The MPB Corpus: A Dataset of Melody, Rhythm, Harmony, and Melody–Harmony Relationships in Brazilian Popular Music

3.1 Harmony

Our model for analyzing harmony divides into two branches, ‘semantic’ and ‘syntactic’. The ‘semantic’ branch accounts for the structures of abstract chord types (or CTs),³ while the ‘syntactic’ branch concerns the relationships between chords and chord types across multiple levels of organization. We now describe the model created for this research at the ‘semantic’ level, the Genera of Chord Types; the ‘syntactic’ level is left for future work.

The Genera of Chord Types (GCT)⁴ is based on 10 archetypal chord types (the basic triads and seventh chords), known as protochords, each representing a set (or genus) of generic chord types whose members include the protochord itself and all its possible variants. Derivation is governed by production rules and the recursive application of three transformations: ADD (addition of a note to a chord type, such as a ‘ninth’ to a major triad), SUB (substitution—e.g., replacing the ‘third’ with the ‘fourth,’ yielding a ‘dominant‑with‑suspended‑fourth’ chord type), and ALT (alteration—chromatic modification of a chord note, yielding, for example, a ‘flattened thirteenth’). The 10 genera are named with the final letters of the Latin alphabet, to avoid confusion with the usual chord naming conventions; uppercase letters refer to major‑mode protochords and lowercase letters refer to minor‑mode ones, as shown in Table 1. For a more detailed description, see Almada (2022).

Protochords and variants of the same genus are identified through two notations: (a) alphanumeric, based on a roughly conventional notation among composers and instrumentalists,⁵ and (b) genealogical, a formula combining the genus symbol with a numerical code identifying the lineage of the respective chord type. By convention, all protochords are numbered with a ‘zero’ to the right of their letter‑symbol, as shown in Table 1.

First‑generation variant CTs are produced when transformational operations are applied to protochords. Consider, for example, protochord z0 (*m7 in alphanumeric notation, the symbol * representing a generic root) and its 10 CT variants, covering three generations, as illustrated in Figure 1. Under recursive application, first‑generation variants in turn become parents of second‑generation ones, and so on: CT z11 results from adding a ninth to z1 (giving *m6.9), and the third‑generation z111 results from adding an eleventh to z11 (giving *m6.9.11). To avoid redundancies and overpopulation, a set of constraints and rules limits the system to 161 variants, distributed across the genera and generations, as presented in Table 2. The main constraints are (for more details, see Almada (2022)):

• The normative order of application of operations is SUB (if possible)–ADD–ALT;

• Operation SUB is allowed only in three cases: *M7$\to$*6 (genus Z), *m7$\to$*m6 (genus z), and *7$\to$*4.7 (genus Y);

• An eleventh can be added only to CTs of genera z, y, x, and w;

• A thirteenth can be added only to CTs of genera Y, W, X, and y;

• A sharp‑eleventh can be added only to CTs of genera Z, Y, W, and V;

• The application of ALT to tensions ‘ninth’ and ‘thirteenth’ is only possible in genera Y, X, and W; and

• The application of ALT to the fifth is only possible in genera Z and V.

Figure 1

As Table 2 shows, these constraints leave some genera quite sparse (such as ‘w,’ with only four members), while others are highly dense, notably the ‘dominant’ genus ‘Y,’ with 59 elements—explained by the combinatorial possibilities arising from its rich set of tensions and their alterations.

The genealogical notation thus describes a variant’s derivative position within its genus through a numerical sequence. The label Y2111 (related to CT *($♭$9$.♯$11)13), for example, is a fourth‑generation variant of genus Y, as the number of digits after the identifying letter indicates. Read from right to left, its position is: first variant (fourth generation) of the first variant (third generation) of the first variant (second generation) of the second variant (first generation) of Y0. For a complete list of CTs and their genealogical relations, see the supplementary material.

For each chord, we also collected its alphanumeric notation and its functional analysis within the context of the piece. Functional categories follow a notation detailed in a lexicon, provided in the supplementary material.

