A Complementary Visualisation Suite for Empirical Performance Analysis: Tempographs, Histograms, Ridgeline Plots, Stacked Bar Charts, and Combination Charts Applied to Beethoven's Piano and Cello Sonatas

Ignasi Sole

arxiv: 2604.18630 · v1 · submitted 2026-04-18 · 💻 cs.SD

A Complementary Visualisation Suite for Empirical Performance Analysis: Tempographs, Histograms, Ridgeline Plots, Stacked Bar Charts, and Combination Charts Applied to Beethoven's Piano and Cello Sonatas

Ignasi Sole This is my paper

Pith reviewed 2026-05-10 06:44 UTC · model grok-4.3

classification 💻 cs.SD

keywords empirical performance analysistempo visualizationBeethoven sonatashistogram smoothingridgeline plotstempographsstacked bar charts

0 comments

The pith

The choice of visualization for musical tempo data is an analytical decision because each graphical form exposes features hidden by the others.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that visualization methods in empirical performance analysis actively shape what can be seen in tempo datasets rather than neutrally displaying them. It introduces a suite of five tools—tempographs, spline-smoothed histograms, ridgeline plots, stacked bar charts, and combination charts—and applies them to bar-level beats-per-minute measurements from two recordings of Beethoven's Op. 5 No. 1 sonata separated by roughly eighty years. The argument is made concrete through a five-panel figure that shows the same data processed by all five methods at once. Each method is defined formally, implemented in open code, and shown to surface specific structural or distributional properties that the remaining methods do not.

Core claim

The paper establishes that the five visualization types are complementary: the tempograph makes moment-to-moment structural parallels visible, the spline-smoothed histogram reveals bimodality and secondary peaks, the ridgeline plot situates individual recordings inside the broader tempo distribution, the stacked bar chart exposes sectional pacing differences despite identical movement averages, and the combination chart integrates mean, variability, and historical markers in one frame. The spline-CDF smoothing procedure, using cubic interpolation with zero-slope boundaries, is offered as a concrete addition to the performance-analysis toolkit.

What carries the argument

A five-tool visualization suite (tempographs, spline-smoothed histograms, ridgeline plots, stacked bar charts, and combination charts) applied to bar-level tempo data, with the spline-CDF smoothing method as the explicit technical contribution.

If this is right

Analysts who rely on a single visualization type will systematically miss at least one class of tempo organization that another type would have shown.
Comparative studies of historical performances can separate overall mean tempo from sectional pacing and from local moment-to-moment correspondences only when multiple chart types are used together.
The spline-CDF procedure supplies a practical way to reduce binning artifacts in histogram-based tempo distributions while preserving boundary behavior.
Open Python and MATLAB implementations make it possible to reproduce or extend the five-panel composite view on any new bar-level tempo dataset.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same complementarity principle could be tested on non-tempo performance parameters such as dynamics or articulation.
The suite might be extended by adding one further chart type that directly visualizes tempo relationships between two performers rather than within a single recording.
If the spline smoothing method is adopted more widely, performance analysts will need explicit checks for over-smoothing on short or highly variable excerpts.

Load-bearing premise

That these particular five visualization types are sufficiently complementary and that the spline smoothing step reveals genuine features without creating or hiding important structure in the tempo measurements.

What would settle it

Apply the same five visualizations to a new set of recordings and check whether at least one method consistently fails to add any new observable feature beyond what the other four already display.

Figures

Figures reproduced from arXiv: 2604.18630 by Ignasi Sole.

read the original abstract

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 and~2; Op.~69; Op.~102 Nos.~1 and~2) 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper demonstrates a practical suite of visualizations for tempo data in music performance analysis with available code, but the spline smoothing method needs checking to ensure it does not create artifacts.

