 starting in measure 33 are all based on the stacked spiral canon with parameters

TI = 1, SI = 7, V = 4, dux = theme1. First a two-voice sub-canon starts with entries in Cb:33, Vc:33 13,

followed by a three-voice sub-canon with entries in Cb:35, Vc:35 13, Va:35 2

3, followed by the full spiral

canon with entries in Vln I:37, Vln II: 37 13, Va:35 2

3, Vc: 38 (handed over to Cb:40).

3.3.2 Section B

Figure 3.4 shows the possible reduced two-voice combinations of theme0 at time distance 5, over the

passacaglia structure. Each vertex is labeled in the formvi ∶ (0,∆t,∆p), where the latter triple indicates

an entry of a combination, andvi is a label for easy reference to a vertex. The ontimes have been reduced


modulo20, the time- distance in beats between any two consecutive entries of(0,0,0), if the dotted arc is

used. Each arc⟨vi, vj⟩ from vertexvi to vj is labeled(∆t,∆p), where∆t denotes the time-component

(mod 20) of the transposition that mapsvi ontovj and∆p its pitch-component modulo 12. A path through

t = 0 (mod 20) t = 5 (mod 20)

t = 10 (mod 20)

t = 15 (mod 20)

(5,7)

(5,11)

(5,11)

(5,11)

(5,7)

(5,3)

(5,11)

(5,3)

v1 ∶ (0,5,11)v0 ∶ (0,0,0)

v2 ∶ (0,10,10)

v3 ∶ (0,10,6)

v4 ∶ (0,10,2)

v5 ∶ (0,15,1)

Figure 3.4: Spiral part 2 section B: stretti withtheme0

the graph corresponds to a chain of two-voice stretti withtheme0 at time-distance5. Three different paths

are possible, modulo the operations of transposition and inversion, based on the three choices for a vertex

at time class10 (mod 20). Each one of those three paths is used exactly once in sectionB, all starting at

v0:

1. the first four occurring entries oftheme0 starting with Vc:42 correspond to vertex path

v0 → v1 → v4 → v5;

2. The first four occurring entries oftheme0 starting with Vln. I:51 correspond to vertex path

v0 → v1 → v3 → v5;

3. The first four occurring entries oftheme0 starting with Cb:60 correspond to vertex path

v0 → v1 → v2 → v5;

The last entry in each of the above paths is extended in the score. This has several functions. First, it

creates a short divertimento which is slightly contrastingto the preceding stretti. Second, it allows for an

overlap with the next path for the first two paths. The prolongation at the end of the last path is most

pronounced and used to return to the calmer character of section A’. Third, the prolongations allow the

entries oftheme0 to start at mutually different pitch classes. Indeed, section B was designed such that

exactly one entry oftheme0 occurs starting on each of the twleve pitch classes, which adds considerable

tension to an already hectic structure. In addition to the aforementioned combinations containing stretti

of theme0 over the passacaglia structure (theme1), extra instances oftheme1 nearly always generate

four- to five-voice textures. This idea gains independence from measure 67 onwards using the four-voice

combination{(1,0,0) , (1,1,3) , (1,2,10) , (1,3,1)}.

3.4. PART 3 43

3.3.3 Section A’

Section A’ opens with the same combinations used in the beginning of section A. The divertimento of

measures[12..27) however returns sooner than in sectionA and is interrupted by chained stretti with

theme2 of the form{(2,0,0) , (2,3,5)} with entries in Vln. I:81, Vln. II:82, Va:83, Vln. I:84, Vln.II:85.

After recapturing the original idea of the divertimento in measure 87, the buildup to the second climax

starts in measure 91. The buildup proceeds similar to the onein section A’, with the (in the current context)

unexpected change of the additional entry oftheme0 in Vln I: 99. Note that this last stretto corresponds to

the stretti used in the beginnings of section A and A’. The extra entry expands the buildup and register even

further, yet compresses the the expected delay between the entries oftheme0 in Va:97 and Vln. I: 99. This

is used to strengthen the appearance of the second climax from measure 103 onwards. Two preceding ideas

are combined here: the bass-line of measure 67 onwards with the stacked canon of measure 37 onwards.

3.3.4 Section B’

The embellished arpeggio of the diminished seventh chord which first appeared in Cb:[49..51], it is a se-

quence oftheme0’s embellished tail initially used for small divertimento as discussed in section 3.3.2. In

section B’, this small device joins the passacaglia in the sense that it is continuously transported through

all voices aliketheme1, with entries in Vc:103, Cb:111, Vln. I:123, Vln II:134, Va:141, Vc:155, Cb:159.

Combined with the now two-voice passacaglia structure, areseveral stretti oftheme0 at time distances of

3,4 and 5.

3.4 Part 3

Part 3 of Spiral is an abbreviated mirror of part 1, in the sense that all melodic material and main structural

features are mirrored chromatically. The first violins are initially divided into left- and right sections to

avoid an extremely high registry for the lower string sections. The respective abbreviations for the first

violin sections are Vln. Ir:1, Vln. Il, other string sections are abbreviated as in section 3.2. The canonic

structure of part 3 is shown in table 3.3 which has a structuresimilar to table 3.2 and explained further

below. Note that all theme entries in table 3.3 refer to the chromatic inversion of the equally numbered

theme. The initial four-voice canon of part 1 is mirrored with entries in Vln. Ir:1, Vln. Il: 6, Vln. 2: 11,

Va: 16. Contrary to the initial canon in part 1, the canon now moves upward, reaching the leading tone

b3 in Vln. 1r:30,36. The downward motion which continues untilthe end of the part commences with the

mirror of the first five-voice canon of part 1 in Vln. 1r: 39. Conceptually connected to the aforementioned

top-tone, a scale descends throughout the string orchestrastarting from b♭3. The pickup of the scale in

Cb.:64 is used to return to a tutti first violin part in bar 80, coinciding with the mirrored entry of the last

embellishment oftheme4 of part 1.

Whereas part 1 cycles through the minor keys of c,a♭ and e, part 2 cycles through the major keys of

C,E and A♭. I.e. besides a conversion ofTongeschlecht2, modulations are in contrary motion. The virtual

mirror axis between pitch classes 3 and 4 is used. The relation between a pitch classpc and its mirrored

counterpartpc−1 can be expressed throughpc−1 = 4+(3−pc) (mod 12) = 7−pc (mod 12). Note that this

relation fixates the tonic triad I modulo Tongeschlecht, similar to the fixation of I in d-minor in the mirror

2In German (or Dutch), the ‘gender’ of a traditional mode is either ‘major’ or ‘minor’[Sch40, p212], referring to the third of themode under consideration.


m. 1 6 11 16 21 26 31 36 41Vln I 0 1 2 3 0 1 2 3 4Vln Il 0 1 2 3 0 1 2 3Vln II 0 1 2 3 0 1 2Va 0 1 2 3 0 1Vc 0Cb

m. 46 51 56 61 66 71 76 81 86Vln I 0 1 2 3 4 0 1 2Vln Il 4 0 1 2 3 4 0 1 2Vln II 3 4 0 1 2 3 4 0 1Va 2 3 4 0 1 2 3 4 0Vc 1 2 3 4 0 1 2 3 4Cb 4 0 1 2 3

Table 3.3: canonic structure of part 3

fugues of Bach’s ‘Die Kunst der Fuge’[Bac87]. Indeed, the mirror axis used in Spiral is the chromatic

adaptation to c modulo Tongeschlecht, of the diatonic mirror axis of the third of the d-minor scale between

the mirror fugues of ‘Die Kunst der Fuge’. I conjecture that the use of a chromatic mirror axis can mirror

the affects experienced by an audience. Indications for this are summarized in table 3.4. Its first two rows

were explained above. As for the last row, note that a helicopter view of the registral expansion in part 1

shows upward motion whereas downward motion can be observedin part 3. Within the greater structure of

property part 1 part 3main key c Ckey pattern (c,a♭,e)∗ (C,E,A♭)∗registral expansion ↱ ↳

Table 3.4: Contrasts between part 1 and 3

Spiral, the chromatic mirror between parts 1 and 3 is used forexpressing opposite intentions. Part 1 affirms

the tonic of c-minor and steers upward to the climactic second halve of part 2, after which part 3 relaxes the

tension while functioning as a Picardian close to the work.

Chapter 4

Fugue in G

This chapter discusses the composition ‘Fugue in G’ for Organ, hereafter referred to as ‘the fugue’. Section

4.1 discusses the design of its theme and countersubject, section 4.2 discusses the structure of the fugue.

4.1 Thematic elements

This section discusses the design of the fugue’s main theme and countersubject in subsections 4.1.1 and

4.1.2 respectively.

4.1.1 Main theme

A requirement for the main theme was the possibility of a fourvoice stretto in which only chords without

doublings occur, similar to Rameau’s canon at the fifth, discussed in section 2.4 on page 2.4. The relative

chord tone approach was applied to the design of stretto in order to meet this requirement. The expanded

chord sequence chosen is shown in the top staff of figure 4.1, its time intervalTI equals3, the scale interval

SI equals7 semitones. The numbers below the chords correspond to the time classes of the corresponding

chords. Except for the omission of relative chord tone4 on time class1 (mod 3), this chord sequence is

equivalent modulo transposition to the Mass’ chord sequence described in section 5.2. The omission results

in a chord sequence in which all three chords of the chord sequence have sizeCS = 4. This allows us

to strength the aforementioned requirement to the requirement of a stretto in which only complete chords

sound. As discussed in section 2.4.2, this boils down to finding a Hamiltonian cycle in the corresponding

dux graphD([4,4,4]). Using algorithm 8 on page 15, the list[1,3,2,0] was chosen as the successive

relative chord tones for each time class. Among the many possible interleavings of three rotated instances

of this list, I chose[1,2,3,0,1,2,3,0,1,2,3,0] as the relative chord tone structure for theunembellished

main-theme,themeu. The relation between[1,3,2,0] andthemeu is shown in table 4.1. Indeed,themeu

is the concatenation of its rows. The three columns represent the three chords in relative chord tone notation

on the three time classes0, 1 and2 in the stretto. Note that the chord at time classi (mod 3) equals the

chord at time class0 (mod 3) rotatedi positions downward. Figure 4.1 showsthemeu on its second staff.

The numbers below this staff correspond to the relative chord tones. The embellished version of the second

staff,theme, is shown in the third staff.

45

46 CHAPTER 4. FUGUE IN G

1 2 30 1 23 0 12 3 0

Table 4.1: relation betweenthemeu and[1,0,3,2]

&

&

&

&

21

21

21

21

∑

∑

Œ œ œ#

˙b

˙0

˙1

œ. œ.

.œb jœ

˙1

˙2

œ œ# œ œ

œb . œ

˙#2

3

˙

œ œ œ

˙#0

˙0

Œ œ

œb . œn .

˙#1

˙1

œ œ.

˙

˙#2

˙2

œ. œ# .

.œ jœ#

˙##0

3

˙

œ. œ#

˙##1

0

œ œ œ#

œ œ# œ#

˙##2

˙1

.œ jœ

œ. œ.

˙##0

˙#2

˙#

˙

˙##1

˙#3

˙

∑

˙###2

˙#0

∑

∑

Figure 4.1: Design of theme and countersubject for ‘Fugue inG’

4.1.2 Countersubject

The countersubject, shown in the bottom stave of figure 4.1, was designed with the following requirements

in mind:

1. It should contrast with the main theme, both melodically and rhythmically. Although modulating,

the main theme clearly has a diatonic structure. The countersubject on the other hand features a

diminished fourth, and its first five beats are repeated one semitone higher. While the main theme’s

meter leaves room for interpretation when heard unisono, its stretti are clearly grouped in3/2 bars as

a result ofTI = 3 (e.g. see[138..170]). The countersubject’s repetitive structure on the other hand

suggests a5/2 bar. The resulting combination of theme and counterpoint thus steers clear from the

latent classical harmony and meter of the main theme.

2. It should allow for several combinations both with (variations of) itself and the main theme. Such

combinations allow for metrical variation, as will become apparent in the sequel. Because of this,

I chose to notate the fugue in a1/2 bar. Instead of the harmonic relative chord tone approach,

the countersubject was designed as a melody with a Sekundgang[Hin70]. This allows for different

interpretations of its notes as either embellishment or chord tone.

The first five bars of the countersubject as displayed in figure4.1 are hereafter referred to ascs0.

4.2. STRUCTURE 47

4.2 Structure

The fugue has a ternary form with sections A, B and A’ respectively discussed in detail in sections 4.2.1,

4.2.2 and 4.2.3. Section A introduces all of the material used in the fugue, section B represents the develop-

ment, A’ the modified reprisal of section A. Section A ends with a plagal cadence on V of G. The dominant

reappears as V7 with added blue note in556.5, followed by a pedal point on V43. It finally resolves to

I in 610. Subsections of A,B and A’ are all marked in the score, table 4.2 provides an overview of the

corresponding bars for easy reference.

section subsection barA A1 1

A2 80A3 138

B B1 170B2 264B3 401

A’ A1’ 474A2’ 496A3’ 565

Table 4.2: (sub-)sections of ‘Fugue in G’

Before discussing each section in detail, we need some notational conventions. Again, a triple such as

⟨cs0, b, p⟩ refers to the translation ofcs0 to barb, such thatp equals the pitch class of its first note. A set

of such triples is referred to as a combination. Let the first member of a combination be a triple with the

smallest bar number, note that it need not be unique. A combination can be reduced modulo transposition

by subtracting from each member’s bar, the bar number of the first member, and from each member’s pitch

class, the pitch class of the first member modulo12. The reduced members of a combination are given

between normal brackets, e.g.(cs0,∆b,∆p).

4.2.1 section A

Section A comprises three subsections A1, A2 and A3, each discussed below.

subsection A1

The exposition of the fugue introduces the main theme in ascending fashion through B:[1..11], T:[11..23],A:[25..35], S:[35..47]. Note that the entrances in the tenor and soprano features all of the relative chord

tones as initially designed, hence the extra length of two beats. The reasons for this are as follows. First,

the entrances in the tenor and soprano are a fourth higher than their respective preceding shorter entries.