The GCT framework also enables a more explicitly transformational perspective on harmony, grounded in the principles of Grundgestalt and developing variation proposed by Schoenberg (Schoenberg, 1975; Frisch, 1984) and further elaborated in Almada (2023b). Harmonic structure can then be interpreted in terms of transformations between abstract chord types rather than relations between surface‑level chord tokens, allowing harmonic syntax to be studied at a higher level of abstraction: the relations between successive chord categories, and the transformations required to derive specific chords from more fundamental protochord structures.

This perspective also suggests compositional applications since the transformations underlying a piece can be repurposed in generative settings to recreate stylistic nuances in a more abstract and controlled manner. Both this compositional direction and the syntactic analysis of the proposed corpus are ongoing lines of research within the project; preliminary results, focused on the works of Tom Jobim, can be found in Almada (2022).

3.2 Pitch and rhythmic structures

The Melodic Filtering Model (MFM) is the component of our system responsible for melody, encompassing pitch and rhythmic structures. It rests on three principles: segmentation, abstraction, and encoding. Segmentation assumes that understanding sequential information requires dividing it into smaller units, each with a certain degree of autonomy—short musical ‘phrases’ or, more neutrally, small groups in the sense of Lerdahl’s grouping structure (Lerdahl and Jackendoff, 1983); see also Snyder (2001). Such segments, analogous to Lerdahl’s groups, are referred to in the model as words. Abstraction allows different structures to be grouped into equivalence classes with respect to a given comparison parameter. Encoding, finally, isolates pitch and rhythmic structures from the melodies analyzed, transforming them into abstract descriptions of these domains.

For melody, each pair of contiguous pitches in a melodic line is encoded as sequences of seven contour gestures, forming an alphabet of c‑letters (Figure 2): note repetitions, stepwise motion (movement of one or two semitones), arpeggios (notes three to five semitones apart), and leaps (six semitones or more), both ascending and descending.⁶

Figure 2

The rhythmic model, briefly presented in Carvalho et al. (2025), is recalled and expanded here. The strategy subdivides a beat (quarter note) into 12 units and associates the possible configurations of IOIs (Inter‑Onset Intervals, the distances between attack points— durations are not considered) occurring within this time window with standard units, called r‑letters in the model, forming an alphabet of 26 elements (Figure 3).

Figure 3

Note that the last four letters in Figure 3 are highlighted. The r‑letters from ‘a’ to ‘v’ proved sufficient to cover almost the entire repertoire of interest, but the MFM, like all frameworks presented in this paper, can also be used to analyze other corpora. An analyst may therefore rely on ‘a’ through ‘v’ and add as many ‘wildcard letters’ as necessary to accommodate less common scenarios; the letters from ‘w’ to ‘z’ are four such suggestions.

Two important metrics are considered in relation to melodic contour and rhythmic structure, both yielding values between 0 and 1: the compensated intervallic economy index (CIEI), which captures the profile of melodic contours within a c‑word from the relative prevalence of more ‘economical’ intervallic movements, and the countermetricity index (CMI). Countermetric rhythms are rhythmical patterns that deviate from or oppose the underlying metrical grid, often by emphasizing offbeats, generating syncopation, or creating tension against the regular pulse. The notion is particularly relevant for Afro‑Brazilian genres, which highly influence MPB, where such rhythmic displacements are integral to the esthetic rather than exceptions; for a detailed discussion, see Sandroni (2001).

To compute the CIEI, each letter in a c‑word $w$ is mapped to a numeric value: $0$ for unison (‘u’), $\pm 1$ for ascending/descending step (‘P’/‘p’), $\pm 3$ for ascending/descending arpeggio (‘A’/‘a’), and $\pm 5$ for ascending/descending leap (‘S’/‘s’). The mean of these attributed values is then computed and its absolute value taken—call this value $|\bar{d}(w)|$—and normalized to the interval $[0,1]$ as:

The maximum is thus attained by a c‑word that perfectly balances upward and downward motions (e.g., ‘pPpP...,’ ‘aAaA...,’ ‘uuu...’), and the minimum by one consisting only of leaps with no balance at all (e.g., ‘sss...,’ ‘SSS...’).

