What does the guitalele like sound: Guitar or ukelele? An application of the K-means cluster method

Sound can be defined as a form of vibrational energy that propagates through a material medium, such as air, water, or solids, in the form of pressure waves that are generated when a sound source produces mechanical vibrations that cause the vibrations of molecules in the medium to move in patterns of compression and rarefaction. The “color” of sound, also called timbre, is a distinctive aspect of musical sound that allows us to differentiate between instruments playing the same note.¹ Its unique character arises from several quantitative elements, including the harmonic spectrum of an instrument, its temporal envelope detailing how sound evolves over time, and the presence of formats or amplified frequency bands. In addition, factors such as inharmonicity, where produced harmonics are not exact multiples of the fundamental frequency, and inherent noise in some instruments further contribute to an instrument’s timbre.

To analyze timbre quantitatively, tools like Fourier analysis are employed to dissect a sound into its frequency components while spectrograms visualize the spectral content’s evolution.² However, despite these analytical methods, timbre perception remains inherently subjective, influenced by various cultural, experiential, and biological factors.^3,4

The relationship between sound and musical timbre is closely linked to the acoustic characteristics of sound waves. While sound encompasses the basic physical properties of a wave, such as frequency and amplitude, timbre refers to the unique and distinctive qualities that enable us to differentiate between various sound sources, like musical instruments or voices. The timbre is a fundamental element of music, playing a critical role in the creation of melody, harmony, and rhythm.

According to a study by Leman and Godøy,⁵ timbre can be considered one of the most important aspects of musical perception. It can be closely linked to timbre and melody, and it is vital for the emotional impact of music because it can be used to create tension, release, and resolution in music.

These properties from a physical point of view can be considered as a function that is determined by the way sound waves propagate and combine in an instrument or sound source, related to several physical aspects of sound, such as waveform that determines whether the sound is softer or more abrupt, harmonic composition that refers to the different frequencies and amplitudes of the waves that make up the sound, in fact, each musical instruments has a different combination of his harmonics,⁶ and sound envelope that describes how the intensity of the sound changes over time, affecting the perceived attack, sustain, and decay of a sound. These factors influence how we perceive and distinguish different musical instruments, even when they play the same note. In particular, in the context of stringed instruments, the timbre is produced by the vibration of the strings, which can be altered by changing the length, tension, or thickness of the strings, the geometry of the instrument, and the interpretation of the receiver.^7–9 

The guitalele is a hybrid instrument between the Guitar and the Ukulele. It is an instrument with the size of a ukulele but very similar to a guitar in its geometry and general composition. The guitalele, similar to the guitar, has six strings but is tuned to a different note (three tones or six semitones above). In the market, the guitalele is traded as an instrument that has a similar or very similar sonority to a ukulele but with the same postures of the guitar, which is also a perception to listing a guitar, while on the other hand, you can have the perception to earn a guitar.

In this work, we seek to explore the timbre of various instruments, focusing specifically on the guitalele. Our objective is to investigate how listeners perceive the guitalele in comparison to these instruments, particularly in terms of its acoustic resemblance to either. By conducting perception studies and acoustic analyses, we aim to quantitatively assess the timbre of the guitalele and determine which instrument it most closely resembles in different contexts. To objectively quantify timbre, we utilize methods such as Fourier analysis. Also, our study extends the analysis by incorporating artificial intelligence techniques, specifically applying a K-means clustering algorithm to harmonic frequency data from three instruments: the guitar, ukulele, and guitalele. The objective is to classify the timbre based on the harmonic content of notes spanning a two-octave range, from C4 to A5 [Fig. 1(a)]. This approach not only provides a statistical answer to our research question but also seeks to bridge subjective perception with objective acoustic analysis.

