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Chapter 3

Figure 3.1.3. A passage from Josquin represented in two-dimensional space.

Figure 3.4.3. Four voice leadings between {C, E} and {D, F}, as represented on the Möbius strip. The amount of rightward motion can be calculated by adding the paths in the two voices; the amount of upward motion can be calculated by subtracting the second path from the first.

Figure 3.5.1. Plotting a phrase from the Allelujia Justus et Palma (a) on the Möbius strip (b) reveals interesting musical structure (c).

Figure 3.5.2. (a) Two passages from Brahms’ Op. 116, No. 5, as plotted on the Möbius strip (b). It is natural to view Y1 and Y2 as variants of X1 and X2, respectively. However, it is clear from the graph that Y1 and Y2 are also mirror images of X1 and X2. This means we can move the pair {X1, X2} off the left edge of the figure so that it coincides with {Y1, Y2} (c). On this reading, Y2 is related to X1, and Y1 is related to X2 (d). Here, “P” and “N” stand for “perfect” and “near perfect” contrary motion.

Figure 3.5.3. There are many ways to notate a C major chord.

Figure 3.6.1. Even dyads and uneven dyads can both be connected to their transpositions by stepwise voice leading.

Figure 3.7.1. Any voice leading can be decomposed into parallel and contrary components. Here, the voice leading (E, B) → (Fs, B) (α) combines the parallel (E, B) →(F, C) (β), which moves both voices up by semitone, with the contrary (F, C) → (Fs, B) (γ), which moves the voices semitonally in opposite directions. Pure parallel motion is represented geometrically by horizontal lines, while pure contrary motion is vertical. This means that the pure contrary component of any voice leading will remain within a cross section of the space (b), containing dyads whose pitch classes sum to the same value.

Figure 3.7.3. There are always two equally good ways to move the contrary component of a voice leading into a particular cross section. Either of the arrows in (b) can represent the voice leadings in (c).

Figure 3.7.4. Individually T-related voice leadings can be moved into the same cross section so that their purely contrary components coincide.

Figure 3.8.3abc. (a) Descending sequences from the first movement of Brahms’ C minor Piano Quartet, Op. 60. (b) These result from moving downward along the equal-tempered lattice at the center of chord space. (Major chords are dark spheres, minor chords light spheres.) (c) There are six basic sequences that can be formed in this way, depending on whether one lowers the root or the third of the initial major triad, and whether the sequence descends by semitone (D1), ascends by seven semitones (A7), or ascends by three semitones (A3).

Figure 3.8.3d. Descending sequences from Schubert (a) and Brahms (b). (c) These result from moving downward along the equal-tempered lattice at the center of chord space (c), passing through the augmented triad without sounding it. (Major chords are dark spheres, minor chords light spheres.) (d) There are six basic sequences that can be formed in this way.

Figure 3.8.5. (a) Two voice leadings in which the total amount of ascending motion perfectly balances the total amount of descending motion. (b) These voice leadings are contained in a cross-section of three-note chord space. The second voice leading, which has voice crossings, bounces off the triangle’s mirror boundary; the first does not.

Figure 3.10.1. Two-note chords are particularly close to their tritone transpositions; three-note chords are particularly close to their major-third transpositions; and four-note chords are particularly close to their minor-third and tritone transpositions.

Figure 3.10.2a. A geometrical representation of the closeness between (a) tritone-related perfect fifths and (b) major-third-related triads. Note that these figures show just a portion of the relevant geometrical spaces, and connect chords by lines if they can be linked by single-semitone voice leading.

Figure 3.10.2b. A geometrical representation of the closeness between (a) tritone-related perfect fifths and (b) major-third-related triads. Note that these figures show just a portion of the relevant geometrical spaces, and connect chords by lines if they can be linked by single-semitone voice leading.

Figure 3.10.4. Nirvana’s “Heart-Shaped Box” switches from major-third root motion to minor-third root motion when it switches from triads to a seventh chord. The upper three voices in the last voice leading of (a) are contained within the common four-voice voice leading in (b).

Figure 3.10.5a. (a) A voice-leading from the “Chopin” movement in Schumann’s Carnaval. Here, an F minor triad moves by major third to an A dominant seventh. (The bass, tenor, and soprano give us a pair of major-third-related triads, with the alto moving semitonally from the doubled third to the seventh.) The next progression moves an A7 chord to an Ef7. Once again the shift from triads to seventh chords accompanies a shift in root motion. (b) A comparison of Schumann’s “Chopin” and Nirvana’s “Heart-Shaped Box.” In Schumann’s progression, the switch to seventh chords occurs at the second chord in the passage: doubling the third of the F minor triad, he uses the semitonal motion Af→G to create an A7 chord. In the Nirvana progression, the switch does not occur until the final chord, where the C of F major is held over to become the seventh of D7.

Figure 3.10.5b. (a) A voice-leading from the “Chopin” movement in Schumann’s Carnaval. Here, an F minor triad moves by major third to an A dominant seventh chord. (The bass, tenor, and soprano give us a pair of major-third related triads, with the alto moving semitonally from the doubled third to the seventh.) The next progression moves an A7 chord to an Ef7. Once again the shift from triads to seventh chords accompanies a shift in root motion. (b) A comparison of Schumann’s “Chopin” and Nirvana’s “Heart-Shaped Box.” In Schumann’s progression, the switch to seventh chords occurs at the second chord in the passage: doubling the third of the F minor triad, he uses the semitonal motion Af→G to create an A7 chord. In the Nirvana progression, the switch does not occur until the final chord, where the C of F major is held over to become the D of D7.

Figure 3.10.6. Triads and seventh chords in mm. 18-20 of the first movement of Schubert's Bf major Piano Sonata (a), mm. 14-15 of the third-movement trio (b), the "Tarnhelm" motive from mm. 37-39 of Scene III of Wagner's Das Rheingold (c), mm. 11-12 of Mozart's C minor Fantasy, K. 475 (d), the eighteenth-century "omnibus" progression (e), and mm. 44–45 of Chopin's Nocturne, Op. 27 No. 2 (f).

Figure 3.12.1. Two collections of paths linking the points {C, E, G} and {D, F, A}. On the left, the three-voice voice leading (C, E, G) → (D, F, A); on the right, the four-voice (C, C, E, G) → (F, A, D, F).

Figure 3.12.2. In circular pitch-class space, it is not immediately obvious which of this chord’s transpositions it is closest to.

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