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Chapter 2

Figure 2.2.2. The passages in (a)–(c) are similar in that they move C to E by four ascending semitones. The passages in (d)–(f) also move C to E, but differently—by eight descending semitones or by sixteen ascending semitones. We can capture what is similar about (a)–(c) by modeling these progressions as paths in pitch-class space. The progressions in (a) to (c) move C to E clockwise by four semitones along the pitch-class circle; (d) and (f) move eight semitones counterclockwise, and (e) moves sixteen clockwise semitones.

Figure 2.3.2. (a-b) Inversionally related passages in Bach’s A minor prelude, WTC, II. (c) Inversion as reflection in pitch space. Here, the note A3 is unaltered by the inversion, so the inversion can be written . All other notes move by twice their distance from A3.

Figure 2.3.3. The chords in each measure are inversionally related, and sound more similar to each other than to either of the chords in the other measure.

Figure 2.4.1. The basic musical object (C4, E4, G4) can appear either harmonically or melodically.

Figure 2.4.2. (a) All of these musical objects represent the C major chord, and are related by some sequence of octave shifts, permutations (or rearrangement of voices), and note duplication. (b) The C major chord can be represented by an unordered set of points in pitch class space.

Figure 2.4.6. Using OPTIC operations to transform musical objects.

Figure 2.4.7. Music-theoretical terms and the symmetry operations to which they correspond. A "chord" is a group of musical objects related by octave shifts, permutations, and note-duplications, while a "multiset" is a group of objects related by octave shifts and permutation. (Thus, when we are talking about multisets, the number of times a note appears is important; when we are talking about chords it is not.) In a "tone row," as defined by Schoenberg, order is important: we are permitted only to shift octaves and introduce note duplications.

Figure 2.5.1. (b) relates to (a) by uniform permutation. To get from (a) to (b) move all notes downward by one staff, shifting the bottom staff to the top. (c) relates to (a) by individual permutation, since it applies different permutations to each chord. Similarly, (d) and (e) relate to (a) by uniform and individual transposition, respectively.

Figure 2.5.2. Three voice leadings between G7 and C.

Figure 2.5.3. Two instances of the same voice-leading schema.

Figure 2.5.4. Voice leadings in pitch space can be represented as paths on a line. (b) Pitch class voice leadings can be represented as paths on a circle. These paths can move in either direction by any distance, and may complete one or more circumferences of the circle.

Figure 2.6.1. Uniformly and individually T-related voice leadings.

Figure 2.6.2. Uniformly and individually I-related voice leadings. Here, pitch-space inversion around E4 (E quarter-tone flat, halfway between Ef4 and E) sends C4 to G4, E4 to Ef4, F4 to D4, G4 to C4 and A4 to Bf3.

Figure 2.6.3. The semitonal voice leadings between major and minor triads. Since inversion preserves the distance moved by each voice, these voice leadings can be grouped into uniformly I-related pairs.

Figure 2.6.4. The opening progressions in Wagner’s Tristan (a), Brahms’ Intermezzo Op. 76, No. 4 (b), and Debussy’s Prelude to “The Afternoon of a Faun.”

Figure 2.6.5. Individually T-related voice leadings in sequential passages from Mozart (a) and Beethoven (b).

Figure 2.7.1. The opening phrase of the Bach chorale “O Herzensangst.” This music can be represented as the sequence of voice leadings (Ef, G, Ef, Bf) → (Ef, Bf, Ef, G) → (Af, Af, Ef, C) → (Af, F, D, Bf). I have ignored the starred (“nonharmonic”) note at the end of the first measure.

Figure 2.7.2. The principle of avoiding crossings tells us that (a) should be no smaller than (b) and (c) should be no smaller than (d).

Figure 2.7.3. Generally speaking, Western composers prefer smaller amounts of motion in more voices (a) to larger amounts of motion in fewer voices (b). The pattern in (a) common in the inner voices in classical music. (c) provides an example from the Bach chorale “Nun lob’ mein’ Seel’ den Herren.”

Figure 2.9.1. Harmonic consistency and efficient voice leading in a range of styles. (a) A common upper-voice pattern for the classical I-IV-I-V-I chord progression. (b) A common jazz-piano “left-hand voicing” for the descending-fifths progression D7-G7-C7-F7. The voicings add ninths and thirteenths and omit roots, fifths, and elevenths, as is common in jazz. (c) Two celebrated examples of Wagnerian chromaticism; the first is a simplification of the opening of Tristan and the second is the opening of The Ring’s “Tarnhelm” motif. (d) Chromatic clusters of the sort often found in late twentieth-century music, particularly in the music of Ligeti and Lutoslawski.

Figure 2.9.3. Harmonic consistency and efficient voice leading in a range of styles. (a) A common upper-voice pattern for the Any chord that is near an augmented triad can be connected to its major-third transposition by efficient voice leading.

Figure 2.9.4. Since both {C, E, Fs, B} and its tritone transposition are close to {B, C, F, Fs} (a-b), they are close to each other. In (c), we retrograde the voice leading in (b) and attach it to (a). Removing the middle chord gives us the voice leading in (d).

Figure 2.9.5. A larger chord can take advantage of the symmetries of a smaller chord, but it requires an additional voice.

Figure 2.9.7. Any chord that is near an inversionally symmetrical chord can be connected to its inversion by efficient voice leading.

Figure 2.9.8. The F half-diminished chord is close to a diminished seventh chord (a). Inverting this voice leading uniformly around A4/Bf4, we get an efficient voice leading from E7 to the same diminished chord. In (c), we retrograde the voice leading in (b) and attach it to (a). Removing the middle chord gives us an efficient voice leading from the F half-diminished chord to E7, a voice leading that plays a central role in Wagner’s Tristan.

Figure 2.9.9. Any chord that is near a chord with pitch-class duplications can be connected to its untransposed form by efficient voice leading.

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