Theme List, Exclusion Criteria, and Data Analysis

Data was collected, computed, and analyzed according to the previous methods and exclusion criteria enumerated extensively in Daniele & Patel 2013. As in the Daniele & Patel studies, the musical materials for the current work were drawn from A Dictionary of Musical Themes, Revised Edition (Barlow and Morgenstern, 1983). Seventy-five (75) musical themes was the cutoff for generating an nPVI frequency plot for an individual composer while fifteen (15) themes was the cutoff for calculating a composer's mean nPVI. The midpoint year (the mathematical average of the birth and death year, representing when the composer was active, abbreviated "ca.") was used to plot composers over time. A k-means clustering test, an unbiased grouping algorithm, was used to classify composers into two categories (k = 2) based on their "midpoint year" and "mean nPVI". A k-means test clustered all composers with a midpoint year before 1790 as "Baroque/Classical Era" (1600-1750/1750-1810) while those after 1790 were considered the "Romantic Era" (1825-1900) and this fits closely with the musicological record (Kmetz et al., 2001). Notably, the k-means test reliably classified Beethoven as a Romantic composer despite him writing most of his compositions during the "Classical period" (Indy, 1970; Solomon, 1998). For frequency plots, the nPVIs of individual themes were grouped based on their values e.g. nPVI = 0-5, 5-10, 10-15, to 140+, and the number of themes within each range was tallied. When plotting the data, the frequency values were then plotted as a percentage (%) of the total themes of that composer (y) vs. the nPVI range (x).

Similar Findings after a Reanalysis of Daniele 2016

One criticism of the original Daniele 2016 manuscript was that Baroque and Classical composers were underrepresented and "Wagnerian" composers from the Romantic era were overrepresented which, together, could cause nPVI values to trend upward in the late 1800's. To decrease this potential bias, ten composers were added to the corpus and new medians were calculated for grouped "Baroque/Classical" composers (nPVI = 32.2, Table 1, Yellow) and "Romantic" composers (nPVI = 39.2, Table 1, Blue). 3 These values were used to demarcate "low" (nPVI < 32.2), "middle" (nPVI > 32.2 and < 39.2), and "high" (nPVI > 39.2) tiers for each composer's theme corpus (Table 1). It should be noted that the intercept point of the German/Austrian and Italian regressions was an nPVI = 41.3 (Daniele and Patel, 2013) which makes an nPVI of 39.2 a reasonable demarcation point for moving away from the "Italian" style.

When nPVI frequencies for "low", "middle", and "high" tiers were plotted against midpoint year the regressions for "low" nPVI (<32.2) and "high" nPVI (>39.2) frequencies were very significant (Figure S1A). Notably, regressions for "high" and "low" nPVI frequency tiers vs. midpoint year were still significant when "New German School" composer data (Wagner, Suppé, Bruckner, J. Strauss, Jr., Bruch, and R. Strauss) was excluded (Figure S1B). Further, this phenomena does not appear to be dependent on the presence of composer data after the ~1825 divide of German music into the "Brahmsian" and "New German" schools (analysis included Mendelssohn, "low" nPVI frequency regression, R² = 0.44, P<0.01, and "high" nPVI tier regression, R² = 0.70, P<0.001, data not shown). As in Daniele 2016, the "middle" tier nPVI frequency vs. midpoint year regressions were not significant in any of these plots. By comparison, a similar analysis on Italian composers (Table 2) found no significant regressions for any nPVI tier when plotted against midpoint year (Figure S2).

I have shown that (1) a significant upward trend in the frequency of "high" nPVI themes can be observed in nPVI data preceding the rise of "Wagnerian" composers and (2) that this trend is not observed in Italian music. But what if "high nPVI" frequencies vs. "midpoint year" from solely Baroque/Classical and solely Romantic composers are plotted (as in VanHandel 2016, Figure S3A and S3B)? My reanalysis found that the frequency of "high" nPVI themes from Baroque/Classical composers and Romantic composers also rises with time (Figure S3A and S3B). This suggests that both data sets have a tendency to trend upward in nPVI with respect to midpoint year.

Finally, using my updated composer list, I tested how nPVI frequency varies under conditions when nPVI increased incrementally with time using a computer simulation from Daniele 2016 (Figure S4A-B). 4 Previously, and in this manuscript with a revised composer list, I found that "middle" nPVI frequency decreases significantly in simulation but did not observe this in the actual composer data (compare to Figure S1). Thus, I concluded that "low" nPVI themes were quickly and dramatically being replaced with "high" nPVI themes around the late 1700s - early 1800s (the "Replacement Hypothesis"). However, when "Wagnerian"/ "New German School" composers were excluded, the regression for "middle" range nPVI proportions vs. midpoint year was no longer significant (compare Figure S4A to S4B). This suggested that the "Wagnerians"/ "New German School" composers, which possess an unusually high frequency of "high nPVI" themes, are shaping these rising nPVI trends (hypothesized in VanHandel 2016). My findings differ slightly from VanHandel 2016 but we both come to the same conclusion; that a comprehensive understanding of historical musicology is important to tease apart patterns seen in nPVI over time.

