Simulacrum and the Abstraction of Tonality (Part 1)

What has become rather famous now in the faux-academic sphere is Jean Baudrillard's 1981 work Simulacra and Simulation, more specifically his idea of the phases of the image, which he describes as phases of the abstraction of a profound reality. 

1. It is the reflection of a profound reality;
2. It masks and denatures a profound reality;
3. It masks the absence of a profound reality;
4. It has no relation to any reality whatsoever: it is its own pure simulacrum. (Baudrillard 1981, 6)

Starting with the profound reality, Baudrillard presents four subsequent stages of the image, distilled and stemming from the preceding stage. Most examples of Baudrillard's theory uses cultural objects to represent these stages. A well known meme example is this one on pumpkins:

In comparison with the four stages: the pumpkin 1) is a representation of itself. Then the pumpkin 2) is turned into pumpkin pie, which is still made from pumpkins, albeit not looking like a pumpkin. Then 3) into a pumpkin spiced latte, which merely engages the taste of pumpkins, without the presence (absence) of a pumpkin. Then 4) with the coffee creamer, the pumpkin has absolutely nothing to do with the creamer itself; it is an abstraction so deep of the pumpkin that it becomes a pure simulacrum, a flavour of coffee creamer.

In general, the concrete image can be transformed into an shadow of an image, an abstraction of the concrete image that gives the image-creator more to play with. If we move towards music and the creation of music, can this same idea be applied to strip down difficult to understand ideas of music into it's fundamental parts, and likewise, build from fundamental parts the difficult ideas of music?

Peter Kivy (1984) writes in Sound and Semblance that, "The artist [...] does not give us a copy or counterfeit [...] rather he re-presents it in his own medium, giving it coherence, designing a pattern." (17) 

I choose to accept this general idea and re-present it (see what I did there) into the general idea of tonality. As the pumpkin is transformed into the pumpkin pie creamer, how are the axioms of music, the true fundamental, real, building blocks of music transformed into the idea of tonality? And then, how do all tonality-based works stem from this transformation?

What I am most interested in with this exploration is not the historical or cognative aspects of tonality; these ideas have been researched by people far smarter and with far greater resources than I and have recived little answer on the function of tonality. It is the very process of the transformation that I am interested in. I am less interested in the idea that Baudrillard had, but instead in the process that one idea/object might be related to other objects. Simulacrums are simulacrums as they contain some essence of the previous iteration closer to reality. 

The current popularity of LLMs and data sets where objects are related by "closeness" and "relation" I think can be easily applied to music. From the abstraction of complex concepts such as tonality, are we able to find how the individual pieces of tonal function, whether that might be pitch, voice-leading, harmony, form, rhythm, or anything like that, would be more or less related to other objects of the same type, thus finding the key pieces that make tonality function. These pieces, objects, are in the realm of an observable and measureable reality. Concepts such as tonality cannot be measured. 

Bibliography

Baudrillard, Jean. 1994. Simulacra and Simulation. Ann Arbor: University of Michigan Press.

Kivy, Peter. 1984. Sound and Semblance: Reflections on Musical Representation. Cornell University Press.

Some thoughts about abstraction

Abstraction is the fundamental method of any theory. To make sense of something, one must abstract it. Thus, I pose this following question about the nature of abstraction:

Can we describe abstraction as the manipulation of the undefinable into definable, and thus make the unorganized into the organized?

Perhaps we can think of it as categorizing the various pieces of food that one would find in stew. Altogether, it is stew. A cohesive entity that is described and functions (when eaten) as one object. However, upon analysis and thus the abstraction of the stew, one could say that it is comprised of several objects. There are pieces of potato, carrot, peas, meat, the liquid, even the dissolved salt and oil can be contributed as pieces of the whole stew. Thus, there are two ways to enjoy the stew. One can think of and interact with the stew as a whole object, and when ordering it at a restaurant, he would say, “Could I get the stew?”, and when consuming it, eat the individual parts of the stew as a whole. The other way one can think of and interact with the stew is as the constitute pieces. He could say, when ordering it at a restaurant, “Could I get pieces of potato, carrot, peas, meat, meat stock, spices, salt, all simmered together in a pot and served hot?” When presented with the concoction of ingredients (parts of the stew), he could then eat it piece by piece, thinking of the individual parts separately, while only vaguely considering the whole stew.

