Volume 13, Number 4, December 2007
Copyright © 2007 Society for Music Theory
A Tetrahelix Animates Bach: Revisualization of David Lewin’s Analysis of the Opening of the F Minor Fugue from WTC IJacob Rosenthal Reed and Matthew Neil BainKEYWORDS: 3D geometry, animation, J.S. Bach, David Lewin, inversion, voiceleading, Cohn flips ABSTRACT: Music theory has long benefited from the use of visualizations to demonstrate analyses. This article presents one example of how music theorists might implement contemporary methods in graphic design and computer modeling. We apply threedimensional geometric modeling and animation to a concise analysis on the opening of Bach’s F Minor Fugue from WTCI by David Lewin. The application models the voiceleading of Cohn flips on Forteset 32 with a triple helix whose structuring tetrahedrons combine chromatic, trichord, tetrachord and octatonic elements. The animations of the music through this model depict the pattern and relationships of the (013) forms described in Lewin’s article but demonstrate how a 3D figure and animation elucidate and amplify his analysis and reveal an unusual aspect of these six measures.

[1] Lewin’s 2D Visualizations [1.1] In his article “Notes on the Opening of the
Figure 1. David Lewin’s isolation of (013) forms appearing in the first six measures of Bach’s (click to enlarge) Example 1. David Lewin’s reduction of the first six measures (click to enlarge) The graph is divided into loweranduppercase 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. [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 ( [1.3] As Lewin alludes, his graph can be easily reconstructed as a Tonnetz. The equilateral triangles in Figure 2 represent (013) forms. Moves along the horizontal axes are moves of interval class 1. Moves on the diagonal axes are either ic2 or ic3. This (013) Tonnetz succeeds in restructuring the trichords as individual pitch classes grouped in a triangular form so that neighboring sets can literally share common tones. The triangulation of trichords provides a more accurate representation of common tone retention between Cohnfliprelated sets. Furthermore, each triangle is adjacent to three and only three other triangles, each of which is one of the three possible Cohn flips respectively. Take, for example, the triangle [1.4] Structurally, the (013) Tonnetz resolves the issues of repetition and order. In addition, under this revisualization the three possible Cohn flips on any given set are no longer forced into an arbitrary hierarchy. Each triangle is bordered by its three Cohnfliprelated sets on its three edges respectively. Despite this the (013) Tonnetz cannot resolve the fundamental issue of register. Neither Lewin’s graph, and descriptions of it, nor the (013) Tonnetz delineate a registration or pitchspecific structure. This slows the transfer from the aural perception of Bach’s music to the visual representation of Lewin’s analysis. In other words, while the progression through the graph is roughly analogous to the music itself, there is no rigorous system of correlation between musical and visual direction. [1.5] A theoretical issue underlying this problem of register is the distinction between pitch space and pitchclass space. When he introduces his graph of Cohn flips on the pcset (013) in the article “Cohn Functions,” Lewin is working in the pitchclass universe, despite occasional use of letter names.^{(6)} When he introduces the graph at the beginning of his article “Notes on the Opening of the [1.6] Thus we searched for geometry that could visualize the interrelation of (013) forms by Cohn flip, demonstrate the general and specific spatial functions described in Lewin’s analysis, and solve problems of register, pitchspecific structure and order. The integration of the first two goals with the third proved the most challenging but also the most fruitful. [2] Development of 3D Geometric Models [2.1] Acknowledging the preclusion of register, this section introduces a modularization of our (013) Tonnetz. The subsequent comparison of this model to some similar examples of geometric modeling will provide context to the reader and further define our method. Theoretically, just as a triadic Tonnetz in equal temperament becomes a torus so may the (013) Tonnetz. However what the resulting torus looks like depends greatly on the limits of the (013) Tonnetz and our method of elevating it out of two dimensions. When we introduced the Tonnetz of Figure 2 we did not specify if