For the CMI, analogously, a value is assigned to each letter of an r‑word $w$ based on its rhythmical identity: ‘n’ is the ‘most countermetric’ r‑letter, with an assigned value of one, and ‘b’ the ‘least countermetric,’ with an assigned value of $0.1$. For the full list of values, see the supplementary material. Let $c(w)$ denote the sum of the corresponding values of the letters in $w$, and ${\mathrm{len}}^{\ast }(w)$ the length of $w$ minus the number of times the r‑letter ‘a’ appears within it. The countermetricity index is then given by

where $\lambda =0.8$ if ${\mathrm{len}}^{\ast }(w)>1$ and the last letter of $w$ is ‘b’, and $\lambda =1$ otherwise. The intuition is that r‑words rich in highly countermetric letters (‘c,’ ‘e,’ ‘g,’ ‘m,’ ‘n,’ ‘p,’ and ‘u’) will have a high CMI, while an r‑word ending with an onset at the beginning of the beat (letter ‘b’) has its CMI reduced by 20% due to the metrical strength of this ending.

3.3 Relations between melody and harmony

Our model establishes hierarchical relations between structural notes and the chords that harmonize them, according to the well‑known chord/scale theory Gonda (1972). The scale of a chord can be seen as a convenient repository of structural notes it may harmonize, which includes not only its arpeggio notes but also the so‑called tensions and, eventually, substitute notes.

Figure 4 illustrates this argument, considering the scale corresponding to the tonic chord (I) in C major, notated in its tetradic version as CM7. On the left of the figure the notes are arranged in scale order. The white ones are the basic structural notes (identified below the staff), which form the arpeggio of the chord—root (1), third (3), fifth (5), and major seventh (M7)—including the sixth (6), which may eventually substitute the seventh. Tensions eligible to integrate the group of structural notes (above the staff) are evaluated by the practical rule of the ‘two semitones’: a note not belonging to the basic arpeggio becomes a candidate harmonizable tension, and thus a structural note, if in the scale it lies two semitones above the arpeggio note immediately to its left. If the distance is only one semitone, the note is excluded from the structural notes group and is generally called an ‘avoided note’.⁷

Figure 4

A verticalized representation of the scale, which highlights the hierarchy of structural notes, is shown to the right of Figure 4 (note F is omitted because it is not structural). From this set of structural notes established for the CM7 chord, functionally contextualized as the I degree, we can list all realizations of the chord that are equivalent in terms of functional position: C (triad), C.9 (triad with a ‘ninth’ added), CM7 (the ‘default configuration’), C6, CM7.9, and C6.9 (the last two incorporating the ‘ninth’ tension). Notice that this equivalence is not necessarily compatible with the hierarchy present in the GCT model, whose goal differs from the one presented here.

From this perspective, we propose a broader understanding of the traditional notion of ‘harmony’ (generally treated as an informal synonym of ‘chords’) as the collection combining the notes that form a given chord with the structural notes present in the melody within that chordal context. Only structural notes are considered in this sum, disregarding any melodic inflections. We formally define an inflection (or ‘non‑harmonic’ note) as a transitional note that, whether or not it belongs to the chord scale, is adjacent by a step to a clearly structural note that follows it, this structural note being called the target of the inflection. Inflections are thus elements outside the chordal context and are not registered as structural. Beyond classifying notes as structural or inflections within a local context (determined by the chords supporting the melodies), we also evaluate them according to the global context (the current key) since melodies in popular music are predominantly diatonic and nondiatonic notes are, to a greater or lesser extent, digressions. This method normalizes melodies: the specific keys involved are not relevant, only the relationships between the notes and their respective diatonic scales.

One graphical representation available within the model is presented in Figure 5. Named NF web (after its shape, which vaguely resembles a spider’s web; ‘NF’ stands for ‘note‑function’), the graph consists of five concentric circles intersected by 12 rays arranged in angular increments of 30°, each ray corresponding to one of the notes of the chromatic scale. The scale degrees (in the figure, the major diatonic scale) represent the global context and are identified with Arabic numerals from 1 to 7, revealing their hierarchical superiority over the nondiatonic degrees on the intermediate axes. The concentric circles describe the local context: the smallest radius corresponds to the level of inflections (orbit 1), and the remaining circles follow the hierarchy of structural notes—the basic notes forming the triad (orbit 2), the components of the seventh/sixth chord (orbit 3), the simple tensions (orbit 4), and finally the more loosely ‘anchored’ altered tensions in the outermost orbit (5).

