Harmonic Tones

We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.

Infinite Canons is an ongoing series of canons with infinite solutions. More specifically, each canon is based on a melodic line that can be combined in any number of voices, at any tempo ratios (rational or irrational), and with each voice moving either forward or in retrograde inversion, while maintaining harmonic consistency. This paper describes the structure of these maximally self-similar melodic lines based on two different constructions: 1) a discrete prime-factorization approach yielding self-similarity under all rational ratios, and 2) a continuous logarithmic approach extending this to irrational ratios. In both cases the vertical interval between voices in a tempo ratio of $λ_i / λ_j$ is given by a homomorphism $φ(λ_i / λ_j)$, which is a constant independent of time. Furthermore, under these constructions retrograde inversion collapses to transposition, allowing for all manner of table canons. These structures are demonstrated with suggestive realizations of several different infinite canons. Future work includes a more complete mathematical treatment, musical applications, and an interactive program that allows users to explore an unlimited number of realizations of these pieces.

Sample retrieval tools can help composers find harmonically compatible material, but querying from a fixed reference sample becomes less informative as arrangements evolve and the harmonic context shifts with each musical decision. We present Reflector, an interactive audio workstation that tracks harmonic combinations as they accumulate on the composer's timeline and adapts retrieval as the arrangement develops. The system is organized around a fixed interval-class oracle: a hand-designed table of weights that scores how pitch-class content combines between sources. An encoder trained entirely on synthetic audio learns to approximate the oracle in a 128-dimensional embedding space, where dot products stand in for compatibility scores at interactive speed. As the composer arranges material on a multi-track timeline, a sweep-line analysis discovers co-sounding regions, computes oracle-weighted centroids, and retrieves against the composite harmonic identity of the session as it evolves. Session centroids projected into a navigable 3-D space reveal structural harmonic relations across the composer's body of work. This paper is a systems account: we give the design rationale for each architectural decision, characterize Reflector's behavior through intrinsic measurements on a working sample library, and describe the implementation. The characterization yields a central finding: the learned embedding preserves the kernel's pairwise judgments while covering the whole library, something the kernel cannot do when used directly as a retrieval rule, because the embedding's normalized geometry cannot express the degenerate solutions that direct scoring favors. The entire pipeline runs locally with no copyrighted training data. Reflector is free, and the training pipeline is open source.

Continuously recorded high-resolution waveform measurements provide rich information about fast power system dynamics. However, they require automated methods to identify events. This problem is addressed by developing a spectrogram-based framework to jointly detect, localize, and classify events in real-world continuously recorded waveforms at the terminal of an Inverter-Based Resource. We recast this problem as a temporal object detection problem on spectrogram images, as they capture the transient and harmonic signatures more explicitly than in raw waveform data. Each time-series waveform is transformed using the short-time Fourier transform, and the resulting per-channel spectrograms are stacked as a tensor for event detection. We benchmark this method against a detector operating directly on raw time-series measurements. Experiments on single-phase disturbances and three-phase faults demonstrate that the proposed spectrogram method consistently improves event detection, localization, and classification over the raw waveform baseline.

Heterogeneous graph built on notes, lyric syllables, and accompaniment events is a natural representation of symbolic music score, providing a substrate for both philological analysis and computational tasks. Music features are therefore well-captured by graph geometry and its properties. This representation has proved effective for analytical tasks as cadence detection, voice separation, and stylistic classification. In the present work, the reduction of harmonic complexity of a music score on graph, by preserving task-relevant information, relation between notes, and graph structure is investigated. A compression scheme for the piano subgraph of vocal-pianistic scores, built on binary wedge partitioning trees, is proposed. The wedges are generated through a fully adaptive greedy algorithm that recursively minimizes the $L^2$-error within a six-dimensional Tonnetz embedding of musical notes. The partitioning process employs a splitting criterion based on harmonic distance, resulting in regions that accurately reflect the intrinsic harmonic relationships among notes. The reconstructed music scores obtained through piecewise-constant functions and the mean values of the notes inside each wedge are used as a new simplified scores human-readable and playable. Some experiments on a corpus of symbolic music scores of three different composers are performed to assess the proposed approach.

The increasing integration of Electric Vehicles (EVs) has imposed a growing harmonic challenge on the power grid. For AC/DC Power Factor Correction (PFC) in single-phase On-Board Chargers (OBCs), Model Predictive Current Control (MPCC) improves the current quality by predicting and tracking the inductor current. However, finite control set MPCC selects switching states, resulting in discrete control actions and a limited optimisation space. Moreover, the MPCC cost function based on instantaneous current tracking error has limited capability to compensate for low-order harmonic disturbances induced by dead time, control delay, and model parameter mismatch. This paper proposes a duty cycle predictive MPCC incorporating a real-time harmonic estimation reference. The proposed method dynamically estimates the low-order harmonic components of the input current and corrects the MPCC reference current, enabling continuous duty cycle control and targeted suppression of dominant low-order harmonics. Simulation results on a single-phase OBC demonstrate that the proposed duty cycle predictive MPCC reduces the steady-state current THD_i from 11.47% to 6.10% compared with the switching state predictive MPCC. With the harmonic reference, the THD_i is further reduced to 2.85%.