This allowed for the shorter theme and countersubject to overlap in e.g. B:11. Second, the extra two bars

22,23 overlap with the start of a small divertimento which allows the entry of the main theme in A:25 at

g1. Finally, the shortened theme and countersubject have equal lengths. This provides a continuation of the

suggested5/2 meter of the countersubject in T:[25..35), by the countersubject in A:[35..45).The exposition’s ending overlaps with a stretto of the (altered) countersubject, part of a divertimento

leading up to the theme entry in S:57 in the dominant. The overlap provides continuity and is created both

by the sequence in which A:⟨cs0,35,8⟩ is transposed by(5,1), and also by the sequential movement in


T:[42..49] according to the pitch pattern(+1,−2)3. The stretto is given below. Note that the translation

from the first to the second member equals the translation from the fourth to the fifth.

{B ∶ ⟨cs1,44,0⟩ ,A ∶ ⟨cs0,45,10⟩ ,B ∶ ⟨cs1,48,8⟩ , T ∶ ⟨cs0,49,1⟩ ,A ∶ ⟨cs0,50,11⟩}

The tail ofcs0 is used in a downward sequence in[54..58), relaxing the time compression resulting from

the stretto. What follows are the last occurrences of the combination of theme and countersubject as shown

in the two bottom staves of figure 4.1, until the reprisal in bar 474. The two entrances referred to are

S ∶ [57..67] and B:[70..80). The different pitch-relation between these entrances avoids the last entry

starting ong. This is achieved by the sequential treatment of the theme’stail in S:[65..70], the last note

again overlapping withS ∶ (cs0,70,1) as in the exposition. The sequential treatment is exploitedin the

reprisal to create an enjambment, a point to which I return insection 4.2.2.

subsection A2

Measures[79 − 113] give the sequential treatment of inverted alternations of two contrasting combinations

through the use of double counterpoint:

• Let cp refer to the embellished chromatic scale fragment in e.g. B:[81..84], adapted from B:[52..58].Combination{(cs1,0,0) , (cs1,1,10) , (cp,2,2)} occurs first with respective entries in T:79, S:80,

B:81 and, through sequential treatment, this occurrence is translated by(4,10), suggesting4/2 bars.

The entire structure reoccurs transposed, in second inversion, with first respective entries in A:94,

B:95, S:96.

• Let cst refer to the countersubject’s tail, e.g. T:[87..89]. The following combination occurs first with

respective entries in T:87, S:88, B:88:

{(cst,0,0) , (cst,1,10) , (cs1,1,0)} .

Through sequential treatment, this occurrence is translated by(3,4) such that the first and last note of

the respective second and first occurrence ofcp1 in this sequential structure overlap and3/2 bars are

suggested. The entire sequential structure reoccurs transposed, in second inversion, with respective

entries in A:102, B:103, S:103. Note how the last note ofcp overlaps with the first note ofcs1 in

S:103. The sequence is expanded this time to four instances of the combination in order to reach a

higher register. The fourth instance is altered inS ∶ 113 to make room for a new combination.

Measures[113..129] are a final buildup towards the climax in130. A new combination is given, comprising

a sequential canon of the head of the main theme withTI = 1 andPI = −5, andcs. In its first instance, the

canon occurs between soprano and alto. The second inversionof this combination starts in119. Instead of

just giving the head, the bass now gives the entire theme. Itstail is imitated by the alto through translation

(2,3). Upon reaching the local top tone a2, a canon at minimum withcs1’s head collapses the climax,

accompanied by transpositions of the main theme’s tail. Measures[129..137] are a prolongation of the

dominant seventh chord rooted at g. It should be understood as an altered IV7 in D, part of the plagal

cadence reaching I (with added sixth) in138.

4.2. STRUCTURE 49

subsection A3

SectionA3 gives the first four-voice stretto with the main theme. It is astacked spiral canon withTI = 3

andPI = 7. Two instances of the main theme are given in each voice at interval4.

4.2.2 section B

Section B represents the development phase of the fugue. It is divided into three subsections, each discussed

separately below.

subsection B1

Subsection B1 is based upon the alternation of two contrasting ideas, each based upon a unique contrapuntal

combination. The first being a combination of the main theme and cs1, the second being a stretto ofcs0,

whence it functions as a divertimento:

1. Combination{(theme,0,0) , (cs1,1,8) , (cs1,2,7) , (cs1,5,7) , (cs1,6,6)} occurs first with respec-

tive entries in A:170, S:171, B:172, S:175, B:176. In its second occurrence marked by T:179,

down a major second modulo12, the soprano and tenor have exchanged voices. Note that the first

two members of this combination have the same relation as thetheme has with the entire coun-

tersubject, all other pairs of this combination are new. Also new is the small overlap between

the entries of the main theme. Indeed, the stretto in subsection A3 featured the sub-combination

{(theme,0,0), (theme,9,9)} instead of{(theme,0,0), (theme,9,10)}. Measures[170..189] re-

turn transposed down a major second in[209..228], the main difference being that the second main

theme now occurs in the soprano through voice exchange. Subsection B1 was designed such that the

first and last entries of the main theme depart from pitch class0 (mod 12).

2. The second idea is based on a stacked spiral canon in whichcs0 is continuously transposed upward a

minor second as is the case with the entire countersubject. The canon’s parameters areV = 3, TI = 2,

PI = 7. It occurs first in[189..205], with a modified return in[230..264). The first occurrence’s

entries are B:189, S:191, T:193. The second occurrence initially starts as the first except for a change

of voices. The initial entries are literally stacked downwards with A:230, T:232, T:234. However,

the unique part of the dux is expanded by two new measures to A:[235..241]. The extra two bars

were designed to allow a fourth voice in an altered spiral canon, namely S:[241..247]. The new dux

is embellished differently from A:248 onwards, similar to the rhythmical variation ofcs1 presented

first in S:171.

subsection B2

Subsection B2 introduces a new idea mostly derived from the countersubject, clearly visible in T:[262..268].The pitch pattern(−1,+4) present in the countersubject, is developed into a combination of a two voice

canon with parametersTI = 1 andPI = 7 (mod 12), accompanied bytheme in later instances. Measures

[304..319] provide a clear instance of this combination with the canon between the outer voices andtheme

in the alto. As the canon’sdux ascends an octave, its melody is sometimes spread over two voices. In bar

276, the soprano is handed over the dux from the alto. Likewise, the alto is handed over the comes from the

tenor exactly one beat later which allows the original canonto recommence in S:280 and A:281 transposed


by (16,9 (mod 12)). This provides a huge upward thrust because of a canon which seemingly spirals a

upward from e1 ♭ in A:264 to b2 in S:292 which is harmonically challenging because of the interpretation

of the+4 skip of the aforementioned pitch pattern as a diminished fourth. The larger combination of the

two canons transposed by(16,9 (mod 12)), each canon possibly accompanied by a theme, occurs twice in

[264..296] and[346..378]. The first exposition of the canon is the only one not accompanied by the main

theme theme. The simpler combination of a single canon and main theme occurs once in[304..319]. The

three aforementioned groups of bars are separated and followed by divertimenti as discussed below.

The first divertimento in this subsection, bars[296..304], is built upon a short canon withPI = 6

(mod 12) based upon the mirrored head of the aforementioned dux in thetwo upper voices. This canon

is a prolongation of the diminished seventh chord{0,3,6,9}, which ‘resolves’ to a b-minor seventh chord

in 304. The canon is accompanied by the main theme’s head in[301..304], also an embellished downward

minor third skip.

The second divertimento in this subsection, bars[320..345], starts with a short spiral canon based on

cs0 between soprano and tenor. Note that the transposition fromdux to comes equals the one used in the

canon between the outer voices starting in S:171 and B:172. The difference in comparison to the latter

combination are the use ofcs0 instead ofcs1, and, the different transposition of thedux in each voice.

Indeed, the transposition from S:320 to S:325 equals(5,−2), whereas the transposition from e.g. S:171 to

S:175 equals(4,−1). The three lower voices of bars[329..344] are a copy of the same three voices of bars

[249..264].The aforementioned canon based upon the mirrored head of theaforementioned dux returns in the third

divertimento of this subsection in307. This time however, it is accompanied by an entire theme withan ex-

tra rest of one beat inserted between its head and tail: B:[377..388]. The tail’s first five beats are imitated a

minor seventh higher in T:[384..388]while the alto continues its sequential treatment of the inverted canon.

Measures[387..401) push upward to the local top-tone a2 in S:401. Having reached b2 already in S:320,

it is rather a failed attempt at a real climax. Its underlyingstructure is clearly visible in bars[393..394].The bass imitates the soprano’s alteration of the canon’s head withTI = 1,PI = −16. Note that the combi-

nations{(S ∶ [395..397],0,0) , (A ∶ [395..397],0,6)}, {(B ∶ [397..399],0,0) , (T ∶ [397..399],0,6)} are

equivalent modulo embellishment. I.e. the structure was designed as a double spiral canon.

subsection B3

Subsection B3 first develops, then gradually returns to the material of bars[44..57]. Initially this is done

through the alternation of two combinations:

• Combination{(cs1,0,0) , (cs0,1,10)} from respectively B:44, A:45 is altered to

{(cs1,0,0) , (cs1,1,10)} in respectively A:400, S:401. Let C0 be the latter combination. LetCe0 be

the combination ofC0 with a copy of itself transposed by(3,1 (mod 12)), e.g. A:400, S:401, B:403,

T:404.

• A:[412..428] is the rhythmically altered transposition of A:[45..60], fully imitated in the soprano

with TI = 1, PI = 10. Let this combination beC1. C1’s dux is only partly imitated in the exposition

with T:[49..56] and A:[50..57].

Subsection B3 opens with the sequential treatment ofCe0 in which every last two entrances of an instance

of Ce0 equal the first two entrances of the next instance ofCe0 . This exposes the new combination clearly

4.2. STRUCTURE 51

while collapsing the registry downward. The sequential treatment is halted by an instance ofC1 with entries

in A:412, S:413. This instance overlaps with the return of bars[409..426] in bar 427, transposed up a

second, accompanied by an alteration of B:[48..58] in B:[434..442]. Instead of another overlap withCe0 ,

the two lower voices now continue with the sequential treatment ofcs0. The counterpoint of B:[434..440]now receives sequential treatment with entries in A:449, S:454 and the entry ofcs0 in A:459. Measures

[459..471) are a slightly modified copy of[45..57), transposed up one major second whereas A1’ marks

the reprisal of bar57 at its original pitch. The structure was designed to create an enjambment which

destabilizes the reprisal marked by the theme entry in S:473 through the following devices:

• The pitch relation between bars[45..57) and the theme entry at S:57, differs from the relation between

[459..471) and the theme entry at S:473;

• Instead of imitatingcs0’s tail as in A:[55..56], the alto recites an expanded version of e.g.[247..254],creating an association with subsection B1;

• The idea of a sequential expansion moving all pitches down one major second of[472..473], is

repeated in[484..485];

• The slight deceleration in485 and continuation a tempo at486.

4.2.3 section A’

Section A’ represents the modified reprisal of section A. Itsinitial bars were already discussed in the previ-

ous subsection.

subsection A2’

Subsection A2’ seemingly starts as the reprisal of subsection A2 at its original pitch. The sequential treat-

ment ofcs1’s head however makes the tenor comes and soprano dux. The second inversion of bars [79..87]

transposed one major second down thus results in bars[496..504]. The entry ofcst in S:504 suggests the

soprano’s leadership in a canon with the tenor. Measures505 however marks the reprisal of bar89 at its

original inversion, one semitone lower. Measures[511..512] represent an expansion of the original se-

quence, moving the pitch an extra major third upward. As the sequence started out one minor second lower,

the result is that the reprisal continues a minor third higher all the way up to bar548. This, in combination

with the below devices, expands the climax in the reprisal:

• The imitation of B:[542..548] now all occurs in all three upper voices with entries in A:544, T:546

and S:548 in a stacked canon withTI = 2 andPI = 3 (mod 12). This pitch interval creates an

enhanced tonal tension.

• The material from bars[129..137 now appear twice: in first inversion in the upper voices of bars

[552..556) and original inversion in the bars following it.

• The soprano reaches the absolute top-tone c3 at V7 with added blue note, followed by a pedal point

starting on V43 in G-major.


subsection A3’

Section A3’ is the downward stacked version of the stretto ofsection A3. After twelve entries of the main

theme, each on a different pitch class, the modified four entries finally reach the tonic through a contrapuntal

cadence in bar610.

Chapter 5

Constraint logic programming stacked

canonic structures

5.1 Introduction

Use cases

Previous chapters introduced the theory regarding the design of stacked canons using relative chord tones

and some of its applications. Requirements to a theme or stacked canon’s dux were stated in the form of

predicates (constraints) in first order logic. The general method for obtaining a solution to such constraints

was to derive an algorithm through logical and mathematicalreasoning[Dij76, Kal90], which produces

the desired solutions. In some scenarios this approach may not be the most efficient expenditure of a

composer’s time. In the preliminary stages of a composition, different configurations or rules for a theme or

dux under development may be examined. The chord sequence, time distances, constraints, or number of

voices are subject to frequent change, resulting in the repeated derivation of different algorithms. Another

scenario is the presence of many constraints. If satisfiable, the derivation of an algorithm may be tedious. If

unsatisfiable, a change in requirements is forced upon the composer as in the first scenario. In yet another

scenario, a composer may not posses the skills required to derive an algorithm for the creation of solutions.

Finally, a composer may wish to manually evaluate all solutions to a set of constraints, in order to select one

according to yet unspecified artistic requirements. The technique ofconstraint logic programmingoffers

an alternative method of generating solutions in such scenarios. All that is required of the composer is the

formulation of constraints in aconstraint logic program(CLP), which requires, besides music theoretical

knowledge, some elementary predicate logic such as taught in a typical first-year computer science or

mathematics class[NK04].

