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Chapter 5

Figure 5.1.1. Two sequences containing the same chords, but in different order.  The first sounds considerably less chromatic than the second.

Figure 5.1.2. Each section of Debussy’s “Voiles” uses a different scale. The switch from whole tone to pentatonic to whole tone moves from greater dissonance to greater consonance and back again.

Figure 5.2.1. One way to combine harmonic and macroharmonic consistency is to choose a set of chords that together contain a relatively small number of notes.  Here, the A minor, F major, and Df major triads contain the pitches of a hexatonic scale.

Figure 5.2.2. In a 2-gap macroharmony, out-of-scale notes can be moved into the scale by semitone, in either direction.

Figure 5.2.3a. Near evenness and gaplessness.  Chords can be transposed along a nearly even scale with minimal distortion ( ).  In a gapless scale ( b), chords outside the scale can be “squeezed” into the scale with minimal distortion.  Here, the E minor triad is squeezed into the C acoustic scale by shifting B down by semitone.

Figure 5.2.3b. Near-evenness and gaplessness.  Chords can be transposed along a nearly even scale with minimal distortion ( ).  In a gapless scale ( b), chords outside the scale can be “squeezed” into the scale with minimal distortion.  Here, the E minor triad is squeezed into the C acoustic scale by shifting B down by semitone.

Figure 5.3.1b. (a) The theme of Bach’s F major two-part invention, along with the number of pitch classes in the first several three- and four-note windows. (b) A table listing the average number of pitch classes per window in the excerpt, for window sizes between 1 and 10 notes.  (c) The same information expressed as a graph.

Figure 5.3.1c. (a) The theme of Bach’s F major two-part invention, along with the number of pitch classes in the first several three- and four-note windows. (b) A table listing the average number of pitch classes per window in the excerpt, for window sizes between 1 and 10 notes.  (c) The same information expressed as a graph.

Figure 5.3.2. Pitch-class circulation in several well-known pieces.

Figure 5.4.1. Pitch-class circulation in Chopin’s Etudes (a) and Debussy’s Preludes (b).  Both collections cover an enormous range, comparable to that of the entire classical tradition (compare Figure 5.3.3). (continued)

Figure 5.4.1. (continued) Pitch-class circulation in Stravinsky’s Rite of Spring (c) and Shostakovich’s Preludes and Fugues (d).  Again, the pieces cover a large range.

Figure 5.4.2. Pitch-class circulation in Schoenberg’s Op. 11 and Webern’s Op. 27 piano pieces.  The individual movements of each piece are much more similar than in Figure 5.4.1.

Figure 5.5.1. Schoenberg’s Op. 11 and John Coltrane’s Giant Steps solo have identical rates of pitch-class circulation (a), even though Coltrane’s solo is tonal (b).

Figure 5.5.3. Debussy’s “La fille aux cheveux de lin7rsquo; (a) and Satie’s “Theme of the Order,” from Sonneries de la Rose + Croix (b).

Figure 5.6.1. Two progressions that use the same pitch classes, but sound very different.

Figure 5.6.2. The same notes are attacked on every sixteenth note, but there is a palpable transformation over the course of the measure.

Figure 5.6.7. The melody in (a) emphasizes the notes E and A, while that in (b) emphasizes E and B.  Though both might be said to be “in E phrygian,” there is an important difference between them.  We might represent this difference by using pitch-class profiles that identify the second-most important note in a mode.

Figure 5.6.9. Stravinsky’s “Petit airs,” from Histoire du soldat uses five “fixed” pitches (G, A, B, D, E) and two “mobile” pitch classes (C/Cs, and F/Fs) (a).  We can represent the music using a four-tiered pitch profile (b), in which the “mobile7rdquo; pitch classes are assigned a lower weighting than the fixed pitch classes.  Here, A is assigned the highest weight, indicating that it functions as a tonic.

Figure 5.6.14. The opening three phrases of Cathedral (one phrase per line), along with the three pitch-class profiles that inspired them.

Figure 5.6.15.

Figure 5.7.1a. (a) Diatonic music with no clear center.  (b) Chromatic music with a clear center.

Figure 5.7.1b. (a) Diatonic music with no clear center.  (b) Chromatic music with a clear center.

Figure 5.8.4a. Nondiatonic scales in nineteenth-century music.  In (a), a fragment of the “Gypsy” scale (G-A-Bf-Cs-D-Ef-Fs) in Grieg’s Lyric piece 7ldquo;Gjetergutt” (“Shepherd’s Boy”), Op. 54 No. 1.  In (b), the acoustic scale in Liszt’s “Angélus! Prière aux anges gardiens”), Années de pèlerinage, Year 3.

Figure 5.8.4b. Nondiatonic scales in 19th-century music.  In (a), a fragment of the “Gypsy” scale (G-A-Bf-Cs-D-Ef-Fs) in Grieg’s Lyric piece 7ldquo;Gjetergutt” (“Sheperd’s Boy”), Op. 54 no. 1.  In (b), the acoustic scale in Liszt’s “Angélus! Prière aux anges gardiens”), Années de pèlerinage, Year 3.

Figure 5.8.8. Messiaen’s “Première communion de la Vierge” (Vingt regards sur l'enfant-Jésus, no. 11) combines an octatonic-infused tonality with atonal upper-register gestures (right hand, m. 1).

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