Wrist-worn photoplethysmography (PPG) enables continuous monitoring of cardiopulmonary physiology, but reliable heart rate (HR) and respiratory rate (RR) estimation in free-living conditions remains challenging due to non-stationary motion artifacts that spectrally overlap with physiological dynamics. Existing signal-processing methods degrade under strong motion, while unconstrained deep learning approaches often lack physiological interpretability and identifiable structure. We propose a Physically-Constrained Harmonic Separation (PCHS) framework that formulates HR and RR estimation from wrist PPG as an analysis-by-synthesis problem, where accelerometer measurements condition artifact separation rather than directly regressing vital signs. A physics-guided harmonic generator decomposes the observed signal into quasi-periodic physiological components and a motion-related residual, enabling HR recovery from the fundamental frequency and RR prediction from respiratory-driven modulations of the harmonic parameters. Robust reconstruction objectives, separation constraints, and uncertainty-aware weighting stabilize the decomposition under motion. Experiments on the motion-intensive PPG-DaLiA dataset demonstrate that PCHS outperforms state-of-the-art methods while yielding interpretable signal decompositions that effectively disentangle physiological activity from motion artifacts.

Implicit neural representation (INR) has emerged as a powerful prior for multi-dimensional data (e.g., multispectral images and videos). However, most INR methods employing periodic activation functions (e.g., Sine) predominantly rely on function composition. This mechanism introduces optimization instability as network depth increases, thereby limiting their performance. Meanwhile, these methods fail to incorporate proper physical priors to effectively alleviate spectrum bias. To address these issues, inspired by the commonalities between deep periodic networks and generalized Fourier series, we propose a novel Calibrated Harmonic Overlaid Implicit Neural Representation (CHOIR). Specifically, we utilize Coordinated Harmonic Superposition (CHS) to replace the conventional function composition used in most INRs, thereby ensuring optimization stability when scaling network depth. Furthermore, we introduce a Perceptual Spectrum Calibration (PSC) to mitigate spectrum bias. This calibration embeds the ubiquitous power-law spectrum prior of natural images and adjusts the globally fixed spectrum towards a physically plausible log-uniform distribution. Extensive experiments on various multidimensional data recovery problems demonstrate that our method achieves superior performance over state-of-the-art approaches. Code is available at https://github.com/chorl0229/CHOIR.

Engine order enhancement is central in automotive sound design, where selective harmonics are synthesized to shape perceptual qualities such as sportiness, refinedness, or power. This paper investigates a neural network-based approach to combustion engine sound modeling that extends conventional engine order analysis and enhancement by deriving synthesis parameters from audio data with machine learning and incorporating stochastic components into the synthesis framework. The system parameterizes engine sounds as a compact representation capturing per-order and broadband timbral variation across the full RPM-torque operating range, while remaining manually tunable and compatible with established automotive audio frameworks. The approach leverages gradient-based optimization and analysis-by-synthesis through an end-to-end differentiable implementation. The resulting synthesis parameter set is directly transferable to conventional DSP implementations for deployment on embedded targets. Spectral metrics and listening tests confirm high reconstruction fidelity, and integration into an established automotive audio development platform (EVx Suite) demonstrates technical feasibility on deployment-ready embedded systems.

We study the assignment of local tonalities to chord sequences, a task useful for harmonic analysis, composition, and jazz-oriented improvisation. Standard dynamic-programming approaches minimize modulations but can introduce unnecessarily many tonal centers. We compare this transition-only objective with pure minimum-vocabulary analysis and with tonal parsimony, which minimizes lexicographically the number of modulations and then the number of distinct tonalities. Although this joint objective is combinatorially hard in general, we give exact algorithms exploiting the fixed 24-tonality major/minor universe. On 31,032 LMD Chords sequences, tonal parsimony preserves the transition optimum while reducing tonal vocabulary in 55.8% of cases. With weighted jazz-substitution closure, it lowers mean tonalities from 3.802 to 3.206 and modulations from 16.728 to 12.141. On 1,555 annotated jazz standards, it improves compatible chord-scale agreement to 95.6%, supporting tractable professional-scale harmonic analysis.

We present Harmonic, a hierarchical state space model (SSM) for language modeling. The architecture stacks three recurrent levels at progressively slower timescales; each level receives the prediction error of the level below as input, rather than its raw hidden state. On enwiki8 with equal token budgets, Harmonic outperforms a comparable Transformer (28M params) by +1.4% at 1K tokens, +6.7% at 8K tokens, and +11.4% at 32K tokens (bpt, lower is better). It also outperforms Mamba at every tested length by 0.7--1.8%. At 64K tokens, both Mamba and Transformer run out of memory on an 80GB H100; Harmonic trains successfully, reaching 6.169 bpt. Results replicate on WikiText-103 (H-TF gap +1.7% to +7.2% across 1K--32K). At 1B parameter scale, replacing all attention layers in TinyLlama 1.1B with HarmonicBlock eliminates the RoPE positional encoding limit: the resulting Hallamonic model maintains stable loss across sequence lengths 1K--8K on two independent clean benchmarks (Lambada and fineweb-edu held-out), while TinyLlama degrades catastrophically past its 2K-token RoPE limit (gap: +9.4 bpt at seq=8K on Lambada). Compute is O(L) per forward pass vs. O(L^2) for attention. Logs: https://github.com/Omibranch/harmonic-logs.