Leaving the (not very good) analogy behind, let’s just look at the question of abstraction.

The stew is easily definable. We can say that there are pieces to the stew and then look at the stew either as the parts or the whole. However, music is not easily definable. Music theorists have, throughout the centuries, given names to different musical objects that they are hearing. This is the definition of abstraction. Notes, for example, as a prime example of this sort of abstraction. What even is a note? A sound? Or perhaps a notion that sound should exist? Is there time involved in sound? What kind of sound should the unadulterated note represent? We have a vague idea of what a note is, but from situation to situation, the idea of the note changes. Many other musical abstractions function in this way as well. Time in music, meter in music, harmony, form, the idea of voices; I can go on and on. Music theorists present that in any of these cases, music, in its essence, can be abstracted into these basic forms. It might not matter that these forms are different in different situations, but at least in this first level of abstraction, ideas can be named. This is the manipulation of the undefinable into the definable. 

Let’s then, give a name to these basic forms of music that all other music theory is supplanted upon. Perhaps we can think about these forms of music as the axioms (to steal something from science) of music theory. They are not mathematical axioms of course, since these are neither perfect laws of music, nor even defined into something that might be able to be proved by perfect, basic laws. They are, in their sense, just basic labels for music. We cannot say that this is music theory yet, it is merely abstraction, labeling pieces of the stew for further analysis.

The second step of abstraction is what people know as music theory. I take the idea of notes, or the idea of harmony, define it as something that would make sense for my application of the theory that I want to do, and then, by combining the basic labels of abstraction, create a theory as the second step. 

One example is tonality. Tonality exists in the second step of abstraction. We can hear that, yes, tonality exists and if I asked a group of people (who listen to music that has tonality) they would probably say, “Oh yes, when that V chord goes to the I chord, I feel a sense of release!” or perhaps “When I hear a major scale end on the 7th scale degree, that makes me feel uneasy!” But these are all ideas that exist on the second step of abstraction. 

How can this be music theory if we are merely asking people to believe that certain phenomena exist? How can anything be provable if we are merely convincing people to believe in a theory, the workings of which are based on faith? 

The issue with the first step of abstraction is that we were wrong in labeling the stew! The objects in the stew don’t exist past our perception; they are not grounded in fact, but instead the opinions of other scholars.

Then you would probably say, well let’s just take tonality apart, let’s find the first step, the basic abstraction, the axioms that exist in tonality. The axioms that can be rooted in fact. What could those be? Is it the fact that voice-leading creates tonality, through the principles of counterpoint? Or could that just be an endogenous fallacy where instead, tonality creates voice-leading? Is the fundamental building block of tonality, harmony? Or is it a collection of non-symmetric pitches when mapped on a 12TET cycle? 

How can tonality be considered provable if the basic abstraction is not provable?

I believe that this is a fundamental issue with music theory. What are the pieces in the stew (basic abstraction); and what even constitutes the stew (second step in abstraction)? It might be in the sciences that, coming back to the original question, we can describe abstraction as the manipulation of the undefinable into definable, and thus make the unorganized into the organized. However, in music theory, the undefinable remains the undefinable, since labeling the parts is a fundamental issue. And a further however, we assume that we can label, and with our hubris in believing that our labels are correct, and the undefinable has been defined, we make the unorganized into the organized. But we are merely organizing opinions about music into higher levels of opinions about music, without ever looking down and saying:

“What are we standing on?”

Music Theory Education and Public Perception: Lockhart's Lament

I just read an essay written by Paul Lockhart titled "The Mathematician's Lament", and after doing a little bit of research, found to be quite the famous and well-read text in the math community. Published online in 2002, it even got turned into a book a couple years later.

Lockhart's Lament, as most mathematicians lovingly call this essay, details a central philosophical question about math pedagogy: is math art or a set of instructions to follow? Lockhart argues that schools teach too much in the latter direction, eliminating the reason math exists in the first place: as a way to explain our curiosities about the world, rather than training students to become experts at assembling IKEA furniture, following a step-by-step guide to solve 'problems'. Lockhart further argues that math is being viewed in our culture as something 'rational thinkers' do, as opposed to what 'poetic dreamers' do. He pushes that people (and pedagogues) just don't understand what mathematicians do and therefore categorizes mathematics incorrectly into the 'rational thinkers' genre, when it really should be treated as an art.