and when it terminated. If the (013) Tonnetz were constructed in just intonation or even in equal tempered pitch space it would theoretically continue indefinitely. If the (013) Tonnetz is modularized into mod12 pitchclass space, then its spatial map in two or three dimensions would consist of only twelve sites. The [2.2] Contemporary music theory contains many examples of the use of geometric modeling and computer animation. NeoRiemannian related work in particular is well suited to such visualization. In his dissertation, “Tonal Intuitions in Tristan und Isolde,” Brian Hyer maps Riemann’s dominant, leittonwechsel, relative and parallel transformations around a hypertorus—a torus with four dimensions. The torus appears again in his article “Reimag(in)ing Riemann.” Unlike Figure 5 Hyer’s torus locates triads on the vertices or sites. Jack Douthett and Peter Steinbach use 3D geometry to visualize voiceleading transformations between and within set classes. Their article, “Parsimonious Graphs: A Study in Parsimony, Contextual Transformations and Modes of Limited Transposition,” maps voiceleading and transpositions with several models including two tori, a “cube dance,” and “power towers.” As with Hyer, Douthett and Steinbach use vertices to represent triads or seventh chords not individual pitches. This leaves the visualization of common tones to the reader but enables moves along edges or between sites on Hyer’s torus to define transformations. [2.3] In his articles “NeoRiemannnian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations” and “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of LateRomantic Triadic Progressions,” Richard Cohn depicts trichords with triangles, using vertices to represent pitch classes. Furthermore, his indepth work investigates the mathematical properties underlying generalized Tonnetz presentations as well as the problems of 2D and toroidal tonnetze. Edward Gollin also conceives of a hypertorus in his article “Some Aspects of ThreeDimensional ‘Tonnetze.’” However, Gollin works with tetrachords situated in a 3D space lattice. Vertices represent pitch classes and “units in a positive direction” along each of three axes represent 4, 7, and 10 semitones respectively. Gollin cleverly constructs a pitchclass space lattice containing tetrahedrons whose edges fall nicely onto three axes. This exemplifies a successful redesignation of points in 3D. While the edges on the surface of the torus in Figure 5 are not as easily described with respect to axes they do represent constant interval cycles. Gollin investigates “the relation of a Tonnetz’s geometry to the group structure of the transforms relating its elements” coming to some conclusions that will prove useful in a discussion of the pedagogical worth of our work.^{(10)} [2.4] As stated earlier, the pitch class modularization of the (013) Tonnetz ignores the very musical concept of registration and thus prevents a rigorous directional correlation to the music itself. An attempt to trace the tenor voice up to the downbeat of measure four and begin the fugue’s answer on any (013) Tonnetz with duplications will demonstrate that allowing these duplications does not fix the problem. None of our preliminary models or Lewin’s 2D model assigns a global direction to register. [2.5] From a geometric perspective, the problem with our preliminary models is that their elevation into a 3D realm is partially flawed. The (013) torus fails to define the Zaxis that adds the third dimension. Bending the right and left ends of 2D Tonnetz around to meet each other bends the Xaxis around a newly added Zaxis. Bending the ends of the resulting rod around to meet each other bends the Yaxis around the Zaxis as well. Yet this change leaves the Zaxis undefined and does not actually redesignate the three Cohn flips to three different dimensions. It simply wraps them around so that their constrained range or repetition is pictured cyclically. The solution to both problems becomes obvious—their integration. The successful model will bend a map of Cohn flips around a Zaxis defined by register thereby creating a 3D geometry with rigorously defined axes that can incorporate a fundamental aspect of Bach’s fugue. [3] A Tetrahelix [3.1] To introduce our figure we will emulate a graphic presented by Carol L. Krumhansl in “The Geometry of Musical Structure: A Brief Introduction and History” and proposed by R. N. Shepard. Krumhansl presents a single helix on which “neighboring tones [3.2] Similar to models of tetrachords constructed by Edward Gollin and Richard Cohn, the volume, not just the surface, of our three dimensional figure is decisively relevant. Richard Cohn’s work, “A Tetrahedral Graph of Tetrachordal VoiceLeading Space,” begins on a similar premise as ours. Cohn takes a 2D graph of voiceleading relations and extends it in three dimensions constructing a large tetrahedron with vertices and additional sites representing 29 T/Itype tetrachord classes. Cohn extensively defines the figure, discusses several results of generalization and investigates some of the inherent peculiarities. Cohn focuses on the theoretical implications of his geometric model—how the location of tetrachord classes within the symmetry of the graph correlate to their inherent voiceleading relations and what certain parallels and discrepancies suggest. As our dismissal of the pitch class tori makes clear, our focus is on developing a model that can be readily applicable to Lewin’s analysis of Bach’s music. [3.3] Figure 6 presents our tetrahelix in more detail. It is composed of tetrahedrons adjoined at faces on alternating sides so that it could potentially spiral endlessly in two directions. Note that most triangles can be said to be on the surface of the model but that some of the triangles, those that adjoin tetrahedrons, are inside the model. In Figure 7 these adjoining faces are highlighted and a closer view is provided. Because our goal is to model Bach’s realization of the voiceleading between (013) forms in pitch space, Figure 6 maps pitches represented by letter names onto the vertices of the tetrahedrons. Since we are mapping pitches and not pitch classes our mapping could continue endlessly with the geometry of the model. However, in this case, our model spans about two octaves as the registral subscripts illustrate. [3.4] In similar work, Elaine Chew and Alexandre R. J. Francois extend
Krumhansl's helix into a spiral array. The outer spiral represents pitch classes. Major and minor triads create triangles connecting pitch classes on this spiral. Spatial points within each triangle create a second inner spiral of chords. In a similar manner “the union of pitch classes in three adjacent major triads [3.5] Returning to our tetrahelix, notice that all outside faces present forms of pcset (013) and consequently the edges connecting the vertices of any given surface triangle represent the appropriate interval content: one minor second, one major second, and one minor third. All inside faces present chromatic trichords and, as will be shown, define a Zaxis with register. [3.6] In “Polyhedra: a visual approach” Anthony Pugh describes a tetrahelix as follows:
What separates our tetrahelix from any other is the way in which it maps pitches. The construction of our tetrahelix, described as follows, integrates the chromatic scale with a voiceleading map of (013) forms. A tetrahedron is constructed and assigned pitches (0123), one to each vertex. Picture triangle (012) as the base lying flat on a table with point (3) above. Next, a second tetrahedron, (1234), is created using triangle (123) as its base. Then another tetrahedron, (2345), is created using triangle (234) as its base. Thus, the structure of the tetrahelix is determined by subsequent additions of tetrahedrons such that each new tetrahedron shares the last three added points, and thus pitches, with the previously added tetrahedron. After at least one octave three ribs, or strands, of a triple helix will be easily visible. Taken individually, they each create the repeating tetrachords mentioned earlier, (0369), (147T) and (258E) respectively. Spiraling together in a triple helix, they produce the chromatic scale. The method of constructing the tetrahelix, as determined by the strict rule for the exact placement of each additional tetrahedron, besides the first, creates the very regular pattern of adjoined faces. These inside triangles, previously mentioned as the only triangles containing chromatic trichords, connect the three strands of the helix. These triangles, and the slow spiral they create between the three tetrachord strands, help create a model that accurately maps all the possible Cohn flips of (013) forms and simultaneously allows Bach’s lines to be correctly visualized with regards to register. This aspect of the fundamental structure of the tetrahelix creates as a skeleton for the geometric modeling of Cohn flips, a chromatic scale capable of depicting registration.