To study the timbre of musical instruments used in the work, it is necessary to record and digitize their sound. For this purpose, a microphone connected to a computer is placed to 5 cm of the sound hole of each instrument, and the pulses produced when playing a specific note are recorded separately for each of the three instruments [see Fig. 1(b)]. The method of exciting the instrument’s strings involves plucking them with a 0.71 mm pick. All notes were performed using L_eff = L/2, where L corresponds to the length of the vibrating string. In our experiment, the notes C, E, G, and A were recorded on all three instruments across two different octaves (from C4 to A5). For each note (on all three instruments), 4 or 5 pulses were recorded within a 15-s interval. The instruments used include an Ibanez guitar 4/4 (model GA6CE-AM), a Mahori Mahogany guitalele (model MAH-50EQ), and a Scorpion soprano ukulele (model SCKUK-N21). Just for comparison, all three instruments used in this experiment have nylon strings because differences have been reported in the sound of a metal-stringed instrument.¹⁰ A USB Microphone Maono Cardioid Condenser (model AU-902) sample rate 44.1–48 kHz was used to capture the pulses. All measurements were carried out in an isolated and closed room. The specific string and fret used for each instrument are summarized in Table I.

To understand the perception of the timbre of stringed musical instruments, a survey was designed using Google Forms, targeting individuals with varying levels of musical experience. A total of N = 35 responses were obtained. In the survey, participants were presented with an audio clip of a guitalele melody featuring a triad of chords. The first question asked participants to listen to this audio clip and then indicate, on a scale of 1–5, whether the perceived sound was more similar to a guitar or a ukulele, with 1 being “completely guitar” and 5 “completely ukulele.” The second question assessed the respondent’s level of musical knowledge, offering three response options: advanced, basic, or none. The third question determined whether the respondents could play any musical instrument, with options to select ukulele, guitar, another instrument, or none. Participants were informed before answering that their responses would be used exclusively for statistical purposes in the current study. The audio files were recorded under the same technical conditions as those used for the acquisition of pulses. In addition, each subject listened to the recordings through their own headphones or speakers, introducing a potential bias that was not considered in this study. It is also important to note that the audio recordings sent to the survey participants correspond to cadences of notes strummed on the Mahori guitalele, so they are more complex sounds than those analyzed in this experiment. The audio recording used in Google Form is attached in the supplementary material.¹¹ 

The results obtained from the survey are shown in the graph in Fig. 2 (up). Here, it is evident that the majority of the sample identified the instrument they heard as a ukelele, while a minority identified it as a guitar. This suggests a broad spectrum of variables, leading to the conclusion that the “guitalele” does not have a unique characteristic timbre. In addition, Fig. 2 (down) displays the correlation matrix of the data obtained from the survey. This matrix indicates that based on the other survey questions, there is no significant relationship between musical knowledge (non-professional), playing an instrument, and perception that would bias the study toward specific populations.

Each musical instrument produces a distinct waveform not only because it is a superposition of waves of different frequencies and amplitudes, corresponding to the harmonics of the instrument, but also due to the presence of transient and sometimes non-harmonic components associated with the vibrating structure of the instrument. These components are particularly evident during the transient attack of a note and are crucial for the character and sonic identity of the instrument. These aspects highlight the complexity of musical timbre, emphasizing that beyond sustained harmonics, the way in which the strings are played and their state of tension, as well as the unique features of the instrument’s vibrating structure, play fundamental roles in producing the distinctive sound of each instrument.^12,13 For the same reason, the waveform is a source that can provide us with information about the nature of the instrument’s sound. Figure 3 displays the waveforms of three instruments playing a C4 note, allowing for a comparison of the tonal richness of each instrument due to the complexity of the observed patterns. In the case of the guitar, the waveform looks quite harmonic, which means that the contribution of the rest of the harmonics is quite lower than the contribution of the fundamental frequency (261 Hz). On the other hand, the Ukulele presents a waveform in which at least a couple of harmonics contribute to generating a more complex pattern than that of the guitar. Finally, the waveform of the guitalele (central graph), in the case of the C4 note, presents an intermediate behavior (with respect to the guitar and the ukulele), since the waveform is clearly governed by the fundamental frequency, but the oscillations in its waveform show that it is being modulated by higher order harmonics. Despite these differences, the information gleaned from the waveforms alone is not sufficient at discerning the distinct timbres or perceptions of each instrument, as the envelopes of each waveform exhibit similar characteristics.

