A Tetrahelix Animates Bach: Revisualization of David Lewin's Analysis of the Opening of the F# Minor Fugue from WTC I

[1] Lewin's 2-D Visualizations[1.1] In his article "Notes on the Opening of the F Minor Fugue from WTCI," Lewin inspects the pattern and function of the seventeen appearances of Forte-set 3-2 (013) within the first six measures of the piece. To demonstrate his analysis Lewin uses "a graph that lays out the various forms of the pcset (013) in a certain format" and a musical example that "lays out consecutive forms of Forte-set 3-2."(1) He then illustrates how an area of the graph of Cohn flips "is surfed by consecutive forms of Forte-set 3-2"(2) within the first six measures. The graph, presented here as Graph 1, is central to Lewin's analysis and he begins by explaining its composition:Graph 1. David Lewin's graph of Cohn flips on Forte-set 3-2Forms of (013) that are displayed as adjacent vertically include the same minor-second dyad. Forms that are adjacent upper-left-and-lower right include the same major-second dyad. Forms that are adjacent lower-left-and-upper-right include the same minor-third dyad.(3)The graph is divided into lower-and-uppercase lettering, where lowercase trichords distinguish forms of pcset (013) that appear in the opening of the fugue. Lewin extracts these and presents them in a separate figure with annotated numbers that "key in to the numbers on the musical example." Lewin's figure and musical example accompany this article as Figure 1 and Example 1.Figure 1. David Lewin's isolation of (013) forms appearing in the first six measures of Bach's F minor FugueExample 1. David Lewin's reduction of the first six measures[1.2] While these visualizations suffice to assist Lewin's analysis, many of their structural characteristics and Lewin's own statements suggest the use of more explicitly geometric models. To begin with, the graph duplicates common tones of the (013) forms appearing in Bach's first six measures. For example, notice that the (G) of the first trichord labeled 1,5,10, the lowest and leftmost form of the figure, is the same (G) in the form labeled 2. While such duplications are rendered harmless when the graph is theoretically conceived in pitch-class space, we find the repetitions make it difficult to imagine the music progressing through the graph. Secondly, Lewin's graph pictures the relation by Cohn flip of each (013) form to three other forms but the format of staggered rows and columns disjoins these relations. Related sets along the diagonals seem farther away than the sets related vertically. This arbitrarily prioritizes the Cohn flips that preserve minor seconds. To describe the graph further Lewin quotes his own article "Cohn Functions," and points out that "the analogous graph for ... harmonic triads would illustrate, in its three different directional bondings, Riemann's relations of our relative major/minor, of our parallel major/minor and of his Leittonwechsel."(4) However, at least in reference to this analysis, Lewin's graph relegates these three directional bondings to an arbitrary hierarchy that is difficult to visually process quickly. Lastly, grouping the trichord sets in brackets visually asserts them as forms of (013) but also creates an ordering problem. The trichord sets in Lewin's 2-D Graph are ordered so that the common tones of the Cohn-flip-related sets are, for the most part, in the same place, first, middle or last. Unfortunately, this ordering rotates the interval content of each set, when read left to right, making the Graph harder to read. Visually depicting the common tone retention of the Cohn flip is only one aim of the graph and in this 2-D case we believe the choice between accessibility and purpose is well made. As the following section demonstrates the need to compromise is alleviated and many of the other structural difficulties are resolved when a more geometric model is used.[1.3] As Lewin alludes, his graph can be easily reconstructed as a Tonnetz. The equilateral triangles in Figure 2 represent (013) forms. …