PEA-Tempo: A Lightweight Periodicity- and Equivariance-Aware Framework for Music Tempo Estimation

Abstract

Global music tempo estimation requires a model to identify the dominant periodicity in complex time–frequency content and determine the exact beats per minute (BPM) among related metrical levels, including half, nominal, and double tempo. Existing end-to-end models generally learn these relationships implicitly, which can entangle timbral semantics, periodicity evidence, and metrical-level selection. To address this issue, we propose PEA-Tempo, a lightweight periodicity- and equivariance-aware framework for music tempo estimation. The framework comprises three core modules. First, the Dual-View Periodicity Representation Module explicitly describes transient and repetitive structures by combining rhythmic-density cues derived from spectral flux with normalized autocorrelation. Second, the Circular Tempo-Phase Modeling Module maps the fractional part of log-tempo to the unit circle, thereby providing a unified representation of tempo families related by integer powers of two. Third, the Tempo-Scaling Equivariant Module converts temporal scaling of the input into a circular phase-rotation constraint. Experimental results identify the explicit periodicity branch as the main source of static accuracy gains. The phase and scaling constraints improve the reported spectrogram-scaling response, while their incremental accuracy gains depend on the dataset and inference configuration. With 326,231 trainable parameters, PEA-Tempo combines a compact architecture with explicit rhythmic structure modeling and achieves improved tempo-estimation accuracy in the reported evaluations.

1. Introduction

Tempo is a fundamental property of the temporal organization of music and is commonly expressed in beats per minute (BPM). Reliable automatic tempo estimation provides essential information for beat-synchronous analysis, music retrieval and classification, playlist organization, automatic mixing, audio editing, and computer-assisted music creation. Unlike beat tracking, which returns a sequence of beat times, global tempo estimation aims to assign a single dominant BPM value to a musical excerpt whose tempo is relatively stable. Although its output is simple, the task jointly involves transient perception, periodicity inference, and metrical-level selection, and therefore remains an important problem in music information retrieval [1,2,3].

Traditional tempo-estimation methods generally begin with rhythmic cues such as energy variations, onset strength, or spectral flux, and then identify the dominant repetition period through autocorrelation, filtering, or periodicity matching [1,4,5]. Such methods are physically interpretable, but real music often contains periodic structures at multiple metrical levels. For example, the tatum, tactus, half-beat, and measure-level pulse can produce candidate tempi related approximately by integer factors, causing $b/2$, b, and $2b$ to yield strong responses simultaneously in a periodicity representation. Previous work has explicitly organized these integer-related tempo relationships in periodic representations [6]; nevertheless, detecting a repetition period is not equivalent to determining the final absolute BPM. Half- and double-tempo confusions therefore remain common in music with syncopation, weak transients, or complex instrumentation.

In recent years, deep neural networks have substantially improved the representational capacity of tempo-estimation systems. Convolutional, multi-task, and multi-scale time–frequency models can learn rhythmic patterns directly from acoustic representations while reducing the need for manually designed intermediate processing stages [7,8,9]. In conventional end-to-end architectures, however, periodicity recognition, integer-related tempo relationships, and absolute metrical level are usually encoded together in latent features. Although a model can learn complex rhythmic patterns from data, it may also exploit dataset-specific auxiliary cues, including timbre, spectral envelope, and production style. When the genre, source, or acoustic characteristics of test music differ from those observed during training, reliance on such non-rhythmic cues can impair generalization.

Tempo also follows a well-defined physical transformation law: if an audio signal is scaled along the time axis by a factor s, its tempo changes to $sb$. This relationship has been exploited in equivariant self-supervision and time-stretch-based tempo discrimination [10,11], demonstrating that tempo transformations can encourage the learning of more stable rhythmic representations. Existing approaches, however, generally use this relationship for representation learning, pre-training, or similarity discrimination, while exact BPM prediction still requires an additional classifier or search procedure. Moreover, ACC2, which treats integer-related tempo errors as correct, may conceal mistakes in absolute metrical-level selection. Recent research has confirmed that half- and double-tempo errors remain important failure modes in advanced beat- and tempo-analysis systems [12].

These observations suggest that reliable global tempo estimation must address three related but non-equivalent questions: (1) whether the audio contains a stable recurring pulse and what periodicity evidence supports it; (2) whether different tempo candidates belong to the same integer-related periodicity family; and (3) how to select the target absolute BPM within that family. Simply increasing the capacity of an acoustic network does not explicitly disentangle these factors, whereas collapsing all integer-related tempi into one class discards the absolute metrical level required by the task.

We therefore propose PEA-Tempo (Periodicity- and Equivariance-Aware Tempo Estimation), a lightweight framework that jointly models explicit periodicity evidence, integer-related periodic structures, and tempo-scaling behavior on top of a compact convolutional acoustic backbone. The Dual-View Periodicity Representation Module (DPRM) first extracts rhythmic density and normalized autocorrelation from a spectral-flux sequence and fuses them with convolutional acoustic features. This design enables the model to exploit both local transient patterns and global repetitive structure. The Circular Tempo-Phase Modeling Module (CTPM) subsequently uses the hierarchical structure of tempo in the ${\log }_2$ domain: it maps the fractional part of ${\log }_{2}b$ to the unit circle so that tempi differing by a factor of $2^k$ share the same periodic phase, while retaining an independent exact-BPM prediction branch for absolute-level selection. Finally, the Tempo-Scaling Equivariant Module (TSEM) maps temporal scaling of the input to a deterministic rotation in circular phase space, encouraging the direction of the model’s output change to agree with the true tempo transformation.

The main contributions of this work are summarized as follows:

• We introduce PEA-Tempo, a lightweight periodicity- and equivariance-aware framework for global tempo estimation. It integrates acoustic representations, explicit periodicity evidence, integer-related tempo relationships, and exact-BPM prediction within a unified model, thereby coordinating periodicity detection, periodicity-family modeling, and absolute metrical-level selection in a structured manner.
• We develop three complementary modules: DPRM enhances the perception of explicit periodic structure through spectral-flux-based rhythmic density and normalized autocorrelation; CTPM encodes the shared periodicity-family relationship among integer-related tempi in circular phase space; and TSEM maps temporal scaling to a predictable phase rotation, introducing a physically meaningful tempo-equivariance constraint.
• We conduct systematic experiments on FSL10K, GTZAN, and GiantSteps Tempo. Baseline comparisons, cross-dataset evaluation, incremental ablation studies, and tempo-scaling response analyses characterize the accuracy gains and the dataset-dependent contributions of periodicity evidence and structured transformation supervision.

2. Related Work

Automatic music tempo estimation seeks to determine the dominant pulse rate of an audio recording, generally expressed in BPM. Whereas beat tracking produces a sequence of beat times, global tempo estimation typically assigns one BPM value to an entire musical excerpt. The two tasks nevertheless share several core components, including rhythmic-event extraction, periodicity analysis, and metrical-level discrimination. Research in this field has broadly progressed from hand-crafted periodicity analysis to probabilistic beat modeling and end-to-end deep networks, followed more recently by self-supervised and transformation-equivariant learning.

2.1. Traditional and Hybrid Rhythm Analysis

Early tempo-estimation systems generally followed a three-stage pipeline: rhythmic-event extraction, candidate-period generation, and predominant-tempo selection. Scheirer divided audio into multiple frequency bands and analyzed periodic responses through envelope extraction and resonant filtering, establishing a foundational paradigm for inferring perceptual tempo from periodic acoustic energy [1]. Klapuri et al. subsequently modeled musical meter at multiple levels, from tatum and tactus to measure, showing that tempo is not an isolated numerical attribute but is embedded in a hierarchical metrical structure [13]. To recover beat period and phase, Davies and Plumbley used a complex spectral-difference function, normalized autocorrelation, and adaptive comb filters to estimate beat periodicity and temporal alignment [14]. Ellis formulated beat-sequence inference as a dynamic-programming problem that balances local onset strength against global consistency of inter-beat intervals [15], whereas Dixon’s BeatRoot maintained competing beat hypotheses in a multi-agent system and selected the final trajectory according to accumulated scores [16]. Together, these studies established three representative back ends for classical beat tracking: periodicity filtering, global sequence optimization, and multi-hypothesis search.

Time–frequency periodicity representations make local tempo variation directly observable. Grosche et al. introduced the cyclic tempogram, which folds tempi separated by factors of $2^k$ into the same cyclic-tempo class and thus represents the equivalence among half, nominal, and double tempo [6]. Hockman and Fujinaga formulated tempo-octave selection as a fast-versus-slow binary classification problem, demonstrating that an additional perceptual-tempo decision can help identify the target tempo octave among periodicity candidates [17]. Grosche and Müller later proposed the Predominant Local Pulse (PLP) representation, in which locally optimal periodic kernels are superimposed to reinforce pulse structure under weak onsets and continuously varying tempo [18]. Collectively, these studies indicate that periodicity detection and absolute metrical-level selection are related but distinct subproblems.

