SUBHARMONIC FREQUENCIES IN GUITAR SPECTRA LEAH M. BUNNELL Bachelor of Science in Mechanical Engineering Cleveland State University May 2020 Submitted in partial fulfillment of requirements for the degree MASTER OF SCIENCE IN MECHANICAL ENGINEERING at the CLEVELAND STATE UNIVERSITY May 2021 We hereby approve this thesis for LEAH M. BUNNELL Candidate for the Master of Mechanical Engineering degree for the Department of Engineering and the CLEVELAND STATE UNIVERSITY College of Graduate Studies _________________________________________________________________ Thesis Chairperson, Dr. Majid Rashidi _____________________________________________ Department & Date _________________________________________________________________ Thesis Committee Member, Dr. Asuquo Ebiana _____________________________________________ Department & Date _________________________________________________________________ Thesis Committee Member, Professor Michael Gallagher _____________________________________________ Department & Date Student’s Date of Defense: May 6, 2021 DEDICATION My late grandmother and grandfather for establishing a wonderful life for their children and grandchildren. My mother and father for raising me and teaching me so many important life lessons. My sister and brother-in-law for giving me great opportunities to learn and grow. My aunt and niece for always supporting me in my endeavors and believing in me. ACKNOWLEDGEMENTS I would like to thank Dr. Majid Rashidi for all of the help and support he provided me during the research process. I am grateful for all of the feedback he provided me regarding my thesis, especially for teaching me how to properly structure a thesis. Not only am I thankful for Dr. Rashidi’s insightful advice, but I am appreciative that Dr. Rashidi took me on as a thesis candidate. After my previous research advisors could no longer continue due to extraneous commitments, Dr. Rashidi welcomed me as his student and has been a wonderful thesis advisor. I would like to thank Dr. Asuquo Ebiana and Professor Michael Gallagher for taking the time out of their busy schedules to hear me present my research. I greatly appreciate the feedback they provided me about my research and any new ideas they helped generate. I would also like to thank the faculty and staff at Cleveland State University. Without their individual contributions, I could not have completed my thesis and degree SUBHARMONIC FREQUENCIES IN GUITAR SPECTRA LEAH M. BUNNELL ABSTRACT Throughout this thesis, evidence is shown that suggests the existence of subharmonic frequencies in the Classical Spanish Acoustic Guitar and Fender Squier Electric Guitar spectra. The classic subharmonic undertone series mimics that of the harmonic overtone series, except that the fundamental frequency is divided by integer values instead of multiplied by integer values. Subharmonics that do not fit the classic subharmonic undertone series criteria are still classified as subharmonics, although subharmonics that do fit the criteria are emphasized. Throughout this manuscript, the author’s original experimental procedures and results are presented. Individual tones were recorded on both guitars on every string from frets zero to twelve and were analyzed for subharmonics via Audacity software. The fundamental and subharmonic frequencies were recorded on Excel and the fundamental frequencies were divided by the subharmonic frequencies, thus yielding ratios. These ratios were used in two different charts that were color-coded based on value and if the ratios were within ten percent of an integer value. These charts suggest the frequent presence of the first and second subharmonics in both guitars, which is especially observed in the higher strings (G, B and high E strings). v TABLE OF CONTENTS Page ABSTRACT ......................................................................................................................v LIST OF FIGURES ...................................................................................................... viii LIST OF SYMBOLS ........................................................................................................x CHAPTER I. INTRODUCTION ...................................................................................................1 1.1 A New Theory.................................................................................................1 1.2 A Theory Supported ........................................................................................2 II. BACKGROUND .....................................................................................................3 2.1 Review of Previous Research .........................................................................3 2.2 Problem Statement ........................................................................................11 III. MATERIALS AND METHODS ...........................................................................13 3.1 Required Materials ........................................................................................13 3.2 Methods.........................................................................................................16 IV. RESULTS AND DISCUSSION ............................................................................21 4.1 Results ...........................................................................................................21 4.2 Discussion .....................................................................................................28 V. CONCLUSION ......................................................................................................31 5.1 Conclusion ....................................................................................................31 REFERENCES ...............................................................................................................32 APPENDIX .....................................................................................................................34 viii LIST OF FIGURES Figure Page 1) Schematic of Four Frequency Versus Time Graphs ............................................4 2) Frequency Versus Time Graph Depicting Frequency Jumps Between I and II, and IV and V, and Biphonation in III ..................................................................5 3) Acoustic Spectra During Modal Singing .............................................................7 4) Acoustic Spectra During Subharmonic Throat ....................................................7 5) Several Graphs Depicting Nonlinearities in the Acoustic Spectra of Fish ........10 6) Acoustic Classical Spanish Guitar from an Unknown Luthier ..........................13 7) Squier Electric Guitar, by Fender ......................................................................14 8) Audacity with Fast Fourier Transform (FFT ......................................................15 9) Schematic View of the Experimental Setup .......................................................16 10) Audacity Data Selection……..…………………………………….…..…...…..18 11) Spectrum Plots with Smaller Window Size (Left) and Larger Window Size (Right) .................................................................................................................19 12) Ratios of Fundamental Frequency to Subharmonic Frequency for the Acoustic Classical Guitar ....................................................................................22 13) Ratios of Fundamental Frequency to Subharmonic Frequency for the Squier Electric Guitar……………………………………………….………..…….23-24 14) Chart Highlighting Near-Integer Fundamental Frequency to Subharmonic Frequency Ratios for the Acoustic Classical Guitar ...........................................25 viii 15) Chart Highlighting Near-Integer Fundamental Frequency to Subharmonic Frequency Ratios for the Squier Electric Guitar ........................................... 26-27 16) Chart Comparing Similar Fundamental Frequency to Subharmonic Frequency Ratios for Notes Actuated with the Thumb or Pick ............................................28 17) Raw Peak Frequency Data for the Acoustic Classical Spanish Guitar ...............34 18) Raw Peak Frequency Data for the Squier Electric Guitar ..................................35 iv LIST OF SYMBOLS Symbol Meaning f……………………………………………………………………….….frequency t………………………………………………...………………………...…….time FFT……………………………………………...…………Fast Fourier Transform Hz……………………………………………………………...………..…….Hertz kHz……………………………………………………..……..…………KiloHertz ms……………………………………………………………....……..milliseconds sec or s……………………………………………………………...………seconds dB…………………………………………………………..…………...….decibels x CHAPTER I INTRODUCTION 1.1 A New Theory What is sound? A simple definition is that sound is a pressure wave that travels through the air. This pressure wave can propagate through a medium and be detected by various entities, such as a person’s eardrum or a microphone. A tone is composed from one fundamental frequency and several overtones that give the note its timbre. In the past, researchers believed that the fundamental frequency was the lowest frequency a tone could produce; however, this theory has been widely disputed in recent years after researchers posited the idea of the subharmonic series. Theoretically, the subharmonic series mirrors that of the harmonic series but in the opposite direction. Instead of the overtones getting subsequently higher in frequency, the subharmonic tones get progressively lower in frequency. Subharmonics are classified as a nonlinearity in the