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Chapter 8

Figure 8.1.1. Intensifying diatonic dominant-tonic chord progressions with chromatic notes.

Figure 8.1.3. Other chromatic embellishments.

Figure 8.1.4. In the nineteenth century, the lowered fourth scale degree is extremely rare. However, it does appear in later music.

Figure 8.2.1. Chromatic voice leading in Schubert’s “Am Meer” (a), Schoenberg’s “Erwartung,” Op. 2, No. 1 (b), Wagner's Tristan(c), Debussy's Pélleas et Mélisande (d), and the last movement of Brahms' Second Piano Concerto, Op. 83 (e).

Figure 8.2.5. In an “augmented sixth” resolution, two voices are separated by ten semitones, and converge semitonally to an octave, doubling one of the notes in a major or minor triad; typically, the augmented sixth lies between what sound like the first chord’s root and seventh, though in (c) this is not the case. If the resolution contains a leading tone resolving upward to the tonic, then the chord can typically function as an altered dominant.

Figure 8.2.6. The voice leadings at the start of Schubert’s “Am Meer” (a) and Schoenberg’s “Erwartung” (b) both exemplify the abstract schema (r, t, f, s) → (f, r, t, f). The voice leading in Tristan is (r, t, f, s) → (r, f, r, t) (c), while in Pélleas it is (s, r, t, f) → (r, t, f, r) (d). All four of these voice leadings are generalized “augmented sixth” resolutions. The voice leading at the last movement of Brahms' Second Piano Concerto (e) employs the schema (f, s, r, t) → (t, f, r, t), in which the converging voices are separated by nine semitones. This is not a generalized augmented sixth.

Figure 8.3.1a. A reduction of Brahms’ Intermezzo, Op. 76, No. 4. Resolutions of the Tristan chord {F, Af, Cf, Ef} and Ef minor triad are identified with the letters “α” and “β” respectively.

Figure 8.3.1b. A reduction of Brahms’ Intermezzo, Op. 76, No. 4. Resolutions of the Tristan chord {F, Af, Cf, Ef} and Ef minor triad are identified with the letters “α” and “β” respectively.

Figure 8.3.1c. A reduction of Brahms’ Intermezzo, Op. 76, No. 4. Resolutions of the Tristan chord {F, Af, Cf, Ef} and Ef minor triad are identified with the letters “α” and “β” respectively.

Figure 8.3.2. Individual T relatedness in Op. 76, No. 4.

Figure 8.3.3. A reduction of Schoenberg’s song “Erwartung,” Op. 2, No. 1.

Figure 8.3.4. (a) An ascending fifths sequence that uses descending stepwise voice leading. (b) A sequence from the development of the first movement of Haydn’s Piano Sonata no. 49 in Ef major, mm. 117-122, which decorates this basic voice-leading pattern. (c) Schoenberg’s sequence, transposed up by semitone for ease of comparison with the earlier examples. In Haydn’s sequence, explicit i@→v resolutions ascend by fifth, linked by secondary dominants; Schoenberg’s progression only hints at the i@ harmonies, subsuming them within extended dominants. Here the dotted line signifies that the G5 in the first measure can be considered “the same” as the G4 in the second measure—or in Schenkerian terms, a “voice transfer.”

Figure 8.4.1. A reduction of the opening of Schubert's D major Piano Sonata, D. 850, Op. 53.

Figure 8.4.3. Cross-phrase mediant progressions in Bach (a) and Mozart (b).

Figure 8.4.4. Descending triadic sequences in Schubert's Quartett-Satz, D. 703, m. 173ff (a) and m. 105ff (b). (The second has been transposed up by minor third to facilitate comparison.) The sequences can be derived by applying "major-third substitution" to a chromatic descending sequence.

Figure 8.4.5. Schubert's sequences as they move along the lattice at the center of three-note chord space. The solid line is the sequence in Figure 8.4.4a, the dashed line is the sequence in Figure 8.4.4b.

Figure 8.5.1. An analysis of Chopin’s Mazurka, Op. 68., No. 4. The harmonies form a series of cycles which are labeled numerically: chords 1a, 1b, and 1c belong to the first cycle, chords 2a, 2c, and 2d belong to the second, and so on.

Figure 8.5.2. In Chopin’s piece, the root of the chord always descends after the other voices. If this rule were violated, unusual harmonies would result. These chords cannot be conceived as stacks of thirds, and are very rare in nineteenth-century music.

Figure 8.5.5. The opening of Chopin’s E minor Prelude, Op. 28, No. 4.

Figure 8.5.7. There are four fundamentally similar sequences that can be formed using these “directions for improvisation,” depending on which of the diminished seventh chord’s notes is lowered.

Figure 8.5.9a. The sequence as it appears in Chopin’s A minor Mazurka (a), Fs minor Mazurka (b), and Df major Nocturne. In each analysis, the letters “a,” “b,” “c,” refer to levels on the cubic structure shown in Figure 8.5.8; the numbers refer to adjacent cubes in the lattice.