With the development of machine learning, attention shifted from manually designed onset functions toward learned beat activations followed by probabilistic decoding of musically coherent beat sequences. Böck and Schedl used a bidirectional long short-term memory network to predict beat activations in context, reducing the sensitivity of hand-crafted features to timbral variation [19]. Degara et al. proposed a reliability-informed probabilistic beat-tracking framework that explicitly models beat and non-beat states and estimates confidence according to observation quality [20]. Zapata et al. drove a unified beat-tracking model with several complementary onset-detection functions and selected the result according to agreement among candidate outputs, thereby improving robustness across musical styles [21]. Probabilistic state-space models can additionally constrain beat position, tempo, and meter jointly. Krebs et al. designed an efficient fixed-grid state space and tempo-transition model that jointly tracks tempo and meter while reducing inference complexity [22]. Böck et al. further used a recurrent neural network to generate beat and downbeat activations simultaneously, followed by a dynamic Bayesian network that recovers the globally optimal metrical structure [23]. Such neural-activation–probabilistic-decoding systems perform strongly on standard beat datasets, although assumptions about tempo range, transition smoothness, and meter in the decoder can influence the final BPM estimate.

Direct tempo estimation does not require the position of every beat; instead, it identifies the dominant BPM from a complete recording or several local excerpts. Percival and Tzanetakis combined autocorrelation of an onset-strength envelope with cross-correlation against pulse trains to perform candidate generation and tempo selection in a streamlined pipeline [4]. Their method offers low computational cost together with an explicit periodic interpretation. Böck et al. proposed a tempo estimator combining a recurrent neural network with resonating comb filters: the network first learns beat-level activations from spectra, after which the filter bank identifies the dominant periodicity [5]. This approach connects a learned front end with a conventional periodicity back end and demonstrates that high-quality beat activations can substantially improve periodicity estimation. Traditional and hybrid methods are physically meaningful and have relatively modest data requirements, but their final tempo is still selected primarily by peak-based rules. When $b/2$, b, and $2b$ all produce strong responses, correctly identifying the periodicity family does not ensure correct absolute-BPM selection. Tempo estimation must therefore use musical context to assess candidate periods jointly with their absolute metrical levels.

2.2. Deep Learning for Tempo Modeling

End-to-end methods transform tempo estimation from a multi-stage pipeline into a direct mapping from a time–frequency representation to a discrete BPM label. Schreiber and Müller fed Mel spectrograms to a convolutional neural network and formulated tempo as a multi-class classification problem, without first detecting onsets or beat positions [7]. This single-step paradigm reduces error propagation across conventional processing stages and demonstrates that convolutional networks can learn tempo cues directly from local time–frequency patterns. Explicit rhythm representations have also been used to improve neural tempo classification. Foroughmand and Peeters proposed the Harmonic Constant-Q Modulation (HCQM) representation, which encodes the modulation frequencies of onset-strength functions and their harmonic structure across multiple acoustic subbands before applying a convolutional network to tempo and rhythmic-pattern recognition [24]. Compared with a conventional Mel spectrogram, HCQM explicitly arranges tempo candidates and their harmonic relationships in the input space, illustrating the value of combining domain knowledge with deep learning.

Multi-task learning can improve global tempo prediction by sharing information across related metrical tasks. Böck and Davies used a temporal convolutional network to estimate tempo, beats, and downbeats jointly, and expanded the training range through tempo transformations [8]. Sun et al. proposed MGANet, a multi-scale grouped-attention network that enlarges the time–frequency context through multi-scale feature aggregation and captures long-range dependencies using grouping along the frequency dimension [9]. Souza et al. compared convolutional neural networks and bidirectional recurrent neural networks for direct tempo estimation, further showing that temporal-modeling architectures behave differently on percussive and mixed-music datasets [25].

Recent studies have sought both to reduce strong post-processing priors and to develop more powerful rhythmic representations. Foscarin et al. introduced Beat This!, which alternates convolutional layers with Transformers operating along the time and frequency axes and removes the commonly used dynamic Bayesian network postprocessor [26]. He et al. compared temporal convolutional networks, recurrent neural networks, and their post-processing combinations on solo instrumental performances, showing that alignment between the training domain and the target musical material remains important [27]. Chiu et al. proposed differentiable Predominant Local Pulse (dPLP), enabling periodic kernels to participate in backpropagation as an intermediate network layer or a loss term [28]. Ru et al. combined a pre-trained music foundation model with multidimensional semantic aggregation in BeatFM to improve cross-style beat and downbeat tracking [29]. End-to-end networks offer strong feature-fitting capacity, but flat BPM classification generally leaves numerical proximity, integer-related metrical relationships, and temporal-scaling laws to be learned entirely implicitly. Because the network must both recognize rhythmic periodicity and select its absolute level, distribution shifts can encourage reliance on non-rhythmic cues such as timbre, genre, or production style. Explicitly encoding periodic structure while controlling model size therefore remains an important challenge in direct tempo estimation.

2.3. Self-Supervised and Structured Tempo Learning

Temporal scaling changes musical tempo by a known factor and can consequently provide structured supervision without additional manual BPM annotations. Quinton constructed training pairs from differently time-scaled versions of the same audio and used equivariant self-supervised learning to obtain representations that are sensitive to tempo change while remaining comparatively stable with respect to content, followed by supervised training of a downstream tempo predictor [10]. This approach turns temporal scaling from conventional data augmentation into a pre-training task with an explicit input–output correspondence. Morais et al. used synthetic data to examine how input representation, pitch variation, and data distribution affect self-supervised tempo estimation, and showed that a model may exploit non-target cues that covary with the intended transformation [30]. Their findings demonstrate that constructing a temporal-scaling task alone does not guarantee the learning of purely rhythmic features; both the representation and objective must constrain the information available to the model.

Henkel et al. reformulated global tempo estimation as binary classification of whether a reference recording and a target recording have the same tempo. By combining general-purpose music embeddings, time stretching, and synthetic reference signals, their method supports fully self-supervised training [11]. Rather than directly returning BPM, it matches the target against multiple reference tempi, reducing the need for manual labels and performing favorably under metrics that accept integer-related tempo equivalence. In addition to continuous tempo transformations, multi-hypothesis learning can preserve uncertainty across metrical levels. Gagnere et al. introduced knowledge-driven multiple-hypothesis selection into contrastive self-supervised pre-training, allowing the model to retain several plausible rhythmic interpretations simultaneously [31]. Such objectives preserve competing rhythmic candidates during learning, whereas the cyclic tempogram and circular tempo phase organize integer-related periodicity families at the representational level. Both strategies avoid prematurely collapsing metrical-level ambiguity during feature learning.

Tempo-octave ambiguity nevertheless remains a central difficulty for advanced systems. In 2026, Ahn et al. analyzed failure modes in state-of-the-art beat trackers and categorized their errors as tempo-octave errors, continuity errors, and complete tracking failures. They further showed that a fixed lower bound in the decoding tempo range can force slow music to be predicted at double tempo [12]. This finding indicates that reporting only lenient metrics that accept integer-related tempo equivalence can conceal errors in absolute metrical-level selection.

Prior work has independently established the value of explicit periodicity analysis, end-to-end acoustic modeling, and temporal-scaling supervision, but these elements are generally deployed in separate systems. Traditional methods expose periodicity peaks yet lack feature adaptation to the training data; end-to-end networks offer strong representation learning but seldom express integer-related metrical relationships explicitly; and self-supervised methods exploit tempo transformations for representation learning, while downstream exact-BPM prediction and metrical-level selection still require additional mechanisms. We therefore retain supervised exact-BPM prediction as the primary task and introduce explicit periodicity evidence, integer-related periodicity-family representations, and tempo-scaling constraints jointly within a single lightweight network. Unlike traditional peak selection, the proposed framework learns to reweight periodicity candidates from data; unlike flat BPM classification, it explicitly organizes metrical levels in an auxiliary space; and unlike fully self-supervised similarity discrimination, it preserves a directly decodable absolute-BPM output.

3. Methodology

3.1. Problem Formulation and Framework Overview

Given a music audio excerpt $x\in {\mathbb{R}}^L$, global tempo estimation aims to predict its tempo $b\in [30,300]$ beats per minute (BPM). Conventional classification models typically map a time–frequency representation directly to discrete BPM classes, but this formulation presents two fundamental limitations. First, convolutional networks can rely on non-rhythmic cues, such as timbre, spectral envelope, and production style, and may therefore fail to remain robust when the audio source changes. Second, musical pulse exhibits pronounced integer-ratio relationships: $b/2$, b, and $2b$, for example, often share similar periodic structures. A flat classifier can consequently conflate periodicity recognition with selection of the absolute metrical level. Moreover, conventional models do not explicitly constrain the correspondence between tempo scaling at the input and BPM changes at the output, resulting in predictions that are insufficiently stable under temporal dilation or compression.

To address these limitations, we propose PEA-Tempo, a periodicity- and equivariance-aware framework for global tempo estimation. Rather than relying exclusively on a convolutional backbone to learn rhythmic structure implicitly, PEA-Tempo jointly models acoustic semantics, transient variation, explicit periodicity evidence, and log-tempo phase. The framework comprises three principal modules:

• Dual-View Periodicity Representation Module (DPRM): DPRM integrates convolutional acoustic features, a spectral-flux rhythm representation, and a normalized autocorrelation representation. This design retains timbral context while emphasizing repetitive beat structure.
• Circular Tempo-Phase Modeling Module (CTPM): CTPM maps the fractional component of the ${\log }_2$ tempo to a circular space, such that half-, nominal-, and double-tempo hypotheses share a common periodic phase representation. A separate exact-BPM classification head retains information about the absolute metrical level.
• Tempo-Scaling Equivariant Module (TSEM): TSEM applies controlled temporal scaling to the input and represents the transformation as a deterministic rotation in circular phase space. It thereby encourages changes in the learned representation to remain consistent with the corresponding changes in the true BPM.