Humans can acquire a highly structured intuitive understanding of musical patterns, yet these patterns often require multiple iterations of reflection and re-listening to internalize fully. To capture such an internalization process, we present a computational model for the learning of jazz harmonic patterns based on library learning. Given a corpus of harmonic progressions, our model searches over a space of programs composed of primitive harmonic relations in order to discover concise generative explanations of the corpus. The model first enumerates possible programs for each piece, and then jointly learns a library of harmonic patterns and refactored programs. To efficiently navigate the vast joint space of programs and libraries, we integrate deductive parsing with library learning on e-graphs. We explore how well our model captures aspects of human musical pattern learning by evaluating the intuitiveness of both programs and libraries, as well as similarities to human-written harmonic derivations.

Aims. This study suggests that the use of multiple perceived pitches arising from a single harmonic complex tone is an active and intentional feature of contemporary popular music. The phenomenon is illustrated through examples drawn from the work of electronic artist Vitalic and others. Methods. Two listening tests were conducted: (1) evaluation of the number of simultaneous pitches perceived from single harmonic tones, and (2) manual pitch transcription of sequences of harmonic tones. Relationships between signal characteristics and pitch perception were then analyzed. Results. The synthetic harmonic tones found in the musical sequences under study were observed to transmit more perceived pitches than their acoustic counterparts, with significant variation across listeners. Multiple ambiguous pitches were associated with tone properties such as prominent upper partials and particular autocorrelation profiles. Conclusions. Harmonic tones in a context of contemporary popular music can, in general, convey several ambiguous pitches. The set of perceived pitches depends on both the listener and the listening conditions.

Music information retrieval distinguishes between low- and high-level descriptions of music. Current generative AI models rely on text descriptions that are higher level than the controls familiar to studio musicians. Pitch strength, a low-level perceptual parameter of contemporary popular music, may be one feature that could make such AI models more suited to music production. Signal and perceptual analyses suggest that pitch strength (1) varies significantly across and inside songs; (2) contributes to both small- and large-scale structure; (3) contributes to the handling of polyphonic dissonance; and (4) may be a feature of upper harmonics made audible in a perspective of perceptual richness.

The ultimate purpose of generative music AI is music production. The studio-lab, a social form within the art-science branch of cross-disciplinarity, is a way to advance music production with AI music models. During a studio-lab experiment involving researchers, music producers, and an AI model for music generating bass-like audio, it was observed that the producers used the model's output to convey two or more pitches with a single harmonic complex tone, which in turn revealed that the model had learned to generate structured and coherent simultaneous melodic lines using monophonic sequences of harmonic complex tones. These findings prompt a reconsideration of the long-standing debate on whether humans can perceive harmonics as distinct pitches and highlight how generative AI can not only enhance musical creativity but also contribute to a deeper understanding of music.

Since antiquity, musical scales have been explained by harmony rather than melody. This view relies on the mathematically designed scales of a few traditions, and was never directly tested. Testing it requires cross-cultural data and a method that judges theories by what they get wrong as well as right. We provide both, modelling scale evolution across 1,314 scales from 96 countries. A Melody model explains the near-universal preference for step-sizes of 1-3 semitones, and matches independent data from melodies, singing, and psychoacoustics. Harmony does far less: it explains the music-theoretic scales, but in those measured from performance it adds only a weak bias towards fourths, fifths, and octaves. Harmony's importance has been overstated, likely due to the historical focus on music-theoretic rather than measured scales. Melody is the primary driver of global scale structure; harmonic constraints are less impactful and mainly reflect musicological theory over musical performance.

Much of Western classical music relies on instruments based on acoustic resonance, which produce harmonic or quasi-harmonic sounds. In contrast, since the mid-twentieth century, popular music has increasingly been produced in recording studios, where it is not bound by the constraints of harmonic sounds. In this study, we use modified MPEG-7 features to explore and characterise the evolution of noise and inharmonicity in popular music since 1961. We place this evolution in the context of other broad categories of music, including Western classical piano music, orchestral music, and musique concrète. We introduce new features that distinguish between inharmonicity caused by noise and that resulting from interactions between discrete partials. Our analysis reveals that the history of popular music since 1961 can be divided into three phases. From 1961 to 1972, inharmonicity in popular music, initially only slightly higher than in orchestral music, increased significantly. Between 1972 and 1986, this rise in inharmonicity was accompanied by an increase in noise, but since 1986, both inharmonicity and noise have moderately decreased. In recent years (up to 2020), popular music has remained much more inharmonic than popular music from the 1960s or orchestral music involving acoustic resonance instruments. However, it has become less noisy, with noise levels comparable to those of orchestral music. We relate these trends to the evolution of music production techniques. In particular, the use of multi-tracking may explain the higher inharmonicity in popular music compared to orchestral music. We illustrate these trends with analyses of key artists and tracks.