Pitch class domains versus relative chord tone domains

A CLP can be solved automatically by means of a constraint solver run on a computer. An important factor

in the success of such an approach is the size of the CLP’ssearch space: the cross product of the constrained

variables’ domains. While modern constraint solvers may effectively prune the search space for some CLPs,

solving a CLP is generally NP-hard. Hence, in a worst case scenario, a constraint solver may need to eval-

53

54 CHAPTER 5. CONSTRAINT LOGIC PROGRAMMING STACKED CANONIC STRUCTURES

uate each of the constrained variables’ possible values. Depending on the problem at hand, such searches

may be intractable even by modern computers. The problem of finding adux for a stacked canon can be

encoded using constraints in which∣dux∣ variables must be assigned a value by the constraint solver.The

search space size whendux is an array of relative chord tones,(∏ t ∶ 0 ≤ t < ∣dux∣ ∶ cs [t (mod ∣cs∣)] ),is typically smaller than the search space size of a specification within the domain of pitch-classes,12∣dux∣.

Example 30 Consider the search space for a CLP which generates Spiral’smain theme discussed in section

3.1.1 on page 37. Fromcs = [3] and ∣dux∣ = 10 it follows that the search space size when using relative

chord tone domains equals310 = 59049, whereas the search space size when using pitch-class domains

equals1210 - more than one million times larger.

Outline

This chapter discusses a computer assisted composition process for stacked canonic structures, using my

composition ‘Missa ad Fugam’ as an example. The tradition ofdemonstrating canon techniques throughout

the musical setting of the Ordinarium Missae which inspiredme to write the ‘Missa ad Fugam’ was estab-

lished by the Franco-Flemish schools of the Renaissance era. Ockeghem’s ‘Missa prolationum’[Par76] is

perhaps the first of all masses based completely on the principle of progressive canon through all move-

ments [SG01]. Other famous examples include compositions such as des Prez’ ‘Missa sine nomine’[JD91]

and Palestrina’s ‘Missa ad Fugam’ [dPS83]. The B-Prolog finite domain constraint solver[Zho10] used in

the aforementioned process was chosen mainly for its speed [Zho06], intuitive syntax, built-in constructs,

and its ability to generate all solutions to a constraint logic program. An outline of the process is shown in

figure 5.1 and explained further below. After choosing parameters for the desired canonic structure(s), such

formulateconstraints

chooseparametersstart

compileto CLP

runconstraint

solver

goodresultfound?

composeno

yes

Figure 5.1: Composition assisted composition process followed in composition of ‘Missa ad Fugam’

as the number of voices, chord-sizes, et cetera, constraints are initially formulated in first order logic using

relative chord tone domains, and subsequently compiled into a quantifier-free B-Prolog constraint logic

program (CLP). I performed the compilation to B-Prolog using a Mathematica[Wol03] notebook, though

other scripting languages can also be used for this purpose.After instructing the constraint solver to search

for solutions to the CLP, the solutions are evaluated by the composer. If solutions are unsatisfactory (or

no solutions are found), the process followed thus far is repeated with changed parameters or constraints.

Otherwise, a satisfactory solution is available for use in the composer’s (manual) composition process.

5.2. ESTHETIC CONSIDERATIONS 55

The application of constraint logic programming to musicalCLPs is extensively surveyed by Pachet,

Roy [PR01] and Anders [And07, pp.35-64]. Its application tothe construction of stacked canons is however

new. After a discussion in section 5.2 of the esthetic principals which influenced the design of the mass,

section 5.3 formulates the constraints on its thematic material. Analytical notes of the mass are given in

section 5.4.

5.2 Esthetic considerations

Besides determining the Ordinarium’s text, the Catholic church expressed its view on the esthetics and role

of sacred music in several publications. Pope Benedict XVI’s ‘The spirit of the liturgy’[Rat00, pp.136-156]

provides a current and instructive (over)view in this regard. What follows is an illustrative summary with

selective quotations.

With the invention of polyphony in the late Middle Ages and Renaissance, artistic invention in both

music and the liturgy starts to assert its rights. “Church music and secular music are now influenced by the

other. This is particularly clear in the case of so-called ‘parody Masses’, in which the text of the Mass was

set to a theme or melody that came from secular music”... “It is clear that these opportunities for artistic

creativity and the adoption of secular tunes brought dangerwith them”... “Music was alienating the liturgy

from its true nature”[Rat00, pp.145-146].

The Council of Trent (1545-1563) decreed that “music shouldbe at the service of the Word; the use

of instruments was substantially reduced; and the difference between secular and sacred music was clearly

affirmed”. During the Baroque and Classical eras, signs of “the dangers to come”in the Romantic era started

to appear. The Romantic era re-introduced the “dangers thatforced the council of Trent to intervene”, ex-

emplified by “self-emancipating subjectivity, virtuoso mentality” and the “vanity of technique”. In an effort

to remove the “operatic”, Pope Pius X declared Gregorian chant and Renaissance polyphony (exemplified

by Palestrina) to be the standard for liturgical music in a Motu Proprio [X03] promulgated in 1903[Rat00,

p.146].

Modernity brought new challenges to sacred music. There is the cultural universalization that the church

is facing while moving outside the Western European boundaries. Regarding this and recent developments

in music the Pope notes that “modern so-called ‘classical’ music has maneuvered itself, with some excep-

tions, into an elitist ghetto, which only specialists may enter- and they do so with what may sometimes be

mixed feelings. The music of the masses has broken loose fromthis and treads a very different path”....

“Pop music”... “ultimately has to be described as a cult of the banal”... “Rock”...“is the expression of ele-

mental passions, as it assumes”... “a form of worship, in fact, opposite to Christian worship.” Crowds are

“released from themselves by the experience of being part ofa crowd and by the emotional shock of rhythm,

noise, and special lighting effects”[Rat00, pp.147-148] The Pope offers no music theoretical solutions to

the apparent dilemma modern composers face in creating sacred music. He notes that renewal should come

from within and offers a summation of some of the esthetic andreligious principles that emerged instead:

1. The music of Christian worship is related to Logos. There must be a clear dominance of the Word.

“That is why singing in the liturgy has priority over instrumental music...”[Rat00, p.149]

2. “There is always an ultimate sobriety, a deeper rationality, resisting any decline into irrationality and

immoderation.”[Rat00, p.150]


3. ‘ ‘The more that human music adapts itself to the musical laws of the universe, the more beautiful it

will be.”[Rat00, p.156] The Pope refers to Pythagorean music in this regard. But what are the music

theoretical implications of this in the present day: does the pope refer to tonality? A definite an-

swer cannot be given but an indication may be derived from hischaracterization of modern classical

music and his closing argument. The Pope notes that our own times are governed by “‘deconstruc-

tionism’ - the anarchistic theory of art. Perhaps this will help us overcome the unbounded inflation

of subjectivity...”[Rat00, p.155] “What in museums is onlya monument from the past, an occasion

for mere nostalgic admiration, is constantly made present in the liturgy in all its freshness... The

artists who take this task upon themselves need not regard themselves as the rearguard of culture.

They are wary of the empty freedom from which they have emerged. Humble submission to what

goes before us releases authentic freedom and leads us to thetrue summit of our vocation as human

beings.”[Rat00, p.155-156] Tonality in some form is indeedwhat goes before us for the greater part

of recorded musical history across most cultures1. The days that the postmen sing Webern’s melodies

on their rounds [MM79, p.543] have yet to come even upon the western world, which suggests a

limited suitability of the associated idioms for a universalizing church.

With the above considerations in mind, I strived for an idiomwith clear ties to Renaissance polyphony,

is tonal, unaccompanied, and characterized by sobriety andmoderation. The Missa ad Fugam is based

entirely on three themes. The basis of the entire mass is a chord sequence of three chords with chord-sizes

cs = [4,5,4], two instances of which are shown in figure 5.2. The second instance (the last three bars) is the

transposition bySI = 7 semitones (mod 12) of the first instance (the first three bars). The scale interval

SI is used for all canons in the Mass. Note that for each dissonant in the chord sequence, its step-wise

downward resolution occurs in a following chord ifSI = 7. Using this chord sequence, the three themesA,

& C wwww wwwwwbwwwwb wwww wwwww

wwww

Figure 5.2: Chord sequence of Missa ad Fugam

B andC are specified in CLPs which partly reflect my interpretation of the above esthetic principles. Each

theme allows a four-voice stacked spiral canon, each pair ofthemes allows a four-voice double stacked

spiral canon, and both three-and four-voice combinations featuring all themes are possible.

5.3 Constraint specifications

This section formulates all constraints which shaped the mass’ three themesA, B andC in first order

logic using relative chord tone domains. As discussed in section 5.1, constraints were compiled into a

quantifier-free B-Prolog 7.1 CLP using a Mathematica notebook, though other scripting languages could

easily be used for the same purpose. Sections 5.3.1 and 5.3.2respectively introduce constants and auxiliary

functions used in the subsequent sections. Sections 5.3.3,5.3.4, 5.3.5 and 5.3.6 respectively formulate

generic boolean functions to constrain similarity within or between themes, the doubling of chord tones, the

resolution of dissonances, and, the occurrence of parallelperfect intervals. Actual constraints on themesA,

1A discussion of tonality can be found in [Scr97, pp 293-308] and a counterpoint in [Den09]

5.3. CONSTRAINT SPECIFICATIONS 57

B andC are respectively formulated in terms of the aforementionedgeneric boolean functions in sections

5.3.7, 5.3.8 and 5.3.9.

5.3.1 Constants

As mentioned in section 5.2, the three themesA, B andC occurring in the mass are designed to occur

in several (double-) stacked spiral canons for at mostV = 4 voices. Our chord sequence discussed in

section 5.2 haschord-sizescs = [4,5,4]. While superfluous in this chapter, the constantTI = ∣cs∣ which

denotes thetime-interval between succesive entries of each configuration, was a useful shorthand in my

Mathematica notebook. Each one of the aforementioned spiraling structures specified in this chapter, is

referred to by a theme configuration: an array containing itsthemes such as[A,B,A,B], in the order in

which they enter the polyphonic structure. Each respectivetheme is itself an array of relative chord tones

of the samelengthL. Our choice forL = 12 is clarified in section 5.3.4. The theme configurations we are

concerned with in this chapter are best explained by their conceptual relative chord tone structure. Table 5.1

shows the resulting relative chord tone structures of two configurations occurring in the Gloria in terms of

its respective theme variables. The top row denotes the timeclasses (mod L) of the spiraling structures.

Below it are the resulting four-voice relative chord tone structures ofconfigurationsC1 = [A,B,A,B] and

C2 = [B,B,B,B], separated by a blank row. Each voicev, 0 ≤ v < ∣C∣ ≤ V , partaking in a configurationCenters the resulting structure afterv ∗ ∣cs∣ rests, to repeat the relative chord tonesC[v]. Note that individual

array variables such asA[0], A[1], . . . , are compiled into the respective B-Prolog variablesA00, A01,. . .

by my Mathematica notebook. Hence our left-padding with a zero, of array indices smaller than10 in tables

such as5.1 in this chapter. This chapterconstrainsonly ascendingstacked spiraling structures based on

0 1 2 3 4 5 6 7 8 9 10 11B[03] B[04] B[05] B[06] B[07] B[08] B[09] B[10] B[11] B[00] B[01] B[02]A[06] A[07] A[08] A[09] A[10] A[11] A[00] A[01] A[02] A[03] A[04] A[05]B[09] B[10] B[11] B[00] B[01] B[02] B[03] B[04] B[05] B[06] B[07] B[08]A[00] A[01] A[02] A[03] A[04] A[05] A[06] A[07] A[08] A[09] A[10] A[11]

B[03] B[04] B[05] B[06] B[07] B[08] B[09] B[10] B[11] B[00] B[01] B[02]B[06] B[07] B[08] B[09] B[10] B[11] B[00] B[01] B[02] B[03] B[04] B[05]B[09] B[10] B[11] B[00] B[01] B[02] B[03] B[04] B[05] B[06] B[07] B[08]B[00] B[01] B[02] B[03] B[04] B[05] B[06] B[07] B[08] B[09] B[10] B[11]

Table 5.1: Conceptual relative chord tone structures ofC1(top) andC2(bottom)

our three themes, in the sense that in a constraint, voice0 is always considered the lowest voice partaking

in a configuration, voice1 the one immediately above it, et cetera. Note however that this convention does

not preclude configurations from being used in descending stacked spiraling structures. Hence, within the

scope of a constraint, configurationC1 refers to the ascending double-stacked spiral canon in which the alto

imitates the bass after2∗ ∣cs∣ time units, and, the soprano imitates the alto also after3∗ ∣cs∣ time units, such

that the alto enters∣cs∣ time units after the bass. After the setup phase (the firstL time-units), the resulting

relative chord tone structure shown in figure 5.1 repeats at each positive multiple ofL. ConfigurationC2refers to the ascending stacked spiral canon with themeB as its dux.


5.3.2 Auxiliary functions

The previous definitions and conventions allow us to refer tothe relative chord tone a voicev has at timet

in configurationC by the auxiliary functionvoice(C, v, t), defined below forv ≤ ⌊t/∣cs∣⌋ ↓ (∣C∣ − 1):

voice (C, v, t) = C [v, t − v ∗ ∣cs∣ (mod L)] .

Similarly, the auxiliary functionchord(C, t) returns an array of the relative chord tones sounding simul-

taneously at timet in configurationC, 0 ≤ t, such thatchord (C, t) [v] equals the relative chord tone

corresponding to voicev:

chord (C, t) = [ v ∶ 0 ≤ v ≤ ⌊t/ ∣cs∣⌋ ↓ (∣C∣ − 1) ∶ voice(C, v, t) ] .

Some constraints are more easily expressed by converting anarray such as returned by our functionchord

to a set, thus filtering out any duplicates occurring in an array. The functionasSet(a) given below returns

the set of the arraya’s entries:

asSet (a) = {a[i] ∣ 0 ≤ i < ∣a∣} .

