The Expectancy Dynamics of Anti-Tonal Twelve-Tone Rows: A Commentary and Reanalysis of von Hippel & Huron (2020) NIELS CHR. HANSEN[1] Aarhus Institute of Advanced Studies & Center for Music in the Brain, Aarhus University ABSTRACT: This commentary provides two methodological expansions of von Hippel and Huron’s (2020) empirical report on (anti-)tonality in twelve-tone rows by Arnold Schoenberg, Anton Webern, and Alban Berg. First, motivated by the theoretical importance of equality between all pitch classes in twelve-tone music, a full replication of their findings of “anti-tonality” in rows by Schoenberg and Webern is offered using a revised tonal fit measure which is not biased towards row-initial and row-final sub- segments. Second, motivated by a long-standing debate in music cognition research between distributional and sequential/dynamic tonality concepts, information-theoretic measures of entropy and information content are estimated by a computational model and pitted against distributional tonal fit measures. Whereas Schoenberg’s rows are characterized by low distributional tonal fit, but do not strongly capitalize on tonal expectancy dynamics, Berg’s rows exhibit tonal traits in terms of low unexpectedness, and Webern’s rows achieve anti-tonal traits by combining high uncertainty and low unexpectedness through prominent use of the semitone interval. This analysis offers a complementary–and arguably more nuanced–picture of dodecaphonic compositional practice. Submitted 2020 August 21; accepted 2020 August 25. Published 2020 October 22; https://doi.org/10.18061/emr.v15i1-2.7806 KEYWORDS: dodecaphony, twelve-tone music, tonality, expectation, entropy, probability, information theory DESPITE Arnold Schoenberg’s (1941/1950) proclamation that “[c]omposition with twelve tones has no other aim than comprehensibility” (p. 103), dodecaphonic (i.e., twelve-tone) music has failed to reach the level of widespread popularity that he and his followers from the Second Viennese School had originally hoped for. As we approach the centennial for the discovery of Schoenberg's “method of composing with twelve tones,” discussions about its comprehensibility continue to resurface in academic circles. Music cognition researchers, in particular, have contributed to this debate and sometimes provided important empirical corroboration (see, e.g., Brown, 2016; Dibben, 1994; Dienes & Longuet-Higgins, 2004; de Lannoy, 1972; Krumhansl, Sandell, & Sergeant, 1987; Ockelford & Sergeant, 2013; Pedersen, 1975). The authors of the current target article, Paul von Hippel and David Huron, made an essential contribution to this endeavor when they presented their initial analysis of anti-tonal (or “contra-tonal”) traits in Schoenberg’s twelve-tone rows in an unpublished conference paper about 20 years ago (Huron & von Hippel, 2000). Although Huron briefly revisited these results in the penultimate chapter of his Sweet Anticipation (Huron, 2006), the recent paper by von Hippel and Huron (2020) provides the first detailed account in journal publication format and, furthermore, consolidates the original findings by expanding them to Anton Webern’s and Alban Berg’s rows and by incorporating a set of control variables and a larger control corpus. A major methodological innovation of this research project is the procedure for quantifying the tonality of twelve-tone rows. Specifically, tonal fit is operationalized as the average of 44 Pearson correlations between the highest-scoring major/minor key profile (see their Figure 2) and sub-segments of cardinality one to eleven—always including the first pitch in the row—of the prime (P), retrograde (R), inversion (I), and retrograde-inversion (RI) forms of each twelve-tone row. In this way, von Hippel and Huron (2020) were able to demonstrate that segments from twelve-tone rows by Arnold Schoenberg (n = 42) and Anton Webern (n = 21) achieved significantly lower tonal fit than segments from randomly © 2020 Hansen. This article is published under a Creative Commons Attribution-NonCommercial 4.0 International License (https://creativecommons.org/licenses/by-nc/4.0/) Empirical Musicology Review Vol. 15, No. 1-2, 2020 generated twelve-tone rows (n = 1,000). There was, on the other hand, no significant difference in tonal fit between the random sample and twelve-tone rows by another famous Schoenberg student, Alban Berg (n = 23), who is known for adhering more closely to tonal traits in his dodecaphonic works (Jarman, 1985). These findings were interpreted to mean that Schoenberg’s and Webern’s twelve-tone rows are in fact “anti-tonal,” i.e. actively composed to avoid tonal features and thus presumably also to circumvent tonal listening schemata. As it provides a methodologically sound empirical basis for discussions about tonality (and absence thereof) in dodecaphonic composition, von Hippel and Huron’s (2020) paper was most certainly worth the long wait. At the same time, their choice of method—which is clearly only one