To delve deeper into our question, we will study the frequency response of each instrument. To achieve this, we will analyze the 4–5 recorded pulses of each note of a specific instrument [see Fig. 4 (above)], comparing them with each other using the cross-correlation function in MATLAB. The objective of this action is to discard pulses that are not representative of the group. After this, with the pulses selected, we will take the first 30 cycles after the maximum amplitude [see Fig. 4 (below), red zone]. This approach is chosen because the initial rise of the pulse to the maximum peak amplitude is strongly influenced by the initial strike of the string, over which we do not have much control other than the skill of the musician playing the string with the pluck. We do not take more than 30 pulses because in several cases, there is a rapid decrease in the amplitude of the pulses, most notably in the ukulele, resulting in a loss of sonic richness over time as the string vibrates predominantly at its fundamental frequency. All data analysis and processing were performed in the MATLAB environment.

Figure 5 illustrates the results of the above analysis for all instruments playing the note C4. In this analysis, the Fourier transform was used to identify the main frequencies that contribute to the waveform of each instrument and their correspondence with a staff in the key to G major. Through this analysis, it can be seen that in the three instruments, the most important contribution is made by the fundamental harmonic, which is expected because the instrument was excited at L_eff = L/2.¹⁴ In the same way, it can be seen that in the three instruments, the odd harmonics (1, 3, 5, 7, …) have much more relevance than the even harmonics (2, 4, 6, 8, …). Another common factor is that in all cases, the contribution of the higher harmonics (n > 10) is much less significant than the first harmonics. In this particular case, it is observed that the guitar has some contributions of frequencies lower than the fundamental frequency that are not observed in such an important way in the guitalele and the ukulele. These contributions were not taken into account in this analysis.

Based on this analysis, this study proposes to examine the dispersion of the peaks of the Fourier transform for each note studied in this system (see Table I) and compare the form that each of them adopts. From Fig. 6, we can see that in all instruments, the first harmonic has a strong presence with respect to all the rest of the harmonics. At the same time, in the three instruments, the odd harmonics have a greater contribution than the even harmonics, except in some particular cases where the second harmonic is the contribution of greater amplitude after the first harmonic as is the case of the Ukulele in G4 and A4 and the guitalele in G5. In all cases, the upper harmonics (n > 10) have a small amplitude compared to the central peak, so their contribution to the waveform is relatively minor. This result is consistent with the audio generated in sound editing programs, where only the contribution of the first 5 or 6 harmonics of an instrument is considered.

Despite the insights provided by preliminary data, it remains unclear how the harmonic distribution of the guitalele is compared to that of the guitar or the ukulele. This ambiguity underscores the need for a more refined analytical method to ascertain the instrument’s acoustic properties. To bridge this gap, we propose a novel approach that examines the Fast Fourier transform (FFT) spectra of notes produced by each instrument. By employing unsupervised artificial intelligence techniques, specifically clustering, we aim to segment this data effectively. This method will allow us to deduce, with quantitative backing, the instrument category each sound sample most closely aligns with.

We plan to utilize the K-means algorithm to categorize the resulting data into distinct groups. This step is crucial for objectively defining the acoustic signature of the guitalele. Our primary objective is to determine whether its timbre more closely resembles that of a guitar or a ukulele, or if it exhibits a unique sonic identity altogether. By leveraging cutting-edge data science techniques, this methodology will clarify the guitalele’s timbral affiliations, providing a solid foundation for comparative analysis and further exploration in musical fidelity.

The K-means algorithm, known for its simplicity and low computational complexity, partitions a dataset into K predefined distinct clusters. The process begins with the arbitrary initialization of K cluster centers. Data points are then assigned to the nearest cluster based on the Euclidean distance metric, and the cluster centers are recalculated as the mean of the assigned points.¹⁵ This iterative process continues until cluster memberships stabilize, indicating convergence. However, the algorithm’s performance is sensitive to the initial center placement and may require multiple runs to achieve optimal clustering. It is also prone to converge to local minima and is influenced by the choice of K, with different values yielding different results. The initial centroid selection, which is random, can significantly affect the final clustering outcome.

