A Multi-Dimensional Analytical Model for Musical Harmony Perception Ning Cong PhD University of York Electronics November 2016 1 Abstract Rules and conventions observed in western music harmony involve a number of psychophysical relationships between musical entities and perception responses in terms of consonance and dissonance concepts (CDC). These well-established relationships have informed a common psychophysical mechanism that can be studied and numerically modelled. In the literature, a number of physiological and psychological based theories have been proposed, but no one single theory is able to fully account for the phenomenon of music harmony perception. This research deems musical consonance and dissonance to be a multi-dimensional concept that is underpinned by several psychoacoustic principles; it is hypothesized that perceived impression of a musical entity/structure is composed of a number of uncorrelated experiences that can be measured on a multi-dimensional space. The psychoacoustic model proposed here contains four types of actively defined dissonance concepts: namely sensory, ambiguity, gloom and tension. Sensory dissonance refers to the (primary and secondary) beats effect due to the physiological functions of auditory pathway organs; the ambiguity dissonance is developed from harmonic-template based theories in which sonorities with ambiguous tonal centres are considered dissonant; the gloom and tension dissonances are two fundamental dimensions of musical emotions that are related to raised and lowered pitch contours described by the melodic expectation theory. The correlation / independence between each type of dissonance concept is statistically analysed based on experimental results from newly conducted listening tests. In the application of this analysis to musical triads, this work shows that: the secondary chord structures (simultaneous chords containing one or two semitone intervals) have the highest level of sensory dissonance; the suspended 4th chords have a higher level of ambiguity dissonance; diminished chords have the highest level of gloom dissonance (followed by minor structures); the augmented chord structures contain the highest level of tension; and, of all chord structures, all the dissonance types are at their lowest level in the major chords . 2 List of Contents Abstract ......................................................................................................................... 2 List of Contents ............................................................................................................ 3 List of Tables ................................................................................................................. 8 List of Figures ............................................................................................................. 10 Acknowledgements .................................................................................................... 14 Declarations ................................................................................................................ 15 Chapter 1 Introduction .............................................................................................. 16 1.1 Object and Scope ................................................................................................ 18 1.2 Methodology ...................................................................................................... 20 1.3 Hypothesis .......................................................................................................... 21 1.3.1 Statement of Hypothesis .............................................................................. 21 1.3.2 Analysis of Hypothesis ................................................................................ 21 1.4 Main Contributions ............................................................................................ 23 1.5 Thesis Structure .................................................................................................. 25 1.6 Model Structure .................................................................................................. 27 Chapter 2 Music Harmony ....................................................................................... 29 2.1 Musical Tone ...................................................................................................... 31 2.2 Between Musical Tones ..................................................................................... 34 2.3 Musical Chord .................................................................................................... 41 2.4 Between Music Chords ...................................................................................... 47 3 2.5 Musical Mode ..................................................................................................... 49 2.6 Chapter Summary ............................................................................................... 53 Chapter 3 Physiological and Psychological Account .............................................. 57 3.1 Physiological-Based Theories ............................................................................ 58 3.1.1 The Theory of Beats .................................................................................... 62 3.1.2 Theory of Harmonic Template .................................................................... 65 3.2 Psychology-Based Theories ............................................................................... 68 3.2.1 Tonal Harmonic Hierarchy .......................................................................... 72 3.2.2 Intervallic Equivalence Theory ................................................................... 74 3.2.3 Tonal Fusion / Numerosity Conjecture Theory ........................................... 75 3.2.4 Melodic Expectation Theory ....................................................................... 76 3.3 Between Physiological and Psychological Theories .......................................... 79 Chapter 4 Psychoacoustic Models ............................................................................ 82 4.1 Acoustic Features ............................................................................................... 83 4.1.1 Musical Notes and Frequencies ................................................................... 83 4.1.2 Frequency Analysis ..................................................................................... 85 4.2 Psychological Measurements for CDC .............................................................. 87 4.3 Sensory Beats based Models .............................................................................. 88 4.3.1 Dissonance Curves ...................................................................................... 88 4.3.2 Estimating Roughness Sensations for Complex Tones ............................... 93 4.3.3 The Predicted Results of Beats-Based Models ............................................ 95 4.3.4 Mathematical Discussion of Beats-Based Computational Method ............. 97 4.3.5 A Conclusion for Beat-Based Models ......................................................... 99 4 4.4 Harmonic-template Based Models ................................................................... 101 4.4.1 Parncutt’s Tonalness Concept ................................................................... 101 4.4.2 Hofmann-Engl Model ................................................................................ 104 4.4.3 Relative Periodicity ................................................................................... 107 4.4.4 The Predicted Result of Harmonic-Template-Based Models .................... 108 4.5 Comparison between Sensory Beats and Tonal Consonance........................... 111 4.6 Summary .......................................................................................................... 114 Chapter 5 Sensory Dissonance ................................................................................ 115 5.1 Secondary Beats ............................................................................................... 116 5.2 An Experimental Study of Dissonance Curve.................................................. 120 5.2.1 Participants’ Profile ................................................................................... 121 5.2.2 Test Procedure ........................................................................................... 122 5.2.3 Analysis of the Test ................................................................................... 124 5.2.4 Results ....................................................................................................... 125 5.2.5 Conclusion of the Experimental Test ........................................................ 128 5.3 Estimation of Sensory Dissonance ................................................................... 129 5.4 Model Prediction Results for Musical Dyads and Triads ................................ 131 5.4.1 Sensory Dissonance for Musical Dyads .................................................... 132 5.4.2 Sensory Dissonance for Musical Triads .................................................... 134 5.5 Summary of Chapter ........................................................................................ 135 Chapter 6 Ambiguity Dissonance ........................................................................... 136 6.1 Noticeable Pitch Components .........................................................................