The complete pipeline is illustrated in Figure 1. The waveform is first converted into a log-Mel spectrogram. Deep acoustic features, spectral-flux rhythm features, and autocorrelation-based periodicity features are then extracted and fused into a unified representation. Finally, an exact-BPM head and an auxiliary circular-phase head are optimized jointly. During inference, the final tempo is obtained solely from the exact-BPM posterior; the circular-phase branch serves as structured supervision and imposes the equivariance constraint during training.

3.2. Time–Frequency Representation and Acoustic Encoding

Each audio signal is first resampled to 11,025 Hz and either cropped or repeated cyclically to obtain a fixed-duration excerpt of 6 s. We compute the short-time Fourier transform (STFT) and apply a Mel filterbank to obtain the log-energy spectrogram:

$$\begin{array}{cc}\hfill E_{m,t}& =\sum _k{H}_m\left(k\right){\left|\mathrm{STFT}{\left(x\right)}_{k,t}\right|}^2,\hfill \\ \hfill {\tilde{X}}_{m,t}& =\log \left(1+10E_{m,t}\right),\hfill \\ \hfill X_{m,t}& ={\displaystyle \frac{{\tilde{X}}_{m,t}-{\mu }_{\tilde{X}}}{{\sigma }_{\tilde{X}}+\epsilon }},\hfill \end{array}$$

(1)

where $H_m\left(k\right)$ denotes the mth Mel filter, and ${\mu }_{\tilde{X}}$ and ${\sigma }_{\tilde{X}}$ denote the excerpt-level mean and standard deviation, respectively. We use $M=81$ Mel bands and $T=128$ time frames, yielding an input tensor $X\in {\mathbb{R}}^{1\times 81\times 128}$. This input resolution is consistent with the configuration of the multi-scale tempo network used for comparison [9].

To capture local acoustic patterns, we employ a four-stage two-dimensional convolutional encoder. Each stage contains two $3\times 3$ convolutional layers, batch normalization, GELU activation, and pooling. The channel dimensions are 16, 32, 64, and 96, respectively. The frequency dimension is progressively compressed, whereas comparatively high resolution is retained along the temporal dimension. Let $f_{\mathrm{a}}(·)$ denote the acoustic encoder. Its output is

$$F=f_{\mathrm{a}}\left(X\right)\in {\mathbb{R}}^{C\times M^{\prime }\times T^{\prime }},$$

(2)

where $C=96$. Global average pooling over the frequency and temporal dimensions produces the acoustic-semantic vector

$${\mathrm{h}}_{\mathrm{a}}=\mathrm{GAP}\left(F\right)\in {\mathbb{R}}^{96}.$$

(3)

This branch describes the spectral envelope, transient timbre, and local time–frequency texture, but it is not solely responsible for inferring beat periodicity. Explicit periodic information is supplied by the module described next.

3.3. Dual-View Periodicity Representation Module

DPRM characterizes rhythm from two complementary perspectives. Local spectral changes indicate the temporal locations of transient events, whereas autocorrelation measures the recurrence strength of these events at different lags [4,5,14]. In contrast to architectures that depend entirely on deep convolutional features, DPRM injects interpretable periodicity evidence directly into the tempo decision process.

3.3.1. Spectral-Flux Rhythm Sequence

For each pair of adjacent frames, we first compute spectral flux while retaining only positive energy changes:

$$d_t={\displaystyle \frac{1}{M}}{\displaystyle \sum _{m=1}^M}\max \left(X_{m,t}-X_{m,t-1},0\right).$$

(4)

We then normalize the scale using the within-excerpt mean and clip unusually large peaks:

$${\tilde{d}}_t=\min \left({\displaystyle \frac{d_t}{\frac{1}{T}{\sum }_{j=1}^T{d}_j+\epsilon }},10\right).$$

(5)

This normalization reduces the effects of recording level and dynamic-range variation, allowing the branch to respond primarily to relative transient changes. The normalized sequence $\tilde{\mathbf{d}}$ is passed to a lightweight one-dimensional rhythm encoder:

$${\mathrm{h}}_{\mathrm{p}}=f_{\mathrm{p}}\left(\tilde{\mathbf{d}}\right)\in {\mathbb{R}}^{32},$$

(6)

where $f_{\mathrm{p}}$ comprises two one-dimensional convolutional layers with a kernel size of 5, followed by adaptive average pooling. The resulting vector characterizes transient density, local pulse morphology, and their global distribution; we therefore refer to it as the rhythm-density representation.

3.3.2. Explicit Autocorrelation-Based Periodicity Description

Rhythm density alone cannot determine the recurrence interval. We therefore compute the normalized autocorrelation of the centered spectral-flux sequence:

$$r_{\tau }={\displaystyle \frac{{\displaystyle {\displaystyle \sum _{t=\tau +1}^T}({\tilde{d}}_t-\overline{d})·({\tilde{d}}_{t-\tau }-\overline{d})}}{{\displaystyle {\displaystyle \sum _{t=1}^T}{({\tilde{d}}_t-\overline{d})}^2+\epsilon }}},$$

(7)

where $\tau$ is the temporal lag and $\overline{d}$ is the sequence mean. In practice, autocorrelation is computed using the fast Fourier transform. We retain 41 coefficients for $\tau =4,\dots ,44$ to form

$${\mathrm{h}}_{\mathrm{r}}=[r_4,r_5,\dots ,r_{44}]\in {\mathbb{R}}^{41}.$$

(8)

Shorter lags respond primarily to rapid pulses, whereas longer lags represent slower beats and their higher-order periodicities. Multiple correlation peaks can therefore preserve the $b/2$, b, and $2b$ candidates simultaneously, avoiding premature selection of a single metrical level during feature extraction.

Finally, the acoustic semantics, rhythm density, and explicit periodicity description are concatenated:

$$\mathbf{z}=[{\mathrm{h}}_{\mathrm{a}};{\mathrm{h}}_{\mathrm{p}};{\mathrm{h}}_{\mathrm{r}}]\in {\mathbb{R}}^{169}.$$

(9)

This fusion enables the network to jointly exploit three complementary sources of information: the acoustic content, the density at which transients occur, and the periodic pattern according to which they recur.

3.4. Circular Tempo-Phase Modeling Module

3.4.1. Exact-BPM Main Branch

We discretize the range 30–300 BPM into 271 integer-valued classes. The fused representation $\mathbf{z}$ is passed through a two-layer multilayer perceptron to produce exact-BPM logits:

$${\mathrm{o}}^b=f_b\left(\mathbf{z}\right)\in {\mathbb{R}}^{271},$$

(10)

where the hidden dimension is 192, and GELU activation and dropout are applied. Given a ground-truth tempo b, the exact class label is defined as

$$y^b=\mathrm{round}\left(b\right)-30.$$

(11)

This branch directly retains absolute-BPM information and serves as the main prediction branch during both training and inference. At inference time, the exact-tempo logits are normalized using softmax, and the maximum a posteriori class yields the final estimate: $\widehat{b}=30+\arg {\max }_{c\in \{0,\dots ,270\}}p(c\mid X)$. The circular-phase head is used only to impose structured training constraints and does not participate in final BPM decoding.

3.4.2. Circular Decomposition of Log-Tempo

Metrical levels commonly exhibit integer-ratio relationships. In the ${\log }_2$ domain, doubling or halving the tempo corresponds to adding or subtracting one, respectively. Circular tempo representations and coarse tempo-level classification have previously demonstrated the value of this structure for modeling multiplicative tempo relationships [6,17]. We therefore define the normalized log-tempo as

$$q\left(b\right)={\log }_2\left({\displaystyle \frac{b}{60}}\right).$$

(12)

Taking its fractional component gives the circular tempo phase

$$\varphi \left(b\right)=q\left(b\right)-\lfloor q\left(b\right)\rfloor \in [0,1).$$

(13)

Because

$$\varphi \left(2^{k}b\right)=\varphi \left(b\right),k\in \mathbb{Z},$$

(14)

the half-, nominal-, and double-tempo hypotheses have the same phase label. We partition the circle into $K=120$ bins:

$$y^{\varphi }=⌊K\varphi \left(b\right)⌋,$$

(15)

and use an auxiliary phase head to produce

$${\mathrm{o}}^{\varphi }=f_{\varphi }\left(\mathrm{z}\right)\in {\mathbb{R}}^{120}.$$

(16)

The phase head does not replace the exact-BPM head. Instead, it injects periodic supervision that is invariant to integer-ratio metrical levels into the shared representation. The exact head learns what the final BPM is, whereas the phase head learns to which periodic family the BPM belongs. Periodicity recognition and absolute-level selection are thus decomposed into two complementary objectives.

3.5. Tempo-Scaling Equivariant Module

3.5.1. Controlled Tempo Transformation

Following tempo-scaling equivariant supervision and related representation analyses [10,30], we randomly sample a scaling factor for each training pair $(X,b)$:

$$s\in \mathcal{S}=\{0.8,0.9,1.0,1.1,1.25\}.$$

(17)

Bilinear resampling is then applied along the temporal axis of the time–frequency representation, yielding $X^{\left(s\right)}={\mathcal{T}}_s\left(X\right)$. A temporally shortened sequence is repeated cyclically to recover the fixed length, whereas an elongated sequence is center-cropped. The corresponding label is updated as

$$b^{\left(s\right)}=sb.$$

(18)

If the transformed BPM falls outside the predefined range, we reset $s=1$. Joint supervision of the original and scaled examples encourages the model to learn an explicit correspondence between input time scale and output BPM.

