Key representation in the spiral array

We represent a musical score as an ordered sequence of elements: , where each element corresponds to the local key of a given bar (or a fixed number of bars) in the piece (see Fig 1). We use a geometric representation in which notes, chords, and keys are represented as points in a helix-type configuration known as the spiral array [45]. Throughout we assume both octave equivalence and enharmonic equivalence, so that pitches an octave apart, and pitches with different names but the same pitch class (such as C and D), are treated as identical. The enharmonic assumption is required by our 12-note MIDI encoding rather than adopted on theoretical grounds; without it the representation would form a torus instead of a helix. We return to this modification of the original spiral array below and in S2 Appendix.

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Fig 1. Example of determining the “local key” of each bar.

The figure shows the first five bars of the piano sonata K. 545 in C by W. A. Mozart, along with the most likely key (or two most likely keys) as determined by the CoE algorithm.

https://doi.org/10.1371/journal.pone.0356548.g001

Using the spiral representation and the center of effect as a model for tonality allows us not only to analyze a large number of musical scores but also to relate their mathematical features to concepts from information theory. Derived from the Tonnetz network in Riemann’s theory [46–48], the spiral representation preserves the hierarchical structure of tonality: key representations are generated from combining chords, and chord representations from combining notes, while representing all these structures in the same space (). Because notes ordered by fifths, the chords built from them, and the keys built from those chords all inhabit this same space, Euclidean proximity in the spiral array is harmonic proximity: tonally close keys—such as a tonic and its dominant—map to nearby points, while harmonically remote keys map far apart. This geometric fidelity, inherited from the Tonnetz, is what makes a distance-based measure of tonal ambiguity musically meaningful, and it underlies the key-uncertainty measure defined below.

Notes in the spiral array are defined as

(1)

where r and h are parameters (see S1 Appendix) and k is an integer representing a specific note. The starting note k₀ is arbitrary; for simplicity, we define k₀ as the C note. Notes k and k + n are separated by n fifths, preserving the harmonic relationships among notes, chords, and keys (e.g., with k₀ = C, k + 1 = G, k + 2 = D and so on). Major and minor chords ( and ) are constructed as linear combinations of notes and major and minor keys ( and ) as linear combinations of chords (see S1 Appendix).

The center of effect is based on the analogy that a set of notes has the equivalent of a center of mass. A set of notes has an effective center of mass in the form of a linear combination of its elements

(2)

Here, the coefficients are normalized () and they represent the importance of each note. These coefficients can be defined in multiple ways (see S2 Appendix) but we choose to use the normalized duration of each note – so that our representation captures some aspects of rhythmic structure. The center of effect (CoE) key-finding algorithm we develop uses the vector for a given set of notes, and defines the most likely key as:

(3)

which corresponds to the key T for which the Euclidean distance to the center of effect of that set of notes is minimized. Here is the set of all possible major and minor keys: . The CoE method also assigns a likelihood to the most likely local key, based on the Euclidean distance above, as well as a likelihood to the nearest alternative keys based on their respective distances (Eq (6)).

The CoE algorithm has proven effective and to perform better than other methods when identifying a key from a small amount of information [45], and has proven useful not only for identifying a key but for other applications such as spelling [42] and passage segmentation, even with post-tonal music by composers such as Messiaen [43]. By construction, the spiral representation is enharmonically inequivalent, meaning that it distinguishes between sharp and flat notes that equal temperament would consider to be the same (e.g., C# and Db). This can, in most cases, be an advantage of the model. However, for our purpose of using a high-order representation, we modified the CoE algorithm to make it compatible with a 12-note representation and with enharmonic equivalence — the convention that two notes with different names but the same pitch are treated as identical, such as C and D, which sound the same on a piano even though they are written differently. The distinction becomes relevant in microtonal and other non-equal-tempered tunings, where the two can take slightly different pitches; under the equal temperament assumed here they coincide, so we treat them as the same (see S2 Appendix).

