This paper uses pitch-class distributions as an analytical tool with which to consider chromaticism in Webern's 31 published works. Previous research shows that tonal music typically has a varied distribution; intuition suggests that the distributions of dodecaphonic music will be much more even. The range of values in the distribution is used as an indication of variety, and VariabilityBased Neighbour Clustering is used to group the works. The results of the analysis draw attention to Op. 10: three movements have idiosyncratic distributions, and the highest-level partition of the corpus is within Op. 10. Expectations regarding distributions for dodecaphonic music are confirmed, but the tonal music diverges from typical tonal distributions.
Webern’s music has benefitted from a wide variety of analytical interest; nonetheless, these
enquiries have typically isolated individual works or movements for detailed consideration. The
most wide-ranging studies are Allen Forte and Kathryn Bailey’s monographs, which cover,
respectively, the atonal music up to and using dodecaphony [
The premise of this paper is distinct from many typical corpus studies [
This paper deploys pitch-class1 distributions, a technique with a significant pedigree in the
analysis of tonal [
There has been some statistical analysis of Webern’s music previously: Herbert Eimert
considered interval content in Op. 28/i [
In this paper, the corpus is restricted to the 31 works with opus numbers. To what extent we can understand this as a single coherent corpus is open to challenge, Webern’s compositions change radically from the Passacaglia to the Cantatas, but they are clearly unified by his creative mind. Movements of multi-movement works are considered individually: each movement is sufficiently harmonically self-dependent that it is more meaningfully analysed on its own terms, much as Bailey understands dodecaphonic movements as having their own prime row-forms [3, p. 12]. In justifying this, she draws on a tonal analogy, which is also instructive with regard to pc distributions. To conflate the pc distributions of two movements in diferent keys might provide some interesting large-scale perspective, but obscures so much detail as to be actively counterproductive, as the patterns will ‘cancel each other out’; the same is true in an atonal context.
This corpus omits the Werke ohne Opus, some of which are now accepted into the canon
[
The movements of the corpus have been encoded in conventional Western music notation
using Sibelius software and exported as XML files, before data collection using music21 [
4All code and data is available at https://github.com/joshua-ballance/Pitch-Class-Distribution. Where copyright allows, scores can be found at musescore.com & imslp.org. ambiguities surrounded the exact placement of dynamic and tempo markings, and whilst the ifrst has little bearing on this project, in practice the second could usually be inferred from proximate musical gestures. At the heart of this project lies the methodological assumption that scores can be viably used as proxies for sounds and so, in order to minimise this diference, various notational alterations had to be made. Obvious examples of this include spelling out artificial harmonics or accounting for piano sustaining pedal, but ornaments are more contentious. The approach adopted here has been to view trills as fundamentally ‘ornamental’, so the G\ trill in Op. 10/i is seen as a single pitch, rather than a prolonged G\–A dyad.
An important part of empirical musicology is interrogating pre-conceived notions, however implicitly expressed. I will therefore present some reasonable hypotheses suggested by the literature, which will frame the data analysis. As with all quasi-scientific approaches, it is important to acknowledge all outcomes, whether they confirm consensus or provide new information. It is easy to discount results that support pre-conceived ideas; in fact, particularly given the dearth of statistical musicology, presenting reproductions of earlier results or extending more limited ideas to a larger corpus is highly significant for the field of knowledge.
Turning to the analytical method, a pc distribution is, as might be expected, the distribution of pc’s in a given piece (tonal distributions are often expressed in terms of scale degrees). These proportions may be calculated in terms of pc instances, or, as in this project, durations. These durations are measured in seconds (or fractions thereof) according to the metronome marks given in the score; gradual changes in tempo (rits. and accels.) are modelled according to Sibelius playback. In summary, therefore, this algorithm counts all of the individual notes in a given movement, records their pc value and duration, and summarises the cumulative proportion of each pc as a proportion of the duration of all the notes. The duration of all the notes is not the duration of the piece: rests are not counted, and each of the notes in a simultaneity is counted individually, therefore providing a greater duration than in the original. For the purposes of the distribution, all pc’s are deemed to be of equal significance, irrespective of their textural or orchestrational presentation. As an analytical method, this is therefore conceptually simple and easy to deploy, and by considering proportions it normalises for the length of works [5, pp. 263–264].
Regarding preconceptions about pc distributions, the literature is clear that tonal works are
characterised by variety. Figure 1 is a distribution using data from Joshua Albrecht & Daniel
Shanahan’s corpus study of 625 major mode and 357 minor mode works [
There is some debate in the literature about what to sample to consider pc distributions.
