Outline of New Section 7.5: onnetz Patterns in Music his section will describe several examples of interesting patterns on the onnetz that occur in a variety of musical compositions. hese patterns show how the onnetz provides a geometric logic for chord progressions, sometimes even when there is no underlying key for the music. Examples from symphonic music here are a number of examples from the book by Richard Cohn, Audacious Euphony: Chromaticism and the riad s Second Nature, (Oxford, 2012). We will refer to this book as an important reference in the section, and as an invitation to readers to explore recent ground breaking scholarship in applications of onnetz 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: A e c (1) As we have indicated, these are all onnetz transformations. On the left of Figure 1, we diagram this pattern of chord progressions. All of these onnetz 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. he 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. hese 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, or0as hours on the chromatic clock. For example, thea chord is obtained by while thegchord is obtained by G +1, B +1, E +0 (3) E 1 D, G +0, B +0 (4) Furthermore, we have marked on the right of Figure 1 all of the pitch classes for the chords shown in (1). his 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: A e c (5) as shown on the left of Figure 2. hese onnetz transformations are all of type,p orl, as they lie on the edges of the hexagons for these pitch classes: A B C E E G A hese 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.
Figure 1: Left: onnetz cycle in Mozart, Symphony in E-flat major, K. 543, finale mm 109 26. Cycle begins with A, ends with E. Right: Pitch class hexagons activated in this Mozart passage. he radial lines point towards the three pitch class hexagons through which all onnetz 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 270 276. Cycle begins with A, ends with c. Right: Pitch class hexagons activated in this Brahms passage. hey 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: G g d D d F f (g ) All of these onnetz transformations either lie on the edges, or pass through, the two pitch class hexagonsd and thena. See Figure 3. hese 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: here are two instruments playing: a cornet and a trombone. he 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 1 L. Bernstein, he Unanswered Question: Six alks at Harvard, Harvard University Press, 1976, p. 343. hese lectures are also available on DVD; the beginning of Lecture 6 contains the passage quoted here.
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 onnetz. hey can be expressed as onnetz transformations in this way: F f D. (6) he second sequence of mappings,d, includes the chorde because that chord is being played by the cornet while the trombone is arpeggiating the chord G. It is also worth noting that the chordsf andd, which are also outlined simultaneously by cornet and trombone, are connected by a onnetz transformation as well. Although the chord progressions in (6) are not typical ones within any fixed key, our onnetz description makes it clear that they do obey a clear tonal logic a tonal logic that is captured concisely and geometrically by the onnetz 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 begins with G g, ends with A g. Chordal motion travels around two hexagons, D then A. Right: onnetz motion in Stravinsky s L Histoire du Soldat. Examples from popular music We will also discuss onnetz diagrams for three popular music compositions. Example 5. Alicia Keys, opening of If I Ain t Got You. his example is interesting in that a spectrogram analysis shows that, as the extended bass notes die out, a new chord is sounding. See the left of Figure 4, where the onnetz transformationc e is explained. Using similar reasoning for several subsequent measures, the opening of this song trace a series of onnetz transformations: C e b D as shown on the right of Figure 4. hese onnetz transformations trace a path through each of the six major and minor chords for the song s key of G-major.
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. riplet 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. ViewingC M7 as an embellishment ofc, the performance highlights the onnetz transformation C e. Right: Moves on onnetz for these two measures, starting withc e. Example 6. Adele Adkins and Daniel Wilson, Someone Like You. he chord progressions of this song also have an interesting symmetrical pattern on the onnetz. he chords in the song s original key ofa-major are A c F 5 D A (7) he chordf 5 is known as a power chord in popular music. It consists of just anf -note and, a fifth above it, a C -note. he chords in (7) are difficult to plot on the onnetz that we have been using. ransposing them to the key of C-major, by adding +3 hours on the chromatic clock, we get the following chords: C e 5 F which are all connected by onnetz transformations. See the left of Figure 5. hey form a cycle, beginning and ending with the tonic chordc. he power chorda 5 consists of an A-note, and a fifth above it, an E-note. Consequently, the arrow in the onnetz diagram points towards the edge between the hexagons A and E. he cycle shown in this onnetz 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 thea 5 chord is arpeggiated in the piano accompaniment. Example 7. Leonard Cohen, Hallelujah chord progression. In his song, Hallelujah, Leonard Cohen uses the following chord progression repeatedly F F in the key of C-major. See the right of Figure 5. he progression is interesting for a number of reasons: (1) Each of the progressions is a onnetz 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). he 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. he 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) he 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.
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. he horizontal edge between a and A is the power chord A 5. Right: Chord progression used in Hallelujah (Leonard Cohen). Starting progression is F. Exercises Some additional onnetz 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, anhelm 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 Section 6.6 to(0,3,4,7,8,11,12).]