SUBHARMONIC 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 series, except that the fundamental 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).

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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 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

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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

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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 ? 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 that give the note its . 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 acoustic spectra and can be either be material or geometric in nature. For example, subharmonics that are an integer fraction of the fundamental are most likely geometric and result from the differential equation that defines the system. On the other hand, subharmonics that are not an integer fraction of the fundamental are most likely material

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and result from the many different materials that are used to make the guitar. Each one of these materials has its own natural frequencies that are sometimes excited during the string actuation.

1.2 A Theory Supported

After several experiments, researchers were able to support their subharmonic theory with physical evidence—they observed frequencies lower than and multiples of the fundamental frequency being present in a tone. Ever since this discovery, a significant portion of research has been characterizing subharmonics in human voices, animal voices, violins and cymbals. However, little to no research has been conducted about subharmonics in guitars. In Chapter II of this paper, several experiments showing the presence of subharmonics in human voice, mammalian and fish vocalization, violin and cymbal spectra are described. A new experiment describing the existence of subharmonics in guitar spectra is presented thereafter in Chapter III.

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CHAPTER II

BACKGROUND

2.1 Review of Previous Research

In the study, “Nonlinear Acoustics in the Pant Hoots of Common Chimpanzees

(Pantroglodytes): Vocalizing at the Edge,” researchers examined the spectra of vocalizations produced by male chimpanzees. Figure 1 outlines the four different types of non-linearities found in this study, as well subsequent studies. Region (I) shows frequency jumps, as each initial frequency almost instantaneously increases to a higher frequency, then instantaneously decreases back to the original frequencies. Region (II) depicts subharmonics, which are frequencies below the fundamental that are less intense as the fundamental frequency. Region (III) displays biphonation, which is the production of two or more frequencies that may be above or below the fundamental frequency.

Region (IV) shows deterministic chaos, as the highlighted section of this region involves random movement of the frequencies.

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Figure 1: Schematic of Four Frequency Versus Time Graphs (Riede).

The chimpanzee vocalizations, called pant hoots, were found to start off as harmonic low frequency calls that were quiet and consistent; however, the end of the vocalization became nonlinear high frequency calls that were loud and inconsistent.

Little to no subharmonic activity was found in the spectra of the early calls, although nonlinear behavior was found in the spectra of the late calls (Figure 2). The fundamental appears as a dark band that increases in frequency from (I) to (II) and decreases from (IV) to (V). Biphonation produces a frequency below the fundamental in (III), which appears as a gray band. An interesting phenomenon that occurred was the shift in frequency between the early linear calls and the late nonlinear calls. The calls with lower fundamental frequencies tended to have fewer subharmonics than the calls with higher fundamental frequencies (Riede).

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Figure 2: Frequency Versus Time Graph Depicting Frequency Jumps Between I and II, and IV and V, and Biphonation in III (Riede).

The most common area of subharmonic research is examining nonlinearities in the acoustic spectra of the human voice. In the “Acoustic Characteristics of Rough

Voices: Subharmonics” study, 389 participants’ voices were examined for the presence of subharmonics in their voices. Out of all the participants, twenty voices had significant presence of subharmonics; although eight of these twenty voices were determined to have typical amounts of jitter and shimmer, two different ways to classify nonlinearities in acoustic spectra. Jitter shows the amount of frequency instability, while shimmer shows the amount of amplitude instability. The voices were analyzed using digital techniques where their jitter, shimmer, and their power and frequency of subharmonics were examined. The researchers determined that rough voices were not only characterized by jitter and shimmer, but also the frequency of subharmonics in the power spectrum

(Omori, et al.).

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During the “Voice Source Characteristics in Mongolian ‘Throat Singing’ Studied with High-Speed Imaging Technique, Acoustic Spectra, and Inverse Filtering” experiment, a Mongolian Throat Singer was examined while singing. Mongolian Throat

Singing is an ancient singing technique that is used to amplify and generate and subharmonics. The Mongolian Throat Singer was examined while singing normally and during bass-type throat singing, a type specifically called Kargyraa. Digital equipment was used to examine the harmonics produced by the singer and it found that subharmonics were indeed produced during bass-type throat singing. There was a significant increase in the number of subharmonics found in bass-type throat singing

(Figure 4) when compared to normal modal singing (Figure 3).