We evaluated the accuracy of our implementation of the CoE (center of effect), including the various weights we introduce in this context, by comparing its output to a gold standard of local keys annotated by hand, bar by bar, for Beethoven’s string quartets [37] and Mozart’s piano sonatas [49]. These annotated datasets are some of the most detailed sets of such information in the literature; it contains information about the key and chord in functional harmony for each bar. Our CoE implementation matches the hand annotations for chords in more than 60% of the bars in the corpus on average, and mismatches are often musically plausible alternatives (see S3 Appendix). For local keys, we obtained more than 80% accuracy for the Mozart piano sonatas, and more than 55% of accuracy for the Beethoven’s string quartets. Additionally, we compared our implementation of the CoE algorithm with the Krumhansl-Schmuckler (KS) key-finding algorithm, finding approximately a 10 percentage point difference in accuracy in favor of our CoE implementation.

Key diversity and uncertainty

Leonard B. Meyer’s information-theoretic account of music holds that musical style is inherently probabilistic, and that listening is an ongoing process of expectation and resolution under uncertainty [50]. Building on this view, we describe the harmonic content of a piece with two information-theoretic quantities defined on its sequence of local keys: key diversity and key uncertainty. Both are Shannon entropies of a probability distribution over keys, and differ only in which distribution and at what scale. We use Shannon’s entropy, the expected information content of a random variable X, , which is low when probability concentrates on a single outcome and high when it spreads evenly across many. Key diversity applies this functional across the bars of a piece, key uncertainty applies it within a single bar.

Key diversity.

Key diversity captures how varied the tonal centers of a piece are: whether it stays on a few keys or ranges widely across many. For each piece of assigned per-bar keys, we build the distribution of keys by their normalized frequency,

(4)

where counts the bars assigned key and the sum runs over the distinct keys appearing in . Key diversity is the entropy of this distribution,

(5)

this feature is low when the piece remains in a few keys and high when many keys occur with comparable frequency.

Key uncertainty.

Key uncertainty quantifies how ambiguous the tonal reference of a bar is, whether its notes point clearly to one key or could belong to more than one. Because distance in the spiral array corresponds to harmonic proximity, this ambiguity is already encoded in the geometry. A bar’s center of effect is a single point, and its placement relative to the surrounding keys provides the following insight: when sits close to one key, the tonal reference is clear, when the center is roughly equidistant from several keys, none of them is favored and the reference is ambiguous. This implies that the probability of to be in a key should be a function to its distance to that key.

To turn this geometric feature into a number, we first need to map the distances from to the candidate keys onto a probability distribution over those keys. The center of effect algorithm returns the keys ordered by distance, with the distance to the r-th nearest key. A natural choice is the Boltzmann (softmax) map, in which probability decays smoothly with distance:

(6)

evaluated over the N = 12 keys nearest to , ordered . The form is borrowed from statistical mechanics, where the distances act as energies and as an inverse temperature, so a key receives high probability when it is harmonically close to and low probability when it is remote—just as a thermal system is found most often in its low-energy states. Among all distributions consistent with the distance vector, this is the least-committal (maximum-entropy) choice: it introduces no assumption beyond “harmonically closer is more probable.” The spiral array’s distances are in arbitrary units, so a scale factor is needed to turn them into probabilities, and that factor is . Rather than tune it to the data, we fix it from the geometry alone, through a single requirement: a bar whose center of effect lands exactly on a key should be assigned that key with near-certainty (p₀ > 0.98). The distances to the competing keys are then nothing other than the fixed key-to-key spacings of the spiral array. This requirement therefore leaves as the only unknown and pins it to a unique value, (S9 Appendix).

