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UC San Diego Electronic Theses and Dissertations UC San Diego UC San Diego Electronic Theses and Dissertations Title Physically-informed Percussion Synthesis with Nonlinearities for Real-time Applications Permalink https://escholarship.org/uc/item/3bt848xs Author Hsu, Jennifer S. Publication Date 2019 Supplemental Material https://escholarship.org/uc/item/3bt848xs#supplemental Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SAN DIEGO Physically-informed Percussion Synthesis with Nonlinearities for Real-time Applications A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Music by Jennifer Hsu Committee in charge: Professor Tamara Smyth, Chair Professor Thomas Erbe Professor Sarah Hankins Professor Scott Klemmer Professor Miller Puckette 2019 Copyright Jennifer Hsu, 2019 All rights reserved. The dissertation of Jennifer Hsu is approved, and it is accept- able in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2019 iii DEDICATION To my family and friends, thank you for your love, support, and encouragement. iv TABLE OF CONTENTS Signature Page....................................... iii Dedication.......................................... iv Table of Contents......................................v List of Figures........................................ ix List of Tables........................................ xi List of Abbreviations.................................... xii List of Symbols....................................... xiii List of Supplementary Audio Examples.......................... xv Acknowledgements..................................... xvii Vita............................................. xix Abstract of the Dissertation................................. xx Chapter 1 Introduction..................................1 1.1 Introduction...............................1 1.2 Research Goals and Objectives....................2 1.3 Nonlinear Percussion Analysis....................4 1.3.1 Pitch Glides..........................4 1.3.2 Low Frequency to High Frequency Energy Cascade.....5 1.3.3 Chaotic Behavior.......................5 1.4 Overview of Percussion Synthesis Methods.............6 1.4.1 FM Synthesis.........................6 1.4.2 Digital Waveguide Mesh (DWM)..............7 1.4.3 Finite Difference Methods (FDMs).............9 1.4.4 Modal Synthesis (MS)..................... 11 1.4.5 Functional Transformation Method (FTM)......... 13 1.5 Dissertation Organization....................... 15 Chapter 2 Percussion Synthesis Models........................ 16 2.1 Introduction.............................. 16 2.2 Timbre Variety with the Physically Informed Control of MS (PhISM) System................................ 16 2.3 Decay Time and Timbre Variety with Coupled Mode Synthesis... 17 v 2.4 A MS Impact Synthesizer....................... 18 2.5 A FDM for the Sound Synthesis of Nonlinear Plates......... 20 2.6 A Circular, Tension-modulated Nonlinear Membrane using the FTM 22 2.7 Experiments with using a DWM for Percussion Synthesis...... 25 2.7.1 DWM for Percussion Synthesis............... 25 2.7.2 Towards a Real-Time 2D DWM............... 27 2.7.3 The 2D DWM as a Parallel Bank of 2nd-Order Sections... 28 Chapter 3 Loopback FM................................ 33 3.1 Introduction.............................. 33 3.2 An Oscillator with Constant Frequency............... 34 3.3 FM Formulation and Frequency Component Analysis........ 35 3.4 Feedback FM Formulation and Frequency Component Analysis.. 36 3.5 Loopback FM Formulation...................... 38 3.6 The Closed-Form Representation of the Loopback FM Oscillator............................... 40 3.7 Parametric Control of Loopback FM with Static Pitch and Number of Harmonics............................... 40 3.8 Time-varying B: Modulation of Pitch and Number of Harmonics with zc(n) .................................. 43 3.9 Time-varying b0 and w0: Modulation of Pitch and Number of Har- monics with z0(n) ........................... 44 3.10 Parametric Control of Loopback FM with Time-varying Pitch and Number of Harmonics........................ 45 3.10.1 Pitch Glides with B(n) .................... 46 3.10.2 Pitch Glides with w0(n) ................... 47 3.10.3 Varying the Number of Harmonics with b0(n) ........ 52 3.11 Loopback FM Application...................... 55 Chapter 4 Percussion Synthesis using Loopback FM Oscillators........... 57 4.1 Introduction.............................. 57 4.2 Percussion Synthesis using Traditional MS............. 58 4.3 Percussion Synthesis with Loopback FM Oscillators......... 61 4.3.1 MS with Loopback FM..................... 61 4.3.2 Sonic Enhancement 1: Adding Further Nonlinearities with Time-Varying 2nd-Order Allpass Filter Processing...... 