In most cases, scholars have sought to find typical distributions of keys. Modulatory passages
can cause issues with the count, and so scholars often sample only beginnings or endings of
works, arguing that these will present more apt distributions [
As a final methodological point, I ofer a note of caution about the implications of the results. It is easy to view quantitative results as unquestionable truths; as Caroline Bassett puts it, Big Data makes claims to provide ‘solutions beyond dispute’ [4, p. 550]. Within defined parameters this may be so, but it is crucial to maintain perspective on what this does not tell us. An important such qualification here is the reliance on texts as source material. Post-structuralist thought has successfully destabilised analysts’ default tendencies to use a score as a proxy for, representation of, or even incarnation of the hallowed ‘music itself’. It is certainly the case that the approach here could analyse recordings or performances; nonetheless, Webern is specific enough with regard to the parameters under consideration (primarily pitch and duration) that scores are acceptable. Even so, significant hesitancy must be applied to temporal modelling. Webern’s specificity is helpful, but any performance will diverge from the score’s temporal indications; indeed, testimony regarding his own performances describes exactly this phenomenon [28, p. 75]. As such, claims about proportions are only indicative, not prescriptive; the values should be seen as suggestive rather than absolute.
The following summary outlines the key results. In Figures 2 & 4 the corpus is ordered chronologically, by date of composition as recorded in the Moldenhauers’ biography [17, pp. 700–705].5 This is largely equivalent to the ordering of opus numbers, though not universally so. The only major uncertainty concerning ordering is for Opp. 3 & 4 for which the dates of composition are unclear. These have therefore been sorted by their eventual position in the published works.
The data gathered for pc distributions can be shown by a bar chart for each piece. In considering the corpus as a whole, however, the most important metric is the distribution spread. This is calculated as the range of values: the diference between the proportions of the most and least common pitch classes. The greater the range, the more variation there is. As mentioned, tonal music is expected to have a high range, corresponding to pc proportion discrepancies; dodecaphonic music, by contrast, would likely have a small range. Figure 2 is a scatter plot of the ranges.6 Clusters will be discussed below, and derive from Figure 4. The general pattern is clear: despite some noise, distribution spreads decline across the corpus; indeed, the correlation between corpus position and range is a moderately strong inverse-correlation of -0.65.7 In summary, therefore, the results are largely as expected.
5For this list see https://github.com/joshua-ballance/Pitch-Class-Distribution. 6For an interactive plot in which each movement is labelled see https://bit.ly/FigureTwoInteractive. 7Correlations are Spearman correlations given to two decimal places.
A further point of interest is which movements are unusually varied. These movements are those with an anomalous range, calculated according to Tukey’s Rule.8 Interestingly, all of these come from Op. 10: movements i, iii, and iv, with spreads of 21.7%, 27.4%, and 17.8%, respectively. Figure 3 shows their distributions; for the first two movements, the distribution is characterised by one anomalously high value above an otherwise typical distribution; for Op. 10/iv, A is anomalously high, and A\ is also noticeably elevated.
Variability-Based Neighbour Clustering can be used to cluster the corpus empirically. It is a hierarchical agglomerative approach that preserves corpus position [first described in 22] [see 24, for an example of its application].9 Figure 4 displays this clustering using complete pc distributions for each movement, rather than relying on spreads.10
The implications of this clustering will be discussed below; at this stage, it is worth drawing out the structure of the various clusters. Unlike other clustering methods, the number of clusters does not have to be pre-determined; rather, this is left to the analyst to discern based on the results of the algorithm, as presented in Figure 4. In this case, the picture is somewhat messy, with several movements clearly dissimilar to their neighbours (e.g. Op. 10/iii & iv, Op. 16/i). Nonetheless, there are several principal clusters that can be discerned. On the highest level, the corpus is partitioned between Opp. 10/iii and 9/ii. Given the anomalous values of Op. 10/iii & iv, this is not hugely surprising. However, even if the three anomalous movements identified above are removed, the highest-level partition is between Op. 9/i & ii, and therefore in the same area. Subdividing these two principal regions further reveals other groupings: in the latter half, Op. 9/ii–Op. 16/i and Op. 17/iii–Op. 31/iii; the first half is less uniform, 8An anomaly is thus defined as a value lying outside 1.5 × IQR below the first or above the third quartile. 9This code is by Folgert Karsdorp (https://github.com/fbkarsdorp/diachronic-text-analysis). 10For legibility, a fully-labelled dendrogram is not presented here, but can be found at https://github.com/joshua-ballance/Pitch-Class-Distribution. but suggests two main clusters, Op. 1–Op. 10/i and Op. 9/iii–Op. 9/i, alongside two of the anomalous movements anomalous movements (Op. 10/iv & iii).
