## Chapter 4

Figure 4.1.1. The opening notes of the first phrases of “Do, Re, Mi,” interpreted chromatically and diatonically. The numbers represent intervals as measured relative to the two scales.

Figure 4.1.2. The “Do, Re, Mi” pattern in other scales.

Figure 4.2.1. Scale degrees and scalar distances. A scalar distance of one step is also called a “second,” a distance of two steps is “a third,” and so on.

Figure 4.2.3. Relative to the C harmonic minor scale, these chords all belong to the same transpositional set class, and are therefore triads. {Af, B, Ef} is not a member of this set class, and is not a triad, even though it is related to {C, Ef, G} by chromatic transposition.

Figure 4.2.4. This music uses the C major scale to measure distance, while also containing notes foreign to the scale (compare Figure 4.1.1).

Figure 4.3.1. (a) An uneven scale, in which the step between D and Bf is much larger than the others. (b) A small but nearly even scale, containing only the notes {C, E, G}.

Figure 4.3.2. (a) Chords that divide the octave completely evenly can be connected to their transpositions by efficient, but parallel voice leading. (b) Slightly uneven chords can be connected to their transpositions by voice leading in which the voices move by different distances, creating the sense of counterpoint. (c) Along a completely even scale, scalar transposition introduces no variation into music. (d) A small amount of unevenness creates slight amounts of variation, which can often be desirable.

Figure 4.4.1. (a) A stack of five perfect fifths falls .9 semitones short of three octaves. (b) A stack of seven perfect fifths overshoots four octaves by 1.14 semitones. (c) A stack of twelve fifths overshoots seven octaves by .25 semitones. (Note that the Bs is not quite the same as Cn.) By eliminating the top note of each stack we can form a relatively even scale with one “imperfect fifth.”

Figure 4.4.2. (a) A stack of four major thirds falls .41 semitones short of an octave. (b) A stack of five minor thirds overshoots an octave by .62 semitones.

Figure 4.4.3. (a) The hexatonic scale combines two augmented triads at a distance of a perfect fifth. (b) The octatonic scale combines two diminished sevenths at a distance of a perfect fifth. (c) The whole-tone scale combines augmented triads at a distance of a major second; it is perfectly even, but contains no perfect fifths.

Figure 4.4.4. Four scales can be represented as nearly even stacks of three major thirds and four minor thirds: diatonic (a), acoustic (b), harmonic minor (c), and harmonic major (d).

Figure 4.4.5. The scales in (a) have steps that are either one or two semitones large, and thirds that are three or four semitones large. Those in (b) have at least one step that is three semitones large. The top three scales in (b) have thirds that are three or four semitones large. The pentatonic scale is enclosed in a box because it is a subset of the diatonic.

Figure 4.4.6. Diatonic fourths and pentatonic "thirds." In (a), from mm. 34-35 of Debussy’s “La fille aux cheveux de lin,” diatonic fourths give way to pentatonic thirds. In (b), from Ligeti’s Piano Concerto (second movement, mm. 60-61) diatonic fifths are superimposed on pentatonic "fourths" (the inversions of pentatonic "thirds"). In (c), Herbie Hancock uses pentatonic thirds in the fifth chorus of “Eye of the Hurricane,” from the album Maiden Voyage. (d) Pentatonic thirds at the end of the chorus of the Decemberists’ “Here I Dreamt I Was an Architect.”

Figure 4.5.1. (a) The transition from the first movement of Mozart’s G major Piano Sonata, K. 283. (b) The passage involves two voice leadings between scales.

Figure 4.5.4a. Spelling often indicates a difference in musical function. Here, the chord {A, C, Ef, Gf} would typically lead back to a Bf major triad, while {A, C, Ef, Fs} would lead to G minor. Spelling can be interpreted as indicating specific voice leadings between scales.