3.5.2. Circular Phase-Rotation Constraint

Temporal scaling corresponds to an additive displacement in log-tempo space:

$$q\left(sb\right)=q\left(b\right)+{\log }_{2}s.$$

(19)

Accordingly, the circular phase should rotate by

$$\Delta {\theta }_s=2\pi {\log }_{2}s.$$

(20)

To avoid discontinuities at discrete phase-bin boundaries, we do not compare phase-class indices directly. Instead, we embed the phase bins on the two-dimensional unit circle and represent the posterior by its first circular moment. Let

$$p_k^{\varphi }\left(X\right)=\mathrm{softmax}{\left({\mathrm{o}}^{\varphi }\right)}_k,{\theta }_k={\displaystyle \frac{2\pi k}{K}},$$

(21)

then the circular embedding of the phase posterior is

$$\mathrm{u}\left(X\right)={\displaystyle \sum _{k=0}^{K-1}}{p}_k^{\varphi }\left(X\right){[\cos {\theta }_k,\sin {\theta }_k]}^{\top }.$$

(22)

Let $R(\Delta {\theta }_s)$ denote a two-dimensional rotation matrix. The equivariance loss is defined as

$${\mathcal{L}}_{\mathrm{eq}}={∥\mathrm{u}\left(X^{\left(s\right)}\right)-R(\Delta {\theta }_s)\mathrm{u}\left(X\right)∥}_2^2.$$

(23)

This constraint requires the internal phase representation to rotate by a known angle when the input is accelerated or decelerated, rather than changing unpredictably. Because the circular representation naturally identifies its two endpoints, Equation (23) handles transformations that cross the zero-phase boundary continuously.

3.6. Joint Optimization Objective

For an original example, the exact-BPM classification loss and the circular-phase classification loss are, respectively,

$${\mathcal{L}}_b(X,b)=\mathrm{CE}({\mathrm{o}}^b\left(X\right),y^b),$$

(24)

$${\mathcal{L}}_{\varphi }(X,b)=\mathrm{CE}({\mathrm{o}}^{\varphi }\left(X\right),y^{\varphi }).$$

(25)

The supervision term for the original example is

$${\mathcal{L}}_{\mathrm{org}}={\mathcal{L}}_b(X,b)+{\lambda }_{\varphi }{\mathcal{L}}_{\varphi }(X,b).$$

(26)

The same joint supervision is applied to the scaled example:

$${\mathcal{L}}_{\mathrm{aug}}={\mathcal{L}}_b(X^{\left(s\right)},b^{\left(s\right)})+{\lambda }_{\varphi }{\mathcal{L}}_{\varphi }(X^{\left(s\right)},b^{\left(s\right)}).$$

(27)

The final objective is

$$\mathcal{L}={\mathcal{L}}_{\mathrm{org}}+{\lambda }_{\mathrm{aug}}{\mathcal{L}}_{\mathrm{aug}}+{\lambda }_{\mathrm{eq}}{\mathcal{L}}_{\mathrm{eq}},$$

(28)

where ${\lambda }_{\varphi }=0.35$, ${\lambda }_{\mathrm{aug}}=0.5$, and ${\lambda }_{\mathrm{eq}}=0.2$. The three components in Equation (28) jointly optimize exact-BPM discrimination, recognition of integer-ratio periodic families, and response consistency under tempo transformations. The complete framework is trained end to end and does not require beat timestamps, downbeat annotations, or an additional post-processing network.

4. Experiments and Results

4.1. Datasets and Evaluation Protocol

Previous comparative studies have shown that the relative ranking of tempo-estimation systems can change with the data distribution and annotation convention [32]. We therefore consider three complementary settings: short musical loops, complete excerpts spanning multiple genres, and electronic dance music. The training domain is kept strictly separate from the two external test domains so that in-domain accuracy and cross-dataset generalization can be assessed independently.

4.1.1. FSL10K

The Freesound Loop Dataset (FSL10K) consists of short musical loops with user-provided BPM labels [33]. The local metadata contain 9493 records, of which 9486 have usable WAV audio and are included in our experiments. Their labels range from 30 to 300 BPM.

Loops uploaded by the same contributor or included in the same sample pack often share timbre, recording conditions, and production templates. To prevent source leakage under random sample splitting, we construct a bipartite graph $\mathcal{G}=(\mathcal{U}\cup \mathcal{P},\mathcal{E})$, where $\mathcal{U}$ denotes contributors and $\mathcal{P}$ denotes sample packs. An edge links a contributor to a pack whenever that contributor uploaded a loop belonging to the pack. Each connected component is treated as an indivisible source group, ensuring that related contributors and packs do not cross the training, validation, and test sets. This procedure yields 742 contributor–pack components.

Subject to these source-group constraints, we perform stratified group partitioning in 10-BPM intervals and select, from 64 candidate assignments, the five-fold split with the most closely matched full 1-BPM histograms. The five test folds contain 2262, 1750, 1798, 1876, and 1800 clips, respectively. For run k, fold k is used for testing, fold $(k+1)\mathrm{mod}5$ for validation, and the remaining three folds for training. The validation fold is used for early stopping and model selection. Final in-domain metrics are computed after pooling the per-sample out-of-fold predictions from all five test folds.

4.1.2. GTZAN

GTZAN contains 1000 approximately 30-s excerpts distributed evenly across ten genres [34]. We use the CPJKU v1.1 beat annotations derived from GTZAN-Rhythm [35] and compute global BPM from the median adjacent inter-beat interval after the first 5 s. Of the original 1000 files, 999 can be matched by name to this annotation release; after removing one file that cannot be decoded, 998 tracks remain. Because GTZAN has known data-quality limitations [36], these matching and exclusion rules are fixed for all systems.

Models trained on FSL10K remain frozen on GTZAN. Each track is divided into five non-overlapping 6-s segments, and their BPM posteriors are averaged to obtain a track-level estimate. GTZAN is used exclusively for external testing and never for training, validation, checkpoint selection, or hyperparameter tuning.

4.1.3. GiantSteps Tempo

GiantSteps Tempo contains 664 Beatport previews of electronic dance music with manually corrected tempo annotations [37]. We use the version-2 annotations and remove three tracks whose BPM value is zero, leaving 661 valid test tracks. All audio files are obtained from the official JKU/Beatport mirror and verified against the official MD5 checksums.

Five uniformly spaced 6-s segments are extracted from each track and combined with the same posterior-averaging strategy used for GTZAN. This setting tests whether a model trained on short loops transfers to long-form electronic dance music and provides a common local evaluation set for Quinton-ESS, Beat This!, and other recent systems.

The datasets employed in this study serve different purposes in the experimental framework, as summarized in

Table 1. Datasets and their roles in the experiments.

Dataset	Valid Samples	Audio Format	Experimental Role	Training
FSL10K	9486	6-s musical loops	Five-fold main study, baselines, and ablations	Yes
GTZAN	998	30-s complete excerpts	Cross-genre zero-shot evaluation	No
GiantSteps Tempo v2	661	∼120-s EDM previews	Standard benchmark and recent-method comparison	No

.

4.2. Implementation Details

Audio is converted to mono and resampled to 11,025 Hz. Each input is a 6-s segment represented by an 81-bin log-Mel spectrogram. We use a short-time Fourier transform window of 1024 samples and a hop size of 512 samples, producing an input tensor of size $1\times 81\times 128$. Tempo estimation is formulated as a 271-class problem with one integer-valued class per BPM from 30 to 300. The model containing only the convolutional acoustic backbone and exact-BPM head is denoted ERM. The configuration that adds only the DPRM is denoted Periodic (ERM+DPRM), and the complete model is denoted PEA-Tempo.

For a long recording, we uniformly select N 6-s segments and fuse their exact-BPM posteriors,

$$\overline{p}(c\mid x)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N}p(c\mid X_n),\widehat{b}=30+\arg \underset{c}{\max }\overline{p}(c\mid x).$$

(29)

Only the exact-BPM posteriors are fused; the circular phase head, a beat tracker, and dynamic Bayesian post-processing are not used. We fix $N=5$ for both GTZAN and GiantSteps.

For PEA-Tempo, its ERM and Periodic variants, and the MGANet and He-TempoTCN adaptations trained from scratch on FSL10K, we use AdamW with an initial learning rate of ${10}^{-3}$, weight decay of ${10}^{-4}$, a batch size of 64, and at most 25 epochs. Training stops when validation ACC1 does not improve for six consecutive epochs, and the gradient norm is clipped to 5.0. The weights of the original-sample and tempo-transformed classification losses are 1.0 and 0.5, respectively; the auxiliary phase-classification and phase-equivariance losses are weighted by 0.35 and 0.2.

Experiments are conducted on an NVIDIA GeForce RTX 4070 GPU with 8 GB of memory. ERM and PEA-Tempo contain 236,927 and 326,231 trainable parameters, respectively. With precomputed feature caches, their cumulative times for five-fold optimization, validation, and final testing are 4.31 and 5.21 min; the mean selected epochs are 8.4 and 9.2. These times cover optimization, validation, and testing with precomputed feature caches; audio decoding and feature-cache generation are excluded. Here, “lightweight” refers to the compact model size of 326,231 trainable parameters. The reported timings characterize computational cost under the cached-feature experimental setting, while complete-pipeline inference profiling is left for future work. DPRM adds only two lightweight one-dimensional convolutions and a 41-dimensional autocorrelation descriptor; its FFT-based autocorrelation has complexity $O\left(T\log T\right)$. CTPM adds a small fully connected head during training, whereas TSEM introduces no additional trainable backbone.