Key uncertainty is the entropy of this per-bar distribution, computed once per bar, and the quantity presented is the median per-piece value over the M bars of the piece:

(7)

where m is the bar index and the inner sum runs over the N = 12 nearest keys of Eq (6). The interpretation follows: a bar whose distribution concentrates on one key has entropy near zero (an unambiguous tonal reference), whereas one spread over several comparably close keys has high entropy (an ambiguous reference). This is tonal ambiguity in the per-bar tonal reference, and not structural or systemic complexity in the sense studied in the complex-systems literature [51,52]. Restricting the sum to the local neighborhood rather than all 24 keys leaves the per-bar values essentially unchanged, because remote keys carry negligible probability mass; we confirm this cutoff invariance, and the robustness of the reported trends to , in S9 Appendix. Summarizing a whole piece by the median of its per-bar values is deliberately reductive: the distribution of per-bar uncertainties is typically broad and often multimodal, so a single scalar cannot capture how tonal ambiguity rises and falls within a piece. We use the median here for comparability across the corpus, but a fuller treatment would follow the per-bar uncertainty profile at higher temporal resolution, and we regard the development of such a dynamic representation as an important direction for future work.

Innovation in key transitions

Following the same insights from Meyer, in his analysis of musical experience: “... the importance of uncertainty in musical communication, the probabilistic nature of musical style, and the operation in musical experience of what I have since learned to be the Markoff process” [53], we use Markov models to represent musical pieces. Although notes in a musical score are known to exhibit long-range dependencies [23,54], several studies have used first-order Markov chains to model musical scores [7,25,55], where the state in a musical sequence depends only on the previous state (or a set of previous states in higher-order models). Markov chains on higher-order representations, such as local keys viewed as harmonic transitions, are likely more useful than Markov Chains on individual notes, which lack contextual information [20]. We implemented a Bayesian Information Criterion (BIC) test and a more adequate permutation test for this type of system that supports our Markov-1 order assumption (see S7 Appendix).

A piece can be seen as a Markov chain, itself the outcome of a Markov process, whose states are the elements (local keys), drawn from an alphabet of keys (12 major + 12 minor). Indexing the keys by , a transition is an ordered pair , and the chain is described by a transition matrix P with entries , the probability of (initial state e₁). Given an empirical score represented as a series of local keys, P is estimated by maximum likelihood:

(8)

where is the count of the transition in the piece and the number of times key i is followed by any key, equal to the frequency of i as a context.

One of our goals is to evaluate how innovative a given piece is when compared with a set of prior works. This comparison is made by assuming that a model Q representing all prior works can be used to generate . We quantify how well Q reproduces P. This quantity is directly related to the Kullback-Leibler divergence, which measures the additional information required to encode a distribution P using a reference distribution Q. For stochastic matrices the KL-divergence rate is defined as (see S5 Appendix):

(9)

where is the stationary distribution of elements in and are the entries of the smoothed historical transition matrix defined below. Here, quantifies the information per step required for the model to reproduce P.

To quantify the innovation of a piece composed in year t, we aggregate all key transitions from pieces composed in years earlier than t into a cumulative historical pool , with counts and . Because an empirical pool assigns zero probability to any transition it has never seen, is regularized by additive smoothing (Dirichlet) with a single parameter :

(10)

where represents the possible states. We then compute the divergence rate as in Eq (9) and refer to this value as the novelty of a piece with respect to all preceding scores. We select a single value by held-out predictive cross-entropy, and we include details of this selection process and a sensitivity analysis in the supplementary materials (S6 Appendix).

Among candidate information-theoretic measures, the KLD rate is the one that isolates novelty relative to history. Mutual information characterizes a sequence’s internal sequential dependency rather than its departure from prior works, while a raw information-content or surprisal score conflates a piece’s own entropy with its novelty. Symmetric divergences such as the Jensen–Shannon divergence quantify mutual dissimilarity with no preferred direction. By contrast, the KLD rate () measures the asymmetric excess information per step required to describe a piece under the historical model Q rather than under its own model P. This directionality mirrors both the forward unfolding of musical transitions and the forward accumulation of historical context, so that novelty relative to the past is precisely the quantity being measured. Other quantitative accounts of harmonic evolution have used instruments of a different kind: one estimates transmission dynamics from time-varying chord-substitution probabilities [27], while another describes tonal organization through the correlation of pitch-class frequency trajectories [36]. Neither provides what we compute here, a directed, per-step measure of how far an individual piece departs from its accumulated past; the correlation in [36], in particular, is symmetric and so carries no preferred direction. also differs from the information content used in previous work to quantify novelty [7], in that KLD rate weights transitions by the asymptotic distribution of the elements (the more repetitive, the less novel) and does not strongly depend on the length of the sequence (see S5 Appendix).