Animation 2 presents the first six measures of Bach’s Animation 2 (click to view animation) Animation 3 (click to view animation) Animation 4 (click to view animation) Animation 5 (click to view animation) Animation 6 (click to view animation) Animation 7 (click to view animation) [3.7] On the tetrahelix, like the 2D tonnetz or the torus, any move from one surface face to another surface face is a Cohn flip and adjacent triangles maintain the same relationships. Again the entire surface of the tetrahelix can be traversed, like the 2D Tonnetz, with literal Cohn flips. In
Animation 3, forms of pcset (013) that are created by Bach’s subject are colored blue and forms of pcset (013) created by the subsequent answer are colored red. When the notes of the subject or answer complete a triangle and thus a form of (013), that triangle is colored red or blue respectively. Here, each vertex that creates a (013) remains lit, despite the notated duration, until the triangle is created, and is only released at the approximate time that the last note of that form is released. To preserve the shape of Bach’s gestures the triangles created remain colored. Notice also that the first triangle colored blue turns a lighter color blue when the subject cadences in measure three and repeats form ( [3.8] In Animation 4 only the (013) forms of the answer and countersubject are highlighted. Animation 5 adds the crosstalk forms created between the answer and counter subject in purple. In Animation 6 the (013) forms of the subject are reinserted so that all forms appearing in measures one through six are represented. Lastly Animation 7 again shows all (013) forms appearing in the first six measures. We remove the tetrahelix base and vertex labels in this animation to emphasize the constellationlike aesthetic created by Bach’s music. Each animation focuses on a different area of the six measures and visualizes a different aspect of Lewin’s analysis. Without demonstrating the entire analysis in these new visualizations we will highlight a few areas where the transfer may not be immediate. [3.9] At the beginning of his analysis Lewin points out that a pattern of “tworanksupandone columnright” is established during the statement of the subject by the first three (013) motives.^{(17)} This pattern, in a sense, skips two forms on his figure, our Figure 1, those labeled with the numbers 5, 9 and 7. Similarly, on the tetrahelix, the first three forms ( [3.10] Next, Lewin introduces the concept of vertical crosstalk explained as the creation of (013) forms between the lines of two voices. The (013) forms created by crosstalk are labeled 10, 11, and 17 and complete his list of seventeen appearances. Form 10 is a restatement of the trichord ( [3.11] Animation 5 highlights the forms created by the answer, countersubject and the vertical crosstalk between the two. The forms created by crosstalk, ( [4] Amplifying the Analysis [4.1] On a basic level the media of computer modeling and animation provide us with new tools with which to demonstrate the analysis. On a deeper level the structure of our geometric model facilitates and augments comprehension of the analysis. [4.2] Lewin’s analysis shows the reader how the opening measures of the [4.3] Another aspect of Lewin’s analysis amplified by computer modeling and animation is the way his forms 4, 7, and 11 fill the gaps created by the subject and answer on the graph or model. Here the difference between the way Lewin’s graph and the tetrahelix animations represent this phenomenon is simple yet drastic. The use of color to distinguish the subject, answer, countersubject and crosstalk creates a realization of the gapfilling that is instantly clearer. On Lewin’s graph the reader tracing the temporal pattern of the subject or answer in his/her mind sees how forms 4 and 11 are first skipped over and then later filled in. However, the pattern, its gaps, and the filling in of those gaps are really only understood after several steps in a logical process. First, the reader must study the graph, then the score and the musical example. Next, the reader must see how the figure is an isolation of the shape made as music of the example surfs the graph. Finally, the reader may see how the forms fill in skipped areas of the figure. On the tetrahelix the music is correlated with the illumination of vertices representing pitches. This allows the viewer to understand immediately movement on the model as visualization of the music. Furthermore, the triangles representing forms become highlighted with color as the form is heard in the music. This illustrates the pattern of the subject and answer as pathways traced in time. Thus, forms skipped by the subject and answer appear immediately as gaps recognizable as uncolored or grey triangles. Subsequently, when a musical gap is filled, a grey triangle is colored visualizing the aural and analytical concept instantly. Consider the gap of the subject filled in by form ( [4.4] In discussing the function of the three crosstalk forms, 10, 11, and 17, our (
