masthead
 

Chapter 7

Figure 7.1.3. The chorale "Bach's chorale "Auf, auf, mein Herz, und du mein ganzer Sinn" (Riemenschneider No. 124).

Figure 7.1.4. Bach’s duet (BWV 803, mm. 22–26) interposes iii between V and I.

Figure 7.1.7. (a) The most popular diatonic harmonic cycles in a selection of 30 major-mode Bach chorales.  (See Tymoczko 2003b.)  All of them conform to the thirds-based model.  (b) The most popular harmonic cycles in the Mozart piano sonatas.  Once again, all conform to the thirds-based model.

Figure 7.2.1. Here, the upper voices are connected by single-step voice leading, while the harmonies move along the descending circle of thirds from tonic to dominant.

Figure 7.2.2a. (a) Third-related triads sound similar, since they share two of their three notes and can be connected by single-step voice leading. (b) One can often replace a diatonic chord with a third-related chord, without much disrupting the harmonic or contrapuntal fabric of a passage.

Figure 7.2.2b. (a) Third-related triads sound similar, since they share two of their three notes and can be connected by single-step voice leading. (b) One can often replace a diatonic chord with a third-related chord, without much disrupting the harmonic or contrapuntal fabric of a passage.

Figure 7.2.3a. Bass-line and third substitution.  In (a) a same I-IV-V-I chord progression appears over two different bass lines.  In (b), root position and first-inversion chords (over the same bass note) play similar harmonic roles.

Figure 7.2.3b. Bass-line and third substitution.  In (a) a same I-IV-V-I chord progression appears over two different bass lines.  In (b), root-position and first-inversion chords (over the same bass note) play similar harmonic roles.

Figure 7.2.5b. Bach’s music often feature’s descending-thirds sequences of pitch classes.  (a) The third movement of the third Brandenburg Concerto, mm. 11-12.  (b) The F major two-part invention, mm. 21-23.  

Figure 7.2.5a. Bach’s music often feature’s descending-thirds sequences of pitch classes.  (a) The third movement of the third Brandenburg Concerto, mm. 11-12.  (b) The F major two-part invention, mm. 21-23.  

Figure 7.2.6a. Two passages in which melodic descending thirds do not clearly determine harmonies.  (a) Bach’s fourteenth Goldberg Variation, mm. 9-10.  (b) The opening of Brahms’ Op. 119, No. 1.  

Figure 7.2.6b. Two passages in which melodic descending thirds do not clearly determine harmonies.  (a) Bach’s fourteenth Goldberg Variation, mm. 9-10.  (b) The opening of Brahms’ Op. 119, No. 1.  

Figure 7.2.7. Ascending motion provides the most efficient voice leading between strongly related triads (top three voices of a). Since tonal phrases often feature descending melodic lines, composers typically have to use at least one non-minimal voice leading per harmonic cycle (b–c).

Figure 7.2.9. Efficient voice leading between strongly related seventh chords descends (top four voices of a).  Harmonic cycles and descending-fifths sequences can therefore be realized with maximally efficient, descending voice leading (b–c).

Figure 7.2.10. These voice leadings are nonfactorizable, because eliminating any voice creates an incomplete triad.

Figure 7.2.13. Nonfactorizable voice leadings in keyboard style.  In each case, a complete triad in the right hand moves to an incomplete chord.

Figure 7.3.1. Five tonal sequences represented on the circle of thirds.

Figure 7.3.2a. Haydn’s E minor Piano Sonata, Hob. XVI/34, mm. 72–75. (b-c) The “down a third, up a step” sequence in Fauré’s Pavane, mm. 2–5 and Bach’s Bf two-part invention, mm. 15–16 (bottom). (d) The “down a third, down a third, down a fifth” sequence in the D major fugue from Book I from Bach’s Well-Tempered Clavier, mm. 9–10.

Figure 7.3.2b1. Haydn’s E minor Piano Sonata, Hob. XVI/34, mm. 72–75. (b) The “down a third, up a step” sequence in Faure’s Pavane, mm. 2–5 (top) and Bach’s Bf two-part invention, mm. 15–16 (bottom). (c) The “down a third, down a third, down a fifth” sequence in the D-major fugue from Book I from Bach’s Well-Tempered Clavier, mm. 9–10.