Why do I bring his essay up? I think there are certain parallels to draw between his lament and the lament of many music theory pedagogues. Funny enough, there's a wonderful anecdote that Lockhart makes in the introduction of his paper, that compares mathematics to music! I will quote it below; I think it makes quite the compelling argument if we tweak it a little bit to encompass music theory more specifically. Read this with more of a sarcastic tone:

"Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."

Quite humorous, but is this not the way music theory is taught in schools? Do we not just learn skills that might apply to future forms of analysis and 'theorizing', but never do those activities from the beginning? As a personal anecdote, I have found that the public perception of music theory, and even the perception of music theory from other musicians is one of this 'skills-based, IKEA furniture' style of thinking. How do I spell a D augmented triad? What 'music-theory' do I need to learn to improve my guitar solos? What scale do I play over a Bb major seven chord? 

As much as these are valid questions, I would hardly categorize them as music theory questions. They are composition questions, performance questions, and more or less, opinion. Should music theory be taught for students to understand actual music theory that theorists are doing? 

Then again, I am not saying that these skills are useless! Of course, as mathematics needs students to understand basic arithmetic, so does music theory need students to understand how to spell chords and conduct roman numeral analysis. But it is the combination of these basic skills and ideas that should be introduced to students at a young age.

Ideas such as 'tonality', which I have found that students, even after studying roman numeral analysis, have no idea why they are providing these numerals to chords. That their tutors that robbed them of the joy and understanding that theorists have thought and defined certain chords having certain functions within a key, which is a key idea to understand all of the common practice!

Along with a knowledge of history, students possess skills, ideas, and facts, about this whole field of music theory! Why, then, do we still teach music theory like cooking class? Why do we take the joy of discovery, the joy of the art of music theory, out of the pedagogy? Teach ideas along with skills in theory class! So many ideas about music that students never get exposed to, ideas about music so prevalent, it's almost unethical for students not to learn about it! Put the music theory back in music theory class! 

Distance Measurements in OP-Space: Mapping tensional profiles

Comparing the discussion on geometric distances in Julian Hook’s new book Exploring Musical Spaces (2023) and the Dmitri Tymoczko classic A Geometry of Music (2011), we can see that there’s been quite the development on geometric distance research in the twelve-ish years between the two monographs. The main point of uncertainty still lies, however, in what ways we can measure the distance between two nodes in any geometric representation of musical space. I will try to explore the options in this post.

Let us look at the ways to measure the distance between sets of pitches in OP-space, since this is the basis of my current research. There should be no difference between using OP-space or any combination of OPTIC spaces. I have chosen to use OP-space since it most easily models individual chords without quality equivalence or inversion equivalence. Therefore, it is possible to model aural perceptions of tonal and post-tonal words closely, since our understanding of harmony allows octave equivalence and honestly, permutation equivalence exists to make my life easier, without thinking about choral inversions. It closely models the Lewinian transformational system that came before it, and this new model will have some of the same limitation as transformational models will too. A new post will come upon the completion of this paper.

Three main ideas are touched upon in the Hook (2023) and two ideas (overlapping with the Hook) in the Tymoczko (2011), to measure distance based on either the maximum distance in one voice, the Euclidian distance between the two nodes, or the ‘taxicab’ distance, the distance that follows the voice leading pathway between the two nodes.

Distance measurement ideas (reproduced from Hook 2023):

And the accompanying figure (Figure 12.4.1):

So, what works the best? Tymoczko (2011), in the appendix, talks about how not one measurement works better than another measurement, and it just depends on what the theorist is measuring for. Therefore, let us use his definition and apply it to this paper. For this paper, I am measuring distance to determine tensional profiles, so let us try all three ways of measuring distance and determine which one best suits our ears in determining the tensional relationships between chords. For our test, let us move away from a two-voice graph, as we have in the figure above, and move to a three-voice graph, as shown below (Figure 11.3.4 in the Hook 2023), to hear the relationships between triads, and not dyads:

Since this graph depicts all three voice triad combinations, related by semitonal voice leading, any combination of three notes fits onto this graph neatly, and most important to our test, it can be measured to determine the distance between any two nodes. It is important to note that this three-dimensional graph is no different than the two-dimensional graph above in the reproduced Figure 12.4.1, the node (alpha and beta) containing the three voices of the chord (Xn, Yn, and now with the addition of the third voice, Zn) are all the exact same in both versions and function the same as well. This graph just contains every single three voice combination, so we can calculate chords and collections aside from major and minor chords. This will be useful in the future, but just not for our upcoming example. When we perform the calculations based on the three methods above, we do the same in the two voiced version and the three voiced version.

