WESTERN SYDNEY UNIVERSITY DOCTORAL THESIS The Cognition of Harmonic Tonality in Microtonal Scales Author: Supervisors: Lillian M. Hearne Dr. Andrew J. Milne Prof. Roger T. Dean A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Music Cognition and Action Group The MARCS Institute 2020 i Acknowledgements First, I acknowledge the traditional custodians of the land upon which I completed my studies: The Darug, Tharawal (also historically referred to as Dharawal), and Eora peoples, and pay my respects to Elders past, present, and future. I would next like to acknowledge the guidance and help of my supervisors An- drew Milne and Roger Dean, who were a pleasure to work with. They were always available to talk when in the office and responded to emails and returned comments on drafts very promptly, and I never felt left in the dark. I’d also like to thank my colleagues from the lab for their help and advice and for their participation in my experiments. My colleague and housemate Ian Colley deserves a special mention here, for participating in every one of my experiments and providing valuable advice with using Max and R as well as in completing a PhD in general. My other housemates Rémi Marchand and Juan A. M. Fuentes and my ex-housemate Patrick Blown also deserve acknowledgement for their help with MATLAB and LaTeX, and my colleague Patti Nijhuis and my mum, Sonja Hearne for listening to my venting about computer issues and such. I’d also like to thank the technical team at MARCS for their help and patience with these issues, and the admin staff for their support throughout. I’d like to acknowledge the members of my musical ensembles Helix Quartet, Cone of Confusion, The Lancer Band, Sydney Harmony and St Mary’s Cathedral Choir who participated in multiple of my experiments; everyone at MARCS, or in AMPS, SMPC, ICMPC and MCM, who watched my presentations and asked ques- tions or discussed my work afterwards; to Marcus Pearce and Psyche Loui and the students in their labs for having me discuss my work with them; and to Paul Erlich, Margo Schulter, Jacob Barton, Juhani Nuorvala, Mike Battaglia, Kite Giedraitis and other mentors and contemporaries in The Xenharmonic Alliance. I’d like to thank my friends John Nicholls, for proofreading one of my papers, and Rachel Singer, for proofreading my papers and thesis; my girlfriend Indigo Ter- ayama for providing helpful distraction in the final week before submission (and continued support and encouragement in the months following); and everyone I’ve spoken to who asked about my research and let me get into any sort of detail! Finally, I’d like to thank Carol Krumhansl and Emery Schubert for their helpful reviews of this thesis. ii Statement of Authenticity The work presented in this thesis is, to the best of my knowledge and belief, original except as acknowledged in the text. I hereby declare that I have not submitted this material, either in full or in part, for a degree at this or any other institution Signed: iii Contents Acknowledgements i Statement of Authenticity ii Abstract xvi 1 Background 1 1.1 Introduction . .1 1.1.1 Tonality and Scale . .1 1.1.2 Thesis Summary . .3 1.2 Background . .4 1.2.1 Sound, spectrum, pitch and tuning . .4 1.2.2 Consonance and Affinity . .7 1.2.3 Probe Tone Experiments . 10 The context: presence or absence of tonal cues . 13 The dependent variable: participant ratings . 15 Participants: musicians or non-musicians . 16 1.2.4 Probe Chord Experiments . 16 Chord type and inversion . 20 Tension and stability . 21 1.2.5 The cognition of music in novel/microtonal tuning systems . 23 2 Experiments 1 & 2 – Tones 26 2.1 Introduction . 26 2.2 Overview of the Experiments and Models . 27 2.2.1 Experimental design . 27 2.2.2 Hypotheses . 29 Experiment 1 . 29 Experiment 2 . 30 iv 2.2.3 Analysis . 31 Bayesian ordinal mixed effects models . 31 Test for tonal hierarchies . 33 Comparison of fit and stability ratings (H1) . 34 Descriptive model . 34 Testing H2 & 3 with Bayesian ordinal mixed effects models . 35 Exploratory analysis . 39 2.3 Experiment 1: Diatonic, Harmonic Minor, Jazz Minor . 39 2.3.1 Method . 40 Participants . 40 Stimulus . 40 Procedure . 41 2.3.2 Results . 42 Test for tonal hierarchies . 42 Comparison of fit and stability ratings . 43 Descriptive model . 45 H2&3: Model 2.1 . 47 Exploratory analysis . 49 2.3.3 Discussion . 52 2.4 Experiment 2: Less familiar scales . 52 2.4.1 Method . 53 Participants . 54 2.4.2 Results . 54 Test for tonal hierarchies . 54 Comparison of fit and stability . 55 Descriptive model . 55 H2&3: Model 2.2 . 59 Combined analysis: Model 2.3 . 61 Exploratory analysis . 63 2.4.3 Discussion . 64 2.5 Conclusion . 65 3 Experiments 1 & 2 – Triads 67 3.1 Introduction . 67 3.2 Overview of the Experiments and Models . 68 v 3.2.1 Hypotheses . 68 Experiment 1 . 68 Experiment 2 . 68 3.2.2 Analysis . 69 Test for tonal hierarchies . 70 Comparison of fit and stability ratings (H1) . 70 Descriptive model . 71 Testing H2 & 3 with Bayesian ordinal mixed effects models . 71 Exploratory analyses . 74 3.3 Experiment 1: Diatonic, Harmonic Minor, Jazz Minor . 74 3.3.1 Method . 75 3.3.2 Results . 76 Test for tonal hierarchies . 77 Comparison of fit and stability ratings . 77 Descriptive model . 79 H2&3: Model 3.1 . 81 Exploratory analysis . 86 3.3.3 Discussion . 88 3.4 Experiment 2: Less familiar scales . 89 3.4.1 Method . 89 Participants . 90 3.4.2 Results . 90 Test for tonal hierarchies . 90 Comparison of fit and stability . 91 Descriptive model . 91 H2&3: Model 3.2 . 92 Combined analysis: Model 3.3 . 96 Exploratory analysis . 98 3.4.3 Discussion . 101 3.5 Conclusion . 102 4 Experiment 3: Intrinsic stability of the triads of 22-TET 104 4.1 Introduction . 104 4.2 22-TET . 105 4.3 Hypothesis . 108 vi 4.4 Method . 108 4.4.1 Participants . 108 4.4.2 Stimuli and Procedure . 109 4.5 Analysis . 109 4.6 Results . 112 4.6.1 Descriptive analysis . 112 4.6.2 Hypothesis test . 113 4.6.3 Bayesian ordinal mixed effects regression model . 115 4.7 Discussion / Conclusion . 123 5 Distributional Analysis of n-dimensional Feature Space for 7-note Scales in 22-TET 125 5.1 Introduction . 125 5.2 Definitions . 126 5.3 Review . 127 5.3.1 Redundancy . 128 5.3.2 Coherence and Evenness . 131 5.3.3 R-ad entropy . 133 5.3.4 Generator complexity . 133 5.3.5 Consonance . 134 5.3.6 Tetrachordality . 134 5.4 Analysis . ..