62 4.3.3 Sonic Enhancement 2: Sense of Space with Commuted Syn- thesis............................. 65 4.4 Musical Parameters for MS with Loopback FM........... 67 4.4.1 Static Number of Harmonics: Oscillators Created with zc;i(n) or z0;i(n) ........................... 67 4.4.2 Static Number of Harmonics: Bi and b0;i ........... 69 4.4.3 Time-varying Number of Harmonics: Bi(n) and b0;i(n) ... 70 vi 4.4.4 Static Sounding Frequency: w0;i ................ 71 4.4.5 Pitch Glides: Bi(n) and w0;i(n) ................ 72 4.4.6 Decay Time: wi(n) ...................... 74 4.5 Musical Parameters for the Time-Varying 2nd-Order Allpass Filters............................. 76 4.5.1 Time-Varying 2nd-Order Allpass Filter with a Single Sinu- soidal Oscillator Input........................ 76 4.5.2 Time-Varying 2nd-Order Allpass Filter with a Single, Static Loopback FM Oscillator Input................ 78 4.5.3 Time-Varying 2nd-Order Allpass Filter with a Single, Time- varying Pitch and Number of Harmonics Loopback FM Oscil- lator Input........................... 79 4.5.4 Time-Varying 2nd-Order Allpass Filter with Multiple Sinu- soidal Oscillator Inputs (MS).................... 79 4.5.5 Time-Varying 2nd-Order Allpass Filter with Loopback FM MS 82 4.6 Musical Parameters for Commuted Synthesis............ 83 4.6.1 Attack Sharpness: Raised Cosine Envelopes......... 83 4.6.2 Attack Noisiness: Filtered Noise Bursts........... 83 4.6.3 Timbre: Acoustic Resonator Impulse Response r(n) .... 83 4.7 Conclusions.............................. 84 Chapter 5 Loopback FM Percussion Synthesis Examples............... 89 5.1 Introduction.............................. 89 5.2 Synthesis Walkthrough Example: the Kick Drum.......... 89 5.2.1 A Simple Kick Drum with Static Pitch and Number of Har- monics............................ 90 5.2.2 Adding a Pitch Glide...................... 91 5.2.3 Varying the Number of Harmonics Over Time........ 94 5.2.4 Applying 2nd-Order Time-varying Allpass Filters for Further Nonlinearities......................... 94 5.2.5 Applying Commuted Synthesis for a Sense of Space.... 95 5.3 Synthesis Examples.......................... 96 5.3.1 Snare Drum.......................... 97 5.3.2 Marimba........................... 98 5.3.3 Wood Block.......................... 99 5.3.4 Tom Tom........................... 100 5.3.5 Circular Plate.......................... 101 5.4 Conclusions.............................. 102 vii Chapter 6 Applications and Conclusion........................ 110 6.1 Introduction.............................. 110 6.2 Qualitative Evaluation......................... 111 6.3 A Real-time Loopback FM Percussion Synthesis Implementation.. 113 6.4 Ideas for a Software Synthesizer................... 113 6.5 Musical Implications......................... 115 6.6 Future Work and Conclusion..................... 116 Bibliography........................................ 118 viii LIST OF FIGURES Figure 1.1: Phase plots of velocity vs. displacement from [49]..............6 Figure 1.2: Magnitude response of a stretched 2D DWM stiff plate from [21]......8 Figure 2.1: Linear and nonlinear thin plate synthesis spectrograms from [7]........ 21 Figure 2.2: Spectrogram showing the synthesized pitch glide of a nonlinear, tension- modulated membrane from [5]........................ 24 Figure 2.3: 3x3 2D DWM structure from [20]....................... 26 Figure 2.4: Poles plotted for the 3 by 3 DWM on the unit circle............. 30 Figure 2.5: Magnitude spectrum for the 3 by 3 DWM.................... 31 Figure 2.6: Magnitude spectrums for second-order sections add up to 3 by 3 DWM magnitude spectrum.............................. 32 Figure 3.1: Magnitude spectrum of FM synthesis..................... 36 Figure 3.2: Magnitude spectrum of feedback FM synthesis................ 38 Figure 3.3: Magnitude spectrum of loopback FM synthesis with B = 0:9.......... 41 Figure 3.4: Magnitude spectrum of loopback FM synthesis with B = 0:5.......... 41 Figure 3.5: Magnitude spectrum of loopback FM synthesis with B = 0:0......... 42 Figure 3.6: Relationship between B and b0........................ 42 Figure 3.7: Magnitude spectrum of loopback FM: zc vs. z0 for large wc and B...... 43 Figure 3.8: Spectrograms of zc vs. z0 with time-varying pitch and number of harmonics. 45 Figure 3.9: Linearly increasing pitch glide for z0(n) and zc(n).............. 48 Figure 3.10: Exponentially decreasing pitch glide for z0(n) and zc(n)........... 49 Figure 3.11: Increasing pitch glide using a square root function for z0(n) and zc(n)..... 51 Figure 3.12: Varying the number of harmonics with linearly increasing b0(n) for z0(n) and zc(n).................................... 53 Figure 3.13: Linearly increasing
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