Figure 4.5.4b. Spelling often indicates a difference in musical function. Here, the chord {A, C, Ef, Gf} would typically lead back to a Bf major triad, while {A, C, Ef, Fs} would lead to G minor. Spelling can be interpreted as indicating specific voice leadings between scales.

Figure 4.6.2. (a) To transform a whole-tone scale into an acoustic, “split” one note into its two chromatic neighbors. (b) To transform an octatonic scale into an acoustic, “merge” a major second into the note at its center. (c) To transform an octatonic scale into a harmonic major or minor, “merge” three notes spanning a minor third into the semitone they enclose. (d) To transform a hexatonic collection into a harmonic major or minor, “split” a semitone into a three-note scale-fragment enclosing it.

Figure 4.7.1. At the end of the B section of Mouvement, Debussy moves from Fs mixolydian to the C whole tone by way of the Fs mode of E acoustic.

Figure 4.7.2. An outline of Prokofiev’s Op. 27, No. 1.

Figure 4.8.1. The subject in Shostakovich's Fugue in E minor, Op. 87 No. 4, as it appears at the opening of the piece and in m. 22.

Figure 4.8.2. The two forms of the subject relate by scalar and chromatic transposition.

Figure 4.8.3. Rehearsal 13 of Debussy’s “Fêtes” (Nocturnes II).

Figure 4.8.4. The D-major theme in the Rondo from Clementi's Piano Sonata, Op. 25 No. 2, originally begins on scale degree 5 in G major, but returns in the transition with Cn replaced by Cs. The two forms of the theme can be related by a combination of scalar and chromatic transposition.

Figure 4.8.5. Three forms of the motive in Bach’s D minor two-part invention. Scale degrees are shown above each example.

Figure 4.8.6a. Interscalar transposition in the opening of Debussy’s “Fêtes” (a), Stravinsky’s Rite of Spring (b), and Shostakovich’s A major fugue (c).

Figure 4.8.6b. Interscalar transposition in the opening of Debussy’s “Fêtes” (a), Stravinsky’s Rite of Spring (b), and Shostakovich’s A major fugue (c).

Figure 4.8.6c. Interscalar transposition in the opening of Debussy’s “Fêtes” (a), Stravinsky’s Rite of Spring (b), and Shostakovich’s A major fugue (c).

Figure 4.8.7a. (a) “Fêtes,” the second movement of Debussy’s Nocturnes, presents the same pattern of scalar intervals in the seven-note dorian mode and the six-note whole-tone scale. (b) Steve Reich’s Variations for Winds, Strings, and Keyboards presents the same pattern of scalar intervals in the six-note diatonic hexachord and the five-note pentatonic scale.

Figure 4.8.7b. (a) “Fetes,” the second movement of Debussy’s Nocturnes, presents the same pattern of scalar intervals in a seven-note dorian scale and a six-note whole-tone scale. (b) Steve Reich’s Variations for Winds, Strings, and Keyboards presents the same pattern of scalar intervals in a six-note diatonic hexachord and a five-note pentatonic scale.

Figure 4.9.1. The voice leading in (a) is strongly crossing free: no matter what octave its voices are in, there will never be crossing. The voice leading in (b) is crossing free but not strongly so, since a crossing is created when the lowest voice moves up by octave (c).

Figure 4.9.2. In a strongly crossing-free voice leading, the voices can be transposed by octave so that each chord is in “close registral position,” spanning less than an octave. Ascending steps in one collection are sent to ascending steps in the other.

Figure 4.9.3. Repeatedly octave-transposing and removing crossings will eventually produce a strongly crossing-free voice leading. (a) Transposing the bass voice up by octave produces a crossing (b). Switching the bass and tenor in the first chord removes the crossing (c). Transposing the tenor up by octave produces another crossing (d). Switching alto and tenor in the first chord removes this crossing (e), producing a strongly crossing-free voice leading. Since removing crossings never makes a voice leading larger, the voice leading in (e) is at least as small as that in (a).

