i i i i Outline of New Section 7.5: Tonnetz Patterns in Music

This section will describe several examples of interesting patterns on the Tonnetz that occur in a variety of musical compositions. These patterns show how the Tonnetz provides a geometric logic for chord progressions, sometimes even when there is no underlying key for the music.

Examples from symphonic music There are a number of examples from the book by Richard Cohn, Audacious Euphony: Chromaticism and the Triad’s Second Nature, (Oxford, 2012). We will refer to this book as an important refer- ence in the section, and as an invitation to readers to explore recent ground breaking scholarship in applications of Tonnetz transformations in music theory. Here are three examples from Cohn’s book: Example 1. Mozart, Symphony in E-flat major, K. 543, finale mm 109–126. Cohn, p. 25, derives the following cycle of chord progressions for this set of measures:

T T T T T T T T T T T T T T A♭ −→ a♭ −→ B −→ E −→ B −→ E −→ G −→ e −→ G −→ C −→ G −→ c −→ G −→ E♭ −→ A♭ (1)

As we have indicated, these are all Tonnetz transformations. On the left of Figure 1, we diagram this pattern of chord progressions. All of these Tonnetz transformations either lie on an edge, or pass through, the following three pitch class hexagons:

E♭ G B (2)

We have marked these pitch classes on the chromatic clock on the right of Figure 1. The three radial lines connecting these chords to the center of the chromatic clock illustrate the 3-fold rotational symmetry of these pitch classes about the center of the clock. These pitch classes, {E♭, G, B}, form an augmented triad—with pitches classes separated evenly by major thirds (4 hours on the chromatic clock). Moreover, each chord in (1) can be obtained from this augmented triad by adding either +1, −1, or 0 as hours on the chromatic clock. For example, the A♭ chord is obtained by

+1 +1 +0 G −−→ A♭ , B −−→ C , E♭ −−→ E♭ (3)

while the G chord is obtained by

−1 +0 +0 E♭ −−→ D , G −−→ G , B −−→ B (4)

Furthermore, we have marked on the right of Figure 1 all of the pitch classes for the chords shown in (1). This collection of 9 pitch classes also has 3-fold rotational symmetry about the center of the chromatic clock. It is interesting that these pitch classes can be generated by applying the Euclidean algorithm from Section 6.5 to 9 ones followed by 3 zeros. Example 2. Brahms’ Concerto for Violin and Cello, Op. 102, 1st mvt, mm 270–276. Cohn, p. 30, derives the following sequence of chord progressions:

T T T T T T A♭ −→ a♭ −→ E −→ e −→ C −→ c −→ A♭ (5)

as shown on the left of Figure 2. These Tonnetz transformations are all of type, P or L, as they lie on the edges of the hexagons for these pitch classes:

A♭ B C E♭ E G A♭

These pitch classes form a hexatonic scale. We have plotted this hexatonic scale on the chromatic clock on the right of Figure 2. Notice that it is a union of two augmented triads: {E, A♭, C} and {E♭, G, B}. Cohn describes how hexatonic scales, and their associated augmented triads, play important roles in 19th century romantic music.

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Figure 1: Left: Tonnetz cycle in Mozart, Symphony in E-flat major, K. 543, finale mm 109–26. Cycle T T begins with A♭ −→ a♭, ends with E♭ −→ A♭. Right: Pitch class hexagons activated in this Mozart passage. The radial lines point towards the three pitch class hexagons through which all Tonnetz transformations pass through (either within a hexagon, or on its edge) in the diagram on the left.

Figure 2: Left: Hexatonic cycle in Brahms, Concerto for Violin and Cello, Op. 102, 1st mvt, mm T T 270–276. Cycle begins with A♭ −→ a♭, ends with c −→ A♭. Right: Pitch class hexagons activated in this Brahms passage. They form a hexatonic scale.

Example 3. Brahms’ Symphony No. 2, 1st mvt, mm 246–270.. Cohn, p. 117, derives the following sequence of chord progressions for this set of measures: T T T T T T T T T T G −→ g −→ B♭ −→ d −→ D −→ B♭ −→ d −→ F −→ a −→ A −→ f♯ (g♭) All of these Tonnetz transformations either lie on the edges, or pass through, the two pitch class hexagons D and then A. See Figure 3. These pitch class hexagons are separated by a fifth. Example 4. L’Histoire du Soldat. In his book on music and language,1 Leonard Bernstein discusses the opening of Stravinsky’s L’Histoire du Soldat as follows: There are two instruments playing: a cornet and a trombone. The cornet by itself is playing a tune that seems to start in F-major, suddenly switches to F-minor, and cadences abruptly in a totally 1L. Bernstein, The Unanswered Question: Six Talks at Harvard, Harvard University Press, 1976, p. 343. These lectures are also available on DVD; the beginning of Lecture 6 contains the passage quoted here.

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unexpected E-major. So, F-major, F-minor, E-major, all in the space of four seconds. Now let’s see what the trombone is doing . . . D-flat major, of all things, with its abrupt cadence in G-major, without so much as a by-your-leave.

On the right side of Figure 3, we show these chord changes on the Tonnetz. They can be expressed as Tonnetz transformations in this way:

T T F −→ f −→ E (6) T T D♭ −→ E −→ G.

