Abstract
In traditional literature, transformational harmony is
considered non-functional (Cohn 1996). This paper introduces a new framework to
view transformational progressions with tensional areas, called levels, that
parallel the system of tension and release seen in functional harmony. These
levels section off phrases into areas of varying levels of tension and lack of
tension. Neo-Riemannian transformations, simplified into common tone relations,
between triads dictate how and what level the phrase on, and what level the
phrase is moving towards. Further application of this theory divides arias into
subsections based on tension and release. Examples from “News has a kind of mystery”
from John Adams’s Nixon in China are used to showcase this novel theory.
Introductory
Information
Introduction to
transformational harmony
Transformational harmony can
be viewed through a Neo-Riemannian lens. An influential book titled “John
Adams’s Nixon in China: Music Analysis, Historical and Political Perspectives”
(Johnson 2011) forms the foundation of the literature for the analysis of this
opera. Johnson quickly describes the necessary information to understand his
analysis for Nixon in China. I have paraphrased it here. There are three basic
Neo-Riemannian transformations. All three transformations take a triad and
changes a note in the triad to alter the quality, root note, or both. The
Parallel transformation, denoted by the symbol ‘P’, shifts the 3rd in the triad
by a semitone, while keeping the fifth constant. Basically, a major chord turns
into a minor chord and vice versa. The Relative transformation keeps the major
third in the triad constant, while moving the other note by a step. It is
easier to think of it as its name: going to the relative major or minor. For
example, from C major to A minor is a relative or ‘R’ transformation. The final
basic transformation is the Leading-tone transformation. Hold the minor 3rd in
the interval constant while moving the other note by a semitone. Since the note
moving will always be to the, or away from the leading tone of the major triad,
the name makes sense. A C major triad transforming to an E minor one would be
an example of ‘L’ transformation. These three are basic transformations because
the triad keep two common tones, while only one note changes. This will be
important later in the analysis. Johnson also speaks to the ‘SLIDE’
transformation, which can be obtained with an ‘LPR’ combined transformation,
applying the basic ones in that order, from left to right (Lewin 1987). This
will be used to turn a major chord into a minor chord a semitone higher. This
is a combined transformation since the basic transformations are combined to
form this transformation. Starting with any triad, you can get to any other
triad just by applying basic transformations to the chord. Since these chords
only relate to chord before it, and not to a key centre, transformational
harmony is non-functional.
The novel theory that this
paper describes does not need the differentiation between the P, R, and L
transformations. Instead, the term basic transformation will be used when two
triads share two common tones. Furthermore, all triads that share one common
tone will be grouped together into the term compound transformations (Cohn
1997). Lastly, SLIDE transformations are different than other compound
transformations because of the aurally distinct half-step motion in the outer
voices (Lewin 1987). The outer voices move up or down by half-steps, causing
the audience to hear parallel fifths. These three types of transformations,
basic, compound, and SLIDE, will be used in the theory instead of P, L, and R.
Introduction to Nixon in
China
Nixon in China shapes American
opera in the twenty first century. A drastic shift from the fantastical plots
of operas in the past, Nixon in China is about what one would expect:
Richard Nixon visiting communist China in 1972. A historic visit, yes, but rather
unorthodox as operas go. The plot of the opera is the real-life event, what the
leaders did on each day of the visit, down to the ballet the American envoy
watched. The dialog between Mao and Nixon is even accurate, or at least
accurately paraphrased, in Act I, Scene 2 (Lord 2006). The opera is the
culmination of nearly five years of work between Peter Sellars, Alice Goodman,
and of course, John Adams. Sellars came to Adams with the idea to write this
opera in 1982, and the two of them agreed to start work in 1985, but only after
Adams has gotten over his fear of writing opera, since he has not composed for
the voice in his career yet (Adams 2008, 134). Alice Goodman, a classmate of
Sellars, wrote the libretto. Goodman, who is well-versed in classical poetry,
wrote a libretto that was, as Adams puts it, “beyond what was usual in an
opera” (Adams 2008, 136). We can discuss the libretto at length, for it is a
very well written work that is able to stand by itself, and combined with Adams’s
writing, makes Nixon in China stand out from the crowd of minimalistic
operas, but that is not the focus of this paper.[1] Nixon
in China premiered at the Houston Grand Opera in 1987 with great
excitement. It received mixed reviews at the premiere, with critics saying that
it will disappear, thanks to the time-sensitive content, but it seems that time
has been the opera’s greatest ally. Nixon in China enjoyed a Met
production in 2011, and subsequently has been the topic of many academic
papers, with entire books written on the opera. Nixon in China is now accepted
as part of the operatic canon.