 

Analysis of a real-life example

Let us take an example from John Adams’ Nixon in China, Act II, Scene 1, from Pat’s Aria “This is prophetic”. The first six chords are as follows:

Eb minor à B7 dominant à Eb minor à E major à E minor à C major

Let us further skip over the four-note chord, B7 dominant for now, since cross-types are a whole other topic, better discussed at length elsewhere. We shall simplify the B7 into a B major chord, eliminating the seventh.

Then we can calculate the distances between all the chords in this progression based on our three methods shown above. First let’s just number our chord relationships for ease:

Eb minor [1] B (major) [2] Eb minor [3] E major [4] E minor [5] C major

And to keep our OP-space, we shall accept octave equivalence and permutations of the chord (any order of voices in our set), so we will seek the smoothest voice leading, as according to the three-voice graph above. And thus:

distMAX:

[1] 1

[2] 1

[3] 2

[4] 1

[5] 1

 

distEuc:

[1] 1

[2] 1

[3] sqrt (6)

[4] 1

[5] 1

 

distVL:

[1] 1

[2] 1

[3] 4

[4] 1

[5] 1

An interesting pattern is formed! The chords that share two common tones, and thus the ones that can be modelled with basic Neo-Riemannian transformations, relation [1, 2, 4, 5], agree in all three systems. They are separated by one semitone. Since these relationships [1, 2, 4, 5] maps onto transformations [L, L, P, L] is it obvious that they share two common tones and therefore the distance between them is the one tone that moves by semitone. Relation three [3], however, is what we’re interested in. In the voice leading distance, it measures Eb minor to E major as 4 semitones of distance, in the Euclidian, square root of 6 (~2.45) semitones, and in the ‘maximum’ calculation, 2 semitones. If we put it in Neo-Riemannian terms, it is a parallel transformation, a SLIDE transformation, and another parallel transformation, P-SLIDE-P.

 

So, which distance calculation is the best (for this analysis)?

What shall we use? Hook (2023) says that most theorists choose to use the voice leading calculation in their work, since it most closely models what we hear. I agree with that. It closely follows the path that the voices need to travel to reach the next chord. It also models Neo-Riemannian transformations very well, since each movement in the voice leading space corresponds to a basic Neo-Riemannian transformation, so it is just like chaining basic transformations together.

If we go the Euclidian path, it also makes some sense. Since it is a straight line in our perfectly organized space, the distance is the shortest path to get from one chord to the next.

The maximum distance does not seem to work well with what we are looking at. Since it is a simplification of the voice leading relationships by only looking at the voice that moves the furthest, it neglects to model situations where the other two (or more) voices are moving lesser amounts, but not holding common tones. Therefore, if we have a chord that holds common tones and a chord that does not, the maximum distance calculation will not make a difference.

The solution is to combine some aspects of the voice leading calculation, for it seems to model the way that the notes travel naturally in our ears, while using the Euclidian path to model distances in the full phrase. In our example above then, the chords that ‘sound’ next to each other, like Eb minor to B (major), will be analyzed using the voice leading distance, since they are truly voice leading from one to the next. But when looking at the distance travelled from Eb minor (first chord) to C major (last chord), since these two chords are not sounding next to each other, we will use the Euclidian path, as it is the straight path to measure the overall distance travelled in a phrase (or whole section). We can then get the best of both worlds without sacrificing musical logic!

 

Issues

Certain issues come up with this analysis. First, there is a difference between OP- and O-space. The fact that the chords are permutable, and I can find the most parsimonious path between the chords means that we are not really measuring the voice leading in this section, but rather the perfect world where the voices go very smoothly. In the above example, the chords are actually rather choppy. This will be expanded upon in the future, where I fine tune this thesis to fit the music a bit better.

The second issue was touched upon very briefly before. How do we do the cross type? How do we move from three notes to four notes? We will, as I said before, discuss this in a later post.