Comparing composers’ harmonic vocabularies

As mentioned above, we can make use of other information-theoretic measures to gain insights from musical pieces with our framework. One of the questions we could ask is: How similar are two composers’ harmonic vocabularies? Which is a way to indirectly quantify or compare composers’ harmonic styles using their harmonic transitions as proxy. This question has no direction (compared to the innovation one), since the resemblance of a to b is the resemblance of b to a.

We build a composer’s harmonic vocabulary by aggregating the key transitions of all pieces by a composer c into pool counts (as in the historical pool of Eq (10), but grouped by composer rather than by year) gives the joint transition distribution

(11)

Because the comparison has no preferred direction, we use the Jensen–Shannon divergence [56,57] rather than the directed ,

(12)

with base-2 logarithms, so that bit and the similarity is bounded with . Unlike , the mixture reference is finite without smoothing, and is a metric [57]. Because a composer’s pool is a finite sample and is biased upward for small samples, we balance the comparison by rarefying every composer to a common number of transitions before computing the similarity (see explanation in S8 Appendix).

To display the results we get from this similarity analysis, we use metric multidimensional scaling (MDS), a method that can be interpreted as a map reconstruction from distances. This method has been effectively applied to a set of intercity distances, and it has shown to recover the actual geographic layout of the cities [58,59]. In our case, we use between composers’ key-transition distributions and let it map each composer’s value to a point, so that nearby points are composers with similar harmonic vocabulary. This means that we would expect for composers that share similar transitions in their harmonic vocabularies to be closer to each other, similarly to a city, composers would be grouped as if they were in neighborhoods (clusters) in a city map. This reconstruction is possible because behaves like an ordinary straight-line (Euclidean) distance: it can be reproduced exactly by points in a space of sufficiently many dimensions (S8 Appendix), of which we plot the two most informative.

As with any map rebuilt from distances alone, the result is fixed only up to sliding, rotating, or mirror-reflecting it. There is no built-in compass, so “north” may point in any direction; only the relative arrangement of composers carries meaning, that is, who lies near whom and how the neighborhoods separate, not the orientation or the axes themselves. The city example has an external map against which the reconstruction can be checked, but the composer layout has none, since the arrangement is what we are estimating. Instead, we report the fraction of the pairwise distances that the two plotted axes preserve (percentage of JSD variance explained by the 2 dimensions), a measure of how faithfully the flattening onto a page reflects the full set of similarities.

As a complement to the MDS map, which compresses all pairwise similarities into two axes, we also report nearest-neighbor glyphs: for each composer we identify its three most similar composers by the exact on the full rarefied matrix, with bubble area encoding similarity and color the historical period (see S8 Appendix).

Corpus

We use a set of MIDI files from the Kunst der Fugue website [60]. This dataset has been used in previous studies [7,14,23,35], it consists of a compilation of approximately 18,000 MIDI files from more than 79 Western composers, spanning the years 1200–1950. We retained only those pieces to which we could assign a year of composition (see S10 Appendix), reducing the dataset to 4,638 MIDI files. And after a second filtering process (Markov-1 permutation test), we used a reduced subset of 3,483 pieces for the novelty values. We processed the pieces using a Julia script to extract the information needed to compute the center of effect – such as the number of bars, pitch, and duration of all notes (see S10 Appendix for details).

Key diversity and uncertainty

A time series of key diversity is shown in Fig 2. Throughout, the time series of yearly values are smoothed with a 20-year sliding window solely for visualization of the long-term trend; all statistics are computed on the per-piece values. We observe a trend toward increasing diversity of keys within a piece over time, consistent with results from previous studies showing that many elements in music tend to be more diverse over time [35]. However, in our study, key diversity does not follow a monotonically increasing trend. Rather, the change in diversity varies across specific periods associated roughly with the different eras of classical music defined by musicologists (Early/Mid Baroque, Late Baroque, Classical, Romantic, and Modern).

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Fig 2. Key diversity.