Prior to the clusterization process, the elbow method was applied. This method is a crucial technique in cluster analysis, especially when the number of clusters is not predefined. It involves plotting the within-cluster sum of squares (WCSS) against the number of clusters, and identifying the point where the rate of decrease sharply changes, resembling an “elbow.”¹⁶ This point is considered to represent the optimal number of clusters, balancing between maximizing variance explained and minimizing the number of clusters.

In this work, the elbow method was meticulously applied to the dataset derived from the FFT analysis of every note’s timbre across three different instruments: guitar, guitalele, and ukulele. Given the dataset’s composition of six variables—frequency (reflecting the harmonic components identified in the FFT analysis), four amplitude measurements (indicating the strength of these components across four attempts), and the instrument type—the method provided a systematic approach to discern the inherent grouping within the data without prior assumptions on cluster numbers.

Furthermore, the dataset underwent preprocessing, which involved transforming categorical variables into numerical values and normalizing the entire dataset. The results, plotted in Fig. 7, indicated a significant change in the slope when the number of clusters reached three. This observation is noteworthy, considering that our study’s set of instruments comprises two types. This finding validates our experimental approach, demonstrating consistency with the data analysis and suggesting a nuanced classification beyond the apparent binary categorization of the instruments.

Considering the aforementioned analysis, it was decided to cluster the complete dataset into three groups, aiming to discern whether the guitalele corresponds to a guitar, a ukulele, or a distinct sound category. The results, shown in Table II, reveal interesting insights that correspond to the clustering of the three musical instruments, where the K-means algorithm was applied with 25 different initial configurations. The results are intriguing as many of the fundamental frequencies of the three instruments are clustered in the same group C₁. This suggests that the sound response of the three instruments is similar, as they are all plucked string instruments playing the same note. The differences, as expected, appear in the higher-order harmonics. For both the guitar and the ukulele, almost all harmonics, excluding in most cases the fundamental frequencies and the harmonics related with G5 and A5 for the guitar, are clustered into two additional distinct groups C₂ and C₃. For the guitar, the majority of harmonics with n > 1 are clustered in C₂, while for the ukulele, a similar pattern is observed but its higher harmonics are clustered in C₃. Therefore, we identify cluster C₂ as group “G” related to the guitar and cluster C₃ as “U” related to the ukulele. In the case of the guitalele, the decomposition of its notes into clusters yields even more interesting results. All the first harmonics (or fundamentals) are, as expected, categorized in C₁, but the majority of the higher harmonics (n > 2) are clustered in either group “G” or “U,” except the harmonics related with G5 and A5. In notes G5 and A5, the clustering tells us that the cluster C₁ is a group that, in this case, groups the fundamental frequency together with higher order harmonics from both the guitar and the ukulele, which could mean that in these high notes, the guitar and the guitalele have a very similar sound. It is important to note that only the first ten harmonics are shown, as, in terms of the frequency response of both the guitalele and the ukulele, harmonics n > 10 present a very small amplitude compared to the first harmonics. Due to the important weight of the first harmonic, in the following analysis, we repeat the procedure, but this time without considering the fundamental frequency when the system chooses the clusters. As in the previous case, the system again generates three well-defined groups and two of these clusters can be associated with both the guitar and the ukulele because they mostly appear in the particular analysis of each instrument (Table III). In the new clusterization, a third cluster appears, which we call C₄, which apparently seems to group mostly the high-frequency and low-amplitude harmonics common in the three experiments for n > 9. This new analysis again shows that for the case of the guitalele (central columns of Table III), the representation of its harmonics consists of a combination of guitar, ukulele, and C₄ for high frequencies. This analysis also shows us that it seems that the lowest notes of the guitalele (C4, E4, A4) are almost entirely associated with the ukulele, while the highest notes (C5, E5, G5, A5) have a predominance of the guitar cluster, which is consistent with the previous analysis for A5 and G5.