The domains of all themes in a configurationC are constrained by the predicatebpDomains(C):

bpDomains (C) ≡ (⋀ v, l ∶ 0 ≤ v < ∣C∣ ∧ 0 ≤ l < L ∶ 0 ≤ C[v, l] < cs [l (mod ∣cs∣)] ) .

The above auxiliary functions are used in subsequent sections for the specification of harmonic- or voice-

leading constraints on the relative chord tone structure resulting from a configuration as boolean functions

of this configuration. These boolean functions form a littlefunction library which is subsequently used

for the specification of constraints for particular configurations such as[A,B,A,B]: the conjunction of a

particular subset of boolean functions applied to this configuration. Using this approach we can quickly

try a different subset of constraints for a particular configuration, which is useful e.g. if the original subset

proves to be unsatisfiable.

5.3.3 Preventing similarity

When specifying multiple themes, we may wish to avoid solutions in which a pair of different themes such

asA, B, are similar, after a rotation byk ∗ ∣cs∣ of one of those themes for0 ≤ k < ⌊L/ ∣cs∣⌋. Specifically,

the predicatebpDissimilar (T0, T1) requires that no pair of length-∣cs∣ sub-arrays of the respective cyclic

arraysT0,T1 exist which start at time-class0 (mod ∣cs) and have all relative chord tones in common.

bpDissimilar (T0, T1) ≡⎛⎝⋀ k0, k1 ∶ 0 ≤ k0 < L

∣cs∣∧ 0 ≤ k1 < L

∣cs∣

∶ ( ⋁ tc ∶ 0 ≤ tc < ∣cs∣ ∶ T0 [k0 ∗ ∣cs∣ + tc] ≠ T1 [k1 ∗ ∣cs∣ + tc])⎞⎠ .

Another similarity we may wish to avoid, is an overly repetitive relative chord tone structure within a theme

itself. The predicatebpNoneRepetitive(T ) specified below, ensures that the circular arrayT has no sub-


arrays of length three in which all three relative chord tones are equal.

bpNoneRepetitive (T ) ≡( ⋀ t ∶ 0 ≤ t < L − 1 ∧ T [t] = T [t + 1] ∶ T [t + 1] ≠ T [t + 2 (mod ∣L∣)])

5.3.4 Preventing chord tone doubling

The predicatebpNoDoublings (C) defined below, enforces that no chord sounding in configurationC has

a doubling of chord tones. This not only results in rich harmonies throughout the configuration, it also

facilitates the regulation of dissonance as discussed in the next section.

bpNoDoublings (C) ≡ (⋀ t, c ∶ L ≤ t < 2L ∧ c = chord(C, t) ∶ ∣asSet(c)∣ = ∣c∣ )

The choices forV , cs, L and the constraintbpNoDoublings are quite intertwined. Indeed,V was set to the

minimum chord-size occurring incs as any larger value necessarily results in relative chord tone doublings

in aV -voice relative-chord tone structure. As for our choice ofL, recall from proposition 25 on page 34 that

usingcs = [4,5,4], duces exists for a stacked spiral canon without parallel octaves or chord tone doublings

for L = ∣cs∣ ∗ lcm ([4,4,4]) = 12 or L = ∣cs∣ ∗ lcm ([4,5,4]) = 60. The choiceL = 12 balances sufficient

length with recognizability as a theme.

5.3.5 Regulation of dissonance

The main principal followed regarding the occurrence of dissonances in the specification of each con-

figuration is thatif a dissonant occurs at timet, then it must resolve in some voice at timet + 1. The

following sections respectively discuss the translation of this principle into constraints on the four-, three-

and two-voice sections of a configuration. The latter three of these sections occur in the setup phase of a

configuration, the first occurs only in the spiraling phase.

Quads

As mentioned in section 5.2, the scale-interval and chord sequence for the Mass were chosen such that the

stepwise downward resolution of a dissonant is contained inthe chord immediately following it. Hence,

our choice ofcs andbpNoDoublings ensure the resolution of all dissonances in quads, except for those

occurring at time class0 (mod ∣cs∣) asV < cs[1] = 5. The predicatebpRequiredRCT (C, tc, rct) is used

in the sequel to require the presence of relative chord tone0 in quads at time class1 (mod ∣cs∣), i.e. the

resolution of relative chord tone3 at time class0 (mod ∣cs∣).

bpRequiredRCT (C, tc, rct) ≡ (⋀ t, c ∶ L ≤ t < 2L ∧ c = chord(C, t) ∶ rct ∈ c )

Triads

The situation concerning the resolution of dissonances in chords within the time interval[2 ∗ ∣cs∣ ..3 ∗ ∣cs∣ − 1)is slightly more complicated, with each successive chord, atriad, being incomplete. From our chord se-

quence it follows that a dissonant occurs between two voicesif their relative chord tone distance exceeds

two. The auxiliary functiondispairs(t) uses this observation in returning a set of sorted arrays representing

dissonant relative-chord tone pairs at timet. With our chord sequence, its return values are{[0,3] , [0,4]}


for time classes0,2 (mod ∣cs∣), joined with {[1,4]} at time class1 (mod ∣cs∣). The auxiliary func-

tion resolution(t, dp) calculates the relative chord tone which must be present at time t + 1, should

the pair of dissonant relative-chord tones of the arraydp sound simultaneously at timet. The shorthand

condition ? valueIfT rue ∶ valueIfFalse was borrowed from the C++ or Java programming languages.

disPairs(t) = {[i, j] ∣ 0 ≤ i ∧ i + 3 ≤ j < cs [t (mod ∣cs∣)] }resolution(t, dp) = dp[1] − (t (mod ∣cs∣) = 0 ? 3 ∶ 2)

The auxiliary predicatedissonancesResolve(C, t) uses the above two functions in specifying that any

dissonant occurring at timet in configurationC, must resolve at timet + 1. The predicate

bpInitialDissonancesResolve(C) in turn requires this explicitly for each triad which is followed by a

triad.

dissonancesResolve(C, t) ≡⎛⎝⋀dp ∶ dp ∈ disPairs(t) ∧ asSet(dp) ⊆ asSet (chord (C, t))

∶ resolution(t, dp) ∈ asSet (chord (C, t + 1))⎞⎠

bpInitialDissonancesResolve(C) ≡(⋀ t ∶ 2 ∗ ∣cs∣ ≤ t < 3 ∗ ∣cs∣ − 1 ∶ dissonancesResolve(C, t) )

Dyads

In order to obtain a tranquil beginning of a part of the Mass, we may require its initial two-voice section

at time interval[∣cs∣ ..2∗ ∣cs∣) to be consonant. The auxiliary predicatebpConsonant(C, t) specifies the ab-

sence of dissonance at timet in configurationC. UsingbpConsonant, the predicatebpConsonantDyads(C)enforces consonant dyads in configurationC.

bpConsonant(C, t) ≡( ⋀ v0, v1, c ∶ c = chord(C, t) ∧ 0 ≤ v0 < v1 < ∣c∣ ∶ ∣c[v1] − c[v0]∣ ≤ 2 )

bpConsonantDyads(C) ≡ ( ⋀ t ∶ ∣cs∣ ≤ t < 2 ∗ ∣cs∣ ∶ bpConsonant(C, t, false) )

5.3.6 Regulation of parallel perfect intervals

While I consider parallel pitch-intervals of the class7 (mod 12) acceptable in sufficiently dissonant con-

texts, their occurrence in consonant surroundings may diminish the independence of voices. The predicate

noParallel(C, t, int) expresses that no two voices may move in parallel relative chord tone intervalint

from time t to t + 1 in configurationC. Using the fact that these pitch-intervals correspond to a relative

chord tone distance of two in our chord sequence,bpNoInitialParallelF ifths(C) expresses that these


pitch intervals must not occur between dyads or triads.

noParallel(C, t, int) ≡⎛⎜⎜⎜⎝⋀ v0, v1, ct, ct+1 ∶ ct = chord(C, t) ∧ 0 ≤ v0 < v1 < ∣ct∣

∧ ct+1 = chord(C, t + 1)∶ ct[v1] − ct[v0] ≠ int ∨ ct+1[v1] − ct+1[v0] ≠ int

⎞⎟⎟⎟⎠bpNoInitialParallelF ifths(C) ≡ (⋀ t ∶ ∣cs∣ ≤ t < 2 ∗ ∣cs∣ − 1 ∶ noParallel(C, t,2) )

As far as parallel pitch-intervals of the class0 (mod 12) are concerned, note that these are already excluded

by bpNoDoublings(C).


5.3.7 ThemeA

This section discusses the specification of a CLP constraining themeA and its solutions.

CLP specification

The CLP for themeA is specified for configurationC0 = [A,A,A,A]: a four-voice stacked spiral canon

with themeA as its dux. The conceptual relative chord tone structure of configurationC0 is shown in table

5.2. The numbers above the horizontal line are the time classes (mod L). The constraint on this configu-

0 1 2 3 4 5 6 7 8 9 10 11A[03] A[04] A[05] A[06] A[07] A[08] A[09] A[10] A[11] A[00] A[01] A[02]A[06] A[07] A[08] A[09] A[10] A[11] A[00] A[01] A[02] A[03] A[04] A[05]A[09] A[10] A[11] A[00] A[01] A[02] A[03] A[04] A[05] A[06] A[07] A[08]A[00] A[01] A[02] A[03] A[04] A[05] A[06] A[07] A[08] A[09] A[10] A[11]

Table 5.2: Conceptual relative chord tone structure of configurationC0 = [A,A,A,A]

ration,themeA, is given below. My notebook compilesthemeA into the B-Prolog CLPthemeA.pl , the

full source code of which is given in appendix A.1.

themeA ≡ bpDomains (C0) ∧ bpNoDoublings (C0)∧ bpNoneRepetitive (A) ∧ bpRequiredRCT (C0,1,0)∧ bpConsonantDyads (C0) ∧ bpInitialDissonancesResolve (C0)∧ bpNoInitialParallelF ifths (C0)

Solutions

Enteringcl(‘themeA‘) into the B-Prolog interpreter, writes all9357 solutions to the filethemeA.txt

within one second on a 2GHz dual core laptop with 1GB of RAM.

Out of all solutions,[3,4,3,2,3,2,1,0,1,0,2,0] was chosen as follows. The solutions were first im-

ported into a Mathematica notebook in which I also defined functions for filtering out solutions which con-

tained repetitions of relative chord tones, or did not contain relative chord tone4 at time-class1 (mod 3).A final selection was based on my manual and subjective assessment of the melodic quality of the theme,

and the parallel thirds provided by the sequence[3,4,3,2,3,2] which absorb parallel dissonances. The

remaining solutions were discarded.

An embellished version of themeA is shown in the top stave of figure 5.3. The numbers below it

correspond to the relative chord tones used, the letter ‘e’ stands for embellishment. The last ‘3’ in measure

13 corresponds to the first note of the transposed repetitionof themeA: the theme repeats with a syncopized

first note. The bottom stave corresponds to the chord sequence introduced in section 5.1. The relative chord

tone structure resulting from configurationC0 and our choice of themeA, is shown in table 5.3.


&

?

C

C

w3

wwww

˙ ˙#4 e

wwwwwb

w3wwwwb

Ó ˙2

wwww

w3

wwwww

˙ ˙e 2

wwww

˙# ˙e 1wwww

&

?

8 .˙ œ0 e

8 wwwww

w1wwww#

.˙ œ#0 e

wwww#

w2

wwwww#

˙ œ# œe 0 e

wwww#

˙ , Ó3

∑Figure 5.3: An embellished form of themeA

0 1 2 3 4 5 6 7 8 9 10 112 3 2 1 0 1 0 2 0 3 4 31 0 1 0 2 0 3 4 3 2 3 20 2 0 3 4 3 2 3 2 1 0 13 4 3 2 3 2 1 0 1 0 2 0

Table 5.3: Relative chord tone structure of configurationC0 = [A,A,A,A]

5.3.8 ThemeB

This section discusses the specification of a CLP constraining themeB and its solutions.

CLP specification

The CLP for themeB is specified for configurationsC1 = [A,B,A,B] andC2 = [B,B,B,B]. Both

configurations and their respective relative chord tone structures were discussed in section 5.3.1 on page

57. The constraint on these configurations,themeB, is given below. My notebook compilesthemeB into

the B-Prolog CLPthemeB.pl , the full source code of which is given in appendix A.2.

themeB ≡ bpDomains (C1) ∧ bpDissimilar (A,B)∧ bpNoneRepetitive (B) ∧ bpNoDoublings (C1)∧ bpNoDoublings (C2) ∧ bpRequiredRCT (C1,1,0)∧ bpRequiredRCT (C2,1,0) ∧ bpConsonantDyads (C1)∧ bpInitialDissonancesResolve (C1) ∧ bpNoInitialParallelF ifths (C1)

Solutions

Enteringcl(‘themeB‘) into the B-Prolog interpreter, writes all25 solutions to the filethemeB.txt

within one second on a 2GHz dual core laptop with 1GB of RAM.The following solution was chosen for

themeB: [3,1,1,0,2,2,1,0,3,2,3,0]. Factors in its choice were:


• its distance to themeA. We define the distance between two themesT0,T1, distance (T0, T1) as:

(↓ r ∶ 0 ≤ r < L

∣cs∣ ∶ [∑ ∶ i ∶ 0 ≤ i < L ∶ T0 [i] = T1 [(i + r ∗ ∣cs∣) mod L] ? 0 ∶ 1])The greater the distance of two themes, the more dissimilar they are. The maximum distance between

themeA and any of the solutions found was 6, among 12 solutions.

• the absence of relative chord tone 4 at time-class 1(mod 3). This further increases the dissimilarity

with themeA and leaves only 2 solutions.

An embellished version of themeB is shown in the top stave of figure 5.4. The numbers below it again

correspond to the relative chord tones used, the letter ‘e’ stands for embellishment. The bottom stave again

corresponds to the chord sequence introduced in section 5.1. The resulting relative chord tone structures of

&

?

C

C

1 w3

1 wwww

˙ œ œ1 e e

wwwwwb

.˙ œ1wwwwb

.˙ œ0 e

wwww

˙ œ œ2 e e

wwwww

.˙ œ2

wwww

&

?

