The Origin and Well-Formedness of Tonal Pitch Structures [PhD Thesis]

1 Introduction and musical background 1
1.1 Questions to address in this thesis . . . . . . . . . . . . . . . . . . 1
1.2 Perception of musical tones . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Critical bandwidth and just noticeable dierence . . . . . 4
1.2.3 Virtual pitch . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.4 Combination tones . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Just intonation and the compromises of temperaments . . . . . . 7
1.3.1 Harmonic series . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Temperament diffculties . . . . . . . . . . . . . . . . . . . 10
1.3.3 Tuning and temperament systems . . . . . . . . . . . . . . 12
1.4 Consonance and dissonance . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Explanations on sensory consonance and dissonance . . . . 13
1.4.2 Different types of consonance . . . . . . . . . . . . . . . . 17
1.5 Tonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 What lies ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Algebraic interpretation of tone systems 25
2.1 Group theory applied to music . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2 Properties of groups and mappings . . . . . . . . . . . . . 27
2.2 Group theoretic and geometric description of just intonation . . . 29
2.2.1 Just intonation in group theoretic terms . . . . . . . . . . 29
2.2.2 Different realizations of the tone space . . . . . . . . . . . 32
2.3 Other geometrical representations of musical pitch . . . . . . . . . 37
3 Equal temperament to approximate just intonation 41
3.1 Short review of techniques of deriving equal-tempered systems . . 42
3.1.1 Continued fractions . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 Fokker's periodicity blocks . . . . . . . . . . . . . . . . . . 45
3.2 Approximating consonant intervals from just intonation . . . . . . 46
3.2.1 Measures of consonance . . . . . . . . . . . . . . . . . . . 48
3.2.2 Goodness-of-fit model . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Resulting temperaments . . . . . . . . . . . . . . . . . . . 54
3.3 Limitations on fixed equal-tempered divisions . . . . . . . . . . . 56
3.3.1 Attaching note-names to an octave division . . . . . . . . . 57
3.3.2 Equal tempered divisions represented in the tone space . . 65
3.3.3 Extended note systems . . . . . . . . . . . . . . . . . . . . 68
3.3.4 Summary and resulting temperaments . . . . . . . . . . . 70
4 Well-formed or geometrically good pitch structures: (star-) convexity 73
4.1 Previous approaches to well-formed scale theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Carey and Clampitt's well-formed scales . . . . . . . . . . 74
4.1.2 Balzano's group theoretical properties of scales . . . . . . . 76
4.2 Convexity and the well-formedness of musical objects . . . . . . . 79
4.2.1 Convexity on tone lattices . . . . . . . . . . . . . . . . . . 80
4.2.2 Convex sets in note name space . . . . . . . . . . . . . . . 83
4.2.3 Convexity of scales . . . . . . . . . . . . . . . . . . . . . . 86
4.2.4 Convexity of chords . . . . . . . . . . . . . . . . . . . . . . 90
4.2.5 Convexity of harmonic reduction . . . . . . . . . . . . . . 92
4.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Concluding remarks on well-formedness . . . . . . . . . . . . . . . 97
5 Convexity and compactness as models for the preferred intonation
of chords 99
5.1 Tuning of chords in isolation . . . . . . . . . . . . . . . . . . . . . 99
5.1.1 A model for intonation . . . . . . . . . . . . . . . . . . . . 100
5.1.2 Compositions in the tone space indicating the intonation . 103
5.2 Compactness and Euler . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.1 Compactness in 3D . . . . . . . . . . . . . . . . . . . . . . 107
5.2.2 Compactness in 2D . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Convexity, compactness and consonance . . . . . . . . . . . . . . 113
5.4 Concluding remarks on compactness and convexity . . . . . . . . 116
6 Computational applications of convexity and compactness 119
6.1 Modulation finding . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1.1 Probability of convex sets in music . . . . . . . . . . . . . 120
6.1.2 Finding modulations by means of convexity . . . . . . . . 125
6.2 Pitch spelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2.1 Review of other models . . . . . . . . . . . . . . . . . . . . 130
6.2.2 Pitch spelling using compactness . . . . . . . . . . . . . . 132
6.2.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.5 Evaluation and comparison to other models . . . . . . . . 141
7 Concluding remarks 145
A Notes on lattices and temperaments 149
A.1 Isomorphism between P3 and Z3 . . . . . . . . . . . . . . . . . . . 149
A.2 Alternative bases of Z2 . . . . . . . . . . . . . . . . . . . . . . . . 150
A.3 Generating fifth condition . . . . . . . . . . . . . . . . . . . . . . 151
Samenvatting 153
Index 171