Time series for key diversity across 4,638 musical scores. The solid line represents the median of the distribution of values within a time bin, while the first and third quartiles are represented by the shaded area. All distributions are computed for values that lie within a sliding window of 20 years. Historical periods are denoted by dashed lines, where the labels stand for: 1) Early/Mid Baroque, 2) Late Baroque, 3) Classical, 4) Romantic, and 5) Modern.

https://doi.org/10.1371/journal.pone.0356548.g002

The increase in key diversity is most evident during the Classical (1750–1820) and Modern (1890–1960) periods. This result for the Classical period agrees with the historical analysis of musical style [61]. Indeed, the Early Classical period saw the development of new instruments that increased the size of orchestras, allowing composers to explore greater tonal modulation and a wider range of musical forms. A similar trend is seen in the Modern period, which is characterized by greater experimentation with modulation and more complex conceptualizations of tonality, or even the rejection of tonality altogether [62].

Fig 3 shows the time series of key uncertainty for the 4,638 scores in our dataset. The overall trend shows an increase in the uncertainty associated with assigning local keys to bars using the CoE algorithm, over the course of 400 years. This increasing trend in key uncertainty may reflect evolving concepts of tonality. Key uncertainty can be interpreted as how ambiguous a tonal center (key) is, and this ambiguity may be related to tonal ambiguity, proximal tonal tension [63] or, in a more colloquial fashion and perhaps easier to think about: key uncertainty is related to tonal spiciness, a food analogy adopted by the musician and YouTuber Adam Neely to describe the property of tonal ambiguity in polytonal music using the circle of fifths as a reference for harmony structure [64]. In our case, the reference is not the circle of fifths, but the spiral array. Long-term changes in key uncertainty are largely consistent with historical features of the Baroque and Classical periods [65]. For instance, in the Baroque period, music was not only polyphonic but also contrapuntal, promoting a higher density of different notes and greater propensity for dissonances. In the Early Classical period the texture of the sound is clearer than in the Baroque period [65], with more emphasis on order and hierarchy, resulting in a homophonic texture with a clear melody above chordal accompaniment – producing a less crowded or ambiguous representation of tonality.

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Fig 3. Key uncertainty.

Time series for the key uncertainty value, the solid line represents the median of the distribution of values while the first and third quartiles are represented by the shaded area. Regions between dashed lines are the historical periods presented in Fig 2.

https://doi.org/10.1371/journal.pone.0356548.g003

While key diversity and key uncertainty represent distinct aspects of a musical piece, the 400-year trend of overall increase holds for both, with the Modern period showing the highest uncertainty and diversity. The relationship between diversity and uncertainty is shown in Fig 4, with a Pearson correlation of . In the same figure, some of the extreme scores are identified and highlighted with circles; these particular pieces are listed in Table 1. This positive but weak correlation likely reflects a partial overlap between the two measures, both related to tonal ambiguity at different granularities, although part of it may instead arise from key-assignment noise, and disentangling these contributions is beyond the present study.

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Table 1. Compositions list. List of the pieces highlighted in Fig 4, the number in the first column corresponds to the label in Fig 4. All pieces can be listened to through the YouTube links provided in the Supporting Information (S10 Appendix).

https://doi.org/10.1371/journal.pone.0356548.t001

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Fig 4. Key diversity vs key uncertainty.

Diversity and uncertainty values for individual pieces classified by their historical period. The ellipses are constructed by computing the covariance error with 95% confidence.

https://doi.org/10.1371/journal.pone.0356548.g004

Simply listening to the pieces highlighted in Fig 4 – which are selected for having extreme values of key diversity and/or key uncertainty – helps provide an intuitive understanding of what these two measures quantify; accordingly, we include audio references (YouTube links) for each piece in Table 1, representing low uncertainty/low diversity, low uncertainty/high diversity, high uncertainty/low diversity, and high uncertainty/high diversity extremes across the corpus. For example, the pieces that are most uncertain and most diverse in tonality are typically by Modern composers (Scriabin and Stravinsky), while the other extreme (least uncertain and least diverse) corresponds to a simple religious hymn (The Great Physician) that was written to be easy to sing and to remember.