Without using throat singing, there are fewer and less intense subharmonic peaks before the fundamental peak and less intense and less defined harmonic peaks after the fundamental peak. When using throat singing, there are a greater number of more intense subharmonic peaks before the fundamental peak and more intense and defined harmonics after the fundamental peak. Throat singing amplifies and defines both subharmonics and harmonics. The original figures are annotated to more easily show the subharmonic, fundamental and harmonic peaks (Figure 3) (Figure 4). The singer’s vocal folds were also examined using endoscopic technology, where it was determined that both the false and true vocal folds were involved in the production of the subharmonics observed in bass-type throat singing (Lindesdad et al.).

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Figure 3: Acoustic Spectra During Modal Singing (Lindesdad, et al.).

Figure 4: Acoustic Spectra During Subharmonic Throat Singing (Lindesdad, et al.).

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Researchers examined the spectra produced by famous rock singer, Freddie

Mercury, while speaking and singing in the study, “Freddie Mercury—Acoustic Analysis of Speaking Fundamental Frequency, Vibrato, and Subharmonics.” The researchers found that Mercury naturally spoke at a frequency within the baritone range, but often sang frequencies in the tenor range; additionally, Mercury’s vibrato was described as being faster than an average singer’s vibrato and more irregular. The researchers also found a significant number of subharmonics present in Mercury’s spectra and hypothesize how Mercury was able to produce subharmonics while singing. They posit that Mercury was somehow able to produce a 3:1 frequency locked pattern on both his true and false vocal folds. This is an unusual singing technique, as most singers only utilize their true vocal folds while singing (Herbst).

Researchers also became interested in nonlinearities produced by various animals, such as primates, other mammals and fish. In the “Subharmonics, Biphonation, and

Deterministic Chaos in Mammal Vocalization” study, researchers studied various vocalizations produced by wild African dogs. The researchers studied the spectra produced by the dog barks and observed several nonlinear behaviors. These nonlinear phenomena included subharmonics and biphonation, as well as other chaotic behaviors.

The researchers also examined the spectra of several other mammals of different taxa, where they also found nonlinear behaviors; furthermore, the researchers present and explain several linear phenomena in the spectra that might be misinterpreted as chaotic behavior for various reasons (Wilden).

Researchers examined the vocalizations of rhesus macaques, macaca mullata and other nonhuman mammals for nonlinear behavior during the study “Calls Out of Chaos:

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The Adaptive Significance of Nonlinear Phenomena in Mammalian Vocal Production.”

When examining the spectra, the researchers found that nonhuman mammals were also capable of producing subharmonics, biphonation and other nonlinear behaviors. The researchers argued that mammals evolved to be able to produce subharmonics so that their calls were distinctive, individualistic and therefore difficult to ignore. Perhaps this idea could also be extended to humans and explain why a large majority of the population enjoy Freddie Mercury’s voice—it is distinctive because of subharmonics and thus difficult to ignore (Fitch).

During the study, “Nonlinear Acoustic Complexity in a Fish ‘Two-Voice’

System,” researches examined the toadfish and its swim bladder which allowed it to produce subharmonics, a previously undocumented phenomenon in fish. The researchers divided a group of fish into a control group and an experimental group: the control group’s swim bladder was not surgically altered, and the experimental group’s swim bladder was surgically altered. The researchers examined the spectra of both groups of fish and found that the control group was able to produce a greater number of nonlinearities, such as subharmonics, than the experimental group (Figure 5) (Rice). The arrows on Figure 5 point out nonlinearities. Graph (a) shows deterministic chaos and subharmonics, and graph (b) is the power series of the linear (gray) and non-linear

(black) portions of graph (a). Graph (c) depicts biphonation, graph (d) displays frequency jumps, graph (e) points out harmonic bifurcation and graph (f) shows an averaged power series for the linear (gray) and non-linear (black) data subsets (Figure 5).

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Figure 5: Several Graphs Depicting Nonlinearities in the Acoustic Spectra of Fish (Rice).

Researchers also became interested in the nonlinearities that instruments could produce after hearing famous violinist Mari Kimura utilize subharmonics in her compositions. Mari Kimura discussed her technique on how to produce subharmonics on the violin and shared the empirical data associated with it in her study, “How to Produce

Subharmonics on the Violin.” She shared that by using various pressures and locations while drawing the bow across the strings, a number of subharmonics could be produced.

The age and type of strings were found to affect the subharmonics that could be produced; for example, twisted strings tend to support subharmonics better than other types of strings (Kimura).

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In the “Subharmonics and Plate Tap Tones in ” study, violins and similarly bowed instruments’ spectra were analyzed for subharmonics. The researchers determined that the loudness and quality of the instrument widely depended on the subharmonic series observed in the instruments’ spectral analysis. The greater number of subharmonics in the instruments’ spectra, the louder and better quality the instrument.