The following discussion will consider the musical and historical implications of these results,
seeking to identify trends and consider their more overt musical relevance. This research is
fundamentally post-hoc analysis, but it is buttressed by Webern’s well-noted sensitivity to the
structural functions of the total chromatic [
Turning first to clustering, Figure 4 suggests that rather than the principal partition in the corpus lying between tonal and atonal music, or indeed the freely atonal and dodecaphonic music, the highest-level clustering partition suggested by this approach instead lies around Opp. 10 & 11, an observation supported by Figure 2. Shrefler has described the instrumental miniatures Opp. 9–11 as a ‘crisis’, inspiring a period of significant experimentation in Op. 12 and the ensuing Lieder in which Webern sought to establish a new style [21, p. 11]. This research certainly confirms her view that the miniatures were an extreme and required a significant break in the ensuing music, an assertion supported by the categorisation of Op. 10/i, iii, & iv as anomalous. In sum, then, this proposes that we can group Webern’s earlier and later practice apart from methods of construction: the first group encompassing tonal and some freely atonal music, leading up to the ‘crisis’; the second covering the new freely atonal style and dodecaphonicism. It is quite remarkable that the greatest change in the chromaticism of Webern’s music came not with the onset of systematic dodecaphony, but rather through his own aesthetic intuition.
As mentioned previously, the distributions of the three anomalous movements (Op. 10/i, iii,
& iv) were characterised by a single particularly prominent note or pair of notes. In each case
this is musically apparent as a pedal: the Celesta G\ trill in the first movement, the E shared
variously between guitar, bells, celesta and harp in the third movement, and the A and BZ in
the viola and clarinet in the fourth movement. The efect of this is to provide a central timbral
and harmonic focal point, a note with a high degree of salience, to use Lerdahl’s term [
As for the start of the corpus, it is notable that the initial cluster splits Opp. 1–3/i and
Opp. 3/ii–5/iii. This grouping suggests little diference between Webern’s tonal music and
his early atonal music, a view reinforced by Figure 2. Indeed, the distributions of these two
tonal works (Figure 5) are quite dissimilar from tonal distributions. Op. 1 has a correlation
of 0.60 with Albrecht & Shanahan’s minor distribution [
Moving forward, the comparative uniformity of the latter period from Op. 9 onwards is
notable, particularly from Op. 17/iii. For context, the range of ranges in Figure 2’s Clusters
1–2 is 17.9 (12.5 without the anomalous Op. 10/i) and in Clusters 3–4 is 10.0; the median
range values, however, are 9.8 (9.5 without Op. 10/i) and 4.7, respectively. As such, whilst
there is a similar amount of variety in the distribution of ranges, the later ranges tend to
be much smaller, indicating significantly less variation in the pc distributions. The
distributions of the dodecaphonic works themselves therefore confirm expectations: though there is
inevitably some variation, this is minimal, and particularly in light of Figure 2 Webern’s claim
that ‘All twelve notes have equal rights’ [25, p. 52] certainly seems apt. This rather
complicates Bailey’s contention that through intersection and Ausfälle Webern limited certain pc’s.
Without context from other composers’ dodecaphonic works it is impossible to discern whether
her supposition that ‘it was not the equality of the twelve notes that was of primary concern
to Webern’ [3, p. 146] is accurate. Indeed, she suggests that this feature emerged as part of
Webern’s style; in fact, the correlation between range and corpus position within the
dodecaphonic works is 0.00, indicating that there was no noticeable change. The final comment on
11For these correlations, the distribution of Op. 1 was ordered according to a D tonic and Op. 2 a G tonic.
12This is between the major profile and Op. 3/i with a ‘tonic’ (pc 0) of G \. Interestingly, the highest
correlation with the minor profile is 0.44, almost identical, and also assumes a ‘tonic’ of G \.
this music is to note that this shift does not come precisely with the onset of the dodecaphonic
technique: there is no new cluster with Op. 17/i. This seems to confirm Shrefler’s picture of
Webern experimenting with, and gradually adopting dodecaphonicism through Op. 15/iv and
the canons of Op. 16 [
My doctoral research is funded by the UK Arts & Humanities Research Council.
I am very grateful to my supervisor, Professor Jonathan Cross, for his help on this research, and to Professor Eric Clarke & Dr Sebastian Wedler, who provided invaluable comments on an early version of this paper.
My thanks to Dr Folgert Karsdorp for providing the code used for clustering in this paper.
Finally, thanks to Dr Mark Gotham, who introduced me to computational musicology.