Figure 5

The NF web thus has 60 possible positions, the intersections of the circles and rays, denoting the possible combinations of local and global contexts. The color of a point at each intersection represents the proportion of the respective local–global combination in the piece being analyzed. It follows that melodies supported by more basic notes tend to occupy ‘internal’ positions in the corresponding web, whereas melodies with more complex relationships between notes and chords (as, for example, in bossa nova) tend to ‘populate’ more ‘external’ ones.

An important metric for the NF model is the melodic anchoring index (MAI), which quantifies, on average, the behavior of a melody in relation to its structural notes, taking as a parameter their degree of ‘anchoring’ in the chords supporting them—that is, how close or distant a given structural note is from its chordal context. The more basic the function performed by the note in the chord, the more ‘anchored’ it will be. Only structural notes are taken into account in the calculation, disregarding the inflections present in the analyzed melody.

To compute the MAI, first recall the possible qualities that a note may have in a local context: ‘root,’ ‘third,’ and ‘fifth,’ the basic triad function notes; ‘tetradic,’ denoting the sixth or seventh degree of the underlying chord; ‘simple tensions,’ the ninth, eleventh, thirteenth, or fourteenth degrees; and finally the ‘altered tensions,’ encompassing the flat/sharp ninth, sharp eleventh, and flat thirteenth. Let $\mathcal{L}$ be the set of these six labels, and, for each $\ell \in \mathcal{L}$, denote by $q_{\ell }$ the number of notes within that category in a given analyzed excerpt. The MAI is a weighted average of the $q_{\ell }$, each weight encoding the ‘anchoring’ of the corresponding category:

Denote by $w_m$ and $w_M$ the smallest and largest of these weights, respectively, and denote by $\bar{q}$ the average of $q_{\ell }$ weighted by $w_{\ell }$, for $\ell \in \mathcal{L}$:

where $\eta$ is the number of inflections within the analyzed excerpt. Finally, the MAI is a normalization of $\bar{q}$ within 0 and 1:

Values close to one indicate a predominance of structural notes within chords, whereas values close to zero indicate the opposite behavior.

3.4 An encoding example

We now summarize the model in action through the analysis of a short, hypothetical excerpt of the kind found in any piece of the MPB corpus (Figure 6a). The analysis protocol follows these steps:

1. Subdivide the melodic line into logical and relatively autonomous segments (Figure 6b).

2. Encode pitch information inside each segment as c‑words: $⟨$Apps$⟩$ (segment 1), $⟨$pPSa$⟩$ (segment 2), $⟨$PPsp$⟩$ (segment 3), and $⟨$pAAAa$⟩$ (segment 4). Note that the transitions between segments are not encoded.

3. Encode rhythmic information inside each segment as r‑words: $⟨$tn$⟩$ (segment 1), $⟨$ul$⟩$ (segment 2), $⟨$pbee$⟩$ (segment 3), and $⟨$pgnc$⟩$ (segment 4). Recall that durations are immaterial in our model, only attack points are considered.

4. Analyze the harmony of the piece, encoding for each chord (Figure 6c): root and bass note (both as pitch classes), chord type (genealogical notation), function (Roman numerals), key (pitch class of the tonic), mode (major or minor), and metric position (bar numbers and fractions, where applicable).

5. Analyze the functions that the notes perform in the harmonic structure, considering both the global (the key) and local (the chords that harmonize the notes) contexts (Figure 6d—functions in global and local context are displayed below and above the staff, respectively). In the local context, a note can be structural (performing the function of root, third, fifth, sixth, seventh, ninth, eleventh or thirteenth) or non‑harmonic—an inflection, in our terminology—performing no chordal function at all and represented by the symbol ‘x’. The NF web related to this excerpt and its MAI are shown in Figure 6e.