The exact meaning of the metaphor aside, it is clear that these two crosstalk forms in some way unite the subject and answer areas of Lewin’s figure. However, notice that on the figure the idea of form 10 suturing the answer area to the subject area again requires a bit of imagination on the reader’s part. The fact that the form is a restatement of the initial trichord ( [4.5] As previously mentioned, in Animations 5 and 6 these crosstalk forms, ( [4.6] Lewin writes the following about the last form of crosstalk, form 17:
Pitch ( [4.7] The most intriguing part of this new visualization of the crosstalk forms relates to Lewin’s endnotes 5 and 6. These two endnotes partly detail the success with which the 3D animations realize Lewin’s analysis and suggest that he might have approved of their use. Lewin’s Endnote 5 reads The “spatial” functions of forms 10 and 11 on the figure are nicely projected, allegorically, by the maximum vertical distance between the voices during the time these forms sound in the music. As one notes on the example, that maximum vertical distance is the seventh Lewin, in describing how significant musical phenomena emerge in the visual field of his graph, is pointing out aspects of the analysis that, only implied by his visualizations, stand out under 3D modeling. [4.8] Seeing the vertical distance between the voices as forms 10 and 11 with Lewin’s visualizations requires crossreferencing the score with his figure. Under 3D modeling the score remains helpful but is not required. Because the model’s skeleton is a chromatic scale the vertical distance between the lines is clearly articulated as the appropriate vertices light at distant ends of the tetrahelix. [4.9] Similarly, it is much easier to see this largest interval of a seventh ( [4.10] In his Endnote 6, Lewin remarks on the spatial allegory of crosstalk form 17 that unites the subject and answer areas of the figure in a clausula gesture at the cadence. The tetrahelix analog of this spatial allegory is clear. The two voices of the fugue lighting the tetrahelix vertices converge onto the final note ( [4.11] Returning to the ‘surfing’ nature of Bach’s Gesture, the geometry of the tetrahelix strengthens another aspect of the analysis. Investigating the details of this surfing motion, Lewin shows how chronologic forms 7 and 9, or forms ( [5] A New Perspective [5.1] 3D animation of the opening of the [5.2] Begin by noticing that, on the tetrahelix, each helix constructs a cycle of IC3s in pitch space not pitchclass space. Straus’ circles present interval cycles in a closed mod12 pitchclass space with octave equivalence. The tetrahelix takes his circles of C3 and pulls them out into the third dimension spiraling them in a pitch space. Of course, the tetrahelix used in all the visualizations provided here only presents about two octaves, but could be extended further. Nevertheless, beyond register, this conversion enables us to see the combination of all three cycles, represented in 3D, into a modular geometric structure. This geometric integration allows a fluid visualization of the musical movement between the cycles about to be discussed. Table 1. Three octatonic collections and the list of (013) forms they each contain (click to enlarge) [5.3] If the octatonic collection can be derived from a combination of any two C3cycles, then analogously, the three permutations of Forteset 828 (0134679T), can be derived from the combination of any two helices on the tetrahelix. Walking the alternating half and whole steps of an octatonic collection on the model traces a lattice between two of the three helices and traverses one of the three twisting sides of the tetrahelix. Each side will be referred to as a combination of helices. Thus, the octatonic collection (0134679T) will be called HC3_{0}HC3_{1}, the collection (124579TE) HC3_{1}HC3_{2}, and the collection (0235689E) HC3_{0}HC3_{2}. Again, these lattice pathways are also visible on the 2D Tonnetz, though, not as explicitly. These three octatonic pitch collections each contain eight of the twentyfour permutations of Forteset 32 (013). These relations are apparent in the structure of the tetrahelix where each triangle is composed of two points from one helix and one from a second helix. Put another way, each triangle lies within one of the three octatonic pathways on the tetrahelix. Table 1 