Figure 4.9.4abc. The minimal voice leading between C major and A minor triads (a), between C major and C minor triads (b), between CØ7 and Ef7 chords (c), and between F diatonic and G acoustic scales (d). All are interscalar transpositions.

Figure 4.9.4d. The minimal voice leading between C major and A minor triads (a), between C major and C minor triads (b), between CØ7 and Ef7 chords (c), and between F diatonic and G acoustic scales. All are interscalar transpositions.

Figure 4.9.6. Suppose a composer decides to connect this F half-diminished seventh chord to some dominant seventh chord by maximally efficient voice leading. There are almost three hundred possible voice leadings to consider. Yet musicians manage to solve problems like this very quickly.

Figure 4.9.7. In order to find the minimal voice leading between two chords, it is necessary to check only the interscalar transpositions between them. Here, the first of interscalar transposition maps the root of CØ7 to the root of F7, the second maps root to third, the third maps root to fifth, and the fourth maps root to seventh. Having chosen a destination for the root, the rest of the voice leading is completely determined by the fact that it is an interscalar transposition.

Figure 4.9.8. The opening phrase of Palestrina7rsquo;s “Adoramus Te” can be interpreted as embellishing a fundamentally crossing-free template.

Figure 4.9.9. The five-voice voice leading on the left is smaller than any of the four-voice alternatives. Here, three voices move by one semitone, whereas the smallest four-voice alternative moves one voice by three semitones and one voice by one semitone. Voice leadings containing doublings cannot be identified using the techniques discussed in this chapter.

Figure 4.10.1a. (a) The scalar or interscalar transpositions between any transpositions of the same two chord types are always individually T-related. (b) The minimal voice leading between C and E major triads can therefore be analyzed as the combination of a one-step descending scalar transposition with a four-semitone ascending chromatic transposition. (c) More generally, the minimal (three-voice) voice leading between any major triads can be depicted as combining chromatic and scalar transposition. As the chromatic transposition increases, the descending scalar transposition increases as well, so that the two forms of transposition cancel out.

Figure 4.10.1c. (a) The scalar and/or interscalar transpositions between any transpositions of the same two chord types are always individually T-related. (b) The minimal voice leading between C and E major triads can therefore be analyzed as the combination of a one-step descending scalar transposition with a four-semitone ascending chromatic transposition. (c) More generally, the minimal (three-voice) voice leading between any major triads can be depicted as combining chromatic and scalar transposition. As the chromatic transposition increases, the descending scalar transposition increases as well, so that the two forms of transposition cancel out.

Figure 4.10.2a. (a) The four interscalar transpositions from the C half-diminished seventh chord to the C dominant seventh chord. (b) The minimal voice leading between the C half-diminished and F dominant seventh chords combines the third of these interscalar transpositions with chromatic transposition upward by five semitones. (c) The minimal (four-voice) voice leading between any half-diminished and dominant seventh chords combines one of these interscalar transpositions with a chromatic transposition. Again, as the chromatic transposition increases, the descending interscalar transposition increases as well, so that the two forms of transposition cancel out.

Figure 4.10.2c. (a) The four interscalar transpositions from the C half-diminished seventh chord to the C dominant seventh chord. (b) The minimal voice leading between the C half-diminished and F dominant seventh chords combines the third of these interscalar transpositions with chromatic transposition upward by five semitones. (c) The minimal (four-voice) voice leading between any half-diminished and dominant seventh chords combines one of these interscalar transpositions with a chromatic transposition. Again, as the chromatic transposition increases, the descending interscalar transposition increases as well, so that the two forms of transposition cancel out.

Figure 4.10.3. The first four resolutions of the half-diminished seventh chord in Wagner’s Tristan. The first two map root to root, while the second two map root to third.

Figure 4.10.4. Three chromatic voice leadings. The first two map the root of the first chord to the root of the second, and can be considered interscalar transpositions from a seventh chord to a triad with doubled root. The last maps the root of the first chord to the fifth of the second.