7

.˙ Œ1

7 wwww

Ó ˙4wwwww

.˙ œ3 ewwww#

˙ ˙#e 2

wwww#

œ œ ˙3 e e

wwwww#

˙ ˙#e 0

wwww#Figure 5.4: An embellished form of themeB

configurationsC1(top) andC2(bottom) are shown in table 5.4. The numbers above the horizontal line again

indicate the time classes[0..12) of the potentially infinite structure.

0 1 2 3 4 5 6 7 8 9 10 110 2 2 1 0 3 2 3 0 3 1 11 0 1 0 2 0 3 4 3 2 3 22 3 0 3 1 1 0 2 2 1 0 33 4 3 2 3 2 1 0 1 0 2 0

0 2 2 1 0 3 2 3 0 3 1 11 0 3 2 3 0 3 1 1 0 2 22 3 0 3 1 1 0 2 2 1 0 33 1 1 0 2 2 1 0 3 2 3 0

Table 5.4: Relative chord tone structures ofC1(top) andC2(bottom)


5.3.9 ThemeC

This section discusses the specification of a CLP constraining themeC and its solutions.

CLP specification

The CLP for themeC is specified for configurationsC3 = [C,A,C,A], C4 = [C,B,C,B], C5 = [C,C,C,C]andC6 = [C,A,B]. The conceptual relative chord tone structures of these configurations are shown in table

5.5. The numbers above the horizontal line again indicate the time classes[0..12) of the potentially infinite

structures. Below the horizontal line are the four structures in the order introduced above, separated by

empty rows. Our constraint on all of these configurations,themeC, is given below. My notebook compiles

0 1 2 3 4 5 6 7 8 9 10 11A[03] A[04] A[05] A[06] A[07] A[08] A[09] A[10] A[11] A[00] A[01] A[02]C[06] C[07] C[08] C[09] C[10] C[11] C[00] C[01] C[02] C[03] C[04] C[05]A[09] A[10] A[11] A[00] A[01] A[02] A[03] A[04] A[05] A[06] A[07] A[08]C[00] C[01] C[02] C[03] C[04] C[05] C[06] C[07] C[08] C[09] C[10] C[11]

B[03] B[04] B[05] B[06] B[07] B[08] B[09] B[10] B[11] B[00] B[01] B[02]C[06] C[07] C[08] C[09] C[10] C[11] C[00] C[01] C[02] C[03] C[04] C[05]B[09] B[10] B[11] B[00] B[01] B[02] B[03] B[04] B[05] B[06] B[07] B[08]C[00] C[01] C[02] C[03] C[04] C[05] C[06] C[07] C[08] C[09] C[10] C[11]

C[03] C[04] C[05] C[06] C[07] C[08] C[09] C[10] C[11] C[00] C[01] C[02]C[06] C[07] C[08] C[09] C[10] C[11] C[00] C[01] C[02] C[03] C[04] C[05]C[09] C[10] C[11] C[00] C[01] C[02] C[03] C[04] C[05] C[06] C[07] C[08]C[00] C[01] C[02] C[03] C[04] C[05] C[06] C[07] C[08] C[09] C[10] C[11]

B[06] B[07] B[08] B[09] B[10] B[11] B[00] B[01] B[02] B[03] B[04] B[05]A[09] A[10] A[11] A[00] A[01] A[02] A[03] A[04] A[05] A[06] A[07] A[08]C[00] C[01] C[02] C[03] C[04] C[05] C[06] C[07] C[08] C[09] C[10] C[11]

Table 5.5: Conceptual relative chord tone structures ofC3(top),C4(2nd), C5(3rd) andC6(bottom)

themeC into the B-Prolog CLPthemeC.pl , the full source code of which is given in appendix A.3.

themeC ≡ bpDomains (C6) ∧ bpDissimilar (A,C)∧ bpNoneRepetitive (C) ∧ bpNoDoublings (C3)∧ bpNoDoublings (C4) ∧ bpNoDoublings (C5)∧ bpRequiredRCT (C3,1,0) ∧ bpRequiredRCT (C4,1,0)∧ bpRequiredRCT (C5,1,0) ∧ bpNoParallelOctaves (C6)

The strengthening ofthemeC with bpNoDoublings (C6) resulted in an unsatisfiable CLP. Hence the sep-

arate conjunctbpNoParallelOctaves (C6) defined below.

bpNoParallelOctaves(C) ≡ (⋀ t ∶ ∣cs∣ ≤ t < 2 ∗ ∣cs∣ − 1 ∶ noParallel(C, t,0) )Solutions

The constraint programthemeC.pl given in Appendix A.3 has 84 solutions. An adaptation of solution

[3,4,1,0,2,0,1,0,3,2,3,2] was chosen for themeC. The main reason for this choice are the first seven


measures which are referred to as themeC1 in the sequel. They allow for an embellishment which resultsin

a repetition of measures[0,1] in measures[6,7] which creates a theme which is more static and stable than

themes A and B and hence a contrast. An embellished version ofthemeC as used in the Credo is shown

in the top stave of figure 5.5. The numbers below it again correspond to the relative chord tones used, the

letter ‘e’ stands for embellishment.

&

?

C

C

1 w3

1 wwww

˙ œb œ4 e e

wwwwwb

˙ ˙1wwwwb

˙ ˙n0 e

wwww

w2

wwwww

˙ œ œe 0 e

wwww

&

?

7

.˙ Œ1

7 wwww

˙ ˙0 ewwwww

.˙ œ3wwww#

œ œ ˙#3 e 2

wwww#

w1

wwwww#

∑

wwww#Figure 5.5: An embellished form of themeC

5.4 Analytical notes

This section provides analytical notes for the individual movements of my ‘Missa ad Fugam’. In reference

to Renaissance music, the score contains no dynamics: theseshould be moderate and in line with the mood

of the text. Also, the mass was written with four soloists in mind with correspondingly large registries.

5.4.1 Kyrie

This section provides a short description of the Kyrie’s structure. It is a simple stacked spiral canon based

on configurationC0 = [A,A,A,A] discussed in section 5.3.7. The respective abbreviations of Bass, Tenor,

Alto and Soprano used throughout this chapter are B,T,A and S. Starting in bar B:1, an embellished form

of themeA is repeated three times in each voice. The text set to each respective repetition isKyrie eleison,

Christe eleison, Kyrie eleison. The coda which starts in B:37, gives one extraKyrie eleisonin each voice

and ends in a cadence on a picardian tonic with sixte ajoutee.

5.4.2 Gloria

This section gives a short description of the actual score ofthe Gloria which is based on the configurations

C1 = [A,B,A,B] andC2 = [B,B,B,B] discussed in section 5.3.8. The conceptual structure of theGloria

is shown in table 5.6. This table contains four sub-tables with headers in the first column. The letters ‘M’

and ‘R’ stand for the measure number and pitchclass of root the chord at the first beat of a measure. The

letters ‘S’,‘A’,‘T’,‘B’ abbreviate the four voices in their usual way. The four center-rows of each sub-table

5.4.A

NA

LYT

ICA

LN

OT

ES

67

M 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46S B B B BA A A A AT B B B BB A A A AR 2 9 4 11 6 1 8 3 10 5 0 7 2 9 4 11

M 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94S B B B BA A B BT B B BB B B BR 6 1 8 3 10 5 0 7 2 9 4 11 6 1 8 3

M 97 100 103 106 109 112 115 118 121 124 127 130 133 136 139 142S A A A AA B B BT B A A AB B B B BR 10 5 0 7 2 9 4 11 6 1 8 3 10 5 0 7

M 145 148 151 154 157 160 163 166 169 172 175 178 181 184 187 190S A A A AA B B B BT A A A AB B B B BR 2 9 4 11 6 1 8 3 10 5 0 7 2 9 4

Table 5.6: Conceptual structure of the Gloria


give the theme entrances in the different voices. The ‘A’ in the second column (M=1) in the Bass row of the

first sub-table for example denotes an entry of themeA in a d-minor seventh chord (R=2).

5.4.3 Credo

This section gives a short description of the actual score ofthe Credo which is based on the configurations

C3 = [C,A,C,A], C4 = [C,B,C,B], andC5 = [C,C,C,C] discussed in section 5.3.9. The conceptual

structure of the Credo is shown in table 5.7. This table is similar to table 5.6. After the dense polyphony

of the Gloria, the Credo opens with a unisono section in whichthemeC is exposed. Starting in B:49, the

plot thickens slightly with two short canons of the form[C,C] with respective entrances in B:49,T:52 and

S:58,A:61. The subconfigurations[C,A] and[A,C] of C3 that follow re-introduce themeA and give a

glimpse of things to come in measures[112..160). Instead of developing this idea further the music re-

turns to the familiar context of themeC with configuration[C,C,C,C] in B:85. Instead of spiraling this

configuration, the tension is relaxed by a canon of the form[C1,C1,C1,C1], whereC1 is the head ofC as

discussed in section 5.3.9.

Starting in B:112, the full double stacked spiral canon[C,A,C,A] is given four times. Again the

tension slightly relaxes by the use ofC1 in B:160, A:166 and S:169. Starting in B:172 full dual stacked

spiral canon[C,B,C,B] is given twice, thus creating a climax in harmonic tension. The relaxing re-

introduction ofC1 in B:196 now finally develops into the full stacked spiral canon[C,C,C,C] from T:205

onwards. The coda returns toC1, starting in T:241, and ends in a three-voice cadence.

5.4.4 Sanctus

The Sanctus is the first part of the Mass in which all three themesA,B andC are combined. The occurrence

of all three themes signifies the religious importance of this part in the Mass. The themes are combined in

the following two ways:

1. The sequenceC,A,B where themesC,A andB respectively enter with the same time distance

TI = 3 andSI = 7 semitones;

2. The sequence/canonC,A,B,A where themesC,A,BA respectively enter with the same time dis-

tanceTI = 3 andSI = 7 semitones;

The above combinations are respectively referred to as 1,2 in the sequel. The conceptual structure of the

Sanctus is outlined in table 5.8. Let, within the current context, a list of the form⟨C,A,B⟩ denote that

themeA enters above themeC and themeB enters above themeA, i.e. the entries are sorted according

to the absolute pitch of the first note of a theme. The possibleabsolute-pitch permutations of combination

1 are{⟨C,A,B⟩ , ⟨C,B,A⟩ , ⟨A,C,B⟩ , ⟨A,B,C⟩ , ⟨B,C,A⟩ , ⟨B,A,C⟩}. Measures[1..67] give all of

these permutations in the given order. This creates diversity and the impression of a triple fugal structure.

In S:70, the entry of themeA extends combination 1 into 2, with an extra entry of themeA. In B:109 this

combination changes voices, which coincides with the commencement of theBenedictus.

5.4.A

NA

LYT

ICA

LN

OT

ES

69

M 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49S CA CT CB C CR 2 9 4 11 6 1 8 3 10 5 0 7 2 9 4 11 6

M 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100S C A CA C C CT C A CB C C C1

R 1 8 3 10 5 0 7 2 9 4 11 6 1 8 3 10 5

M 103 106 109 112 115 118 121 124 127 130 133 136 139 142 145 148 151S C1 A A AA C1 C C C CT C1 A A A AB C C C CR 0 7 2 9 4 11 6 1 8 3 10 5 0 7 2 9 4

M 154 157 160 163 166 169 172 175 178 181 184 187 190 193 196 199 202S A C1 B BA C1 C C C1

T B BB C1 C C C1

R 11 6 1 8 3 10 5 0 7 2 9 4 11 6 1 8 3

M 205 208 211 214 217 220 223 226 229 232 235 238 241 244 247 250 253S C C C C1

A C C C C11

T C C C C1 C21

B C C C C1

R 10 5 0 7 2 9 4 11 6 1 8 3 10 5 0 7 2

Table 5.7: Conceptual structure of the Credo


M 1 4 7 10 13 16 19 22 25 28 31 34 37S A BA B B CT A CB C C AR 2 9 4 11 6 1 8 3 10 5 0 7 2

M 40 43 46 49 52 55 58 61 64 67 70 73 76S A AA C C CT B A AB A B BR 9 4 11 6 1 8 3 10 5 0 7 2 9

M 79 82 85 88 91 94 97 100 103 106 109 112 115S A AA C C BT A A AB B B CR 4 11 6 1 8 3 10 5 0 7 2 9 4

M 118 121 124 127 130 133 136 139 142 145 148 151 154S A A A AA B B BT A A AB C C CR 11 6 1 8 3 10 5 0 7 2 9 4 11

Table 5.8: Conceptual structure of the Sanctus

5.4.5 Agnus Dei

The Agnus Dei is based on the mirrored versions of themesA,B,C, referred to as themeA−1, theme

B−1 and themeC−1 in this section. Each theme is mirrored chromatically, i.e.respecting the exact dis-

tance in semitones between consecutive pitches of the corresponding original. As a result, the mirrored

equivalent of some relative chord tonerct at indext of a theme,mirrored(t, rct), corresponds to rel-

ative chord tonecs [t (mod ∣cs∣)] − 1 − rct in the chromatically mirrored chords-sequence, defined for

0 ≤ rct < cs [t (mod ∣cs∣)]. Two transpositions of the mirrored chord sequence are given in figure 5.6.

Note that the scale interval equals5, the inverse (mod 12) of the original scale interval of7 semitones

defined in section 5.2. Also note that the original chord sequence was chosen such, that each dissonant in

& C wwww wwwww wwwwb wwwwb wwwwwb wwwwb

Figure 5.6: Two repetitions of the mirrored chord sequence

its mirror eventually resolves in some voice, if all chords are complete. Indeed, relative chord tone3 at time

class0 or 1 eventually ‘resolves’ to relative chord tone3 at time class2, relative chord tone4 at time class

1 eventually resolves to relative chord tone2 at time class0, and, relative chord tone3 at time class2 even-

tually resolves to relative chord tone1 at time class2. Hence, for each configurationC0,⋯,C6 constrained

5.4. ANALYTICAL NOTES 71

in sections 5.3.7 through 5.3.9, we have that the absence of chord tone doublings, and, the regulation of

parallel perfect intervals or dissonance in the spiraling phase, are all transferred to its mirrored counterpart.