The researchers also determined that the properties of the wood used to make the instruments and back plates also influenced the quality and loudness. This implies that the material out of which the instruments were made influences the number of subharmonics found in their spectra (Hutchins).

During the “Subharmonic Generation in Cymbals at Large Amplitudes” study, researchers examined the frequencies, harmonics and subharmonics that cymbals produced. When a cymbal was hit, it vibrated at a low frequency with large amplitudes; however, the energy associated with the low frequency vibrations was converted to produce higher frequencies and modes via chaotic vibrations at large amplitudes. To further study this phenomenon, the researchers drove the cymbal to vibrate just below its natural frequency and observed the resulting frequencies. They discovered both the natural frequency fundamental mode and either a half or a third of the lower drive frequency in the acoustic spectra. The researchers pose that nonlinear behavior, such as subharmonics, are the reason why cymbals can convert low frequency energy at large amplitudes into higher frequencies and modes (Wilbur).

2.2 Problem Statement

Many experiments involving subharmonics have been completed; however, there are currently no experiments examining the subharmonic series present in guitars.

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Throughout this original experiment, the spectra of various guitars were analyzed and examined for any nonlinear behavior, such as subharmonics. Equipment that could produce acoustic spectra graphs, such as Audacity, were used to analyze the spectra a guitar produces when playing various tones while using various techniques.

Furthermore, different guitars were analyzed to determine if any physical characteristics or properties of the instrument aid in its subharmonic production. Characteristics such as string age and string type were analyzed to show their significance to subharmonic production. These results, if fruitful, could be used to design a guitar that can produce subharmonics more frequently and easily.

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CHAPTER III

MATERIALS AND METHODS

3.1 Required Materials

The required materials in this experiment include both hardware and software.

The required hardware is a Classical Spanish Guitar from an unknown luthier (Figure 6), a Fender Squier electric guitar (Figure 7), a Fifine brand USB microphone and a laptop.

Figure 6: Acoustic Classical Spanish Guitar from an Unknown Luthier.

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Figure 7: Squier Electric Guitar, by Fender.

The software used in this experiment is Audacity, an open office audio analyzing software. Audacity (pictured in Figure 8) has the ability to record and view the resulting spectra as a waveform or spectrogram, which is often called a Fast Fourier

Transform or FFT. A quiet place to record the guitars is also required for this experiment to produce the most accurate and precise results since background noise contributes to errant frequencies in the spectrogram.

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Figure 8: Audacity with Fast Fourier Transform (FFT).

In order to set up this experiment, a new Audacity project should be opened on the computer. The computer should be positioned conveniently so the user can see the screen and easily reach the keyboard and mouse. The microphone should be connected to the computer using the USB port and should be positioned so that it is close to the guitar and pointing towards the strings or sound hole. The individual performing the experiment may choose to sit on a chair or stand while recording. If the individual prefers to sit, they should be sure to sit close to the microphone to help reduce undesired noise in the data.

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Guitar, acoustic or electric without amplifier

Computer with Audacity audio editing software

Fifine brand microphone, connected to computer via USB Figure 9: Schematic View of the Experimental Setup.

3.2 Methods

When setting up to record, it is important to select the USB microphone from the input dropdown menu since computers have several internal and external microphones that could yield inaccurate or imprecise results. When setting up the USB microphone, the front of the microphone should be pointed towards the sound hole of the classical acoustic guitar or towards the middle of the Squier electric guitar’s strings. A distance of a few centimeters is optimal since it ensures the microphone mostly senses the sound waves produced by the guitar, thus helping eliminate background noise. The red dot near the top left of the screen is selected to start recording, then the desired note is strummed three times at approximately the same volume. Each of the three trials are allowed to fully ring out before starting the next trial. The file is saved as an Audacity project, which creates a folder and an FFT file that share the same file name; the FFT file is useful for future analysis while the folder is not.

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The first set of data was taken using the Acoustic Classical Spanish Guitar pictured in Figure 6. This guitar is strung with Ernie Ball 2409 Ernesto Palla Nylon Ball

End Classical Acoustic Guitar Strings. The high E, B and G strings are composed of nylon while the D, A and low E strings are composed of nylon wound with steel. The strings are approximately seven years old and are in fair condition, as the strings lack any oil, dirt or rust. The second set of data was taken using the Squier Electric Guitar pictured in Figure 7. This guitar is strung with Ernie Ball 3223 Super Slinky Nickel

Wound Electric Guitar Strings. All six strings on the guitar are composed of Nickel, with the low E, A and D strings wound with Nickel. The strings are approximately five years old and are in fair condition, as the strings lack any oil, dirt or rust.