Innovation in key transitions

We compute a novelty value for each piece using the Kullback-Leibler divergence rate from Eq (9), described in greater detail in the Materials and Methods section. The novelty value is computed using two alternative key representations: the original key and the transposed key. The original key representation refers to the absolute key in a given bar (e.g., C, E, G for major keys and a, e, d for minor keys) while the transposed key corresponds to the key relative to a given reference – in our case, the tonic (i.e., the main key) of the piece. We use these two representations to control for the fact that tonic keys underwent substantial change over the course of the dataset. The transposed novelty measure controls for this gradual variation in tonic keys when computing novelty.

For the representation of novelty after transposition, we use Roman numerals, as in harmonic analysis, mapping every key sequence to a sequence of Roman numerals in which I denotes the global tonic key of the piece (e.g., if C major is the reference: C, E, G I, III, V; and if A minor is the reference: a, C, E i, III, V, see S4 Appendix for details). This mapping allows us not only to study sequences of Roman numerals (functional harmony), but also to transpose all pieces and analyze them within the same frame of reference, preventing transitions with the same harmonic relationship from being counted as distinct.

Results for innovation values in both the original and transposed representations are shown in Fig 5. Lower innovation values for transposed pieces are expected: for example, the transition E A in a piece with tonic key E becomes I IV, which is the same as in any other major piece where the tonic transitions to the subdominant, making the piece in E less novel than it would score in the original key representation. The overall trend for original and transposed keys appears to be the same, with lower novelty values for the transposed pieces as expected. We confirmed the authenticity of this global trend with a statistical test (see S5 Appendix).

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Fig 5. Innovation values.

Novelty or innovation values for original key representation (yellow) and transposed or functional harmony representation (blue) for the 3,483 pieces. Values are computed in the same fashion as in Fig 2 with an overlapping window of 20 years.

https://doi.org/10.1371/journal.pone.0356548.g005

Fig 5 shows a notable decline in the novelty of harmonic transitions during the Classical period. Although perhaps surprising, this feature has been previously reported, based on a smaller set of pieces and with a representation that considers chords with the same harmonic function as distinct [7].

Another surprising observation is that one of the least innovative composers according to this measure is W. A. Mozart. However, it is well known that Mozart’s style, like Haydn’s, is an archetype of the Classical style, in which clarity, balance, and transparency are hallmarks of his work [66,67]. This conventionality has a compositional basis: the galant idiom was assembled from a shared repertory of stock harmonic schemata [68], so a measure of transition novelty relative to the accumulated historical pool scores its most representative practitioners lowest. It was not until his late active years that he explored chromatic harmony; one example is his well-known String Quartet in C major, also known as the “Dissonance” quartet. Further observations can be made from the results in Fig 5, such as the shift in novelty between the Classical and Romantic periods, where Beethoven (number 7, as indicated in the figure) played an important role in a movement that was consolidated by composers including Schubert and Liszt (numbers 8 and 9, respectively). The increasing trend in harmonic innovation after the second half of the Classical period represents a tipping point that led to continued increases in innovation during the Romantic and Modern periods, which are known for their developments in musical form and harmonic language.

We also computed novelty values separately for pieces in major and minor keys (see Supporting Information, S5 Appendix). We found that although the trends in both major and minor tonalities are similar, they differ considerably in the functional harmony (transposed) representation, with a clear separation during the Classical period. This could indicate that exploration of modulation occurred more frequently in minor keys. This result may reflect the fact that minor tonalities have three different scale patterns compared with major tonalities, which have only one; these three patterns prove a richer space for harmonic modulation in minor keys compared major keys.

A list of the five compositions with the lowest novelty values for each period of time is presented in Table 2. The key sequences in these pieces are highly repetitive, consisting of the most common keys in music theory, such as the tonic (I), the dominant (V) and the subdominant (IV).