read the letter

This paper puts forward a set of five visualization methods for bar-level tempo data from classical recordings and shows how they work together on Beethoven sonata examples, with public code. The spline-smoothed histogram is the part that stands out as potentially new, but it also raises the biggest question. The work does a solid job laying out each tool with formal descriptions and then applying them all to the same pair of recordings, one from the 1930s and one from 2012. The five-panel figure makes the point concrete: the tempograph catches structural timing parallels that averages hide, the ridgeline plot shows where each recording sits in the broader distribution, the stacked bars highlight sectional differences even when overall means match, and the combination chart pulls in historical context. Having working Python and MATLAB implementations available is a real plus for anyone who wants to apply the same approach to other datasets. The central claim that these tools are complementary holds up in the examples given. Different plots do surface different features of the tempo variation. The soft spot is the spline-CDF smoothing used for the histograms. The method interpolates the cumulative distribution with cubic splines under zero-slope conditions to get a smoother density estimate. On data with abrupt tempo changes, which are common in older recordings, this can easily create oscillations or even negative density values near the tails. The paper presents it as revealing true bimodality that binning misses, but without additional checks—such as comparing against kernel density estimates or running the method on simulated data with known distributions—it is difficult to know whether the secondary peaks are real or just smoothing artifacts. That directly affects how much weight to give the complementarity argument for this particular tool. The dataset spans several sonatas across decades, though the detailed worked example sticks to two recordings. The citation pattern seems light on prior visualization work in performance analysis, but the focus is more on demonstration than on claiming broad theoretical advance. This paper is aimed at people doing empirical work on musical timing who already handle similar data and are looking for better ways to explore it visually. A reader in that niche would find the code and the side-by-side comparisons useful to adapt or build on. It deserves a serious referee because the contribution is concrete, the code is there to inspect, and the examples are fully worked. The smoothing issue is fixable with some extra validation. I would recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper claims that visualization choice in empirical performance analysis is analytical rather than neutral, since different graphical forms expose distinct features of the same tempo dataset while any single form conceals others. It presents a suite of five tools (tempographs, histograms with spline-CDF-smoothed PDFs, ridgeline plots, stacked bar charts, and combination charts), supplies formal descriptions, characterises their analytical properties, provides public Python/MATLAB code, and demonstrates complementarity on bar-level BPM data from two recordings of Beethoven Op. 5 No. 1 (Casals/Horszowski 1930–39 and Isserlis/Levin 2012), with a five-panel composite figure and extension to five sonatas spanning 1930–2012.

Significance. If the claimed complementarity holds and the spline-CDF method does not fabricate features, the work supplies a practical, reproducible toolkit that encourages analysts to cross-check multiple views rather than rely on single-summary statistics. Public code availability and the concrete worked example on historical recordings are clear strengths that would facilitate adoption in performance studies.

major comments (1)

[Spline-CDF smoothing description] Spline-CDF smoothing section: the central complementarity argument rests on the claim that cubic-spline interpolation of the empirical CDF (with zero-slope boundaries) reveals bimodality and secondary peaks that binning suppresses. However, the manuscript provides no validation that the resulting PDF is free of oscillations or negative densities, which cubic splines on finite bar-level samples can produce, especially near tails or with abrupt tempo changes typical of 1930s recordings. Without such checks or comparison to kernel density estimates, it is unclear whether the 'revealed' features are data-driven or method-induced, directly affecting the load-bearing claim that this tool adds distinct analytical value.

minor comments (2)

[Abstract and figure caption] The abstract states that all five tools are applied simultaneously in a composite figure, yet the text does not specify panel ordering, axis scaling, or colour conventions; adding these details would improve reproducibility of the visual argument.
[Abstract] The paper mentions 'full implementation code is publicly available' but does not provide a DOI or repository link in the abstract; this should be added for immediate access.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for pinpointing the need for explicit validation of the spline-CDF smoothing method. We address this concern below and will incorporate the suggested checks into a revised manuscript.

read point-by-point responses

Referee: [Spline-CDF smoothing description] Spline-CDF smoothing section: the central complementarity argument rests on the claim that cubic-spline interpolation of the empirical CDF (with zero-slope boundaries) reveals bimodality and secondary peaks that binning suppresses. However, the manuscript provides no validation that the resulting PDF is free of oscillations or negative densities, which cubic splines on finite bar-level samples can produce, especially near tails or with abrupt tempo changes typical of 1930s recordings. Without such checks or comparison to kernel density estimates, it is unclear whether the 'revealed' features are data-driven or method-induced, directly affecting the load-bearing claim that this tool adds distinct analytical value.