Adams’s harmonic language is
atonal, but Nixon in China is not a serial opera. This opera sits on a
tightrope when it comes to definitions. Adams’s language is one that uses
chords that only relate to each other through transformations, and not a tonal
centre. His usage of this style is quite rare time period. Late romantic
composers often used transformational harmonic techniques when they expanded
away from tonal sections, but never for an entire work. He shows us what tonal
harmony could have been if it wasn’t for the mainstream adaptation of atonal
ways of writing. His minimalistic language greatly contrasts other new works of
this era too; there are no lush harmonies or sweeping lyric lines to be found
in this opera, and he doesn’t write in a film music tradition. He does not try
to extend the romantic, rather, his development of tonal harmonies takes a
completely different path, which is of course, minimalism. One can say that
Adams minimalizes unnecessary harmonies that serve to only isolate the listener
from the melody. Therefore, we can simplify his style of musical composition as
melody focused, with the harmony serving to only support the melody and shape
the scene. With this framework in mind, we can explore how Adams writes his
harmonies to fit his vocal lines.
_________
We
have now discussed all the foundational information needed to understand the
topic of this paper. With a non-functional chord progression that is the
foundation of progressions in Nixon in China, why does one still hear
tension and release? Through the analysis of three types of transformations
(basic, compound, and SLIDE), I will demonstrate that transformational phrases
still have quasi-functional tension and release. These sections of heightened
tension, called “levels of tension” in this analysis, can be viewed further as
formal dividers for entire arias in Nixon in China.
Theory and Analysis
Tension in transformational
phrases
Figure 1: “Cycle chart” Tensional levels and types of transformations that move between them
The
roman numerals represent the levels of tension. Level I has no tension, while
level II and III have more tension than the previous level. Therefore, phrases
always move from level I to level II, sometimes to level III, and always back
to level I again. This cycle forms every phrase, and multiple cycles form
entire sections and in larger form, the whole piece. This diagram also has
three types of transformations that represent the movement between levels. When
chords undergo these types of transformations, the level of tension either
stays the same, increases, or is released, in the case of level III to level I.
A basic transformation causes the level of tension to stay constant. A compound
transformation causes the level to increase by one, for example, a series of
chords that are in level I to move to level II. And finally, the SLIDE
transformation releases tension from either level II or III to level I.
In
the next section, a phrase will be analysed from Nixon in China that has
every type of transformation and moves through all three levels of tension.
This will form the example that supports this novel theory.
Theorem example: “News has a
kind of mystery” from Act I, Scene 1 of Nixon in China (m. 347-417)
Figure 2: Chord progression of "News has a kind of mystery" from Act I, Scene 1 (m. 347-417)
This is a sequence of triads that
contains transformational relationships between each chord. The labels at the top
are just for easy reference to the sections below on each individual type of
transformation. The levels of tension at the bottom show the movement between
the levels of tension, from level I to level III, with a resolution back to
level I. From first glance, one can tell that each subsequent chord always
shares one note with the chord before it. During no change of level, they share
two. In addition, basic transformations always switch modalities, from major to
minor and vice versa. In a compound transformation, modality is kept constant,
major staying major or minor staying minor. Therefore, there is always a modal
contrast between an increase in level or a constant progression. Only a SLIDE
transformation has a change in modality and a change in level. This is useful
in deciphering a previously unseen example, to try to figure out the level
changes within it.
Example
1 shows no change in tension. The Ab major chord goes to an F minor chord and
vice versa for the first two transformations. These two chords share two common
tones, or using Neo-Riemannian terminology, an R transformation. Then the Ab
major to C minor, an L transformation, occurs but new chord does not mean there
is a change in tensional level, for they still share two common tones.
Example
2 shows an increase in tension. From Ab major to C major, which by the way, is
modally similar, signals an increase in tension due to the two chords sharing
only one common tone. The next change, from C major to Ab major, is a return to
level I, not an increase in tensional level since we are just undoing the
change from before.[2] The next tensional increase is from C major to E major, sharing once again,
just one common tone, E. Now at the third level of tension, tension must be
released.
