Figure 6.2.1. The Allelujia Justus et Palma from Ad Organum Faciendum. Open and closed noteheads represent the two different musical voices.
Figure 6.2.3a. (a) The seventh phrase of the Allelujia Justus et Palma, traced out on the Möbius strip. (b) The phase contains four “nearly stepwise” voice leadings connecting a perfect fifth to an octave (or perfect fourth to a unison).
Figure 6.2.3b. (a) The seventh phrase of the Allelujia Justus et Palma, traced out on the Möbius strip. (b) The phase contains four “nearly stepwise” voice leadings connecting a perfect fifth to an octave (or perfect fourth to a unison).
Figure 6.3.1. The opening of Josquin’s “Tu pauperum refugium.” Beneath the staff, I have shown how the music contains a series of efficient three-voice voice leadings between three-note chords.
Figure 6.3.2. In Josquin’s music, fourths typically resolve to thirds (a), diminished triads sometimes appear in first inversion (b), and cadences sometimes anticipate the functional harmony of later centuries (c).
Figure 6.3.3. There are only a few three-note chords containing only perfect consonances, and they are spread throughout three-note chord space (a). (Note that we are viewing chord space from above, looking down through the triangular faces; compare the side view in Figure 3.8.2, where the triangular faces are on the top and bottom.) As a result, there are only a few ways to connect them with efficient voice leadings (b), particularly if one wants to avoid parallel octaves.
Figure 6.3.4. Doubled consonances (a) and triads (b).
Figure 6.3.8. Two strongly crossing-free voice leadings between C and F, represented on the circle.
Figure 6.3.9. Four voice leadings from the opening of Josquin’s “Tu pauperum refugium” plotted in three-note chord space (a) and on the diatonic circle of thirds (b).
Figure 6.3.10. (a–b) Two common chord progressions, as they are often played on a guitar. (c–d) The notes produced by these fingering patterns. The top three voices articulate strongly crossing-free voice leadings that can be represented on the triadic circle.
Figure 6.3.12. Fourth progressions allow a composer to harmonize a wide range of stepwise melodies in a “3+1” fashion, with the upper voices moving by strongly crossing-free voice leadings and the bass moving from root to root. Root motions by second and third are comparatively less flexible, either because they do not harmonize many stepwise melodies or because they can create forbidden parallels.
Figure 6.3.13. (a) A basic medieval two-voice cadential pattern, in which two voices converge onto an octave by stepwise motion. (b–c) Three-voice harmonizations producing a common medieval cadence (with the Fs), a vii°6-I cadence (with the Fn), or a phrygian cadence (with Fn and Df). (d) A three-voice harmonization resembling a V-I cadence. (e) A failed harmonization producing parallel fifths. (f–h) Four-voice extensions of the preceding schemas. Only the V-I form (g) cadences on a sonority containing only perfect consonances.
Figure 6.3.14. Parallel motion within the diatonic scale (a-c) and in Messiaen’s “Mode 2” (d–e). Underneath each interval I have written its chromatic size in semitones. The brackets identify chromatic motion that is not parallel.
Figure 6.4.1. (a) The first phrase of Bach’s chorale “Herr Christ, der ein’ge Gott’s sohn” (No. 303 in the Riemenschneider edition) and a reduction of the upper voice counterpart. (b) A simple model of the chord progressions it contains. Chords can move freely rightward, but can move leftward only by following the arrows.
Figure 6.4.3. The first two phrases of one of Bach’s harmonizations of “A Mighty Fortress is Our God” (Riemenschneider No. 20).
Figure 6.5.1. A harmonic summary of Schumann’s “Chopin.” Each whole note in the summary corresponds to a measure in the actual work.
Figure 6.5.2. The last measure of “Chopin” can be interpreted as a four-voice passage in which each voice moves by semitone.
Figure 6.6.1. (a) Relative to the diatonic scale each voice moves by the smallest possible distance. (b) Relative to the chromatic scale, there are gaps that can be filled in. The resulting progression appears in the nineteenth-century song “You tell me your dream, I’ll tell you mine.”
Figure 6.6.2. The second phrase of Chopin’s E major prelude. Beneath the excerpt I provide simplified voice leadings that illustrate the major- and minor-third systems.
Figure 6.6.3. An outline of the modulations in the scherzo to Schubert’s C major String Quintet. Local tonics are shown with open noteheads. Of the eleven modulations, nine involve the major- and minor-third systems; only two employ traditional pivot chords.
Figure 6.7.1. Accompanying chromatic chords with scales. In (a), the scale does not contain the chromatic chords; in (b–c) it does. The music in (b) augments the diatonic scale with the altered notes, while in (c), the altered notes replace their diatonic forms.
Figure 6.7.2. (a) In Strauss' Till Eulenspiegel (R7), the harmony Bf-Df-E-Gs serves as an altered dominant chord resolving semitonally to the tonic F. The scale in the bass, however, is a simple Af major that does not contain the En of the harmony, resulting in a fleeting dissonance as the scalar F-Ef clashes with the harmonic En. (b) In Wagner’s Parsifal (mm. 668-9, Schirmer vocal score p. 40), a diminished seventh chord is harmonized with a melody touching on all the chromatic notes except Fs and G. The effect is of a wash that creates a sense of motion, but does not clearly suggest any familiar scale. (c) In m. 86 of the first movement of Mozart’s Piano Sonata K. 533, a familiar augmented sixth chord gives rise to an unusual ("gypsy") scale. This scale is derived by combining the notes of the C diatonic scale with the alterations belonging to the German augmented sixth.
Figure 6.7.3. Twentieth-century scalar composers might use acoustic scales to harmonize altered chords. The result is subtly different from the three strategies described in Figure 6.7.1, and it produces a melodic texture without augmented seconds or consecutive semitones.
Figure 6.7.4. (a) Whole-tone and octatonic scales in Ravel’s String Quartet. (b) The underlying progression, and its diatonic logic (c).
Figure 6.7.5. Bill Evans using the octatonic scale in his Town Hall recording of "Turn Out the Stars."