Traditional baselines use the standard implementations or batch-analysis outputs archived with the project, with per-sample results aligned by sound ID. Percival14 and Percival14-Essentia denote our reproduction and the Essentia implementation, respectively [4,38]. madmom RNN+ACF uses pretrained neural beat activations and autocorrelation to infer global tempo [39]; it is distinguished from the resonating-comb-filter backend of Böck et al. [5]. Samples for which a traditional system produces no valid estimate are not numerically imputed and are excluded only from the common-subset comparison.

For learning-based baselines, the MGANet and He-TempoTCN adaptations are trained on FSL10K using the same 6-s input, 30–300-BPM output space, and optimization protocol. Quinton-ESS is initialized from the official pretrained TCN and fine-tuned end to end on the corresponding fold. Beat This! uses the official checkpoint with DBN post-processing disabled; global BPM is obtained from the median adjacent inter-beat interval in the range 0.2–2.0 s, and a clip is marked as having no prediction when no valid interval can be formed. On the external datasets, all local systems receive the same budget of five 6-s segments. The FSL10K-trained or fine-tuned systems use no target-domain data for training, tuning, or checkpoint selection; Beat This! retains its official pretraining provenance.

We distinguish three comparison settings: training from scratch on the same FSL10K folds (ERM, Periodic, PEA-Tempo, and the MGANet and He-TempoTCN adaptations); official pretraining followed by FSL10K fine-tuning (Quinton-ESS); and inference with an official pretrained checkpoint (Beat This!). The first setting supports comparisons under matched local training conditions. The latter two describe transfer performance with different training histories. Published GiantSteps scores are reported separately as benchmark context.

The comparison methods span three methodological families. Degara12, Zapata14, Percival14, and madmom RNN+ACF represent traditional or hybrid periodicity analysis [4,5,20,21]; Schreiber CNN, the Böck–Davies multi-task model, the MGANet adaptation, Beat This!, and the He-TempoTCN adaptation represent supervised learning and beat-transfer approaches [7,8,9,26,27]; and Quinton-ESS and the corrected Henkel-SDNet represent self-supervised tempo learning [10,11].

4.3. Evaluation Metrics and Statistical Analysis

ACC1 and ACC2 follow the standard protocol for tempo-induction systems: ACC1 accepts an approximately 4% relative error, whereas ACC2 additionally accepts common integer-related metrical levels [2,3]. Per-sample predictions, correctness decisions, and error terms are retained in accordance with the transparent evaluation principles advocated by MIR evaluation tools [40].

Let $y_i$ and ${\widehat{y}}_i$ denote the reference and predicted tempos. ACC1 is defined as

$$\mathrm{ACC}1={\displaystyle \frac{100}{N}}{\displaystyle \sum _{i=1}^N}\mathbb{I}\left({\displaystyle \frac{|{\widehat{y}}_i-y_i|}{y_i}}\le 0.04\right).$$

(30)

ACC2 additionally accepts the metrical factors $\{1/3,1/2,1,2,3\}$ [3]:

$$\mathrm{ACC}2={\displaystyle \frac{100}{N}}{\displaystyle \sum _{i=1}^N}\mathbb{I}\left(\underset{r\in \{1/3,1/2,1,2,3\}}{\min }{\displaystyle \frac{|{\widehat{y}}_i-r{y}_i|}{r{y}_i}}\le 0.04\right).$$

(31)

The ACC2-only error is the percentage of examples that satisfy ACC2 but not ACC1 and therefore reflects predictions at an alternative metrical level accepted by the relaxed metric. Because ACC2 also accepts one-third and triple tempo, we do not equate this quantity with a strict half/double-only octave error. MAE is the mean absolute difference between predicted and reference BPM. The tempo-scaling experiment additionally reports response accuracy, ${\log }_2$-tempo response error, and posterior phase error.

For FSL10K, confidence intervals are obtained with 10,000 paired bootstrap replicates at the contributor–pack connected-component level. Track-level paired bootstrap is used for GTZAN and GiantSteps. The main five-fold evaluation uses one training run per configuration per fold. We additionally evaluate the four incremental configurations using three random seeds on the fixed FSL10K fold-0 split, with the same seed set across configurations. Checkpoints are selected using only FSL10K validation data, and results are reported as the mean and sample standard deviation across seeds.

For the additional multi-seed paired comparisons, the same evaluation units are resampled for both models and all seeds within each bootstrap replicate, and metric differences are averaged across seeds. These intervals characterize test-sample uncertainty for the evaluated runs, complementing the across-seed standard deviations.

4.4. Baseline Comparisons

The out-of-fold predictions from all five test folds are pooled, and only the 9364 samples for which every traditional and learned system has a valid output are retained. PEA-Tempo obtains 67.30% ACC1 and 76.59% ACC2, outperforming every traditional method. Relative to the best traditional scores, these values represent gains of 10.73 and 5.66 percentage points. Adding DPRM to ERM already raises ACC1 and ACC2 to 66.16% and 74.99%, while the complete PEA-Tempo achieves the highest values for both metrics in this pooled comparison. Periodic obtains the lowest MAE, further indicating that explicit periodic evidence reduces absolute BPM deviations.

Table 2. Comparison with traditional methods and our variants on the five-fold FSL10K out-of-fold common subset ($N=9364$). Best values for each metric are shown in bold. ↑ indicates that higher values are better, while ↓ indicates that lower values are better. The same notation is used in the following tables.

Method	Year	Type	Parameters	ACC1 ↑	ACC2 ↑	ACC2-Only ↓	MAE/BPM ↓
Zapata14 [21]	2014	Traditional	–	56.11	67.29	11.18	21.13
Degara12 [20]	2012	Traditional	–	56.26	67.08	10.82	21.12
Percival14 [4]	2014	Traditional	–	55.88	70.93	15.05	21.80
Percival14-Essentia [4,38]	2014	Traditional	–	56.57	70.43	13.86	20.74
madmom RNN+ACF [5,39]	–	Hybrid	–	48.64	68.57	19.93	30.32
ERM	Ours (baseline)	Learned	236,927	64.06	71.55	7.49	15.02
Periodic (ERM+DPRM)	Ours (variant)	Learned	326,231	66.16	74.99	8.83	14.19
PEA-Tempo	Ours	Learned	326,231	67.30	76.59	9.29	14.65

reports the five-fold pooled comparison, whereas

Table 3. Comparison with recent learning-based methods on FSL10K fold 0 ($N=2262$). PEA-Tempo and the MGANet and He-TempoTCN adaptations are trained from scratch under matched local training conditions. Quinton-ESS uses official pretraining followed by FSL10K fine-tuning, whereas Beat This! uses an official pretrained checkpoint without FSL10K fine-tuning. These settings distinguish comparisons under matched local training conditions from transfer evaluations using pretrained models.

Method	Year	Parameters	ACC1 ↑	ACC2 ↑	MAE/BPM ↓
MGANet adaptation [9]	2021	3,746,725	55.22	66.53	19.61
Quinton-ESS [10]	2022	32,942	31.48	41.07	21.86
Beat This! small0 [26]	2024	2,099,944	31.83	48.19	50.75
He-TempoTCN adaptation [27]	2025	105,471	54.07	68.21	21.61
PEA-Tempo	Ours	326,231	56.68	71.79	20.48

fixes the same 2262 fold-0 test clips for all locally evaluated recent methods. To ensure a fair comparison, the fold-0 results in Table 3 are interpreted separately from the pooled five-fold results in Table 2. On the fixed fold-0 test set, PEA-Tempo obtains the highest ACC1 and ACC2, reaching 56.68% and 71.79%, respectively. It improves ACC2 over the MGANet adaptation by 5.26 percentage points (95% CI: [0.73, 6.72]) and improves ACC1 and ACC2 over the He-TempoTCN adaptation by 2.61 and 3.58 points, respectively. PEA-Tempo uses only approximately 8.7% as many parameters as MGANet and 15.5% as many as Beat This!, indicating that the observed gains are not simply attributable to increased model capacity.

Quinton-ESS and Beat This! perform less strongly on FSL10K. The dataset consists primarily of short loop audio, whereas Beat This! requires a predicted beat sequence from which global BPM can be summarized; it fails to produce a valid tempo for 235 test clips. Quinton-ESS is initialized from the official pretrained TCN and then fine-tuned, and its result highlights the effects of pretraining domain, tempo representation, and downstream adaptation on transfer to loop audio. Overall, explicitly modeling periodic structure and tempo transformations is well suited to this short-loop setting.

4.5. Cross-Dataset Generalization

Checkpoints trained and selected exclusively on FSL10K are transferred directly to GTZAN and GiantSteps Tempo. The external data are not used for training, validation, or hyperparameter tuning. We report the fold-0 checkpoint and posterior averaging across five source-disjoint-fold checkpoints as separate inference configurations. Their comparison measures the effect of checkpoint averaging; variation across training partitions and random seeds requires separate analysis.

4.5.1. Results on the GTZAN Dataset

Table 4. Zero-shot results on GTZAN. Bold values indicate the best result within each inference configuration.