Authors: We agree that the manuscript currently lacks explicit validation for the spline-CDF procedure and that this weakens the claim of distinct analytical value. The description in the methods section presents the cubic-spline interpolation of the empirical CDF with zero-slope boundaries and demonstrates its output on the Beethoven data, but does not include checks for oscillations, negative densities, or side-by-side comparison with kernel density estimation. We will add a new validation subsection that (1) confirms the interpolated PDF remains non-negative over the observed tempo range for both recordings, (2) reports the maximum absolute second derivative as a quantitative measure of oscillation, and (3) overlays the spline-CDF PDF against Gaussian KDEs computed with cross-validated and rule-of-thumb bandwidths. Any discrepancies, especially near the tails or at abrupt tempo shifts in the 1930 recording, will be discussed. These additions will allow readers to judge whether the reported bimodality and secondary peaks are robust data features rather than interpolation artifacts, thereby strengthening rather than altering the complementarity argument. revision: yes

Circularity Check

0 steps flagged

No circularity: visualizations applied directly to external data

full rationale

The paper introduces five visualization methods (tempographs, spline-smoothed histograms, ridgeline plots, stacked bar charts, combination charts) and demonstrates their complementarity by applying them to bar-level BPM data from external historical recordings (Casals/Horszowski 1930-39 and Isserlis/Levin 2012). No derivations reduce claims to fitted parameters, self-definitions, or author prior work by construction. The spline-CDF method is presented as a tool with explicit implementation details rather than a prediction derived from its own outputs. Central claims rest on empirical comparison of the same dataset across panels, which is independent of any internal fitting loop or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper relies on standard statistical visualization practices and publicly available recording data; no free parameters, axioms beyond domain assumptions about tempo measurement, or invented entities are introduced.

pith-pipeline@v0.9.0 · 5636 in / 1227 out tokens · 50500 ms · 2026-05-10T06:44:32.355907+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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work page 1996
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A Manual Bar-by-Bar Tempo Measurement Protocol for Polyphonic Chamber Music Recordings: Design, Validation, and Application to Beethoven's Piano and Cello Sonatas

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work page internal anchor Pith review Pith/arXiv arXiv 2026
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Coexisting Tempo Traditions in Beethoven’s Pi- ano and Cello Sonatas: AK-means Clustering Analysis of Recorded Performances, 1930–2012,

I. Sole, “Coexisting Tempo Traditions in Beethoven’s Pi- ano and Cello Sonatas: AK-means Clustering Analysis of Recorded Performances, 1930–2012,” arXiv preprint, submitted April 2026. [Online]. Available:https:// arxiv.org/submit/7492708/preview

work page arXiv 1930
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Tempo, Duration and Flexibility: Tech- niques in the Analysis of Performance,

J. A. Bowen, “Tempo, Duration and Flexibility: Tech- niques in the Analysis of Performance,”Journal of Musi- cological Research, vol. 16, pp. 111–156, 1996

work page 1996

[2] [2]

Performance Analysis and Chopin’s Mazurkas,

N. Cook, “Performance Analysis and Chopin’s Mazurkas,”Musicae Scientiae, vol.11, no.2, pp.183–207, 2007

work page 2007

[3] [3]

Leech-Wilkinson,The Changing Sound of Music: Approaches to Studying Recorded Musical Performance

D. Leech-Wilkinson,The Changing Sound of Music: Approaches to Studying Recorded Musical Performance. London: CHARM, 2009. [Online]. Available:https:// www.charm.kcl.ac.uk/studies/chapters/chap5.html

work page 2009

[4] [4]

A Manual Bar-by-Bar Tempo Measurement Protocol for Polyphonic Chamber Music Recordings: Design, Validation, and Application to Beethoven's Piano and Cello Sonatas

I. Sole, “A Manual Bar-by-Bar Tempo Measurement Protocol for Polyphonic Chamber Music Recordings,” arXiv:2604.15278 [cs.SD], April 16, 2026. [Online]. Avail- able:https://arxiv.org/abs/2604.15278

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

Coexisting Tempo Traditions in Beethoven’s Pi- ano and Cello Sonatas: AK-means Clustering Analysis of Recorded Performances, 1930–2012,

I. Sole, “Coexisting Tempo Traditions in Beethoven’s Pi- ano and Cello Sonatas: AK-means Clustering Analysis of Recorded Performances, 1930–2012,” arXiv preprint, submitted April 2026. [Online]. Available:https:// arxiv.org/submit/7492708/preview

work page arXiv 1930

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Sole,Tempo and Performance Style in Record- ings of Beethoven’s Cello Sonatas, 1930–2012