The third issue comes when we get the distance numbers. When graphed, these relationships are just numbers that rise and fall according to the relationship between the chords. But if we wanted to see how the whole phrase functions, in what way can we manipulate this data to give us the ‘mode’ of the rising and falling distance? Should we also calculate the distance between the first chord and every other chord? Should we look at how close the phrase is to tonal ideas and map it that way? Many avenues are still to be explored in the next post.

Tension And Transformational Harmony: “News Has A Kind Of Mystery” From John Adams's Nixon In China (1987) (Part 3)

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.

Harmonic Analysis of Three Arias in John Adams’s Nixon in China (Part 2)

            As Timothy Johnson demonstrates in his book, Adams avoids the traditional functional harmony in favour of chord progressions than can be approached with a Neo-Riemannian harmonic language. He uses chords, both triads and sevenths chords, to represent the emotions of the characters and the evolving attitude of the characters towards each other. To avoid discussing too much of the plot, I will focus solely on the music, and not the text. Adams uses non-functional chords in such progressions that it mimics functional harmony. In certain sections some chords can be called the functional tonic and dominant, while in others, chords serve as different levels of “terraces”, either stepping up tension or releasing it. As with all theories, there are a few holes in Johnson’s analysis of Nixon in China, some of which will be addressed to the best of my ability. I do not believe that Adams composed with this “system” in mind, mostly because it hasn’t been invented yet when he wrote the opera; the theory attempts to fit music, just as grammar attempts to fit language. Some things just do not fit neatly within the Neo-Riemannian context.

Click here to view the Score and Flowchart

Notes for “News has a kind of mystery”

            The first aria, Nixon’s aria “News has a kind of mystery” starts on an Ab major triad. For ten measures (mm. 374 – 386), this chord remains before changing to an F minor chord, a relative transformation. Since this is a basic transformation, I consider is a “non-tension” movement. In the Johnson book, he talks about how the Ab and the Fm are from separate hexatonic systems, but I think that the systems are good for categorizing the different transformations, just not so good at analyzing the chords that are a part of the systems. The “set up” not only makes little sense contextually, but also, more importantly, make little sense aurally. The sequence sounds, in a functional way, like I – vi (and to iii when it moves to Cm later). These two chords are all basic transformations from Ab major, and therefore should considered mere colourations of the “tonic” or what I’m going to call “home” triad.

            In mm. 409, the Ab triad goes through a leading tone and then a parallel transformation, a complex transformation, into C major. This, followed by a crescendo, sounds to me like a real chord change, into a new tonic, effectively making the Ab major triad the tension chord resolving into the C major. The C major then goes through another LP transformation into E major, and C becomes the new tonic before finally going to the new home centre, F minor and Db major. From E major to F minor, the chord goes through a SLIDE transformation, keeping only one note constant. This always sounds like a modulation to me. The system of keeping a home centre around two or three chords continues in this aria, and modulations or tension building occurs on compound transformations.

            Section F (starting mm. 542) has a curious section that uses a new technique to achieve tension. Bichord dissonance set in contrasting registers. I can only describe this section as a flowing tension and release, as the chords are connected by many complex transformations. This does not fit with the Neo-Riemannian understanding of the rest of the aria, and I believe that is why Johnson did not include this section in his analysis. There is another section like this one at measure 616, which Johnson just wrote “transitional section” and moved on. The bichords Abm/Db into Bbm/Gb are true non-functional chords, that only adds colour to the melodic line, which stays solely in Db major. The chord change only then occurs in mm. 553, at the F#m chord. From that point onwards, the Fm chord at mm. 560 only serves to reposition the melody. Then the C#m add major 7 chord melts into Ab+ and Ab, which is the home of the next section. I believe Adams wrote this section to colour the existing melody. The F#m at mm. 553 heightens the tension before falling back to “home” at the Fm chord. Then, the C#m add major 7 is the real climax of this section, falling twice, first halfway to the Ab+, losing the bass note, and then finally back to Ab major. Adams uses tension as levels, not tension and release, which will also appear in the next aria.

Notes for “I am the wife of Mao Tse-tung”

            “I am the wife of Mao Tse-tung” uses the dominant and subdominant transformations. A dominant transformation changes a minor chord into its “tonic” as if the chord was the dominant, for example, Bbm into Eb. The subdominant transformation does the opposite.