Model	Inference Configuration	ACC1 ↑	ACC2 ↑	ACC2-Only ↓	MAE/BPM ↓
ERM	Fold-0 checkpoint	55.01	72.65	17.64	26.89
Periodic (ERM+DPRM)	Fold-0 checkpoint	57.82	75.55	17.74	25.45
PEA-Tempo	Fold-0 checkpoint	59.12	77.56	18.44	25.92
ERM	Five-checkpoint ensemble	56.01	72.55	16.53	25.69
Periodic (ERM+DPRM)	Five-checkpoint ensemble	59.42	78.26	18.84	25.58
PEA-Tempo	Five-checkpoint ensemble	60.02	79.16	19.14	25.22

reports zero-shot results on 998 GTZAN tracks. Each track is represented by five uniformly spaced 6-s segments whose posteriors are fused. The first three rows use the fold-0 checkpoint, and the final three rows use five-checkpoint ensembles.

With a single checkpoint, PEA-Tempo improves ACC1 and ACC2 over ERM by 4.11 and 4.91 percentage points; the corresponding 95% confidence intervals are [2.10, 6.21] and [2.81, 7.01]. It also improves over Periodic by 1.30 and 2.00 points. The five-checkpoint PEA-Tempo ensemble reaches 60.02% ACC1 and 79.16% ACC2, gains of 4.01 and 6.61 points over the corresponding ERM ensemble, while reducing MAE from 25.69 to 25.22 BPM. It further improves ACC1 and ACC2 over the Periodic ensemble by 0.60 and 0.90 points. Explicit periodicity modeling provides the main cross-genre transfer gain. CTPM–TSEM adds positive ACC1 and ACC2 point estimates in both GTZAN inference configurations; the ensemble increments have confidence intervals spanning zero. The higher ACC2-only rate indicates that the remaining errors are increasingly concentrated at integer-related metrical levels rather than at unrelated periods.

4.5.2. Results on the GiantSteps Tempo Dataset

As shown in

Table 5. Local results on the common 661-track GiantSteps Tempo test set. All systems receive five segments; training provenance and the number of checkpoints differ across rows.

Method	Year	Inference Configuration	ACC1 ↑	ACC2 ↑	ACC2-Only ↓	MAE/BPM ↓
MGANet adaptation [9]	2021	Fold-0 checkpoint	80.94	91.23	10.29	12.52
Quinton-ESS (FSL10K transfer) [10]	2022	Transferred model	10.74	10.74	0.00	26.03
Beat This! small0 [26]	2024	Official checkpoint	80.03	85.78	5.75	11.60
He-TempoTCN adaptation [27]	2025	Fold-0 checkpoint	82.45	94.55	12.10	13.03
ERM	Ours	Fold-0 checkpoint	79.58	90.77	11.20	13.51
Periodic (ERM+DPRM)	Ours	Fold-0 checkpoint	79.73	90.47	10.74	12.81
PEA-Tempo	Ours	Fold-0 checkpoint	82.00	92.74	10.74	12.25
ERM	Ours	Five-checkpoint ensemble	82.30	90.62	8.32	11.33
Periodic (ERM+DPRM)	Ours	Five-checkpoint ensemble	85.93	94.40	8.47	9.67
PEA-Tempo	Ours	Five-checkpoint ensemble	86.54	95.16	8.62	9.72

, on the same 661 tracks and five-segment budget, the five-checkpoint PEA-Tempo ensemble obtains the highest ACC1 and ACC2 in the local comparison. Relative to the ERM ensemble, ACC1 and ACC2 increase by 4.24 and 4.54 points and MAE decreases by 1.61 BPM. The PEA-Tempo ensemble exceeds the single-checkpoint He-TempoTCN adaptation by 4.09 and 0.61 points in ACC1 and ACC2, respectively. This comparison combines a method difference with checkpoint averaging; the PEA-Tempo fold-0 checkpoint scores below He-TempoTCN on both accuracy metrics. Compared with the Periodic ensemble, PEA-Tempo adds 0.61 points in ACC1 and 0.76 points in ACC2, while MAE differs by only 0.05 BPM. These ensemble increments concern hit rates, while Periodic retains a slightly lower MAE. For the fold-0 checkpoint, PEA-Tempo obtains 82.00% ACC1, 92.74% ACC2, and 12.25 BPM MAE. Compared with Periodic, ACC1 and ACC2 each improve by 2.27 percentage points, while MAE decreases by 0.56 BPM. Relative to ERM, the corresponding improvements are 2.42 and 1.97 percentage points, with a 1.26 BPM reduction in MAE. Together with the positive single-checkpoint accuracy increments on GTZAN, these results show higher ACC1 and ACC2 point estimates for the joint CTPM–TSEM configuration beyond explicit periodicity modeling.

As shown in

Table 6. Representative published GiantSteps Tempo results, shown as benchmark context with differing training and evaluation protocols.

Method	Year	ACC1 ↑	ACC2 ↑
Schreiber CNN [7]	2018	82.10	97.10
Böck–Davies multi-task [8]	2020	86.00	96.50
Quinton-ESS [10]	2022	47.00	88.60
Henkel-SDNet corrected [11]	2024	72.80	94.90
PEA-Tempo, five-checkpoint ensemble	Ours	86.54	95.16

, among the listed results, PEA-Tempo achieves the highest ACC1, exceeding the Böck–Davies multi-task model and Schreiber CNN by 0.54 and 4.44 percentage points. Relative to the recent self-supervised Quinton-ESS and corrected Henkel-SDNet systems, its ACC1 improves by 39.54 and 13.74 points and its ACC2 by 6.56 and 0.26 points. Its ACC2 remains 1.34 and 1.94 points below Böck–Davies and Schreiber CNN, respectively, showing that the current advantage is concentrated in strict absolute-tempo identification, with room to improve the coverage of integer-related periodic families. Across the three evaluated datasets, the complete PEA-Tempo model improves ACC1 and ACC2 over plain ERM. The five-checkpoint PEA-Tempo ensemble improves ACC1 over the ERM ensemble by 4.01 points on GTZAN and 4.24 points on GiantSteps, indicating that the learned improvements transfer across short loops, mixed-genre recordings, and electronic dance music.

4.6. Ablation Studies

4.6.1. Incremental Analysis of the Core Modules

Starting from the shared acoustic backbone, we incrementally add DPRM, CTPM, and TSEM, as summarized in

Table 7. Incremental ablation of the core modules on FSL10K fold 0. ✓indicates that the corresponding module is included in the ablation setting.

Configuration	DPRM	CTPM	TSEM	ACC1 ↑	ACC2 ↑	ACC2-Only ↓	MAE/BPM ↓
Acoustic backbone	–	–	–	53.89	65.34	11.45	20.11
+DPRM	✓	–	–	55.78	69.52	14.10	19.62
+ DPRM + CTPM	✓	✓	–	56.15	69.89	14.68	19.93
Complete PEA-Tempo	✓	✓	✓	56.68	71.79	15.12	20.48

. All configurations use the same input, exact-BPM classification head, data split, and optimization settings. DPRM improves ACC1 and ACC2 by 1.89 and 4.18 percentage points, respectively, while reducing MAE by 0.49 BPM, indicating that the explicit spectral-flux, autocorrelation, and rhythmic-density features provide a substantial contribution to static tempo recognition. Adding CTPM further improves both ACC1 and ACC2 by 0.37 percentage points. Adding TSEM yields additional gains of 0.53 and 1.90 points in ACC1 and ACC2, respectively, resulting in the highest ACC1 and ACC2 values in this ablation. These results show that CTPM and TSEM provide complementary improvements in classification accuracy on top of DPRM. However, the accuracy gains are accompanied by increases in ACC2-only error and MAE. Relative to DPRM alone, the complete model improves ACC1 and ACC2 by 0.90 and 2.27 percentage points, respectively, while MAE increases by 0.86 BPM.

The pooled five-fold ablation results are summarized in

Table 8. Five-fold ablation results for the principal configurations.

Configuration	DPRM	CTPM	TSEM	ACC1 ↑	ACC2 ↑	ACC2-Only ↓	MAE/BPM ↓
Acoustic backbone	–	–	–	64.17	71.66	7.50	14.99
+ DPRM	✓	–	–	66.24	75.05	8.80	14.16
Complete PEA-Tempo	✓	✓	✓	67.40	76.66	9.26	14.61

. DPRM improves ACC1 and ACC2 over the acoustic backbone by 2.07 and 3.39 percentage points, respectively, while reducing MAE by 0.83 BPM. Adding CTPM and TSEM further improves ACC1 and ACC2 by 1.16 and 1.61 points over DPRM, with 95% paired confidence intervals of [$-0.46$, 3.97] and [$-0.05$, 4.49], respectively. Both intervals include zero. Meanwhile, MAE increases by 0.45 BPM from 14.16 to 14.61 BPM, and ACC2-only error rises from 8.80% to 9.26%. These results indicate a metric-dependent trade-off: the complete model achieves the highest ACC1 and ACC2, whereas DPRM alone yields the lowest MAE among the ablated variants.

The paired performance differences between PEA-Tempo and the Periodic baseline across the different evaluation settings are summarized in

Table 9. Paired differences for PEA-Tempo minus Periodic with 95% bootstrap confidence intervals. Accuracy differences are in percentage points, and MAE differences are in BPM. Positive ACC1/ACC2 differences and negative MAE differences favor PEA-Tempo.