Figure 8.5.9b. The sequence as it appears in Chopin’s A minor Mazurka (a), Fs minor Mazurka (b), and Df major Nocturne. In each analysis, the letters “a,” “b,” “c,” refer to levels on the cubic structure shown in Figure 8.5.8; the numbers refer to adjacent cubes in the lattice.

Figure 8.5.9c. The sequence as it appears in Chopin’s A minor Mazurka (a), Fs minor Mazurka (b), and Df major Nocturne. In each analysis, the letters “a,” “b,” “c,” refer to levels on the cubic structure shown in Figure 8.5.8; the numbers refer to adjacent cubes in the lattice.

Figure 8.5.11. Chopin's F minor Mazurka and E minor Prelude are related by way of a tritone substitution. The mazurka is based on a semitonally descending sequence of seventh chords (top line), while the prelude uses a descending-fifths sequence. One can transform each sequence into the other by replacing every other chord with its tritone transposition.

Figure 8.5.12a. Precursors to Chopin’s procedures in (a) Mozart’s Symphony No. 40, movement 1, start of the development section; and (b) Beethoven&rquo;s Piano Sonata Op. 54, movement 2, mm. 64ff.

Figure 8.5.12b. Precursors to Chopin’s procedures in (a) Mozart’s Symphony no. 40, movement 1, start of the development section; and (b) Beethoven&rquo;s Piano Sonata Op. 54, movement 2, mm. 64ff.

Figure 8.5.13ac. (a-c) In his 1898 operetta “The Serenade,” Victor Herbert uses chromatic voice leading to connect dominant sevenths by way of various intermediaries. The examples are from nos. 3a (Schuberth vocal score, p. 35), 4 (p. 46), and 5b (p. 59). (d) In “Shreveport Stomp,” Jelly Roll Morton uses descending stepwise voice leading to connect dominant sevenths.

Figure 8.5.13d. (a-c) In his 1898 operetta “The Serenade,” Victor Herbert uses chromatic voice leading to connect dominant sevenths by way of various intermediaries. The examples are from nos. 3a (Schuberth vocal score, p. 35), 4 (p. 46), and 5b (p. 59). (d) In “Shreveport Stomp,” Jelly Roll Morton uses descending stepwise voice leading to connect dominant sevenths.

Figure 8.6.1a. (a) Suppose you want to move from a Tristan chord to a dominant seventh by efficient voice leading. Which voice leading would you choose? (b) The most efficient (four-voice) voice leadings from the Fø7 chord to each of the twelve dominant sevenths.

Figure 8.6.1b. (a) Suppose you want to move from a Tristan chord to a dominant seventh by efficient voice leading. Which voice leading would you choose? (b) The most efficient (four-voice) voice leadings from the Fø7 chord to each of the twelve dominant sevenths.

Figure 8.6.3. In moving from a Tristan to half-diminished seventh, Wagner often passes through one of the intervening chords in four-note chord space.

Figure 8.6.4. Unfortunately, the actual voice leadings in Tristan often have crossings.

Figure 8.6.5. Other passages in Tristan that seem to embellish a crossing-free substrate: mm. 10-11 (a), mm. 22-23 (b), and m. 81 (c).

Figure 8.6.6. Suppose we would like to combine an Fø7→E7 progression with melody that ascends chromatically by minor third. One possibility is to use an interscalar transposition by ascending step (a), producing ascending motion in all four voices. Another possibility is to begin with an efficient voice leading (b), which is then embellished by a voice crossing (c).

Figure 8.6.7. When we remove Wagner’s voice crossings, we typically end up with a highly efficient voice leading.

Figure 8.6.8. The voice-leading schemas in the Prelude return many times throughout the opera. These three passages are taken from Act I, scene iii: (a) p. 51/98, (b) p. 49/96, (c) p. 49/95. (Page numbers refer to the Schirmer vocal score and Dover full score, respectively.)

Figure 8.6.9. “Tritone substitution” applied to Tristan chords in Wagner’s opera.

Figure 8.6.10. Measure 36ff of the prelude mirrors the form of the opening: an initial voice leading is transposed exactly, whereupon the opening chord type is resolved in a new way. Here the initial chord is a French augmented sixth and the third voice leading is individually T-related to the first (b). Note that when this music repeats at the end of the prelude, Wagner uses Dn instead of Ds in the second-to-last measure of (a).

Figure 8.6.11. A reduction of two passages from the Tristan prelude, both of which move along the lattice at the center of four-note chord space. Labels below each chord again show how the music moves through the connected hypercubes. Where Chopin moves steadily forward, Wagner moves backward and forward in a stuttering manner.

Figure 8.6.12. A passage from Act III, Scene I, p. 233/494 that can also be described using Chopin’s sequence.

Figure 8.6.14. Two additional resolutions of the Tristan chord from later in the opera: Act II, Scene 1, p. 121/239 (root to seventh) (a), and Act II, Scene 2, p. 141/287 (root to fifth) (b).

Figure 8.8.1. In Strauss’ Salome, extensive chromatic voice leading is accompanied by equally chromatic melodic motion, leading to a saturated pitch-class space in which all twelve notes are constantly in play.

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