Figure 7.3.2b2. Haydn’s E minor Piano Sonata, Hob. XVI/34, mm. 72–75. (b) The “down a third, up a step” sequence in Faure’s Pavane, mm. 2–5 (top) and Bach’s Bf two-part invention, mm. 15–16 (bottom). (c) The “down a third, down a third, down a fifth” sequence in the D-major fugue from Book I from Bach’s Well-Tempered Clavier, mm. 9–10.

Figure 7.3.2c. Haydn’s E minor Piano Sonata, Hob. XVI/34, mm. 72–75. (b) The “down a third, up a step” sequence in Faure’s Pavane, mm. 2–5 (top) and Bach’s Bf two-part invention, mm. 15–16 (bottom). (c) The “down a third, down a third, down a fifth” sequence in the D-major fugue from Book I from Bach’s Well-Tempered Clavier, mm. 9–10.  

Figure 7.3.5. The “down a third, up a step” sequence in the third movement of Mozart’s first Piano Sonata, K. 279. (b) When the sequence returns at the start of the development, Mozart alters it so that it begins with a pair of descending fifths.  (c) This change is a minimal perturbation when represented along the circle of thirds.

Figure 7.3.6. An interpretation of two passages from Bach’s Af major fugue from WTC II.  (a) Measures 6-7 involve the “down a third, up a step” sequence, while mm. 13-15 involve the descending fifths sequence (b).  In the original music, the middle voice of (a) and the top voice of (b) involve sixteenth-note figuration; thus the chromatic descent moves from soprano (a) to middle voice (b).

Figure 7.3.7. A sequence-like passage from the F minor fugue in book I of Bach’s Well Tempered Clavier, mm. 16-18.  Harmonically, the passage involves an unsystematic collection of “strong” progressions.

Figure 7.3.8a. (a) The “down a third, up a step” sequence can be derived from more familiar sequences by exchanging root position and first-inversion chords.  Here, the four forms on the bottom staff, are derived from the descending-fifth and descending-step sequences in the upper staff. (b) Measures 62-66 from Contrapunctus X in Bach’s Art of the Fugue.  Here, root substitution over a fixed bass line changes the descending fifths progression into the “down a third, up a step” progression.

Figure 7.3.8b. (a) The “down a third, up a step” sequence can be derived from more familiar sequences by exchanging root position and first-inversion chords.  Here, the four forms on the bottom staff, are derived from the descending-fifth and descending-step sequences in the upper staff. (b) Measures 62-66 from Contrapunctus X in Bach’s Art of the Fugue.  Here, root substitution over a fixed bass line changes the descending fifths progression into the “down a third, up a step” progression.

Figure 7.3.9(a-d). Figure 7.3.9.  (a-d) There are four basic forms of the “up a step, down a third” sequence, depending on which inversions are used.  An example of each is provided.  (e) Haydn’s D major Piano Sonata Hob. XVI/42, II, mm. 11-12.  (f) The opening of the Crucifixus, from Bach’s B minor Mass (BWV 232).  (g) Brahms' F minor Piano Quintet, Op. 34, I, mm. 8-9. (Note that the rhythm, dynamics, and musical context all suggest that the sevenths should not be understood as suspensions.)  (h) Bach’s G major Fugue, Book II of the Well-Tempered Clavier, mm. 66-69.

Figure 7.3.9(e-h). Figure 7.3.9.  (a-d) There are four basic forms of the “up a step, down a third” sequence, depending on which inversions are used.  An example of each is provided.  (e) Haydn’s D major Piano Sonata Hob. XVI/42, II, mm. 11-12.  (f) The opening of the Crucifixus, from Bach’s B minor Mass (BWV 232).  (g) Brahms' F minor Piano Quintet, Op. 34, I, mm. 8-9. (Note that the rhythm, dynamics, and musical context all suggest that the sevenths should not be understood as suspensions.)  (h) Bach’s G major Fugue, Book II of the Well-Tempered Clavier, mm. 66-69.

Figure 7.4.4b1. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.4b2. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.4b3. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.4c1. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.4c2. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.4c3. Using voice leading to calculate distances between keys.  (a)  For major keys, the distances are simply the voice leading distances between the relevant diatonic collections.  (b) For distances between major and minor, we calculate the size of the voice leadings from the major scale to each minor scale, and take the average.  Here, the average distance between C major and the three A minor scales is 1.  (c) For minor scales, we take the average of the three voice leadings in the most efficient pairing of the scales in one key with those in the other.  Here, the average distance for the best pairing between A minor and E minor scales is two.