Evaluation	$\mathbf{\Delta }$ACC1 [95% CI]	$\mathbf{\Delta }$ACC2 [95% CI]	$\mathbf{\Delta }$MAE [95% CI]
FSL10K pooled five-fold	1.16 [$-0.46$, 3.97]	1.61 [$-0.05$, 4.49]	0.46 [0.02, 0.85]
GTZAN fold 0	1.30 [$-0.50$, 3.11]	2.00 [0.10, 3.91]	0.47 [$-0.49$, 1.46]
GTZAN ensemble	0.60 [$-0.70$, 1.90]	0.90 [$-0.60$, 2.30]	$-0.36$ [$-1.04$, 0.31]
GiantSteps fold 0	2.27 [0.00, 4.54]	2.27 [0.30, 4.24]	$-0.56$ [$-2.06$, 0.88]
GiantSteps ensemble	0.61 [$-0.76$, 1.97]	0.76 [$-0.45$, 1.97]	0.05 [$-0.89$, 1.00]

. The table shows that PEA-Tempo generally yields positive gains in ACC1 and ACC2, while the changes in MAE are less consistent across datasets and evaluation settings, indicating that improvements in tempo accuracy do not always translate into lower absolute BPM error.

To assess the reproducibility of the incremental gains, we repeat the four configurations with three random seeds on the fixed FSL10K fold-0 split.

Table 10. Multi-seed evaluation, reported as mean ± SD.

Dataset	Configuration	ACC1 (%) ↑	ACC2 (%) ↑	MAE (BPM) ↓
FSL10K	ERM	$53.90\pm 0.72$	$65.40\pm 0.85$	$20.20\pm 0.31$
	+ DPRM	$56.60\pm 0.65$	$70.80\pm 0.71$	$19.70\pm 0.28$
	+ DPRM + CTPM	$56.20\pm 0.78$	$70.20\pm 0.90$	$20.00\pm 0.35$
	Complete PEA-Tempo	$56.90\pm 0.69$	$71.60\pm 0.76$	$20.30\pm 0.32$
GTZAN	ERM	$55.20\pm 0.81$	$72.80\pm 0.94$	$26.80\pm 0.48$
	+ DPRM	$57.70\pm 0.74$	$75.50\pm 0.82$	$25.50\pm 0.39$
	+ DPRM + CTPM	$58.10\pm 0.86$	$76.00\pm 0.88$	$25.80\pm 0.45$
	Complete PEA-Tempo	$59.00\pm 0.77$	$77.30\pm 0.80$	$25.90\pm 0.41$
GiantSteps	ERM	$79.50\pm 0.91$	$90.60\pm 0.75$	$13.50\pm 0.42$
	+ DPRM	$79.90\pm 0.88$	$90.70\pm 0.79$	$12.90\pm 0.37$
	+ DPRM + CTPM	$80.60\pm 0.96$	$91.40\pm 0.84$	$12.70\pm 0.40$
	Complete PEA-Tempo	$81.70\pm 0.83$	$92.50\pm 0.72$	$12.40\pm 0.35$

shows higher mean ACC1 and ACC2 for PEA-Tempo than Periodic on all three datasets, with larger mean increments on the external datasets. All six corresponding accuracy confidence intervals in

Table 11. PEA-Tempo minus Periodic, averaged across seeds. Accuracy differences are in percentage points; MAE differences are in BPM. Positive ACC1/ACC2 differences and negative MAE differences favor PEA-Tempo.

Dataset	$\mathbf{\Delta }$ACC1 [95% CI]	$\mathbf{\Delta }$ACC2 [95% CI]	$\mathbf{\Delta }$MAE [95% CI]
FSL10K	$+0.30[-0.70,1.30]$	$+0.80[-0.40,2.00]$	$+0.60[-0.10,1.30]$
GTZAN	$+1.30[-0.40,3.00]$	$+1.80[-0.20,3.80]$	$+0.40[-0.30,1.10]$
GiantSteps	$+1.80[-0.40,4.00]$	$+1.80[-0.10,3.70]$	$-0.50[-1.30,0.30]$

include zero, indicating uncertainty in the magnitude and direction of these incremental differences.

To quantify uncertainty in these increments, we compare PEA-Tempo and Periodic using paired bootstrap confidence intervals on the same evaluation samples.

Table 11 shows positive mean accuracy differences on all three datasets. Mean MAE is lower for PEA-Tempo on GiantSteps and for Periodic on FSL10K and GTZAN, with all three MAE intervals including zero. In the pooled five-fold FSL10K evaluation, the MAE difference is $+0.46$ BPM $[0.02,0.85]$ in favor of Periodic (Table 9). The positive accuracy differences therefore coexist with a metric-dependent trade-off rather than a consistent reduction in absolute BPM error.

4.6.2. Tempo-Scaling Equivariance of TSEM

As shown in

Table 12. Spectrogram-domain tempo-scaling results. Metrics are macro-averaged over the four non-identity scaling factors.

Configuration	DPRM	CTPM	TSEM	ACC1 ↑	ACC2 ↑	Response Acc. ↑	Log-Response MAE/Oct. ↓	Phase MAE/Cycle ↓
Acoustic backbone	–	–	–	51.82	61.17	57.73	0.1920	0.0833
+ DPRM	✓	–	–	57.06	66.82	65.29	0.1656	0.0798
Complete PEA-Tempo	✓	✓	✓	57.93	68.53	66.83	0.1659	0.0762

, we apply factors $\{0.8,0.9,1.1,1.25\}$ to the cached test log-Mel spectrograms using the same temporal bilinear resampling, repetition, and center-cropping operation as in training. Transformed labels are $sb$, and examples outside 30–300 BPM are excluded for each factor. This experiment measures consistency under the specified spectrogram transformation. Response accuracy measures whether ${\widehat{b}}^{\left(s\right)}$ is within 4% of $s\widehat{b}$; phase error is computed from the exact-BPM posterior.

DPRM raises response accuracy by 7.56 points. Adding CTPM and TSEM increases transformed ACC2 by 1.71 points and response accuracy by 1.54 points, while reducing phase error from 0.0798 to 0.0762 cycles. Log-response MAE is similar (0.1656 versus 0.1659 octaves). These measurements characterize the response to the spectrogram-domain transformation used in training.

As shown in

Table 13. Waveform-domain tempo-scaling evaluation, macro-averaged over factors $\{0.8,0.9,1.1,1.25\}$.

Configuration	ACC1 (%) ↑	ACC2 (%) ↑	Response Acc. (%) ↑	Log-Response MAE (Oct.) ↓	Phase MAE (Cycle) ↓
ERM	50.90	60.80	56.40	0.2010	0.0870
+ DPRM	55.80	66.00	63.50	0.1740	0.0830
+ DPRM + CTPM	55.60	66.20	63.20	0.1760	0.0820
Complete PEA-Tempo	56.40	67.10	64.60	0.1730	0.0800

, to test whether the learned scaling response transfers beyond spectrogram resampling, we apply pitch-preserving time stretching with Rubber Band to the test waveforms, recompute log-Mel features, and evaluate the existing checkpoints without further fine-tuning. We use playback-rate factors $\{0.8,0.9,1.1,1.25\}$, with factor s corresponding to reference tempo $sb$. All configurations use the same eligible examples and segment-selection protocol. Examples with transformed reference tempos outside 30–300 BPM are excluded for the corresponding factor. Metric definitions follow the spectrogram-domain evaluation.

Compared with the configuration containing DPRM and CTPM, PEA-Tempo improves response accuracy by 1.40 percentage points and reduces phase MAE from 0.0820 to 0.0800 cycles. These results suggest that TSEM’s contribution extends to the evaluated waveform-domain transformation, complementing the spectrogram-domain findings.

4.6.3. Multi-Segment Posterior Fusion

On GTZAN, five-segment posterior fusion raises ACC1 by 2.20, 1.90, and 2.10 points for the single-fold acoustic backbone, Periodic, and complete PEA-Tempo models, respectively. The five-checkpoint PEA-Tempo ensemble gains 2.91 points. This training-free strategy aggregates local evidence from short segments into a more stable track-level estimate.

4.7. Qualitative Analysis

To clarify the failure modes underlying the aggregate accuracy gains, we distinguish between metrical-level errors that fail ACC1 but satisfy ACC2 and errors that fail both metrics. The following examples examine both types of correction, together with newly introduced errors and shared failures. Sensitivity to tempo scaling is evaluated separately through the controlled transformations in Section 4.6.

4.7.1. Representative Cases

As shown in

Table 14. Representative corrections and failure cases.

Category	Audio	Reference	Backbone	+DPRM	PEA-Tempo	Observation
Correction	Messy Calipso loop	99	200	100	100	Periodic branch suppresses the double-tempo peak
Correction	Typewriter loop	100	200	100	100	Main period recovered for nontraditional percussion
Correction	Hardcore loop	200	100	150	200	PEA selects the correct fast metrical level
New error	Sound 474322	124	125	125	250	Posterior switches to the double-tempo peak
New error	Sound 72151	124	125	125	250	Correct family but incorrect absolute level
Shared failure	Rain loop	32	150	120	156	Weakly periodic ambience lacks stable pulses

, the first two cases show that DPRM can reorganize competing 100/200-BPM peaks when timbral cues are unreliable. The 200-BPM hardcore example illustrates how the joint CTPM–TSEM constraint can assist fast-level selection. The two 124-BPM cases explain the increase in ACC2-only error: circular phase identifies the correct periodic family, but the prediction can still switch between integer-related metrical levels.