The variable examined in the first set of data are the different strings; the method of strumming a string with the thumb remains constant while the string and fret number varies. Data is taken sequentially from frets zero to twelve for the low E, A, D, G and B strings. The low E, A, D and G strings’ data is taken without silencing strings that are not in use while the B string’s data is taken with silencing strings that are not in use.

Upon analyzing the low E, A, D and G strings’ spectra, it is discovered that lower unsilenced string frequencies consistently appear in the FFTs. This indicates that the vibration of the desired string is exciting unwanted vibration in lower strings, thus producing more noise in the data. The raw frequency data from the low E, A, D and G strings is further examined for non-variable subharmonic frequencies that are similar to lower strings’ fundamental and/or harmonic frequencies. Once these frequencies are located, they are removed from the data set and no further analysis is done to them. In order to reduce non-subharmonic lower frequency noise in the data for the B and high E

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strings, the unused strings are silenced by firmly placing the hand and/or fingers over the lower strings while strumming the desired note.

The first set of data is analyzed using a combination of Audacity and Microsoft

Excel. The dropdown menu on the left side of the screen allows the type of graph to be changed from a waveform to a spectrogram or FFT. Using the same menu, the spectrogram settings can be altered to zoom in on a desired range of frequencies. Once the settings are changed so the fundamental and possible subharmonic frequencies are in view, the cursor is used to select a small section of the spectrogram immediately after the initial strum, which appears as a thick vertical band (Figure 10). It is important that the region immediately after the initial strum is selected since subharmonics are usually unpredictable and can die out quickly; therefore, analyzing the data immediately after the initial strum ensures that all possible subharmonics are included in the analysis.

Figure 10: Audacity Data Selection.

After highlighting this region, “plot spectrum” is selected under the “analyze” tab at the top of the screen. A small intensity versus frequency graph should appear in front of the main Audacity screen. The frequencies of the subharmonics are found by hovering the cursor over a peak and recording the frequency displayed below. This process is

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repeated for all subharmonic and fundamental peaks on the spectrum graph, and the results are tabulated in a Microsoft Excel workbook. Once the subharmonic and fundamental peak frequencies are inputted for frets zero through twelve for the low E, A,

D, G, B and high E strings, the subharmonic frequencies are compared to the fundamental frequencies on a separate chart.

Figure 11 contains examples of the spectrum plot obtained for an open A string using window sizes 4096 and 16384. The larger the window size, the finer the resolution of the resulting spectrum plot and the peak frequencies are more precise. This means that using a larger window size can clarify peaks that were initially difficult to interpret. A larger window size will also widen the horizontal frequency axis, allowing even higher and lower frequencies to be visible. Therefore, using a larger window size can also reveal additional peaks that were originally cut out of frame when using a smaller window size.

Figure 11: Spectrum Plots with Smaller Window Size (Left) and Larger Window Size (Right).

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The ratios between the fundamental frequencies and subharmonic frequencies were calculated thereafter.

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CHAPTER IV

RESULTS AND DISCUSSION

4.1 Results

The experimental work presented in the previous chapter resulted in creating five different charts: Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. Each one of these charts depict the ratios of fundamental frequency to subharmonic frequency for the given string and fret. Each column of each chart is labeled with the string name and fret number. The string names are the low E, A, D, G, B and high E. Please note that the high E string is represented with a lowercase “e” so that it is not confused with the low E string. The fret numbers are indicated by the number immediately following the string name and are integer values from 0 to 12. Please note that the zeroth fret indicates an open string, meaning that the string was not fretted while being strummed. The color gradient is assigned based on the value of a cell, according to the key at the bottom of the respective figures. Figure 12, Figure 13 and Figure 16 have color gradients that color the ratios based on their values, even if the ratios are integers or close to being integers.

Figure 16 shows a side-by-side comparison of fundamental to subharmonic frequency ratios for strings actuated via thumb and pick. Figure 14 and Figure 15 have color

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gradients that only color near-integer values, which are defined as numbers that fall within ten percent of an integer.

Figure 12: Ratios of Fundamental Frequency to Subharmonic Frequency for the Acoustic Classical Guitar.

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Figure 13: Ratios of Fundamental Frequency to Subharmonic Frequency for the Squier Electric Guitar.

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Figure 14: Chart Highlighting Near-Integer Fundamental Frequency to Subharmonic Frequency Ratios for the Acoustic Classical Guitar.