amongst many possible ones—invites for replication using alternative procedures. This may challenge or indeed further substantiate their claims regarding anti-tonality in Schoenberg and Webern (and tonality in Berg for that matter). To that end, below, I offer two methodological expansions of von Hippel and Huron’s (2020) analysis. First, I present a close replication of their analysis using a modified operationalization of tonal fit that considers all sub-segments of twelve-tone rows on equal terms rather than having a bias towards initial and final tones of the rows. Second, I quantify the sequential expectancy dynamics of twelve-tone row forms using a computational model of melodic expectation. As will be evident, some of these analyses are consistent with von Hippel and Huron’s (2020) findings while others deviate slightly and thus may offer a more nuanced picture of anti-tonal and tonal twelve-tone composition in Schoenberg’s, Webern’s, and Berg’s oeuvres. CONSIDERING ALL POSSIBLE SUB-SEGMENTS OF ROWS As mentioned above, von Hippel and Huron’s (2020) operationalization of tonal fit averages 44 correlations corresponding to the highest numerical Pearson correlation with any of the 12 major or 12 minor key profiles (cf. Krumhansl & Kessler, 1982) for the 1 to 11 first tones of the P, I, R, and RI forms of each row. Because peripheral (i.e., initial or final) tones of the row are more likely than medial tones to be featured in sub-segments computed in this way, this automatically leads to an imbalance in favor of the former over the latter. This resulting row-initial and row-final bias may indeed be justified in terms of general psychological mechanisms where primacy and recency effects are well-established (Lewandowsky & Murdock, 1989) and in musical practice where themes are often identified by their beginning, as exemplified by dictionaries of musical themes (e.g., Barlow & Morgenstern, 1983). Musical serial order effects also appear, for example, in the reconstruction of order for verses in church hymns (Maylor, 2002). Yet, the current operationalization of tonal fit seems somewhat at odds with dodecaphonic principles of equality between all twelve pitch classes, such as expressed in Josef Matthias Hauer’s (1920) “law of the twelve tones” which inspired subsequent dodecaphonic compositional practice. Thus, it seems relevant to test if von Hippel and Huron’s (2020) findings hold true with an alternative operationalization of tonal fit that considers all possible sub-segments of the rows. Methods All the twelve-tone rows used in the original study by von Hippel and Huron (2020) were obtained in the humdrum **kern format. Specifically, 42, 21, and 23 rows composed by Arnold Schoenberg, Anton Webern, and Alban Berg, respectively, were downloaded from the website of the Center for Computer Assisted Research in the Humanities (London et al., 2002). The authors of the target article, moreover, provided a link to the 1,000 random rows used in their study. All available files were imported into R (R Core Team, 2018) and processed there. First, a **kern spine was added to the random row files who previously only contained a **pc spine with numerals 0-11 to designate 12-tet pitch classes. The **kern spine was created by (arbitrarily) mapping 0 to C, 1 to C#, 2 to D etc. Second, an arbitrary quarter-note duration was added to all note events (this was required for import into the computational model used below). Third, retrograde (R), inversion (I), and retrograde inversion (RI) forms of each row were generated from the imported prime (P) forms. Specifically, the order of pitches of the P form was reversed to obtain the R form. To obtain the I form, the intervals of the P form were first inverted after which the new row was transposed up or down in its entirety to comprise the exact same pitch set as the P form (i.e., all pitches from C3 to B3). For consistency, the **pc spine was updated 129 Empirical Musicology Review Vol. 15, No. 1-2, 2020 accordingly. Finally, the RI form was obtained by reversing the order of pitches in this newly obtained I form. Next, for replication purposes, von Hippel and Huron’s (2020) original measure of tonal fit was computed. This was done by extracting the 11 sub-segments of cardinality 1 to 11 (i.e., comprising the first tone, the first two tones, the first three tones etc.) of each of the four P, R, I, RI forms of each twelve-tone row. The Krumhansl-Schmuckler key-finding algorithm (Krumhansl & Schmuckler, 1986; Krumhansl, 1990) was then applied to each of these 44 sub-segments to identify the highest scoring Pearson correlation with any of the 12 major and 12 minor key profiles (Krumhansl & Kessler, 1982). Tonal fit was finally calculated by averaging these 44 values. Finally, an alternative operationalization of tonal fit was computed that equally considers all possible sub-segments of rows, thus avoiding the bias for peripheral over medial tones in von Hippel and Huron's (2020) original measure.