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Table 2. Compositions with lowest novelty values. List of compositions with the lowest novelty values on each period of time, and their first 10 elements. Full sequences are included in Supporting Information (S5 Appendix).

https://doi.org/10.1371/journal.pone.0356548.t002

The pieces with the highest novelty values for each period are listed in Table 3. Inspecting the local keys of the first 10 bars of these pieces, we find many keys that do not appear among the common ones in Table 2, with the exception of the piece by Schubert, which starts with common keys before transitioning to less common ones (see S5 Appendix).

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Table 3. Compositions with highest novelty values. List of compositions with the highest novelty values on each period of time and their first 10 elements. Full sequences are included in Supporting Information (S5 Appendix).

https://doi.org/10.1371/journal.pone.0356548.t003

Comparing composers’ harmonic vocabularies

Applying the symmetric similarity of Eq (12) to the composers of Figs 2, 3 and 5 produces the similarity matrix in Fig 6 (left), and embedding the resulting distances by MDS produces the map in Fig 6 (right). The two plotted axes account for 43% of the variance, a value that reflects the intrinsically high-dimensional nature of harmonic-vocabulary similarity rather than a poor embedding, and is in the range typical for such data [58,69]; we therefore read the map together with the full set of distances (S8 Appendix). In our results, composers group by historical period, with one systematic exception: Scott Joplin lies nearest the Classical composers rather than his early-twentieth-century contemporaries, both on the map and among nearest neighbors in the full space.

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Fig 6. Composer harmonic similarity (selected composers).

Left: support-normalized similarity between composers’ key-transition distributions, ordered chronologically. Right: classical multidimensional-scaling map of ; closer points are more similar in harmonic vocabulary, and composers are grouped by historical period. The MSD plot preserves 43% of the pairwise variance.

https://doi.org/10.1371/journal.pone.0356548.g006

This harmonic conservatism may also account for Joplin’s low novelty value relative to his contemporaries: he is the fourth least innovative composer in our corpus (Fig 5), despite working in the early twentieth century. Because the directed measure of Eq (9) scores each piece against the accumulated historical pool, which by that time already contained the Classical and Romantic repertoire his key transitions resemble, Joplin’s harmonic vocabulary registers as largely familiar rather than innovative, in contrast to contemporaries such as Scriabin and Stravinsky.

The two-dimensional MDS map preserves only 43% of the pairwise variance, enough to recover the coarse grouping by period but not the exact ordering of each composer’s closest neighbors. To expose that ordering we read it directly from the full rarefied matrix and plot, for six representative composers, the three most similar composers to each (Fig 7). For Joplin the three nearest are Romantic composers, not Classical ones, even though the map seats him beside the Classical cluster. This reinforces the result we get from the map, that states that Joplin sits away from his early-twentieth-century contemporaries and toward earlier harmonic practice, while the nearest-neighbor glyph identifies that practice, at the level of individual neighbors, as Romantic. Both agree with his documented adoption of European classical and Romantic harmonic idioms in classic ragtime [70,71]. This conservatism is specifically harmonic. Ragtime’s defining innovation was rhythmic, a syncopated style set over conventional European harmonic and formal schemes [72], so a key-transition measure scores Joplin as familiar on the axis where ragtime was least novel and does not register the rhythmic dimension that carries its originality.

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Fig 7. Top-three most similar composers for six representative composers.

Each hub (outlined) is surrounded by its three nearest composers by on the full rarefied matrix, with bubble area similarity (scaled to the global range, so bubbles are comparable across panels) and color denoting musical period. Joplin’s three nearest neighbors are Romantic, resolving a relation not legible from the 43% of variance captured by the MDS map in Fig 6.

https://doi.org/10.1371/journal.pone.0356548.g007

A second relation legible in the nearest-neighbor glyphs is the proximity of J. S. Bach to Buxtehude, which aligns with a documented line of influence: in 1705 the young Bach travelled on foot from Arnstadt to Lübeck to study Buxtehude’s art [73,74], and Buxtehude’s contrapuntal and ostinato idioms are widely held to inform Bach’s own [74]. The agreement is suggestive rather than direct, since our similarity is computed on bar-level key transitions rather than the organ and fugal textures through which that influence is usually traced.