Figure 6

In the case of an actual piece, the score, in its nominal form, is first transcribed into MIDI format using music notation software. The segmentation described in step 1 is implemented by inserting an identifying note into a register well above the melody (E6 in the present work); the presence of this note marks the end of a segment. Step 2 is then performed automatically by processing the corresponding MIDI file with a Python script. We attempted to automate step 3 as well, but particularities of the MIDI protocol made this unfeasible, so the encoding is performed manually; attempting the procedure with the MusicXML format instead is a possible direction for future work. Steps 4 and 5 are performed entirely by manual annotation. In particular, the harmonic analysis of each chord is the most critical component of the process and, to the best of our knowledge, cannot currently be reliably automated.

Throughout this process, occasional errors in source scores are identified and corrected during the analysis. Typical examples include the reinterpretation of chord symbols that, though notated so as to simplify harmonic execution (particularly on the guitar, a central instrument in MPB), represent a different chord in context; the incorporation of structurally relevant melodic notes into the underlying chord; and the correction of inaccurately notated harmonies. For examples of such corrections, see supplementary material.

5.1 Melodic rhythm

The model for melodic rhythm was extensively presented in Almada (2023a) and briefly explored on an excerpt of this dataset in Carvalho et al. (2025), whose discussion we now recall.¹⁰ Figure 7 presents, for each corpus examined, the frequency of each r‑letter. The data corroborates the practical observation that the r‑letter ‘b’ (characterized by a single attack akin to a ‘rhythmic tonic degree’) predominates in nearly all repertoires. The presence of ‘j’ is also notable in the works of Chico Buarque, Caetano Veloso, Milton Nascimento, Gilberto Gil, and Rita Lee, the last exhibiting the greatest percentage, i.e., nearly twice the average for this r‑letter.

Figure 7

Information about r‑letters can alternatively be conveyed using the metric profile, the distribution of rhythmic attack points across a micrometric grid dividing each beat (taken here as a quarter‑note unit) into 12 segments (cf. Figure 3); these data are shown in Table 4 for the corpora studied. Fundamentally connected to rhythm, the metric profile is recognized as a crucial factor in highlighting the Brazilian ‘lineage’ in MPB. Several observations stand out: (1) most repertoires predictably feature position 1 as the most frequent, with the exception of Rita Lee, where position 7 dominates; (2) the corpora of Jobim, Chico Buarque, Djavan, João Bosco, and Gilberto Gil closely align with each other, given their similar frequency of attacks at the first position (about $1/3$ of instances); (3) position 7, which divides the beat in half, sees significant activity in the Milton Nascimento corpus, considerably higher than the rest, and most notably in Rita Lee, marking the highest rate at 48.2%; (4) the positions that segment the beat into three equal sections (5 and 9) are prominently represented in Tom Jobim (7.4% and 7.5%) and Ivan Lins (10.7% and 10.9%), respectively, significantly more than in other repertoires, followed closely by João Bosco (6.7% and 7.1%).

Table 5 outlines the concurrence of countermetric r‑letters. The emphasis on the countermetric r‑letter ‘e’ suggests a jazz influence in the rhythmic structure of Ivan Lins and Edu Lobo. Milton Nascimento also uses this r‑letter frequently, which stems not only from jazz (a strong influence) but possibly from folk music, sacred music, or even Beatles songs; for Caetano Veloso, and notably Rita Lee, one might instead argue for rock and rhythm‑and‑blues. The collections of Jobim, Chico Buarque, João Bosco and, to a slightly lesser degree, Djavan and Gilberto Gil clearly indicate a closer relation to samba, represented by the dominance of the r‑letter ‘n.’ Overall, the substantial use of countermetric r‑letters (around one‑third of all occurrences) signals a high level of rhythmic syncopation, consistent with the CMI index displayed in Table 6: for Chico, Bosco, Djavan, Gil and Jobim it is higher than $0.53$, probably due to the strong presence of sambas in these repertoires.