Example
3 shows the release of tension. From E major to F minor there is a SLIDE
transformation. The middle note of E major, G#, is kept constant while the
outer voices move up by half-steps. This gives F minor, and a release in
tension.
Application to determine
form from “News has a kind of mystery” from Act I, Scene 1 of Nixon in China
(m. 347 – 509)
A
further application of this theory is to determine small scale form. This can
be shown in the rest of the aria.
Figure 3: Chord progression of "News has a kind of mystery" from Act I, Scene 1 (m. 347 - 509) detailing subdivisions of form
This
diagram shows the progression of triads in the full first section of the aria
“News has a kind of mystery”. With the level theory, this section can be
divided into a further four sections by where the SLIDE transformations are.
Each new line is an increase in level, also labeled on the right. Therefore,
the SLIDE transformations correspond to a cadential gesture, signalling the end
of sections and phrases. The text of this aria also corresponds to these cadential
gestures, with sentences ending at the SLIDE transformations without fail.
This shows the cycles of
tension and release in Nixon in China, and one can see that these cycles
almost mimic what functional harmony would have achieved in terms of tension
and release. This is the height of this novel theory at the moment. In the
conclusion, I will discuss the further applications of this theory, and what
work I think can be done to increase the relevance of the tensional levels in
transformational harmony.
Conclusion and Further
Questions
A
large gap in this theory is not including seventh chords in the cycle chart, in
figure 1. Sevenths chords make up large part of the harmonic language in the
opera, and of course, in pieces that use transformational harmony as a whole. There
is literature that connects triads to sevenths chords, but I have not found a
connection to fit the sevenths chords into the cycle chart. I think the work of
Hook (2007) and Childs (1998) might provide the resources needed to further the
integration of sevenths chords into the new theory. Hook provides a new
transformation, the modified L, or L prime. The basic L transformation, which
lowers the root into its leading note, is the foundation of this
transformation. Hook explains that “…for any major or minor triad X, L′(X) is
by definition the unique major-minor or half-diminished seventh chord that
contains all the notes of L(X). Thus L′ maps a C major triad to a C-sharp
half-diminished seventh chord (which contains the notes of the E minor triad, L
of C major), and maps a C minor triad to an A-flat major-minor seventh chord
(Hook 2007, 2).” To move between seventh chords, we must look at another paper,
this time by Childs (1998, 185-189). He writes about two sets of transformations,
the S family, and the C family. He frames his work in the idea of smooth voice
leading, and these transformations are connections between sevenths chords
(Forte 4-27) with the least amount of note movement. He explains:
This system consists of two
distinct families of operations. The larger family is that of the S transforms,
which involve holding two pitches constant while the other two move by half
step in similar motion. Like the neo-Riemannian operations, each of these six
transformations results in a change of mode and is involutional in nature. The
individual transformations are labeled with a subscript that indicates the
interval class between the two pitches being held constant and a parenthetical
subscript that indicates the interval class of the two pitches that move. The
second family is that of the C transforms, which involve contrary motion for
the non-fixed pitches. The subscripts for the three members of this family
follow the same labeling convention. Since the C transforms maintain chord
quality, only C6(5) is an involution. C3(2) and C3(4) are each other's inverse.
(Childs 2007, 185)
In addition to sevenths
chords, Adams uses bichords extensively in this opera as well. The dissonance
and multiple voices that bichords have poses a problem to the current
framework. I have no research on this subject and I suppose to fit this into
the framework of the cycle chart, there needs to be new research on how the
voices move and relate to each other; to connect everything on a common tone
basis.
Further application to other
pieces that use transformational harmony is needed to expand this theory.
In “News
has a kind of mystery” in Nixon in China, we have seen how small form can be
derived from the presence of SLIDE transformations, and how those SLIDE
transformations are apart of a cycle of rising and falling tension. Through the
observation of basic and compound transformations, we have seen that phrases go
through sections of rising tension, and then falling tensions through SLIDE
transformations. With this framework, we have added tension to transformational
harmony, harmony that was traditionally viewed as non-functional, and
therefore, without the usual tension and release of the tonic, subdominant, and
dominant. We have created a quasi-functional system where transformational
harmony can also go through the same tension and release that functional phrases
go through.
