Rhythmic segment analysis: Conceptualizing, visualizing, and measuring rhythmic data

We present vega-mir, an open-source Python library that bundles nine information-theoretic and statistical metrics for the analysis of symbolic music corpora behind a small, tested, citable API, and demonstrates two of them at corpus scale in case studies not addressed by the upstream Cygnus paper. Of the nine metrics, three (Shannon entropy, Kullback-Leibler divergence, Zipfian fits) were deployed in the companion Cygnus arXiv preprint; two (network analysis on chord-transition graphs and spectral analysis of rubato curves) are deployed in full case studies here; the four remaining (multi-dimensional Gini, chi-squared stationarity, Higuchi fractal dimension, interval distribution) are validated against analytic anchors and exercised as sanity checks on a bundled 8-composer dataset. The two case studies yield two main observations. First, on the fourteen MAESTRO composers with N >= 10 pieces, the PageRank value of the gravity-centre node correlates with the marginal Kullback-Leibler distance at rho = 0.61 (Spearman, composer-level jackknife N = 14); the categorical gravity-centre identity takes five distinct values across the corpus but is not itself correlated with marginal KL (rho = 0.13, p = 0.21). Second, on the 247-piece Bach multi-master corpus (Schiff, Gould, Richter), Gould holds the highest periodicity ratio of the three performers, not the lowest, inverting the cliché that low scalar rubato reads as "metronomic": Gould's rubato is small in amplitude but structured in time, with a median dominant period of 66 beats against Schiff's 102 and Richter's 104.

The choice of visualisation in empirical performance analysis is not a neutral presentation decision but an analytical one: different graphical forms reveal different features of the same dataset, and reliance on any single type systematically conceals what the others expose. This paper presents and argues for a suite of five complementary visualisation tools; tempographs, histograms with spline-smoothed probability density functions, ridgeline plots, stacked bar charts, and combination charts. These are applied to bar-level beats-per-minute data from recordings of Beethoven's five piano and cello sonatas (Op.~5 Nos.1 and2; Op.~69; Op.~102 Nos.1 and2) spanning 1930--2012. Each tool is described formally, its analytical properties characterised, its implementation detailed in working Python and MATLAB code, and its specific contribution demonstrated on a worked example using two recordings of Op.~5 No.~1 (Casals/Horszowski 1930--39 and Isserlis/Levin 2012) separated by eight decades. A five-panel composite figure applies all five tools to the same two recordings simultaneously, making the complementarity argument concrete: the tempograph reveals moment-to-moment structural parallels invisible in aggregate statistics; the spline-smoothed histogram exposes bimodality and secondary peaks suppressed by binning artefacts; the ridgeline plot positions both recordings within the full distributional space; the stacked bar chart shows divergent sectional pacing concealed by identical movement means; and the combination chart integrates mean tempo, variability, and historical reference marks in a single view. The spline-CDF smoothing method, applied to histogram data via cubic spline interpolation with zero-slope boundary conditions, is presented as a novel contribution to the performance analysis toolkit. Full implementation code is publicly available.

Pitch tracking systems typically couple a per-frame fundamental frequency ($F_0$) estimator with a temporal smoothing stage to obtain continuous trajectories. Conventional Viterbi smoothers enforce first-order continuity but lack long-term temporal awareness and could lock into octave errors across corrupted frames. We propose Relative Interval Networks (RIN), a trajectory smoothing framework that reconciles per-frame pitch estimates with data-driven multi-hop pitch differences. We extract robust relative pitch intervals across arbitrary frame offsets using Variable-Q Transform cross-correlation. We formulate pitch smoothing as an $L_1$-norm optimization problem and prove its equivalence to a minimum cost circulation problem, solved efficiently via linear programming. Evaluations across speech, singing, and instrumental datasets show that RIN substantially improves weak estimators, matches or outperforms Viterbi decoding at a comparable computational cost, and provides superior robustness under certain acoustic degradation.