            Although we have not seen 7^th chords yet, in this aria, they serve as an added layer of suspension, usually played by a separate instrument than the rest of the chord. The A section of this aria has a Neo-Riemannian section, where the voice leading makes conventional sense. The Bb to Bbm to Eb7 to Ebm goes though parallel transformations at the major to minor switch, aurally sounding like a depression to scare you into the dominant which is actually more of a IV sound to me. The pull is still from the Eb (IV) to the Bb (I) and not, as you would probably have guessed a V to a I, Bb to Eb. The basic transformation of the major to minor, is still only a colour change.

By the end of the aria, during the instrumental coda (mm. 961-end), the chords D7, E7 and the home chord, Bb are used to represent different levels of tension. The D7 and E7, both only sharing one note with Bb serve as tension towards the Bb home chord. Since E7 also has a tritone relationship with Bb, it sounds more jarring in relation to D7. In addition, the way the bass always starts on the 1 of the chords during this section makes the D7 sound like it wants to move to the E7, and in turn, to Bb. Therefore, it can be said the D7 serves as a subdominant function chord, and the E7 and Bb serves as dominant and tonic.

A quirk of Adams in this aria that we haven’t discussed yet is at mm. 832, the dominant sound of the Bb chord, with the E7 chords. The tritone distance makes the E7s sound very jarring, and it can be seen as a secondary dominant to the “resolution”: a Gb chord at mm. 852. This isn’t a V-I though, Adams still steps down to it, through a P and L transformation, after the high note. This climax, you would think, needs a climatic harmony to compliment it., but with the transformations, Adams “side steps” your ear, and modulates into a new “home”. This technique of stepping down into a resolution happens very often throughout the entire, and often at traditional cadential points.

Notes for “I am old and cannot sleep forever”

            I am going to diverge once again from the Johnson. He analyses this aria from the melodic solo line and then moves to describe the function of the bichords and how they serve to harmonise the very dissonant melody. I do not believe that the bichords serve a function in this aria. There is a background dissonance from the lower half of the bichord, that being the chord on the bottom, the lower range, serves no functional purpose. When listening to it, all the “dissonance” that Johnson has analysed sounds only like noise, and not additional notes of the chord. In addition, the bichords share many of the same notes as the main chord, and it can be viewed as extensions. Basically, the bichords colour the melody. Therefore, we can analyse the A section as three distinct colours, without tension between the three. E minor, G# minor and C minor are used to propel the melodic line along, but the shared common tone doesn’t allow for, at least from what I can hear, much tension and release.

            The C section, at the tail end of the opera, uses Eb+/Cm chords, or just a C minor major 7^th chord as the “home” chord. And honestly, I cannot figure the rest of it out. From mm. 907 to the end, just the last couple of bars, I really don’t know what Adams was trying to achieve. We know that the opera is ending, and that Chou En-Lai is struggling with his insomnia and his legacy. The muddy mess of colour and chord that populate the final few bars of this opera sound just like that. No function, no purpose, just… noise? It’s still a mystery.

Additional Notes

            For the flowchart, I decided to just mark the flow of the chords out with the form and sections marked in, since there’s very little cadential action in these arias. In addition, I did not use roman numeral analysis for these arias since that would make very little sense harmonically, due to the whole lack of traditional functional harmony.

            There are still some sections in the flow chart that doesn’t make functional sense. The A prime section in the capo of “News has a kind of mystery” where the E half diminished chord moves to the F# minor then to the D of the B prime section makes little sense to me. I labeled it “transitional period” but that’s just to sidestep writing nothing there. The quartal chord throws me off. And so does the 7^th chord since I’ve been ignoring the 7^th chords in general. I think quartal theory and a deeper dive into the newer papers on 7^th chord in a Neo-Riemannian context will be helpful to understand Adams’s compositional technique there. Of course, the very last section of the opera is still foggy to me. And then there’s the transformations. I have found that basic transformations sound little like real functional chord changes, but then then question arises on what counts as chord function. I have always labeled a chord as the “home” or tonic, but that’s just the chord that begins the aria and therefore, what I start analysing on. Then, perhaps, just like Johnson’s theory, this one is also kind of wrong and kind of wonky at certain areas. Things don’t always fall in place in a functional context, like I thought it would. And unlike a scientific paper, I’m not going to try to make the data fit my erroneous hypotheses.