5.2 Melodic contour

Figure 8 shows the frequency of each c‑letter within each of the corpora examined. Although all repertoires converge toward ‘stepwise motion’ as the preferred type of melodic motion, suggesting a shared stylistic rule, it is intriguing that in the corpora of João Bosco, Caetano Veloso, Chico Buarque, Djavan, Edu Lobo, Gilberto Gil, Tom Jobim, and Ivan Lins the most frequent c‑letter is ‘p’ (descending step). Whether the predominance of descending stepwise motion characterizes Brazilian popular music more broadly is an interesting avenue for further exploration. In the Milton Nascimento corpus, by contrast, the predominant melodic motion by a wide margin is note repetition (‘u’), reflecting the notable stasis that prevails in the composer’s melodic construction and is a defining trait. The Rita Lee repertoire is similar, differing only in that the percentage of ‘u’ is lower, closely followed by the c‑letter ‘p.’

Figure 8

The CIEI measures, in essence, the degree of melodic stability. The values obtained, shown in Table 6, broadly support the general notion that a kind of compensatory law may be at work in melodic lines, which would ideally tend toward rectification, with leaps and dips cancelling each other out: all repertoires exhibit relatively high and very similar indices. The most common number of attack points in each melodic segment, also shown in Table 6, converges toward six or seven melodic articulations, in agreement with previous studies on melody segmentation (Snyder, 2001).

5.3 Harmony

Figure 9 illustrates the relative frequency of the 10 genera of chord types in all corpora analyzed. An initial interesting finding is the convergence of almost all corpora on the prominence of genus Y, which undoubtedly reflects shared tonal syntactic properties and, more specifically, the presence of primary and secondary dominant preparations in these repertoires. Genus V is strongly present in Caetano Veloso, Gilberto Gil, and Rita Lee, consistent with their greater adherence to the pop‑rock esthetic compared to the other MPB composers; in Milton Nascimento’s corpus, which also presents a considerable proportion of triads, this is probably due to the predominance of modal syntax in the songs.

Figure 9

Figure 10 illustrates the 10 most common functional categories in all corpora analyzed. Not surprisingly, I and V (in this order) are significantly more common than the other categories in every corpus; IV, II, and VI (this last only in Ivan Lins) share the third position of the rankings, followed by different options, especially secondary dominants. This suggests that the basic functions, performed by central diatonic pillars, solidly ground the ‘harmonic mass’ of MPB music shared by all the composers, while rarer chromatic, eccentric chords—particular ‘spices’—can be associated with individual stylistic preferences, a point to be properly examined in the continuation of the research.

Figure 10

5.4 Relationship between harmony and melody

The relative frequency of each note‑function in the corpora analyzed is shown in Figure 11. We first observe a certain uniformity in the distribution of inflections, which correspond on average to one‑fifth of the melodic notes. Considering the classes of structural notes separately: (a) Triads (1, 3, 5) exhibit a similar distribution across all corpora, with some apparent exceptions such as Djavan, Edu Lobo, Jobim, and Ivan Lins, whose confirmation through statistical tests is left for future work; (b) tetrads (6, 7) show a relative homogeneity among all repertoires, probably due to the frequency of the chord genus Y noted above in the evaluation of harmonic attributes; (c) simple tensions (9, 11, 13, 14) are likewise homogeneous across the repertoires, with the exception of Djavan, Ivan Lins, and Tom Jobim, which balances, in a way, their lower use of triadic notes; and, regarding (d) altered tensions ($\mathrm{♭}$9/$\mathrm{♯}$9, $\mathrm{♯}$11, $\mathrm{♭}$13)¹¹, unlike the simple‑tension class, this one is less distinctive, showing relatively similar distributions across all repertoires, with the exception of Edu Lobo and Tom Jobim, again compensating for their lower use of triadic notes.

Figure 11

In summary, the corpora of Djavan, Edu Lobo, Ivan Lins, and (especially) Tom Jobim display a higher percentage of tensions than the other repertoires, suggesting that they may form a special subgroup characterized by more complex melody–harmony relationships. We expect to examine this hypothesis more rigorously from a clustering analysis perspective in future work.