4.7.2. Visualization of Error Structure

Figure 2, Figure 3 and Figure 4 analyze three complementary levels: local candidate reranking, the global distribution of prediction errors, and controlled transformation responses.

For the first two clips, whose reference tempos are 99 and 100 BPM, ERM ranks 200 BPM first and therefore makes a double-tempo error. Periodic moves the dominant peak to 100 BPM after DPRM is introduced. PEA-Tempo further concentrates most of the probability mass within 98–101 BPM and suppresses the competing 200-BPM peak. The explicit autocorrelation and spectral-flux evidence can therefore rerank candidates when the acoustic backbone has already detected the correct tempo but ranks it below an integer-related alternative. For the fast 200-BPM hardcore loop, ERM and Periodic select 100 and 150 BPM, whereas PEA-Tempo promotes 200 BPM to the dominant hypothesis, illustrating the complementary contribution of phase and scaling constraints.

The figure also retains representative failures. For both 124-BPM clips, ERM and Periodic concentrate near 124–125 BPM, whereas PEA-Tempo promotes 250 BPM. The correct periodic family is recognized, but the posterior switches from the original to the double-tempo level, consistent with the increase in ACC2-only error. In the final weakly periodic 32-BPM clip, stable transient intervals and a clear autocorrelation peak are absent, and all three systems predict far from the reference. Structural periodicity priors cannot replace reliable low-level pulse evidence when that evidence is missing.

ERM forms pronounced horizontal high-density bands around 80–150 BPM, suggesting that different reference tempos are compressed toward frequent regions of the training distribution. After DPRM is added, the main density moves closer to the red diagonal, particularly in the mid-tempo range. PEA-Tempo further strengthens the high-density trajectory along the diagonal over a wider range, consistent with its higher ACC1 and ACC2. Residual bands near the half- and double-tempo lines indicate that metrical-level ambiguity remains in some examples. In the pooled five-fold ablation results (Table 8), PEA-Tempo reduces the proportion of predictions failing ACC2 from 28.34% to 23.34% compared with ERM. Meanwhile, the ACC2-only rate changes from 7.50% to 9.26%. Together with the improved ACC1, these results show better overall tempo recognition and greater coverage of the metrical alternatives accepted by ACC2, while identifying absolute metrical-level selection as a direction for further improvement.

Periodic retains higher ACC1 than ERM at every non-identity scaling factor, demonstrating that explicit periodic features mitigate degradation under time-scale changes. PEA-Tempo generally achieves the highest accuracy between 0.8× and 1.1×, with particularly clear margins over ERM at 0.8× and 1.1×. Under the stronger 1.25× acceleration, Periodic is slightly more accurate than PEA-Tempo, indicating that substantial acceleration can still trigger a metrical-level switch.

All systems have zero phase-response error for the identity transformation by construction. Across the non-identity transformations, the macro-average circular error decreases from 0.0833 cycles for ERM to 0.0798 for Periodic and 0.0762 for PEA-Tempo. Together with PEA-Tempo’s transformed macro-average ACC1 of 57.93%, this result indicates improved posterior-phase consistency for the evaluated spectrogram resampling. ERM and Periodic also use tempo augmentation during training, so the comparison measures the added phase objectives under this transformation. The response remains asymmetric between deceleration and acceleration.

4.7.3. Matched-Sample Comparison with Recent Methods

Figure 5 compares track-level predictions from ERM, MGANet-2021, Quinton-ESS-2022, Beat This!-2024, He-TempoTCN-2025, and PEA-Tempo on the same FSL10K fold-0 examples. The solid black line denotes the reference BPM, the green region denotes the $\pm 4\%$ ACC1 interval, and gray dashed lines show the half- or double-tempo positions when they fall within the evaluation range. N/A denotes that no valid estimate was produced.

In the first three cases, PEA-Tempo is the only method whose prediction falls within the ACC1 interval. For the 140-BPM drum loop, PEA-Tempo predicts 140 BPM, whereas ERM, MGANet, Quinton-ESS, Beat This!, and He-TempoTCN predict 100, 96, 120, 250, and 70 BPM. For the 90-BPM synthesizer loop, PEA-Tempo predicts 90 BPM, while all five comparison systems fall between 120 and 140 BPM. For the 95-BPM piano loop, PEA-Tempo predicts 96 BPM, while the comparison systems predict 90, 120, 120, 47.6, and 128 BPM. These clips originate from different contributors. For the 90-BPM synthesizer loop (FSL10K ID 370351), all five baseline estimates fail both ACC1 and ACC2, whereas PEA-Tempo satisfies both. The baseline values of 120–140 BPM fall outside the accepted metrical families, so this example concerns an incorrect periodicity estimate rather than only half/double-tempo selection. The 140-BPM drum loop and 95-BPM piano loop each have four baselines failing both metrics; He-TempoTCN at 70 BPM and Beat This! at 47.6 BPM, respectively, still satisfy ACC2. These two examples combine periodicity errors with metrical-level ambiguity. These examples illustrate corrections of both errors outside the metrical alternatives accepted by ACC2 and errors in absolute metrical-level selection. DPRM provides explicit transient and repetition cues, while CTPM supplies auxiliary phase supervision shared by half- and double-tempo hypotheses.

The final 200-BPM example exposes a representative limitation. ERM, MGANet, Beat This!, and He-TempoTCN all predict 200 BPM, whereas PEA-Tempo selects the 100-BPM half-tempo level. Thus, although the proposed method outperforms every comparator in the first three examples, it does not fully resolve absolute metrical-level selection in fast music.

5. Conclusions, Limitations, and Future Work

5.1. Conclusions

This study addressed three difficulties in global music tempo estimation: insufficient explicit periodic evidence, confusion among integer-related metrical levels, and unstable responses to time scaling. We proposed PEA-Tempo, a lightweight periodicity- and equivariance-aware framework that incorporates structural tempo priors through DPRM, CTPM, and TSEM rather than relying on a larger acoustic backbone. DPRM combines spectral-flux rhythmic density, normalized autocorrelation, and convolutional acoustic semantics to preserve candidate periods at different lags. CTPM decomposes log-tempo into an integer level and a circular phase shared by integer-related tempo families. TSEM maps input time scaling to a deterministic circular phase rotation, encouraging the learned response to follow the physical tempo transformation. On the contributor–pack-disjoint five-fold FSL10K evaluation, PEA-Tempo improves ACC1 and ACC2 over the acoustic backbone by 3.24 and 5.00 percentage points, with paired-bootstrap confidence intervals excluding zero. The frozen model also retains cross-dataset gains on GTZAN and GiantSteps Tempo, where the five-checkpoint ensemble improves ACC1 over the ERM ensemble by 4.01 and 4.24 points. Incremental ablations identify DPRM as the main source of the static accuracy gain. CTPM–TSEM improves the reported spectrogram-scaling response and yields positive pooled ACC1 and ACC2 increments over DPRM, with confidence intervals spanning zero. The joint CTPM–TSEM configuration yields higher single-checkpoint ACC1 and ACC2 point estimates than Periodic on both GTZAN and GiantSteps. The three-seed evaluation also shows higher mean ACC1 and ACC2 on all three datasets, although all corresponding confidence intervals include zero. MAE differences vary across evaluation settings, reflecting a trade-off between tolerance-based accuracy and absolute BPM error. The remaining half- and double-tempo cases highlight an opportunity to further refine absolute metrical-level selection within the proposed framework.

5.2. Limitations

First, PEA-Tempo improves ACC1 and ACC2, but its FSL10K ACC2-only error increases from 7.50% to 9.26%. Circular phase helps identify integer-related periodic families, but does not by itself guarantee the choice of the exact BPM within the levels accepted by the relaxed metric. Further improving absolute metrical-level discrimination is therefore an important direction for extending the present framework. Second, the training data consist predominantly of short musical loops. Although external testing demonstrates transfer to complete tracks, performance on weakly pulsatile or freely timed genres such as classical and jazz remains lower than on music with stable rhythmic patterns. Finally, the present study predicts a single global BPM and does not address gradual tempo changes, abrupt local tempo shifts, or multiple perceptually plausible metrical interpretations within a track.

5.3. Future Work

The additional multi-seed evaluation covers three random seeds on the fixed fold-0 split, while the main five-fold evaluation uses one training run per configuration per fold. Training variability is therefore assessed over a limited set of runs; the reported bootstrap intervals characterize test-sample uncertainty conditional on these runs. Extending repeated training across all folds would provide a more comprehensive assessment of the incremental differences. Future work will first target absolute metrical-level selection. A direct approach is to freeze the acoustic backbone and periodic branch, introduce a lightweight selector over half-, original-, and double-tempo candidates, and optimize it jointly with strict ACC1 and a level-aware objective. Another direction is to retain multiple level hypotheses alongside the circular family posterior and rerank them using genre, rhythmic density, and long-term context instead of committing to a hard decision from a single short segment. The training data can also be expanded to include more weakly percussive, classical, jazz, and solo-instrumental music, with source-isolated cross-dataset protocols used to evaluate robustness to genuine distribution shifts. Future comparisons should retrain additional recent baselines under a unified split and input protocol. Finally, DPRM, CTPM, and TSEM could be extended to local tempo curves and beat tracking so that the circular tempo representation jointly describes time-varying period, phase, and metrical level.