T T The second sequence of mappings, D♭ −→ E −→ G, includes the chord E because that chord is being played by the cornet while the trombone is arpeggiating the chord G. It is also worth noting that the chords f and D♭, which are also outlined simultaneously by cornet and trombone, are connected by a Tonnetz transformation as well. Although the chord progressions in (6) are not typical ones within any fixed key, our Tonnetz description makes it clear that they do obey a clear tonal logic—a tonal logic that is captured concisely and geometrically by the Tonnetz diagram on the right side of Figure 3.

Figure 3: Left: Paired hexagonal motion in Brahm’s Symphony No. 2, 1st mvt, mm 246–270. Motion T T begins with G −→ g, ends with A −→ g♭. Chordal motion travels around two hexagons, D then A. Right: Tonnetz motion in Stravinsky’s L’Histoire du Soldat.

Examples from popular music We will also discuss Tonnetz diagrams for three popular music compositions. Example 5. Alicia Keys, opening of If I Ain’t Got You. This example is interesting in that a spectro- gram analysis shows that, as the extended bass notes die out, a new chord is sounding. See the left of T Figure 4, where the Tonnetz transformation C −→ e is explained. Using similar reasoning for several subsequent measures, the opening of this song trace a series of Tonnetz transformations:

T T T T T T C −→ e −→ b −→ D −→ a −→ C −→ G

as shown on the right of Figure 4. These Tonnetz transformations trace a path through each of the six major and minor chords for the song’s key of G-major.

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Figure 4: Left: Spectrogram of first two measures of If I Ain’t Got You. Bottom left arrow points to fundamental for a C note. Triplet above it points to fundamentals for notes in the set {E, G, B} for minor chord e. Arrow on bottom right indicates that the C note has completely faded out, while triple arrow above it indicates emphasis by the performer/composer, Alicia Keys, on the notes in this e chord. Viewing CM7 as an embellishment of C, the performancehighlights the Tonnetz transformation T T C −→ e. Right: Moves on Tonnetz for these two measures, starting with C −→ e.

Example 6. Adele Adkins and Daniel Wilson, Someone Like You. The chord progressions of this song also have an interesting symmetrical pattern on the Tonnetz. The chords in the song’s original key of A-major are A c♯ F♯ 5 D A (7) The chord F♯ 5 is known as a power chord in popular music. It consists of just an F♯-note and, a fifth above it, a C♯-note. The chords in (7) are difficult to plot on the Tonnetz that we have been using. Transposing them to the key of C-major, by adding +3 hours on the chromatic clock, we get the following chords: T T T T C −→ e −→ A5 −→ F −→ C which are all connected by Tonnetz transformations. See the left of Figure 5. They form a cycle, beginning and ending with the tonic chord C. The power chord A5 consists of an A-note, and a fifth above it, an E-note. Consequently, the arrow in the Tonnetz diagram points towards the edge between the hexagons A and E. The cycle shown in this Tonnetz diagram appears to cycle around the minor chord a. In fact, at the end of measure 7 as well as other measures, a C note does make a brief appearance within the lyrics while the A5 chord is arpeggiated in the piano accompaniment. Example 7. Leonard Cohen, Hallelujah chord progression. In his song, Hallelujah, Leonard Co- hen uses the following chord progression repeatedly

T T T T T T F −→ a −→ F −→ C −→ G −→ C −→ G

in the key of C-major. See the right of Figure 5. The progression is interesting for a number of reasons: (1) Each of the progressions is a Tonnetz transformation; (2) aside from the use of a, the motion goes from sub-dominant (F) to tonic (C), then back and forth between tonic and dominant (G). The ending on the dominant G reflects the fact that, within the song as a whole, this chord sequence is always used to lead into the tonic chord C. The chord changes within this Hallelujah chord progression, and its placement in the song, is perfectly consistent with the fundamentals of music theory that Cohen alludes to in the song’s lyrics. (3) The use of the minor chord, a, is a brief nod to the relative minor. It is used at various poignant moments in the lyrics, a typical use of a minor chord, especially in popular music.

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Figure 5: Left: Cyclic progression of chords in Someone Like You (Adele Adkins and Daniel Wilson), transposed to C-major. Starting, and ending chord, is C. The horizontal edge between a and A is the “power chord” A5. Right: Chord progression used in Hallelujah (Leonard Cohen). Starting T progression is F −→ a. Exercises Some additional Tonnetz patterns will be discussed in exercises for this section, including • Beethoven, Sonata for Violin and Piano, Op. 25, mvt 2, mm 38–54. (Cohn p.27). • Liszt, Grande fantasie symphoniqe, mm 379–439. (Cohn p. 100). • Chopin, Ballade, O. 23, mm 68–167. (Cohn p. 98). • Brahms, Ein deutsche requiem, mvt 2, 261–271. (Cohn p.97). • Liszt, Il Penseroso, mm 1–9. (Cohn p. 73). • Wagner, Tanhelm progression from his Ring cycle. (Cohn p. 23). • Stravinsky, Rite of Spring violin chords. (Bernstein, p. 342). and we will also include some further examples from popular music and jazz. Some additional exercises will be on some further details mentioned briefly in the section’s text, e.g., • Suppose you want to create a set of 9 distinct pitch classes from the 12 possible pitch classes on the chromatic clock. Show that the Euclidean algorithm from Section 6.5 yields the pitch classes shown on the right of Figure 1. • Supply the rest of the chord changes that map pitches from the augmented triad {E♭, G, B} to the chords shown in (1). • Find the dissonance score for the hexatonic scale shown in Figure 2. [Hint: apply the method of Sec- tion 6.6 to (0, 3, 4, 7, 8, 11, 12).]

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