The conceptual structure of the Agnus Dei is shown in table 5.9. The mirrored configurations are intro-

duced in the order of their respective originals in the Mass.Indeed, the Agnus Dei opens with configuration

[A−1,A−1,A−1,A−1], the mirror ofC0 used in the Kyrie. From measure37 onwards,[A−1,B−1,A−1,B−1]is introduced, followed by[B−1,B−1,B−1,B−1]. Their respective originalsC1, C2 were introduced in the

Gloria, in the same order. Starting in measure193, configurations[C−1,C−1,C−1,C−1] and[B−1,C−1,B−1,C−1]are respectively derived fromC5 andC4, both of which were introduced in the Credo. From measure220

onwards, mirrored configurations from the Sanctus are used.

72

CH

AP

TE

R5.

CO

NS

TR

AIN

TLO

GIC

PR

OG

RA

MM

ING

STA

CK

ED

CA

NO

NIC

STRU

CT

UR

ES

M 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49S A−1 A−1 A−1 A−1 A−1

A A−1 A−1 A−1 B−1

T A−1 A−1 A−1

B A−1 A−1 B−1

R 2 7 0 5 10 3 8 1 6 11 4 9 2 7 0 5 10M 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100S A−1 A−1 A−1

A B−1 B−1 B−1 B−1 B−1

T A−1

A−1

A−1

A−1

B B−1 B−1 B−1 B−1

R 3 8 1 6 11 4 9 2 7 0 5 10 3 8 1 6 11M 103 106 109 112 115 118 121 124 127 193 196 199 202 205 208 211 214S B−1 B−1 B−1 C−1 B−1

A B−1 B−1 C−1 C−1

T B−1 B−1 C−1 B−1

B B−1 B−1 C−1 C−1

R 4 9 2 7 0 5 10 3 8 1 6 11 4 9 2 7 0M 217 220 223 226 229 232 235 238 241 244 247 250 253 256 259 262 265S B−1 B−1 A−1 A−1

A C−1 C−1 C−1 C−1

T A−1

A−1

A−1

A−1

B B−1 B−1 B−1

R 5 10 3 8 1 6 11 4 9 2 7 0 5 10 3 8 1M 268 271 274 277 280 283 286 289 292 295 298 301 304 307 310 313 316S C−1 C−1 C−1

A C−1 A−1 A−1 A−1 A−1

T C−1 B−1 B−1 B−1

B A−1

A−1

A−1

A−1

R 6 11 4 9 2 7 0 5 10 3 8 1 6 11 4 9 2

Table 5.9: Conceptual structure of the Agnus Dei

Chapter 6

Summary, conclusion and outlook

This chapter summarizes the preceding chapter in section 6.1. A conclusion is provided in section 6.2, after

which section 6.3 provides an outlook on future research.

6.1 Summary

Chapter 1 discussed the development of canon- and fugue-techniques and their connection in stretto-fugues

such as found in Bach’s ‘Kunst der Fugue’. Examples 3 through5 showed that larger polyphonic structures

are sustainable by a main theme which can appear in many different canons, calledstretti. Hence, techniques

to effectively design such themes require the availabilityof efficient techniques for the creation of several

types of canons. In search of such techniques, chapter 2 provides a theoretical basis for the remainder of

the thesis. An analysis of the establishedcounterpointingand intervallic approaches to the construction

of stacked canonsshowed that these provide limited harmonic control and are computationally complex.

While efficient and in complete control of harmony, Morris’ Tonnetz approach targets serial stacked canons

ad minimum and does not consider voice-leading constraints. A style-independent, constructive approach

usingrelative chord tones, chord sequencesandchord sequence modulationswas presented along with its

connections to graph-theory in address of these issues. My analysis of Rameau’sCanon at the Fifthfrom

hisTraite de l’harmoniein section 2.4 introduced the concept ofrelative chord tones, which identify a pitch

class of a chord by its relative position within this chord, with a natural numbers. Two constraints discussed

in the Traite were integrated in the newly proposed relative chord tone model: first, the requirement of

complete chords in a stacked canon, and second, the proper preparation and resolution of sevenths. The

analysis explained Rameau’s choice ofdux and chord sequence in terms of the conjunction of these con-

straints. Using my definition ofrestless dux graphson page 17, the problem of obtaining complete chords

is reduced to the Hamiltonian cycle problem in a dux graph. The problem is solved in the strictly linear

algorithm 8. The problem of finding adux according to the conjunction of the aforementioned constraints

is reduced to the definition of a generating function in section 2.4.2. Again, the construction of adux using

such a function is a linear exercise.

A general approach to the incorporation of voice leading constraints is sketched in section 2.5, by the

detailed discussion of prohibitions of first parallel octaves, and second, parallel fifths. After formulating the

least upper bound on the maximum number of voices in a stackedcanon without parallel octaves in terms

73

74 CHAPTER 6. SUMMARY, CONCLUSION AND OUTLOOK

of the chord sequence’schord sizesin inequality 2.2, the problem of finding adux for such a stacked canon

with a maximum number of voices is reduced to the Eulerian cycle problem in a restless dux graph in the

proof preceding proposition 15. Proposition 19 proves thatthe conjunction of either constraint reduces the

aforementioned least upper bound only by a single voice. My definition of dux graphs with rests allows

the definition of two near lineardux generation algorithms which respectively satisfy the first(algorithm

17) and either constraint (algorithm 18). The problem of constraining the inversion of chords was sketched

in terms of the incorporation of a constraint which prohibits six-four chords in section 2.6. This section

also briefly discussed the constrained use of six-four chords as in e.g. the cadential six-four chord, which

may enable interested readership to further work out thorough bass considerations in the relative chord tone

model.

Two methods for avoiding repetitiveness, a possible dangerinherent to basing a stacked canon on a

predetermined chord sequence, are discussed in section 2.7. First, the basic idea of thechord sequence

modulationdiscussed in section 2.7.1 is to reinterpret the relative chord tones of an incomplete chord in a

stacked canon, as relative chord tones of another chord, while fixing the pitches of the incomplete chord.

While proposition 23 proves that this approach neither introduces nor removes parallel octaves if certain

conditions are met, the same cannot be said for parallel fifths. The second method for providing variety,

exploits the choice of a chord sequence with chord sizes greater than the number of voices. This method

has the advantage that the underlying chord sequence remains completely predetermined, only the subsets

of chord tones in various sections of the resulting stacked canon change at the composer’s discretion. The

latter method is discussed in section 2.7.2. Algorithm 26 provides a linear method for the creation of a

‘short’ dux based on such chord sequences, which by proposition 25 can infact be quite long if the subsets

of relative chord tones chosen have cardinalities which arerelatively prime. By concatenating several of

such ‘short’duces, all based on different subsets of the relative chord tones of the same chord sequence, a

longdux results which results in variety in both melody and harmony.

The compositions written in demonstration of the main theoretical part of my thesis are discussed in

chapters 3 through 5. The common factor in these compositions is the predominant use of stretti based on

the respective composition’s themes, to the extent that is has become an organizing principle. Chapter 3

discusses my compositionSpiralwhich is based on several themes, the main two of which were derived as

relative chord tone sequences using results from chapter 2.The chapter also discusses several techniques

related to the organization of polythematic stacked canonic structures in larger scale composition. The first

movement was designed in terms of severalharmoniolas, in which neither of the aforementioned themes

yet appear in a stretto, although the movement is concluded by a stretto related to the preceding harmo-

niola. The second movement is designed as a passacaglia structure based on the aforementioned second

theme, onto which many distinct combinations of various themes are superimposed, without violating the

constraints I chose. The final movement is a liberal chromatic mirror of the first movement, testing my

conjecture that the use of a chromatic mirror axis can mirrorthe affects experienced by an audience.

Chapter 4 discusses myFugue in G, the first theme of which was derived as a Hamiltonian cycle inthe

underlying dux graph. As a result, it allows four-voice stretti without chord tone doublings or parallel oc-

taves in which dissonance is strictly regulated thanks to the chosen chord sequence. The contrasting second

theme was constructed as a counterpoint with manySekundgangen, which allows for different interpreta-

6.2. CONCLUSION 75

tions of its notes as either embellishment or chord tone. Hence this theme too allows several distinct stretti.

As a result, the fugue contains many stretti of its two themes. Hence the chapter provides further discussion

on the organization of polythematic canonic structures in larger scale composition.

Chapter 2 presented a methodology for the design of stacked canons using relative chord tones and

some of its applications. Requirements to a theme or stackedcanons dux were stated in the form of predi-

cates (constraints) in first order logic. The general methodfor obtaining a solution to such constraints was

to derive an algorithm through logical and mathematical reasoning, which algorithm produces the desired

solutions. In some scenarios this approach may not be the most efficient expenditure of a composers time.

In the preliminary stages of a composition, different configurations or rules for a theme or dux under de-

velopment may be examined, resulting in the repeated derivation of different algorithms. Another scenario

is the presence of many constraints. If satisfiable, the derivation of an algorithm may be tedious. If un-

satisfiable, a change in requirements is forced upon the composer as in the first scenario. In yet another

scenario, a composer may not posses the skills required to derive an algorithm for the creation of solutions.

Finally, a composer may wish to manually evaluate all solutions to a set of constraints, in order to select

one according to yet unspecified artistic requirements. Chapter 5 proposes a method which usesconstraint

logic programs(CLPs) to efficiently search for the thematic material of polythematic stacked canonic struc-

tures in relative chord tone domains. Constraints which describe polythematic stacked canonic structures

are stated in first-order logic, compiled into a CLP, and solved by a computer. It was established that for

typical chord-sizes, the use of relative chord tone domainsresults in an exponential reduction of the search

space size in comparison to the use of pitch class domains. Inpractice this can be the difference between

a tractable and intractable CLP, even by fast computers. Thecomposition process for myMissa ad Fugam

is discussed in demonstration of this technique. The Mass exhausts the capabilities of three themes, all of

which are based on a chord sequence which is similar to the chord sequence chosen for the aforementioned

Fugue in G. Each one of the Mass’ themes allows a four-voice stacked spiral canon, each pair allows a

four-voice double stacked spiral canon, and both three-andfour-voice combinations featuring all themes

are possible.

6.2 Conclusion

Table 2.9 on page 36 summarizes the conclusion of my thesis, in the sense that the techniques for generating

stacked canons summarized in section 6.1 are efficient by objective mathematical standards provided by

complexity theory, yet highly productive. Indeed, proposition 28 shows that the number of sub-canons of

a stacked canon,2V −1 − 1, is exponential in the stacked canon’s number of voicesV . Yet the construction

of the encompassing stacked canon can be accomplished in (near) linear time and space with the provided

algorithms (depending on the constraints imposed by the composer). In combining propositions 15 and 28,

table 2.9 showsdoubleexponential growth of the number of distinct stretti without parallel octaves a theme

designed as a stacked canon allows in terms of the chord sizesof the underlying chord sequence. In fact,

this number is so large for chord sizes of e.g. four or five (which are quite common in contemporary music),

that it is hard to imagine a composition which exhausts them all.

76 CHAPTER 6. SUMMARY, CONCLUSION AND OUTLOOK

6.3 Outlook

This thesis represents a first step in the definition of efficient methods for the design of canons, in the sense

that stacked (spiral) canons - and thereby also rounds - where researched in some depth, along with meth-

ods for their organization in larger structures. This however leaves many other types of canons such as non

stacked-, mensuration-, proportion-, inversion-, and retrograde canons as relatively open problems, though

Melson already provides some insights into mensuration- and proportion canons[Mel].

As for the model and methods proposed in this thesis, furtherwork could be done first on extensions

of the proposed model of relative chord tones, second on treatment within this model of other problems

in music theory, and third, on a more balanced integration with other compositions techniques in future

compositions:

1. The proposed model of relative chord tones does not formally define embellishments, which makes

some stacked canons difficult to explain in the present model. Indeed, adux can be defined such

that a melody tone can be understood as a local ornamentationin some chords (e.g. a suspension or

passing tone), while it is a chord tone in others. While extensions of this kind add generality and

more expressive power to the model, they also increase its complexity. This added complexity may

however be necessary for the analysis of (some) existing compositions.

2. Subjects such as functional harmony (or thorough bass practice) and counterpoint as commonly

taught at conservatories could benefit from the definition ofrules in terms of relative chord tones.

The skeleton of a harmony exercise in relative chord tone notation for example quite clearly reveals

the violation of many types of constraints, so often hidden from students’ view of a graphical score.

I also expect this to allow for a more constructive and formaltreatment of many known problems in

music theory, as demonstrated in this thesis.

3. In an ideal world, one receives driving lessons from an experienced driving instructor, well after the

invention of the automobile. While working on this thesis, Ihad to simultaneously invent the methods

discussed in the preceding chapters while demonstrating them in compositions. It not only takes

time to invent such methods, they also need to be mastered, which requires practice and experience.

Thankfully, the effort I put into this thesis and the associated compositions provided me with some of

both, which I expect to put to use in future compositions in which several techniques can be balanced.

Finally, the theory presented in chapter 2 may seem too steepa climb into formalism for readers with a

limited background in mathematics. I hope to serve those by writing up a more pedagogical version of the

main findings in future work.