presents the three octatonic collections and lists the (013) forms that they each contain. [5.4] Returning to Bach, the progression of (013) forms appearing in the subject delineates a distinct pattern with regard to the divisions in Table 1. The five appearances, ( [5.5] Certainly, it is easy to see octatonic relationships wherever Forte set 32 (013) is present, often without any musical impetus or relevance. Admittedly, these octatonic patterns have minor significance in any satisfactory analysis of Bach’s fugue. Note also, that the 2D Tonnetz could easily be extended to permit the continuation of octatonic lattices and a chromatic scale can be traced through them. However, with regard to this specific concept and visualization, we are more interested in demonstrating the potential of the geometric model than in enhancing the analysis. To that extent, the helical model better depicts the cyclical nature of the tetrachordal and octatonic elements, their internal relationships and their realization within the music. If the tetrahelix can reveal this noticeable pattern in these six measures of Bach’s music then it may prove useful in the investigation of the music of more modern composers. Nevertheless, without implying that Bach was composing with trichord sets, tetrachord cycles or octatonic collections in mind, these patterns add another perspective on the nature and effect of the chromaticism present in the opening of the [6] Conclusions and Some Prospects for Further Research [6.1] Threedimensional animation and the tetrahedral triple helix elucidate Lewin’s analysis of the opening of Bach’s [6.2] This article is not intended as pedagogical theory. Nor is it intended to advance a specific lesson, model or method for practical classroom use. However, the general nature of our media and design of our model correspond to established theories and techniques in the fields of music education and education in general. Wideranging evidence has shown that “the listening condition that include[s] both visual and kinesthetic elements in addition to the aural component [is] most effective in enhancing children’s aural perception of musical form.”^{(26)} Entire methodologies of music education such as Kodaly or Dalcroze are founded on such concepts. In an article discussing reference maps and their effects on prose learning, educational psychologists Abel and Kulhavy state that “maps consistently have been found to improve memory for related prose” and that “maps add a dimension to pictorial adjuncts by depicting logical relations among individual features.”^{(27)} However, referring to past experiments they reiterate that “the mere presence of extratextual information fails to account for increased learning when maps and prose are combined” but that “what matters is how the map information is provided.”^{(28)} In previous work, Kulhavy and Schwartz explained that “figural characteristics work to maintain relational context, at least in the case in which a recall task demands performance that is spatially isomorphic with the original stimuli.”^{(29)} Concluding their study, Abel and Kulhavy write:
[6.3] This concept is readily seen in common tools of music educators worldwide: listening maps, form charts and call charts. Music educator Samuel Miller writes: “A ‘listening map’ is a graphic representation that symbolizes the essential features of a musical selection in a visual format. It is a valuable teaching aid [6.4] As demonstrated in section 4, our revisualization operates in two ways to make the analysis more accessible. It reduces the number of things the reader must keep in his/her head to visualize the theory by synthesizing the components of Lewin’s analysis. With one figure we can see the complete (013) graph, the musical line with register, and a graph of the forms in the subject, answer, countersubject and crosstalk. Secondly, the animation weds the music and the analysis, portraying a second synthesis between the abstract concepts and the actual notes. Admittedly all the theory cannot be grasped at the speed at which even this slow fugue passes by and much of the analysis must be read prior to understanding the model. Nevertheless, both of these operations facilitate connections that, however easy for Lewin, may be more difficult for others. Lewin writes:
[6.5] This relation serves as an excellent example of a place where the realtime animation helps readers hear things Lewin hears easily. Our model, as an extension of Lewin’s visualizations, adds meaning to the analysis in exactly the same way. The overall meaning Lewin imparts to the analysis is that Bach’s realizations of (013) forms creates an intricate pattern on a map of Forte Set 32 as related by Cohn flips. This pattern is both aesthetically pleasing and rife with voiceleading and inversional relationships. Thus, as the example shows, and as detailed in section 4, our even fancier spatial map brings out these relationships and the way they fill the region of the tetrahelix. As Gollin writes at the end of his work on 3D Tonnetze:
[6.6] The design and realization of the 3D geometric animations presented here required us to make many decisions with regard to language, labeling, audio/visual relationships, timing, duration, camera angle, color, and most obviously geometric structure. Although we are confident that our choices were informed and find our results satisfactory, we fully acknowledge that many other ways to visualize Lewin’s analysis exist. Beyond this, more work remains available within the [6.7] Outside of the
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Rogers, Nancy and Michael Buchler. 2003. “Square Dance Moves and TwelveTone Operators: Isomorphisms and New Transformational Models.” Music Theory Online 9.4: 1–12. Straus, Joseph. 2005. Introduction to PostTonal Theory, 3rd ed. Upper Saddle River, NJ: Pearson Prentice Hall. Straus, Joseph. 2005. Introduction to PostTonal Theory, 3rd ed. Upper Saddle River, NJ: Pearson Prentice Hall. Footnotes1. Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 235. 2. Ibid., 236. 3. Ibid., 235. 4. Ibid., (emphasis mine). 5. Cohn, “NeoRiemannnian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations,” 14. 6. Lewin, “Cohn Functions,” 189. 7. Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 235. 8. Ibid., 238 (emphasis mine). 9. Ibid. 10. Gollin, 195. 11. Krumhansl, 9. 12. Chew, “Slicing It All Ways: Mathematical Models for Tonal Induction, Approximation and Segmentation Using the Spiral Array,” 7. 13. Chew and Francois, “Interactive Multiscale Visualizations of Tonal Evolution in MuSA.RT Opus 2,” 1. 14. Ibid., 3. 15. Pugh, 53. 16. In “Square Dance Moves and TwelveTone Operators: Isomorphisms and New Transformational Models” Nancy Rogers and Michael Buchler show a correlation between “musical and square dance transformations.” No 3D geometry is used, unless, of course, the reader performs the dances, but simple animations demonstrate the square dance moves and their twelvetone analogs. 17. Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 236. 18. Ibid., 237. 19. Ibid. 20. Ibid. , 238 (emphasis mine). 21. Ibid. 22. Ibid. 23. Ibid. 24. Ibid., Notes, 239. 25. Straus, Introduction to PostTonal Theory,
154–155. 26. Gromko and Russel, “Relationships among Young Children's Aural Perception, Listening Condition, and Reading of Graphic Listening Maps,” 334335. 27. Abel and Kulhavy, “Maps, Mode of Text Presentation, and Children’s Prose Learning,” 263. 28. Ibid., 265. 29. Kulhavy and Schwartz, “Mimeticism and the spatial context of a map,” 418. 30. Miller, “Listening Maps for Musical Tours,” 28. 31. Bletstein, “Call Charts: Tools from the past for Today's Classroom,” 54. 32. Lewin, Notes, 237–238. 33. Gollin, 204 Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 235. Ibid., 236. Ibid., 235. Ibid., (emphasis mine). Cohn, “NeoRiemannnian Operations, Parsimonious Trichords, and Their “Tonnetz” Representations,” 14. Lewin, “Cohn Functions,” 189. Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 235. Ibid., 238 (emphasis mine). Ibid. Gollin, 195. Krumhansl, 9. Chew, “Slicing It All Ways: Mathematical Models for Tonal Induction, Approximation and Segmentation Using the Spiral Array,” 7. Chew and Francois, “Interactive Multiscale Visualizations of Tonal Evolution in MuSA.RT Opus 2,” 1. Ibid., 3. Pugh, 53. In “Square Dance Moves and TwelveTone Operators: Isomorphisms and New Transformational Models” Nancy Rogers and Michael Buchler show a correlation between “musical and square dance transformations.” No 3D geometry is used, unless, of course, the reader performs the dances, but simple animations demonstrate the square dance moves and their twelvetone analogs. Lewin, “Notes on the Opening of the F Minor Fugue from WTCI,” 236. Ibid., 237. Ibid. Ibid. , 238 (emphasis mine). Ibid. Ibid. Ibid. Ibid., Notes, 239. Straus, Introduction to PostTonal Theory,
154–155. Gromko and Russel, “Relationships among Young Children's Aural Perception, Listening Condition, and Reading of Graphic Listening Maps,” 334335. Abel and Kulhavy, “Maps, Mode of Text Presentation, and Children’s Prose Learning,” 263. Ibid., 265. Kulhavy and Schwartz, “Mimeticism and the spatial context of a map,” 418. Miller, “Listening Maps for Musical Tours,” 28. Bletstein, “Call Charts: Tools from the past for Today's Classroom,” 54. Lewin, Notes, 237–238. Gollin, 204
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