Figure 7.4.7.  Estimated modulation frequencies in Bach’s Well Tempered Clavier and the piano sonatas of Haydn, Mozart, and Beethoven.  The left column indicates the directed chromatic interval of root motion from source key to target key.  (For example, a modulation from F major to G major or B minor to Cs major is represented by the number two 2.)  Under each composer’s name, the “min” and “maj” columns refer to the modality of the target key. The two or three largest values in each column are in boldface.

Figure 7.5.2. (a) On the circle of thirds, we move from C to G by way of E minor.  (b) If we reverse the order of the voice leadings C→B and E→D, we can move from C to G by way of the “suspension chord” C-D-G.  By scrambling every adjacent pair of voice leadings on the circle of thirds, we produce a lattice of squares each sharing an edge with their neighbors.

Figure 7.5.6. The diatonic chord cube contains all the sonorities that can resolve to CEG by either a single or double suspension.

Figure 7.5.7. The sonorities on the chord lattice provide waystations allowing composers to break large movements into smaller steps.  Instead of moving directly from DGB to CEG, as in (b), a composer can use nonharmonic tones to smooth out the journey (c).  The path in (a) depicts the music in (c).   Historically, the F in the bass voice of (c) originated as a nonharmonic passing tone, but was eventually granted harmonic status as part of a seventh chord on G.

Figure 7.5.8. Four ways to use suspensions to decorate a series of descending first-inversion triads, represented on the chord cube.  The first produces the ubiquitous 7-6 suspension, while those in (b) and (c) interpose an additional (root position or second-inversion) triad between consecutive first-inversion triads; the path in (d) creates a double suspension that sounds like an incomplete seventh chord.

Figure 7.5.9. (a) Philidor’s “Art of Modulation” (Sinfonia V, Fuga, mm. 36-38) alternates between 7-6 and 4-3 suspensions, utilizing paths a and b in Figure 7.5.8.  (b) The Prelude to Grieg’s Suite “From Holberg’s Time” uses the unusual double suspension represented by path (d).

Figure 7.6.1. Mozart’s variations on “Ah, vous dirai-je, Maman,” (K. 265), along with Cadwallader and Gagné’s Schenkerian analysis.

Figure 7.6.2. The first movement of Mozart’s Piano Sonata K. 279, mm. 25–30. A traditional theorist might consider the I@ to be a “passing chord,” since it violates the expectation that I@ goes to V.  By treating it as the product of linear motion (as shown in b), we obtain a syntactical progression from an apparently nonsyntactical surface.

Figure 7.6.3. These “neighboring” and “passing” chords are contrapuntally unobjectionable, though the resulting chord progressions are harmonically unusual.

Figure 7.6.4. Three progressions that are contrapuntally unobjectionable, but which rarely appear in baroque or classical music: a root-position V-IV (a), a major-key I-iii-V-I (b), and the analogous progression in minor (c).

Figure 7.6.6. The opening of the slow movement of Beethoven’s Sonata in C minor, Op. 10, No. 1, along with two ways of parsing its structure.  In the traditional tonal analysis (top) five harmonic cycles are concatenated like beads on a string.  In the Schenkerian reading (bottom) harmonies are nested recursively.  (For example, the progression IV-IV6-V#-I, which belongs to the fourth harmonic cycle, is taken to represent a single IV chord on level 2.)  Schenkerians believe that these sorts of recursive structures, which cut across the articulation into harmonic cycles, can be reliably inferred from a piece’s contrapuntal structure.  Ultimately, this recursive embedding proceeds until entire pieces are reduced to one of just a few basic templates, each resembling a I-V-I progression.

Figure 7.6.7. If we were to consider every note to be harmonic, we would confront an array of unusual sonorities, such as the A-D-E on beat 2.  By eliminating nonharmonic tones, we reveal a “deeper level” of musical structure, in which triadic harmonies progress in familiar ways.  The violation of our expectations (e.g. that harmonies should be triadic) is what motivates our reductive analysis.

Website Terms and Conditions and Privacy Policy
Please send comments or suggestions about this Website to custserv.us@oup.com        
cover