Appendix A

B-Prolog sources

A.1 themeA.pl

%bpVarDec([A,A,A,A],themeA)

themeA(Vars) :- Vars=[A00,A01,A02,A03,A04,A05,A06,A07 ,A08,A09,A10,A11],

%bpDomains([A,A,A,A])

[A00,A03,A06,A09] :: 0..3,

[A01,A04,A07,A10] :: 0..4,

[A02,A05,A08,A11] :: 0..3,

%bpNoneRepetitive(A)

(A00#=A01)#=>(A01#\=A02),

(A01#=A02)#=>(A02#\=A03),

(A02#=A03)#=>(A03#\=A04),

(A03#=A04)#=>(A04#\=A05),

(A04#=A05)#=>(A05#\=A06),

(A05#=A06)#=>(A06#\=A07),

(A06#=A07)#=>(A07#\=A08),

(A07#=A08)#=>(A08#\=A09),

(A08#=A09)#=>(A09#\=A10),

(A09#=A10)#=>(A10#\=A11),

(A10#=A11)#=>(A11#\=A00),

(A11#=A00)#=>(A00#\=A01),

%bpNoDoublings([A,A,A,A])

all_distinct([A00,A03,A06,A09]),



%bpRequiredRCT([A,A,A,A],1,0)

(0 #\/ 0#=A01 #\/ 0#=A04 #\/ 0#=A07 #\/ 0#=A10),

77

78 APPENDIX A. B-PROLOG SOURCES

%bpConsonantDyads([A,A,A,A])

%bpConsonant([A,A,A,A],3)

abs(A00-A03)#=<2,


abs(A01-A04)#=<2,


abs(A02-A05)#=<2,

%bpInitialDissonancesResolve([A,A,A,A])

%dissonancesResolve([A,A,A,A],6)

((0 #\/ 0#=A06 #\/ 0#=A03 #\/ 0#=A00)#/\(0 #\/ 3#=A06 #\/ 3#= A03 #\/ 3#=A00))#=>

(0 #\/ 0#=A07 #\/ 0#=A04 #\/ 0#=A01),

%dissonancesResolve([A,A,A,A],7)

((0 #\/ 0#=A07 #\/ 0#=A04 #\/ 0#=A01)#/\(0 #\/ 3#=A07 #\/ 3#= A04 #\/ 3#=A01))#=>

(0 #\/ 1#=A08 #\/ 1#=A05 #\/ 1#=A02),

((0 #\/ 0#=A07 #\/ 0#=A04 #\/ 0#=A01)#/\(0 #\/ 4#=A07 #\/ 4#= A04 #\/ 4#=A01))#=>

(0 #\/ 2#=A08 #\/ 2#=A05 #\/ 2#=A02),

((0 #\/ 1#=A07 #\/ 1#=A04 #\/ 1#=A01)#/\(0 #\/ 4#=A07 #\/ 4#= A04 #\/ 4#=A01))#=>

(0 #\/ 2#=A08 #\/ 2#=A05 #\/ 2#=A02),

%bpNoInitialParallelFifths([A,A,A,A])

(A03-A06#\=2)#\/(A04-A07#\=2),

(A04-A07#\=2)#\/(A05-A08#\=2),

labeling(Vars).

:- open(’themeA.txt’,write,Stream),

findall(V,themeA(V),S),write(Stream,S), close(Stream ).

A.2. THEMEB.PL 79

A.2 themeB.pl

%bpVarDec([A,B,A,B],themeB)

themeB(Vars) :- Vars=[

A00,A01,A02,A03,A04,A05,A06,A07,A08,A09,A10,A11,

B00,B01,B02,B03,B04,B05,B06,B07,B08,B09,B10,B11

],

%bpDomains([A,B,A,B])

[A00,A03,A06,A09,B00,B03,B06,B09] :: 0..3,

[A01,A04,A07,A10,B01,B04,B07,B10] :: 0..4,

[A02,A05,A08,A11,B02,B05,B08,B11] :: 0..3,

%bpPreset(A,[3,4,3,2,3,2,1,0,1,0,2,0])

A00#=3,A01#=4,A02#=3,A03#=2,A04#=3,A05#=2,A06#=1,A0 7#=0,A08#=1,A09#=0,A10#=2,A11#=0,

%bpDissimilar(A,B)

A00#\=B00#\/A01#\=B01#\/A02#\=B02,

A00#\=B03#\/A01#\=B04#\/A02#\=B05,

A00#\=B06#\/A01#\=B07#\/A02#\=B08,

A00#\=B09#\/A01#\=B10#\/A02#\=B11,

A03#\=B00#\/A04#\=B01#\/A05#\=B02,

A03#\=B03#\/A04#\=B04#\/A05#\=B05,

A03#\=B06#\/A04#\=B07#\/A05#\=B08,

A03#\=B09#\/A04#\=B10#\/A05#\=B11,

A06#\=B00#\/A07#\=B01#\/A08#\=B02,

A06#\=B03#\/A07#\=B04#\/A08#\=B05,

A06#\=B06#\/A07#\=B07#\/A08#\=B08,

A06#\=B09#\/A07#\=B10#\/A08#\=B11,

A09#\=B00#\/A10#\=B01#\/A11#\=B02,

A09#\=B03#\/A10#\=B04#\/A11#\=B05,

A09#\=B06#\/A10#\=B07#\/A11#\=B08,

A09#\=B09#\/A10#\=B10#\/A11#\=B11,

%bpNoneRepetitive(B)

(B00#=B01)#=>(B01#\=B02),

(B01#=B02)#=>(B02#\=B03),

(B02#=B03)#=>(B03#\=B04),

(B03#=B04)#=>(B04#\=B05),

(B04#=B05)#=>(B05#\=B06),

(B05#=B06)#=>(B06#\=B07),

(B06#=B07)#=>(B07#\=B08),

(B07#=B08)#=>(B08#\=B09),

(B08#=B09)#=>(B09#\=B10),

(B09#=B10)#=>(B10#\=B11),

(B10#=B11)#=>(B11#\=B00),

(B11#=B00)#=>(B00#\=B01),


%bpNoDoublings([A,B,A,B])

all_distinct([A00,A06,B03,B09]),






%bpNoDoublings([B,B,B,B])

all_distinct([B00,B03,B06,B09]),



%bpRequiredRCT([A,B,A,B],1,0)

(0 #\/ 0#=A01 #\/ 0#=A07 #\/ 0#=B04 #\/ 0#=B10),

(0 #\/ 0#=A04 #\/ 0#=A10 #\/ 0#=B01 #\/ 0#=B07),

%bpRequiredRCT([B,B,B,B],1,0)

(0 #\/ 0#=B01 #\/ 0#=B04 #\/ 0#=B07 #\/ 0#=B10),

%bpConsonantDyads([A,B,A,B])

%bpConsonant([A,B,A,B],3)

abs(B00-A03)#=<2,


abs(B01-A04)#=<2,


abs(B02-A05)#=<2,

%bpInitialDissonancesResolve([A,B,A,B])

%dissonancesResolve([A,B,A,B],6)

((0 #\/ 0#=A06 #\/ 0#=B03 #\/ 0#=A00)#/\(0 #\/ 3#=A06 #\/ 3#= B03 #\/ 3#=A00))#=>

(0 #\/ 0#=A07 #\/ 0#=B04 #\/ 0#=A01),

%dissonancesResolve([A,B,A,B],7)

((0 #\/ 0#=A07 #\/ 0#=B04 #\/ 0#=A01)#/\(0 #\/ 3#=A07 #\/ 3#= B04 #\/ 3#=A01))#=>

(0 #\/ 1#=A08 #\/ 1#=B05 #\/ 1#=A02),

((0 #\/ 0#=A07 #\/ 0#=B04 #\/ 0#=A01)#/\(0 #\/ 4#=A07 #\/ 4#= B04 #\/ 4#=A01))#=>

(0 #\/ 2#=A08 #\/ 2#=B05 #\/ 2#=A02),

((0 #\/ 1#=A07 #\/ 1#=B04 #\/ 1#=A01)#/\(0 #\/ 4#=A07 #\/ 4#= B04 #\/ 4#=A01))#=>

(0 #\/ 2#=A08 #\/ 2#=B05 #\/ 2#=A02),

%bpNoInitialParallelFifths([A,B,A,B])

(B03-A06#\=2)#\/(B04-A07#\=2),

(B04-A07#\=2)#\/(B05-A08#\=2),

labeling(Vars).

A.2. THEMEB.PL 81

:- open(’themeB.txt’,write,Stream),

findall(V,themeB(V),S),write(Stream,S), close(Stream ).


A.3 themeC.pl

%bpVarDec([C,A,B],themeC)

themeC(Vars) :- Vars=[

A00,A01,A02,A03,A04,A05,A06,A07,A08,A09,A10,A11,

B00,B01,B02,B03,B04,B05,B06,B07,B08,B09,B10,B11,

C00,C01,C02,C03,C04,C05,C06,C07,C08,C09,C10,C11

],

%bpDomains([C,A,B])

[A00,A03,A06,A09,B00,B03,B06,B09,C00,C03,C06,C09] :: 0..3,

[A01,A04,A07,A10,B01,B04,B07,B10,C01,C04,C07,C10] :: 0..4,

[A02,A05,A08,A11,B02,B05,B08,B11,C02,C05,C08,C11] :: 0..3,

%bpPreset(A,[3,4,3,2,3,2,1,0,1,0,2,0])

A00#=3,A01#=4,A02#=3,A03#=2,A04#=3,A05#=2,A06#=1,A0 7#=0,A08#=1,A09#=0,A10#=2,A11#=0,

%bpPreset(B,[3,1,1,0,2,2,1,0,3,2,3,0])

B00#=3,B01#=1,B02#=1,B03#=0,B04#=2,B05#=2,B06#=1,B0 7#=0,B08#=3,B09#=2,B10#=3,B11#=0,

%bpNoneRepetitive(C)

(C00#=C01)#=>(C01#\=C02),

(C01#=C02)#=>(C02#\=C03),

(C02#=C03)#=>(C03#\=C04),

(C03#=C04)#=>(C04#\=C05),

(C04#=C05)#=>(C05#\=C06),

(C05#=C06)#=>(C06#\=C07),

(C06#=C07)#=>(C07#\=C08),

(C07#=C08)#=>(C08#\=C09),

(C08#=C09)#=>(C09#\=C10),

(C09#=C10)#=>(C10#\=C11),

(C10#=C11)#=>(C11#\=C00),

(C11#=C00)#=>(C00#\=C01),

%bpNoDoublings([C,A,C,A])

all_distinct([A03,A09,C00,C06]),






%bpNoDoublings([C,B,C,B])

all_distinct([B03,B09,C00,C06]),




A.3. THEMEC.PL 83



%bpNoDoublings([C,C,C,C])

all_distinct([C00,C03,C06,C09]),



%bpRequiredRCT([C,A,C,A],1,0)

(0 #\/ 0#=A04 #\/ 0#=A10 #\/ 0#=C01 #\/ 0#=C07),

(0 #\/ 0#=A01 #\/ 0#=A07 #\/ 0#=C04 #\/ 0#=C10),

%bpRequiredRCT([C,B,C,B],1,0)

(0 #\/ 0#=B04 #\/ 0#=B10 #\/ 0#=C01 #\/ 0#=C07),

(0 #\/ 0#=B01 #\/ 0#=B07 #\/ 0#=C04 #\/ 0#=C10),

%bpRequiredRCT([C,C,C,C],1,0)

(0 #\/ 0#=C01 #\/ 0#=C04 #\/ 0#=C07 #\/ 0#=C10),

%bpNoParallelOctaves([C,A,B])

(A00-C03#\=0)#\/(A01-C04#\=0),

(A01-C04#\=0)#\/(A02-C05#\=0),

(A02-C05#\=0)#\/(A03-C06#\=0),

(A03-C06#\=0)#\/(A04-C07#\=0),

(A04-C07#\=0)#\/(A05-C08#\=0),

(A05-C08#\=0)#\/(A06-C09#\=0),

(A06-C09#\=0)#\/(A07-C10#\=0),

(A07-C10#\=0)#\/(A08-C11#\=0),

(A08-C11#\=0)#\/(A09-C00#\=0),

(A09-C00#\=0)#\/(A10-C01#\=0),

(A10-C01#\=0)#\/(A11-C02#\=0),

(A11-C02#\=0)#\/(A00-C03#\=0),

labeling(Vars).

:- open(’themeC.txt’,write,Stream),

findall(V,themeC(V),S),write(Stream,S), close(Stream ).

Bibliography

[And07] Torsten Anders.Composing Music by Composing Rules: Design and Usage of a Generic Music

Constraint System. 2007.

[Bac87] Johann Sebastian Bach. Die kunst der fuge bwv 1080; herausgegeben von christoph wolff, 1987.

[Bai88] Kathryn Bailey. Canon and beyond: Webern’s op. 31 cantata.Music Analysis, 7(3):pp. 313–348,

1988.

[BK05] W. Bathe and K. Karnes.A Briefe Introduction to the Skill of Song. Music theory in Britain,

1500-1700. Critical editions. Ashgate, 2005.

[BR93] J. Burmeister and B.V. Rivera.Musical Poetics. Music Theory Translation Series. Yale Univer-

sity Press, 1993.

[BW64] Friedrich Blume and Piero Weiss. Bach in the romanticera. The Musical Quarterly, 50(3):pp.

290–306, 1964.

[Cal92] Seth Calvisius.Melopoeia sive melodiae condendae ratio, quam vulgo musicam poeticam vo-

cant, ex veris fundamentis explicata. Erfurt, 1592.

[Den09] A. E. Denham. The future of tonality.Brit J Aesthetics, 49(4):427–450, October 2009.

[Dij76] Edsger W. Dijkstra.A discipline of programming. Englewood Cliffs, N.J. : Prentice-Hall, 1976.

[Dij96] Edsger W. Dijkstra. Een scheve schaats. circulatedprivately, March 1996.

[Dij02] Edsger W. Dijkstra. Ewd1300: The notational conventions i adopted, and why.Formal Asp.

Comput., pages 99–107, 2002.

[dPS83] G.P. da Palestrina and H. Stern.Missa ad fugam: fur Chor SATB (SAATB). Hanssler Verlag,

1983.

[Eul36] L. Euler. Solutio problematis ad geometriam situs pertinentis. Comment. Academiae Sci. I.

Petropolitanae, 8:128–140, 1736.

[Fle83] Fleury. Deux problemes de geometrie de situation.Journal de mathematiques elementaires,

pages 257–261, 1883.

[Fux42] Johann Joseph Fux.Gradus ad Parnassum (German translation with commentary byL. C. Mi-

zler). L. C. Mizler, Leipzig, 1742.

