SPAR-Timbre: Unlocking Non-linear Dynamics in Music Audio with Symmetric Projection Attractor Reconstruction

SPAR-Timbre: Unlocking Non-linear Dynamics in Music Audio with Symmetric Projection Attractor Reconstruction Mateusz Solinski1,2[0000→0002→6461→1738],WentaoHao 2[0009→0005→4207→4838],and Elaine Chew1,2[0000→0002→8342→1024] 1School of Biomedical Engineering and Imaging Sciences, Faculty of Life Sciences and Medicine, King’s College London, London, United Kingdom [email protected] https://cosmos.isd.kcl.ac.uk/ 2Engineering Department, Faculty of Natural, Mathematical and Engineering Sciences, King’s College London, London, United Kingdom Abstract. This study investigates the use of the Symmetric Projection Attractor Reconstruction (SPAR) method as a novel approach to visualising and analysing musical sounds and the dynamic changes in their signals. Rooted in the principles of deterministic chaos, the SPAR method reconstructs attractors in N-dimensional phase space to reveal unique geometrical patterns that distinguish di!erent instruments, unveiling complex non-linear structures. SPAR is relatively robust to noise as it constructs the underlying attractor structure. Using an unsupervised machine learning method (hierarchical clustering), we identify distinct attractor shapes for string and wind instruments. String instruments exhibit more symmetric attractors, whereas wind instruments show greater variability and asymmetry. The clustering accuracy for distinguishing between string and wind instruments was 93%, and 71% for di!erentiating between string, woodwind, and brass instruments, demonstrating the e!ectiveness of this method for automatic timbre classification. The study further highlights significant variability in attractor shapes in one-second-long windows of the sound of an instrument playing a given pitch. The observed rapid transitions between stable and chaotic states underscore the complexity and dynamic nature of sound signals. Additionally, a comparison of the attractors between sounds generated using the VST (Virtual Studio Technology) libraries (BBC Orchestra, Apple Studio Strings) and recorded samples for violin was performed to show di!erences associated with the sound source. This research demonstrates the potential of the SPAR method for sound analysis, providing valuable insights for music information retrieval and serving as an input option for machine learning models. Keywords: Timbre ·Sound visualisation ·Attractors ·SPAR. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 784 2M.Solinskietal. 1Introduction Sound, often defined by its frequency spectrum, is a complex phenomenon that results from the combination of waves of varying lengths, amplitudes, and phases. These waves interact in intricate ways, leading to the modulation of non-linear properties that contribute to the dynamic structure of sound. Timbre, or tone colour, refers to the characteristic quality of sound that enables us to distinguish between di!erent sound sources, such as musical instruments or voices, based on these complex wave interactions. The human ear is remarkably adept at recognising timbre, with listeners able to reliably identify the instrument producing a sound by its unique spectral properties [1]. In Carol Krumhansl’s experiment with short music clips, some listeners have been able to identify artists and song titles from slivers of sound only 300400ms long [2]. In the international computer music community, David Wessel was a seminal figure who pioneered research on timbral spaces since the early 1970s, representing and creating a controller for timbre manipulation in two-dimensional space based on dissimilarity judgements [3]. Such perceptual and cognitive representations of timbral space are complemented by acoustical modelling using audio descriptors [4]. In the field of Music Information Retrieval (MIR), the automatic recognition and analysis of sound properties are essential for applications such as music classification, sound synthesis, and instrument identification. A variety of methods have been developed to analyse sound, ranging from spectral analysis (such as spectrograms and cepstral analysis) to time-domain features (such as amplitude envelopes and zero-crossing rates). Recent advancements have incorporated machine learning techniques, leveraging traditional features alongside convolutional and recurrent neural networks to enhance sound classification [5]. While these approaches provide valuable insights, they often overlook the complex, non-linear dynamics inherent in sound signals. This complexity can be better captured through methods that focus on the geometry of sound signals in phase space. In this context, chaotic systems and non-linear dynamics, often used in analysing self-organising systems, o!er a promising framework for understanding the deeper geometry of sound signals. Previous studies, such as those by Pietro Di Lorenzo, have demonstrated that di!erent instruments produce distinct patterns in phase space, which can be di!erentiated through clustering techniques [6]. Here, we introduce the use of the Symmetric Projection Attractor Reconstruction (SPAR) method to visualise and analyse the complex dynamic patterns of musical sounds. Grounded in the physics of deterministic chaos, this method visualises and analyses attractors derived from pseudo-periodic signals. Using Takens’ embedding theorem, the SPAR method creates attractors for a defined embedding dimension. Originally applied to problems in medical research, such as ECG signal classification, respiratory pattern di!erentiation, and genetic mutation detection, the SPAR method has proven its versatility and e!ectiveness in extracting meaningful features from complex signals [7] [8] [9] [10] [11]. This paper presents a novel way to visualise and analyse sound signals, applying the Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 785 Title Suppressed Due to Excessive Length 3 Fig. 1. Examples of three attractors obtained from one-second-long sound signals of clarinet, horns, and violins. (Lower row) rand ωdensity and outline profiles calculated for these attractors. SPAR method to musical sound samples to distinguish between di!erent instruments. 