I. Sole,Tempo and Performance Style in Record- ings of Beethoven’s Cello Sonatas, 1930–2012. PhD dissertation, University of Aberdeen, 2025. [Online]. Available:https://abdn.primo.exlibrisgroup.com/ discovery/fulldisplay?context=L&vid=44ABE_INST: 44ABE_VU1&docid=alma9918611382805941

work page 1930

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Sole,PhD Appendix: Tempo Dataset and Coded Graphs

I. Sole,PhD Appendix: Tempo Dataset and Coded Graphs. GitHub, 2024. [Online]. Available:https:// github.com/isolepinas/PhD-Appendix

work page 2024

[8] [8]

B. W. Silverman,Density Estimation for Statistics and Data Analysis. London: Chapman & Hall, 1986

work page 1986

[9] [9]

On the Origin of Karl Pearson’s Term ‘Histogram’,

D. Riaño Rufilanchas, “On the Origin of Karl Pearson’s Term ‘Histogram’,”Estadística Española, vol. 59, no. 192, pp. 29–35, 2017

work page 2017

[10] [10]

Pop Culture Pulsar: Origin Story of Joy Division’sUnknown PleasuresAlbum Cover,

J. Christiansen, “Pop Culture Pulsar: Origin Story of Joy Division’sUnknown PleasuresAlbum Cover,”Scientific American, Feb. 18, 2015

work page 2015

[11] [11]

Listening and Responding to the Evidence of Early Twentieth-Century Performance,

D. Leech-Wilkinson, “Listening and Responding to the Evidence of Early Twentieth-Century Performance,” Journal of the Royal Musical Association, vol.135, no.S1, pp. 45–62, 2010

work page 2010

[12] [12]

Seaborn: StatisticalDataVisualization,

M.L.Waskom, “Seaborn: StatisticalDataVisualization,” Journal of Open Source Software, vol. 6, no. 60, p. 3021, 2021

work page 2021

[13] [13]

Noorduin,Beethoven’s Tempo Indications

M. Noorduin,Beethoven’s Tempo Indications. PhD dis- sertation, University of Manchester, 2016. [Online]. Available:https://www.escholar.manchester.ac.uk/ uk-ac-man-scw:302884

work page 2016

[14] [14]

Matplotlib: A 2D Graphics Environ- ment,

J. D. Hunter, “Matplotlib: A 2D Graphics Environ- ment,”Computing in Science & Engineering, vol.9, no.3, pp. 90–95, 2007

work page 2007

[15] [15]

Data Structures for Statistical Comput- ing in Python,

W. McKinney, “Data Structures for Statistical Comput- ing in Python,” inProceedings of the 9th Python in Sci- ence Conference, S. van der Walt and J. Millman, Eds. Austin, TX: SciPy, 2010, pp. 51–56

work page 2010

[16] [16]

SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,

P. Virtanen et al., “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,”Nature Methods, vol. 17, pp. 261–272, 2020

work page 2020

[17] [17]

Jupyter Notebooks, a Publishing For- mat for Reproducible Computational Workflows,

T. Kluyver et al., “Jupyter Notebooks, a Publishing For- mat for Reproducible Computational Workflows,” inPo- sitioning and Power in Academic Publishing, F. Loizides and B. Schmidt, Eds. Amsterdam: IOS Press, 2016, pp. 87–90

work page 2016

[18] [18]

An Interdisciplinary Review of Music Performance Analy- sis,

A. Pati, A. Lerch, C. Arthur, and S. Gururani, “An Interdisciplinary Review of Music Performance Analy- sis,”Transactions of the International Society for Music Information Retrieval, vol. 3, pp. 221–245, 2020. DOI: 10.5334/tismir.53

work page doi:10.5334/tismir.53 2020

[19] [19]

Patterns of Expressive Timing in Performances of a Beethoven Minuet by Nineteen Famous Pianists,

B. Repp, “Patterns of Expressive Timing in Performances of a Beethoven Minuet by Nineteen Famous Pianists,” Journal of the Acoustical Society of America, vol. 88, no. 2, pp. 622–641, 1990

work page 1990

[20] [20]

TempoandCharacterinBeethoven’sMusic,

R.Kolisch, “TempoandCharacterinBeethoven’sMusic,” The Musical Quarterly, vol. 77, no. 1, pp. 90–131, Spring 1993

work page 1993