Figure 12 depicts the NF webs of two songs with very distinct MAI values: Rita Lee’s Saúde (0.817) and Jobim’s Angela (0.416). The two graphs make the reason for such a discrepancy clear: in the former, most occurrences of NFs are distributed on the inner circles, denoting simpler relations between notes and chords, whereas Jobim’s piece occupies a larger number of positions, especially in the more external circles, revealing more varied, eccentric, and complex relations.

Figure 12

5.5 Statistical assessment of rhythmic profiles

To illustrate one possible application of the proposed dataset, we present a brief statistical assessment of the rhythmic profiles of the selected composers. The purpose is not to provide a comprehensive musicological investigation (this is left for future work) but rather to demonstrate how the dataset can be employed to quantitatively compare stylistic characteristics across composers. Rhythm was chosen as a representative example because it is widely recognized as one of the defining features of Brazilian popular music, playing a central role in the characterization of genres, compositional styles, and individual musical identities, and because the visualizations presented above suggest noticeable differences in the distributions of rhythmic patterns across composers, motivating the question of whether these differences are statistically significant. Similar analyses may be carried out using the remaining musical descriptors available in the corpus.

Let $d(x_i,x_j)$ denote the Jensen–Shannon distance (Lin, 1991) between the frequency distributions of a musical descriptor of two musical pieces, given by

where $D_{KL}$ denotes the Kullback–Leibler divergence between two probability distributions and $x_i$ and $x_j$ denote the normalized frequency distributions of the musical descriptor for musical pieces $i$ and $j$, respectively. To quantify the degree of separation between composers, we define the statistic

where $c_i$ and $c_j$ are the corresponding composers of pieces $i$ and $j$, and $N_W$ and $N_B$ are the numbers of within‑composer and between‑composer pairs of musical pieces, respectively.

The rationale behind this statistic is straightforward. If the distribution of the employed feature contains little stylistic information, the average distance between pieces by different composers should be comparable to that between pieces by the same composer, yielding values of $R$ close to one. Conversely, if composers exhibit distinctive distributions of the chosen feature, pieces by the same composer are expected to be more similar to one another than to pieces by different composers, resulting in $R>1$. Therefore, $R$ provides a simple and interpretable measure of the extent to which feature distributions discriminate between composers, while naturally accounting for the variability observed within each composer’s repertoire.

To assess whether the observed value of $R$ within the dataset, denoted by $R_{obs}$, could reasonably arise by chance, a permutation test can be performed (Good, 2005). Under the null hypothesis that the chosen musical feature is unrelated to the composer’s identity, the composer labels assigned to the musical pieces are exchangeable, so samples from the null distribution for $R$ can be generated by repeatedly permuting the labels at random, preserving the number of pieces associated with each composer, and recalculating the statistic after each permutation. Comparing $R_{obs}$ with the resulting empirical null distribution then yields the $p$‑value, estimated as the proportion of permuted statistics greater than or equal to the observed value. More specifically, the Monte Carlo estimate is given by

where $B$ denotes the number of permutations and $R_{\mathrm{perm}}$ the statistic obtained under each random permutation. This estimator avoids zero $p$‑values and provides an unbiased Monte Carlo estimate of the exact permutation $p$‑value (Good, 2005).

For this analysis, the selected musical feature was the distribution of r‑letters across musical pieces. The observed statistic was $R_{obs}=1.072$, and none of the $B=10,000$ permuted statistics exceeded it; Figure 13 shows the empirical null distribution of $R$ together with the observed value. The estimated permutation $p$‑value was smaller than $10^{-4}$, so the null hypothesis that r‑letter distributions are independent of composer identity can be rejected at the adopted simulation resolution: the observed differences between composers are unlikely to have arisen by chance alone. These results indicate that r‑letter distributions carry statistically significant stylistic information and illustrate the potential of the proposed dataset for quantitative studies of composer similarity and discrimination.

Figure 13

Extending this analysis to other musical descriptors, and relating the resulting patterns to clustering and other exploratory techniques, is an important direction for future work, with the potential to yield further musicological insight into stylistic organization in Brazilian popular music.