2DataandMethods 2.1 Sound samples 10-second sound samples were generated in Logic Pro (Apple, US, v. 11.1.2) using the BBC Symphony Orchestra VST (Virtual Sound Technology) plug-in (v. 1.7.0; Spitfire Audio, UK). The samples are generated using sounds of 14 orchestral instruments from three categories: strings (violins, violas, cellos, basses), woodwinds (piccolos, flutes, oboes, clarinets, bassoons), and brass (horns, trumpets, tenor trombones, bass trombones, and tubas). All instruments were set to play the same pitch, A3 (220 Hz), except the instruments whose pitch was out of their range, thus, double bass (set to A1, 55 Hz), bass trombone and tuba (A2, 110 Hz), and piccolo and violin (A4, 440 Hz). All samples were exported to an audio file in .wav format (with a standard sampling of 44,1 kHz). The dataset was supplemented by audio samples produced by a professional violinist (Schoeps Stereo-Set MK 4 microphones), as well as a single violin sample generated using a di!erent virtual instrument (Studio Strings from Apple’s Logic Pro). This allowed for a comparison of attractor patterns across violin sounds from di!erent sources. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 786 4M.Solinskietal. 2.2 Symmetric Projection Attractor Reconstruction (SPAR) The SPAR method was used to construct attractors from the audio signals, highlighting di!erences in phase space properties between instruments and capturing dynamic changes in one-second windows of the same instrument. The complete description of the SPAR method can be found in [8]. Briefly, the method reconstructs an attractor in an N-dimensional phase space, which serves as a non-linear representation of the time series. The first step involves setting the attractor dimension N (here, N = 10), which is also the number of points from the time series that create a single point in the phase space. These points are separated by a constant delay, usually related to the length of the cycles in the time series; in this implementation, the delay is calculated as 1/N ·ω, where ωis the fundamental frequency of the sound sample. The reconstructed multidimensional phase space is then projected onto a 2-dimensional map while preserving its geometric properties. Finally, a heat map is generated to represent the density of points along di!erent trajectories in the attractor. Next, descriptors are obtained from the heat maps. The heat maps are analysed in polar coordinates by generating radial (r)andangular(0→ε→2ϑ) density profiles according to distance from the centre of the attractor and based on angle from a reference, respectively, as well as an outline profile, which is obtained from the maximum radius rwith non-zero density for each angle ε. The example of the attractors created from one-second windows and the associated profiles for three di!erent instruments are shown in Figure 1. 2.3 Hierarchical clustering Attractors from di!erent instrument classes (string, woodwind, brass; BBC Orchestra) are compared and clustered using hierarchical clustering, and visualised as a dendrogram (using Euclidean distance as the metric and the Ward variance minimisation algorithm). Three parameters, identified in the preliminary analysis as the most distinctive, are extracted from the density profiles: (a) the index of rwith maximum density, (b) the number of zero-crossings in the εdensity plot, and (c) the attractor outline profiles. 3Results The results reveal a high degree of variation in attractor shapes, both between sounds of instruments and between one-second windows of sound from the same instrument. Figure 3 (left) shows the dynamical changes in the attractors of one-second windows of the double bass sound. In most windows, we observe a highly symmetrical rosette shape. Figure 4 (left) shows the attractors extracted from the sound of the clarinet. We observe changes in rdensity between consecutive windows. In contrast, the attractors based on the sound of the trumpet (Figure 4, right) usually depict chaotic trajectories (except windows 1 and 9). The attractors from the sound of the violin (Figure 3, right) are characterised Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 787 Title Suppressed Due to Excessive Length 5 Fig. 2. Examples of three attractors obtained from one-second-long sound signals of clarinet, horns, and violins. (Lower row) rand ωdensity and outline profiles calculated for these attractors. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 788 6M.Solinskietal. 1 2 3 4 5 6 7 910 10 seconds 8 1 2 3 4 56 7 8 910 10 seconds Double bass Violin Fig. 3. (1-10) Changes in attractor dynamics during di!erent one-second windows of the double bass (left) and violin (right) sound signals. (Bottom) The attractors created from the whole ten-second windows. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 789 Title Suppressed Due to Excessive Length 7 by changes in rdensity and by the stability of the trajectories. The dynamical changes of the attractors can also be visualised as a video available online 1. Cluster analysis is applied to attractors from the full 10-second samples for each instrument (see examples at the bottom of Figures 3 and 4) to encapsulate all the dynamic behaviours within one attractor. The dendrogram based on three parameters extracted from the density profiles is shown in Figure 2. Using a threshold around 1, we identify three distinct clusters: one containing predominantly string instruments (along with horns), and the other two groups consisting of wind instruments (brass and woodwind). The accuracy for classifying the instruments into three groups is 0.71, while the distinction between string and wind instruments shows a higher accuracy of 0.93. We compared the sound of the violin playing A4 (440 Hz) from four di!erent sources: two virtual instruments (VST) plug-ins, one real audio recording, and a synthetically generated sine wave. This comparison aimed to explore similarities and di!erences in their corresponding attractors. For each source, a 1-second segment—visually selected from a 10-second signal for best pattern clarity—was analysed. The resulting attractors and their corresponding profiles are shown in Figure 5. All violin attractors exhibit a characteristic rosette-like pattern, though it is less distinct for the BBC Orchestra sample, likely due to the combined sound of multiple violins. This di!erence is also evident in the profile plots: while the recording and the solo instrument sample from Studio Strings show similar profiles, those from the ensemble are more irregular, especially in the ε density. Nevertheless, the di!erences between the profiles presented for di!erent violin sounds are much more similar to each other than the profile based on attractors from di!erent instruments, as observed in Figure 1. As expected, the sine wave attractor forms a circular shape, and its profiles remain nearly constant across both r and ε. The animated evolution of attractors for the violin sounds was presented as a video available online 2. 4Discussion In this study, we explored the dynamic structure of sounds using the SPAR method, an approach grounded in the study of chaotic systems. The analysis of sound signals through SPAR allowed us to visualise and di!erentiate the unique phase-space attractors of various musical instruments, revealing distinct geometrical patterns related to their timbre. The clustering analysis revealed that the most significant di!erence in attractor shapes occurs between string and wind instruments (woodwind and brass). Attractors from string instruments tend to be more symmetric, exhibiting a characteristic rosette shape, which produces distinct fluctuations in the εdensity profile and the attractor outline. Consequently, the number of zero crossings in these profiles emerged as the most distinct feature for string instruments. In 1https://youtu.be/LHUKXcta1PA 2https://youtu.be/Bj3V-XV5ybc Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 790 8M.Solinskietal. Clarinet Trumpet 1 2 3 4 5 6 7 910 10 seconds 8 1 2 3 4 5 6 7 910 10 seconds 8 Fig. 4. (1-10) Changes in attractor dynamics during di!erent one-second windows of the clarinet (left) and trumpet (right) sound signals. (Bottom) The attractors created from the whole ten-second windows. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 791 Title Suppressed Due to Excessive Length 9 Fig. 5. Examples of four attractors obtained from one-second-long sound signals of the violin (audio recording, Studio Strings VST plug-in, and BBC Orchestra, string section plug-in), and a sine wave (440 Hz). (Lower row) rand ωdensity and outline profiles calculated for these attractors. contrast, the attractors from wind instruments are generally less symmetric and show greater interand intra-instrument variability in shape. SPAR reconstructs the underlying attractor geometry of the signal, which can reveal hidden temporal and spectral dependencies beyond traditional frequencydomain analysis (like Fourier or wavelet transforms). The results from one-second windows highlighted the rich and non-linear dynamic nature of attractors and their properties in phase space. The SPAR method o!ers new and visual ways to characterise timbre that can serve as input for machine learning models. The comparison of violin sounds from various sources and configurations highlights distinct variations in attractor shape and complexity between solo and ensemble (section) performances. These distinctions likely reflect subtle timbral nuances contributed by individual instruments within the ensemble. This study, while providing valuable insights into the application of the SPAR method for sound analysis, has several limitations. First, we used only one sample per instrument (except for violin). Expanding the analysis to include a wide range of pitches, expressive nuances like variations in physical control of instruments, and a broader set of sound libraries (more instruments, more instances of each instrument) will be necessary to gain a more robust understanding of the dynamic behaviours of the attractors. Second, the dimension of the phase space was arbitrarily set to 10, without optimisation. According to Takens’ theorem, the optimal embedding dimension for a given time series can be determined by minimising the time-delayed mutual information. Future work should focus on exploring the e!ect of di!erent embedding dimensions on the resulting attractor properties to assess the impact of phase space dimension on the analysis. Proc. of the 17th International Symposium on CMMR, London, UK, Nov. 3-7, 2025 792