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Figure 15: Chart Highlighting Near-Integer Fundamental Frequency to Subharmonic Frequency Ratios for the Squier Electric Guitar.

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Figure 16: Chart Comparing Similar Fundamental Frequency to Subharmonic Frequency Ratios for Notes Actuated with the Thumb or Pick.

4.2 Discussion

Figure 12 is a chart displaying the fundamental frequency to subharmonic frequency ratios for each string from fret zero (open) to fret twelve for the classical acoustic guitar. Figure 13 is a chart displaying the fundamental frequency to subharmonics frequency ratios for each string form fret zero (open) to fret twelve for the

Squier electric guitar. The ratios are colored according to their values; the key is on the bottom of Figures 12 and 13 and it displays the relationship between color and value.

Mostly reds, oranges and yellows are visible when viewing Figures 12 and 13, meaning that the majority of fundamental frequency to subharmonic frequency ratios are between one and three. This observation is consistent with Mari Kimura’s violin study, as she also found that the first and second subharmonics were more prevalent than lower

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subharmonics. In general, more subharmonics are present on higher strings, such as the

G, B and high E strings, which is consistent with Tobias Riede’s study involving chimpanzee vocalizations. Perhaps the reason why more subharmonics are present on higher strings than on lower strings pertains to the lower strings being bound in steel while the higher strings are not.

Furthermore, the ratio values are bolded and italicized when they are within ten percent of an integer value. These values are especially visible in Figures 14 and 15, where they are highlighted, and all other non-integer values are not colored. Figures 14 and 15 highlight the fundamental frequency to subharmonic frequency ratios that are within ten percent of an integer value. The color used to highlight the ratios depends on the integer value and can be found in the key on the bottom of Figures 14 and 15.

Multiple near-integer ratios occur on each string, but not necessarily every fret. Mostly yellows and oranges are visible when viewing Figures 14 and 15, meaning that the majority of near-integer subharmonic ratios are between two and three, which is also consistent with Mari Kimura’s violin study. Higher strings tend to have more near- integer ratios on almost all frets while lower strings do not. Additionally, higher strings have repeated near-integer ratios on almost every fret, thus indicating the consistent presence of the first and second subharmonic frequencies, which is consistent with both

Mari Kimura’s violin study and Tobias Riede’s chimpanzee vocalization study.

Figure 16 shows a side by side comparison of fundamental frequency to subharmonic frequency ratios generated by thumb and pick for selected frets. Both the thumb data set and pick data set contain subharmonic frequencies, which is why ratios for both data sets could be generated. In other words, the method of plucking the string does

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not disrupt the presence of subharmonic frequencies. The ratios were arranged so that similar values would be adjacent to each other and therefore easier to recognize. There are many ratios that are shared by both data sets, especially for higher strings. This indicates that the thumb data set and the pick data set contain similar subharmonic frequencies, which further shows that the method of plucking the string does not disrupt subharmonic activity, especially for higher strings.

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CHAPTER V

CONCLUSION

5.1 Conclusion

Evidence of subharmonics was published in several studies involving the human voice, mammalian and fish vocalizations, violins and cymbals. Of these studies, Mari

Kimura’s violin study and Tobias Riede’s studies characterized the nature of subharmonics; Kimura’s study revealed that subharmonics are most commonly one half or one third of the fundamental frequency, while Riede’s study showed that higher notes tend to have more subharmonic frequency activity than lower notes. These observations also hold true for the experiment described throughout this paper. Evidence of subharmonic frequencies in Acoustic Classical Spanish Guitar as well as Squier Electric

Guitar spectra exists, regardless if the frequencies are generated via thumb or pick. The majority of subharmonics generated have a frequency that is either one half or one third of the fundamental frequency and have both near-integer and non-integer fundamental frequency to subharmonic frequency ratios; however, near-integer ratios have a greater and more frequent presence in higher string data.

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5. Acoustical Society of America, 28 August 2018. Web. 19 April 2019.

Wilden, I.; Herzel, H.; Peters, G. & Tembrock, G. “Subharmonics, Biphonation, and

Deterministic Chaos in Mammal Vocalizations.” The International Journal of

Animal Sound and its Recording, volume 9 issue 3, pages 171-196. Taylor &

Francis Online, 13 April 2012. Web. 19 March 2019.

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APPENDIX

Figure 17: Raw Peak Frequency Data for the Acoustic Classical Spanish Guitar.

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Figure 18: Raw Peak Frequency Data for the Squier Electric Guitar.

35