85

86 BIBLIOGRAPHY

[Gau96] Robert Gauldin. The composition of late renaissance stretto canons.Theory and Practice, 21,

1996.

[GJ90] Michael R. Garey and David S. Johnson.Computers and Intractability; A Guide to the Theory

of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1990.

[Gos97] Alan R. Gosman. Stacked canon and renaissance compositional procedure.Journal of Music

Theory, 41(2):289–317, 1997.

[Gos00a] Alan Gosman. Rameau and zarlino: Polemics in the ”traite de l’harmonie”. Music Theory

Spectrum, 22(1):pp. 44–59, 2000.

[Gos00b] Alan R. Gosman.Compositional Approaches to Canons from Ockeghem to Brahms. PhD thesis,

Harvard University, 2000.

[Hin70] Paul Hindemith.Unterweisung im Tonsatz / Paul Hindemith. Schott’s Sohne, Mainz :, 1970.

[Hon12] A. Honarmand. Canon and dodecaphony in the music of alfred schnittke. Honar–Ha-Ye-

Ziba:Honar-Ha-Ye-Namayeshi Va Mosighi, 3(43), 2012.

[JD91] Josquin and Nigel Davison. Missa sine nomine / josquin des prez ; edited by nigel davison.

[music], 1991.

[Kal90] Anne Kaldewaij.Programming: the derivation of algorithms. Prentice-Hall, Inc., Upper Saddle

River, NJ, USA, 1990.

[Mea93] Andrew Mead. Webern, tradition, and ”composing with twelve tones...”.Music Theory Spectrum,

15(2):pp. 173–204, 1993.

[Mel] E.A. Melson.Compositional Strategies in Mensuration and Proportion Canons, Ca. 1400 to Ca.

1600. McGill University, Montreal.

[MM79] Hans. Moldenhauer and Rosaleen. Moldenhauer.Anton von Webern, a chronicle of his life and

work / Hans Moldenhauer and Rosaleen Moldenhauer. Knopf : distributed by Random House,

New York :, 1st american ed. edition, 1979.

[Mor97] T. Morley. A Plaine and Easy Introduction to Practicall Musicke. London, 1597.

[Mor88] Robert D. Morris. Generalizing rotational arrays.Journal of Music Theory, 32(1):pp. 75–132,

1988.

[Mor95] Robert D. Morris. The structure of first-species canon in modal, tonal and atonal musics.Integral,

9:33–66, 1995.

[Mor05] Robert D. Morris. Aspects of post-tonal canon systems. Integral, 18/19:189–221, 2004/2005.

[Nei64] O. W. Neighbour. Schoenberg’s canons.The Musical Times, 105(1459):pp. 680–681, 1964.

[NK04] R. Nederpelt and F.D. Kamareddine.Logical Reasoning: A First Course. Texts in Computing.

King’s College Publications, 2004.

BIBLIOGRAPHY 87

[Par76] M. Park. The Missa Prolationum of Johannes Ockeghem: A Performing Edition. Kent State

University. Graduate School. Master’s theses. School of Music. 1976.

[Per61] Vincent Persichetti.Twentieth century harmony : creative aspects and practice /by Vincent

Persichetti. Norton, New York, 1961.

[Per71] George Perle. Webern’s twelve-tone sketches.The Musical Quarterly, 57(1):pp. 1–25, 1971.

[PR01] Francois Pachet and Pierre Roy. Musical harmonization with constraints: A survey.Constraints,

6(1):7–19, 2001.

[Rat00] Joseph Cardinal Ratzinger.The spirit of the liturgy. Ignatius Press, San Francisco, 2000.

[RG71] Jean-Philippe Rameau and Philip Gossett.Treatise on harmony / Jean-Philippe Rameau ; trans-

lated with an introduction and notes by Philip Gossett, volume 3. Dover Publications, New York

:, 1971.

[Roe03] John Roeder. Beat-class modulation in steve reich’s music.Music Theory Spectrum, 25(2):pp.

275–304, 2003.

[Rog68] John Rogers. Some properties of non-duplicating rotational arrays.Perspectives of New Music,

7(1):pp. 80–102, 1968.

[Sas03] Humphrey F. Sassoon. Js bach’s musical offering andthe source of its theme: Royal peculiar.

The Musical Times, 144(1885):pp. 38–39, 2003.

[Sch40] G. Schilling.Lehrbuch der allgemeinen Musikwissenschaft oder dessen, was Jeder, der Musik

treibt oder lernen will, nothwendig wissen muß: nach einer neuen Methode zum Selbstunterricht,

und als Leitfaden bei allen Arten von praktischem wie theoretischem Musikunterricht bearbeitet.

Groos, 1840.

[Sch63] A. Schoenberg.30[i.e. Dreissig] Kanons. Barenreiter, 1963.

[Scr97] Roger. Scruton.The aesthetics of music. Clarendon Press, Oxford :, 1997.

[Sea89] Michael Searby. Ligeti’s chamber concerto - summation or turning point? Tempo, (168):pp.

30–34, 1989.

[SG01] Stanley. Sadie and George Grove.The new Grove dictionary of music and musicians / edited by

Stanley Sadie. Grove, London :, 2nd ed. edition, 2001.

[Sim67] Christopher Simpson.A compendium of practical musick in five parts [microform] : teaching,

by a new and easie method ... / by Christopher Simpson. Printed by William Godbid for Henry

Brome, London, 1667.

[Sto08] Nolan Stolz.Contrapuntal Techniques in Schoenbergs Fourth String Quartet. Eunomios, August

2008. online article.

[Tan03] Andranik Tangian. Constructing rhythmic canons.Perspectives of New Music, 41(2):pp. 66–94,

2003.

88 BIBLIOGRAPHY

[Tar12] Richard Taruskin.Oxford History of Western Music. http://ccrma.stanford.edu/-

˜jos/mdft/ , 2012. online book.

[Tho00] Margaret E. Thomas. Nancarrow’s canons: Projections of temporal and formal structures.Per-

spectives of New Music, 38(2):pp. 106–133, 2000.

[vG12] J.L. van Geenen. On designing stacked canons with relative chord tones.Journal of Mathe-

matics and Music: Mathematical and Computational Approaches to Music Theory, Analysis,

Composition and Performance, 6(3):187–205, 2012.

[Vuz91] Dan Tudor Vuza. Supplementary sets and regular complementary unending canons (part one).

Perspectives of New Music, 29(2):pp. 22–49, 1991.

[Vuz92a] Dan Tudor Vuza. Supplementary sets and regular complementary unending canons (part three).


[Vuz92b] Dan Tudor Vuza. Supplementary sets and regular complementary unending canons (part two).


[Vuz93] Dan Tudor Vuza. Supplementary sets and regular complementary unending canons (part four).


[Wol02] C. Wolff. Johann Sebastian Bach: The Learned Musician. Oxford University Press, 2002.

[Wol03] Stephen Wolfram.The Mathematica book (5. ed.). Wolfram-Media, 2003.

[X03] Pope Pius X. Tra le sollecitudini.Motu Proprio, 1903.

[Zho06] Neng-Fa Zhou. Programming finite-domain constraint propagators in action rules.Theory Pract.

Log. Program., 6(5):483–507, 2006.

[Zho10] Neng-Fa Zhou.B-Prolog User’s Manual (Version 7.4). Afany Software & CUNY & Kyutech,

2010.

http://www.oxfordwesternmusic.com

Glossary

Chord the record type for chords. 18, 90

LU the length of theunique part of the dux. 7

L the length of the dux in a (stacked) canon. 7

Pcc constrains the harmony of a stacked canon to chords ofV different pitches. 15

Pnf constraint which excludes the occurence of fourths betweenthe bass and another voice in a stacked

canon. 28

Pnpar (rcti) constraint which excludes the occurence of parallel relative chord tones at the relative chord

tone intervalrcti. 25

Psc summarizes the relation between the relative scale tones ofthe voices in astackedcanon. 30

Pspr constraint which requiressevenths to beprepared andresolve stepwise. 16

SI scaleinterval of a (stacked) canon. 7, 92

Scale the record type for scales. 18

S ascale. 7

TI thetime interval of a (stacked) canon. 7, 92

V the number ofvoices in a polyphonic structure. 7, 91

A (D (cs)) refers toD (cs)’s arcs. 17

D (cs) refers to a dux graph for a chord sequence with chord sizescs. 17, 89, 90

V (D (cs)) refers toD (cs)’s vertices. 17

asSet (a) returns a set representation of the arraya. 58

bpConsonant(C, t) requires the consonance of theme configurationC at timet. 60

bpConsonantDyads(C) requires the initial two-voice section of theme configurationC to be consonant.

60

bpDissimilar (T0, T1) constrains a degree of similarity between themesT0 andT1. 58

89

90 Glossary

bpDomains (C) constrains the domains of all themes in theme configurationC. 58

bpInitialDissonancesResolve(C) requires dissonances to resolve in the initial triads of theme configuration

C. 60

bpNoDoublings (C) constrains the chords in theme configurationC have no doublings. 59

bpNoInitialParallelF ifths(C) requires that no parallel fifths occur between dyads or triads. 61

bpNoneRepetitive (T ) constrains a degree of repetitiveness in themeT ’s relative chord tone structure.

59

bpRequiredRCT (C, tc, rct) requires the chords in theme configurationC to contain relative chord tone

rct at time classtc. 59

chord (C, t) returns the array of the relative chord tones sounding simultaneously at timet in configuration

C, 0 ≤ t, such thatchord (C, t) [v] = voice (C, v, t). 58

chord aChord array which represents the chord sequence of a stacked canon. 18

cs an array which contains thechord-sizes of the chord sequence. 17, 89

disPairs(t) returns a set of sorted arrays representing dissonant relative-chord tone pairs at timet. 60

dissonancesResolve(C, t) requires that any dissonant occurring at timet in theme configurationC re-

solves at timet + 1. 60

edg (D (cs))) boolean function which is trueiff D (cs) is aEuleriandux graph. 21

noParallel(C, t, int) requires the absence of a parallel relative chord tone interval int in the transition

t → t + 1 in theme configurationC. 61

rct a rectangular array, such thatrct[v, t] corresponds to the relative chord tone sounding in voicev at

time t. 14, 16

resolution(t, dp) returns the relative chord tone which must be present at timet + 1, should the pair of

dissonant relative-chord tones of the arraydp sound simultaneously at timet. 60

tc thetimeclass (mod TI). 16

theme0 Spiral’s main theme. vii, 19, 20, 37

theme1 Spiral’s second theme. vii, 38

voice (C, v, t) returns the relative chord tone a voicev has at timet in theme configurationC. 58, 90

canon a compositional technique in which derivations of a single melody are overlappingly imitated. 1, 91

chace a canonic form, popular in thirteenth and fourteenth century France, in which once voice ‘hunts’ the

other. 2

chord sequencea chord progression, the repeated transposition of which can be used as a model for the

harmonic structure of a stacked canon. 13, 29, 89, 90

Glossary 91

chord sequence modulationreinterpret the relative chord tones of an incomplete chordin the canon, as

relative chord tones of another chord, while holding the pitches of the incomplete chord invariant. 30,

91

chord size the cardinality of a chord’s pitch class set. 18, 89

circle canon a canon which can be be repeated ad libitum at the same pitch. 1, 92

CLP Constraint Logic Programming. 53

combination the overlapping of two or more themes. 40, 47, 91

comes a voice entry following a dux in a canon or fugue. 1

counterpointing approach a method for the creation of canons based on writing counterpoint upon coun-

terpoint. 9

dux a leading voice entry in a canon or fugue. 1, 7, 89, 91, 92

dux graph a directed graph which’ vertices represent relative chord tones and arcs voice leadings between

relative chord tones. 16, 17, 23, 89

Eulerian a directed graph is Eulerianiff it contains a closed path covering all arcs. 21, 27, 91

Eulerian cycle a cycle in a directed Eulerian graph which covers all arcs. 21, 27

Eulerization the pruning of vertices and adjacent arcs from a directed graph, such that it becomes Eulerian.

21

GCL the Guarded Command Language[Dij76, Kal90]. 23

Hamiltonian cycle a cycle in a directed graph which covers all vertices. 17

harmoniola a harmonic passage forV voices in which the concatenation of the individual voices corre-

sponds to the dux of a canon forV voices. 9, 11, 35, 39

hoquetus the dovetailing of sounds and silences by means of the staggered arrangement of rests[SG01]. 2

intervallic approach a method for the creation of canons based on the conversion ofconsonant harmonic

intervals to melodic intervals. 9

pes ground or ostinato. 1

pivot chord within the context of a chord sequence modulation: incomplete chord which allows chord

sequence before and after the modulation to overlap. 31

reduced members the reduced members of a combination identify the entries ofthe combination modulo

transposition. 40, 47

92 Glossary

relative chord tone an integer which identifies the index of a chord tone within its chord or transpositions

thereof. 14, 89, 91, 92

relative chord tone approach a method proposed in this thesis for the creation of stacked canons, which

specifies the dux as a series of relative chord tones. 13

relative scale tonean integer which identifies the index of a scale tone within its scale. 30, 89

rota see circle canon. 1

rotational array an array of pitch classes, the rows of which are rotatations of each other. 11, 14, 29

round see circle canon. 1

Sekundgang the stepwise organisation of a melody’s contour[Hin70]. 46

species counterpointan approach to strict counterpoint that proceeds methodically from note-against-note

settings of the cantus firmus to more complex combinations ofparts[SG01]. 9

spiral canon a canon in which each voice consists of transposed repetitions of the dux. 7

stacked canona special kind of canon in which each comes is a transpositionby a fixedtime interval

TIandscaleintervalSI of its immediate predecessor. 7

stretto the canonic overlap of a theme with itself. 2

theme configuration an array containing theme-entries such as[A,B,A,B], in the order in which they

enter a polyphonic structure. 57, 89, 90

Tonnetz a lattice representing tonal space. 12

Copyright and use of this thesis · PDF fileconstructive approach using relative chord tones,...

Documents

Transcript of Copyright and use of this thesis · PDF fileconstructive approach using relative chord tones,...