Evidence for the "Wagnerian" influence

I developed a new method to study the "metric" of a composer's style, namely the distribution and spread of nPVI for a composer's entire theme corpus. After the addition of ten composers to the corpus (to supplement with more Baroque/Classical and more non-"Wagnerian" Romantic composers) I find the same results as in Daniele 2016. I observed the frequency of themes with "low" nPVIs (< 32.2) drops dramatically with time and themes with "high" nPVIs (> 39.2) concomitantly rises with time. Themes in the "middle" nPVI range (between 32.2 and 39.2) again showed no significant trend. (Figure S1). I also observed that (1) a similar analysis of Italian composers revealed no significant linear regressions for any frequency tier (Figure S2), (2) that the proportion of "high" nPVI themes rises in isolated data sets of both Baroque/Classical and Romantic composers (Figure S3) and (3) a simulation of incremental nPVI increases with time differs significantly from the observed results though, importantly, this result depends on the presence of "Wagnerian" composers (Figure S4).

While the data without "Wagnerian" composers show significant trends (suggesting high "Germanic" nPVIs are quickly being adopted around 1760), the fact that the simulation results are indistinguishable from the observed results when "Wagnerian" composers are excluded from the analysis suggests the "Wagnerians" strongly shape these trends. This is a clear weakness of the nPVI proportion "metric" as a smaller window for "middle" range nPVIs (revised to 32.2-39.2 from 32.2-43) can make subtle trends more difficult to study.

The "Rhythmic Fingerprint"

In the process of writing this manuscript I observed that the methods in both Daniele 2016 and VanHandel 2016 were unable to distinguish major composer classes using mean nPVI or nPVI proportions alone. For instance, a k-means test could not distinguish "Brahmsian" from "Wagnerian" composers while an ANOVA could (VanHandel, 2016). However, an ANOVA failed to distinguish between "Baroque/Classical" and "Brahmsian" composers (VanHandel, 2016). Thus, I developed a new analytical tool to quantify and classify composer types based on their rhythmic distributions ("rhythmic fingerprint").

Initial analysis suggests that it wasn't until Mozart that similarities are seen between rhythmic "fingerprints" in my composer panel (Table 3). Surprisingly, Mozart's "fingerprint" is significantly different from Haydn's despite the two distributions appearing similar (D = 0.14, P < 0.01). 7 This suggests that Haydn had not yet abandoned the "Italianate" style for the "German" style but that Mozart might have been an early adopter. Musicological analysis of this time also suggests Mozart's style was considered "Germanic" (Morrow, 1997: 111, 135).

A true watershed figure in this transition, however, is Beethoven. When I compared Beethoven's nPVI distribution to that of earlier composers I observed a striking frequency shift to writing higher (more "Germanic") nPVI themes (Figure 1 & 2 and Supplementary Movie). A visual analysis of these distributions showed that composers who wrote after Beethoven had a significant proportion of "high" nPVI themes (and similar "fingerprints"). This particular "fingerprint" seems to start with Beethoven and persist through to Schubert, Mendelssohn, Schumann, and Brahms (Figures 1 & 2). A K-S test proved that these distributions were very similar (Table 3) and this is most apparent when distributions are overlaid (Figure 3A). The K-S test could also reliably identify similarities in the "Wagnerian" composers (Table 4 and Figure 3B). Notably, this test was especially useful when comparing the "fingerprint" of J. Strauss, Jr. with Beethoven since the two distributions look similar but are actually statistically quite distinct (Table 3).

An unexpected result of this analysis was the complete lack of similarity between Franz Schubert's "fingerprint" and any other composer (by K-S test). 8 This amazingly prolific composer died very young (age 32) and left 1,500+ works to his name, many of which were discovered and re-popularized after his death (Kreissle von Hellborn et al., 1869; Liszt and Suttoni, 1989). A potential line of study, for instance, could be on whether the cause of Schubert's apparently unique rhythmic style was due to the corpus used (see London, 2013) or potentially, to some other factor(s).

Finally, both these tools should enable future research with the nPVI to uncover complex stylistic trends and allow researchers to better identify and understand these shifts by quantifying their rhythmic underpinnings.