Estonian Academy of Music and Theatre

Jorge Gómez Rodríguez

Constructing Tonality in Molto Adagio

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy (Music)

Supervisor: Prof. Mart Humal

Tallinn 2015

Abstract

The focus of this paper is the analysis of the work for string ensemble entitled And

Silence Eternal - Molto Adagio (Molto Adagio for short), written by the author of this

thesis. Its originality lies in the use of a harmonic approach which is referred to as

System of Harmonic Classes in this thesis. Owing to the relatively unconventional

employment of otherwise traditional harmonic resources in this system, a preliminary

theoretical framework needs to be put in place in order to construct an alternative

tonal approach that may describe the pitch structure of the Molto Adagio. This will be

the aim of Part 1. For this purpose, analytical notions such as symmetric pair, primary

dominant factor, harmonic domain, and harmonic area will be introduced before the

tonal character of the piece (or the extent to which Molto Adagio may be said to be

tonal) can be assessed. A detailed description of how the composer envisions and

constructs the system will follow, with special consideration to the place the major

diatonic scale occupies in relation to it. Part 2 will engage in an analysis of the piece

following the analytical directives gathered in Part 1. The concept of modular tonality,

which incorporates the notions of harmonic area and harmonic classes, will be put

forward as a heuristic term to account for the use of a linear analytical approach.

To my father

Contents Introduction……………………………………………………………………………i

1 The System of Harmonic classes (SHC) as the Source of my Compositional Method . 1

1.1 What is Tonal? ............................................................................................................... 2

1.2 The System of Harmonic Classes .................................................................................. 8

1.2.1 Content and Aim of the System of Harmonic Classes ......................................... 8

1.2.2 Primary Dominant Factor and Harmonic Domain............................................. 12

1.2.3 Harmonic Area and Tonal Modules ................................................................... 17

1.2.4 Harmonic Areas and Vertical Sonorities ........................................................... 21

1.3 Construction of the SHC .............................................................................................. 24

1.3.1 Left-Right Symmetry in the Major Diatonic Scale ............................................ 24

1.3.2 Modal Layers ..................................................................................................... 25

1.3.3 Modal Interchange ............................................................................................. 27

1.3.4 Harmonic Classes............................................................................................... 31

1.4 The Major Diatonic Scale as a Harmonic Class .......................................................... 35

2 Analysis of Molto Adagio ............................................................................................ 37

2.1 Compositional Background ......................................................................................... 40

2.2 Musical form, harmonic and motivic material ............................................................. 41

2.2.1 Triadic Grid ........................................................................................................ 42

2.2.2 Motivic Material ................................................................................................ 44

2.2.3 Harmonic Material ............................................................................................. 47

2.2.4 Linear Approach ................................................................................................ 50

2.3 Tonal layers in Molto Adagio and their analytical representation ............................... 65

3 Conclusion ................................................................................................................... 67

4 Bibliography ................................................................................................................ 74

5 Töö lühikokkuvõte ....................................................................................................... 76

6 Appendix: And Silence Eternal .................................................................................. 80

i

Introduction

In order to guide the analysis of Molto Adagio, the main goal of the present thesis, this

paper will offer a description of the compositional method that lies behind it. Briefly

put, it consists of a hierarchically organized group of synthetic scales symmetrically

constructed upon the major diatonic scale; there is no prioritization of intervals, which

means: 1) any type of sonority can be built using any intervallic arrangement possible

(seconds, thirds, fourths, and their inversions, whichever the consonant or dissonant

quality that these intervals may have), and 2) any combination of scales derived from,

and including, the combination of the major scale and its inversion, can constitute a

harmonic resource (either in succession or in juxtaposition) without exceeding the

restrictions that exist within that method. This method I have labeled the System of

Harmonic Classes.

Before engaging in a discussion concerning the music, how was it written and what

choices were made, the author needs to account for the claim that the work possesses,

overall, a tonal orientation and it is guided by tonal principles such as function (the

principle that establishes a necessary relation between chords) and hierarchy (the

principle that tells us which of these sonorities stands as a primary reference). I am

aware that in the context of my own method, which uses musical materials in an

organized but, overall, free manner, to claim that a sonority possesses functionality

may seem to be stretching this concept too far. And yet, it is my proposal that a given

sonority (representing a scale, as I will show), by progressing to another sonority

(representative of another scale) at a critical juncture in the music, can be said to

resolve onto that subsequent sonority.1 Function, always in the sense Riemann gave to

this word,2 is perhaps the most telling example of the difficulty to establish a clear-cut

division between my system (which informs my compositional practice) and a more

rigorous, common, traditional conception of tonality. I understand function as a fluid

concept in which a musical object progresses to another musical object providing a

1 Doubtless, in cases such as this other factors are also relevant to bring to the listener an impression of

necessity in a musical harmonically consistent flow. These may include dissonance, careful voice

leading, and dynamics at structurally significant points. 2 “There are only three kinds of tonal functions (significance within the Key), namely, tonic, dominant,

and subdominant.” (Riemann 1893, 1900; Introduction).

ii

meaningful structural or harmonic relation. Problematic as this may be, it forms the

basis of my harmonic thinking and compositional practice, and, I quickly state, it is

also limited to it: this means that no theoretical claims of general validity will be

made, but only the potential significance or connection, in the context of music theory,

of the concepts provided in this thesis.

The theoretical premises from where my system is derived require at this point an

important precision. Since the early stages of my exploration of harmony as an

expressive means, a number of works and a great deal of scholarly research have been

drawn to my attention. What appeared to be an old fashioned enquiry into the theory

of harmony took all of a sudden a new twist when the work of the neo-Riemannian

music theorists, mostly American authors, was introduced to me by Professor Humal

of the Estonia Academy of Music. Their research not only put an end to the isolation

in which I had been working but promptly made me realize that I had been thinking

along the very same lines! Riemann’s theories are nevertheless transformed in the

hands of the “neo-Riemannians”,3 actively involved in geometrical models and

smooth chromatic motions that connect logically almost anything that can be written

with the twelve notes onto almost anything else, even to the extent of using computer

models to generate analytical results that sometimes contradict cherished theoretical

assumptions.4 Dualism is still problematical in that, by those not acquainted with this

specialist field, it still retains the aged aura of Hauptmann and Oettingen acoustic

theories or, worse still, Riemann’s speculations about “undertone” series.5 For me

dualism presupposes a symmetric view of musical resources, specifically chords in

Riemann and neo-Riemannian accounts, and in that everybody agrees; what I find

3 Some of whom do not accept the “dualist” and “neo-Riemannian” labels, but who nonetheless accept

gladly their theoretical Riemannian ascendancy. 4 See R. Cohn (1998), Kopp (2002), D. Tymoczko (2011). I will expand on these authors in due time. 5 Riemann himself finally dropped this idea (something usually overlooked), although admittedly late

in his career; see Klumpenhouwer, Dualist Tonal Space and Transformation in Nineteenth-Century

Musical Thought (2002) Klumpenhouwer reminds us that dualism is now, at best, an accepted

synonym for the hierarchic equivalence of major and minor. See also Rehding: “[…] after 1905,

Riemann agreed to do away with arguments based on undertones altogether, provided that overtones

were no used either in the arguments of music theory [!]. with this last stage, where acoustical

undertones were no longer necessary as a conceptual crutch, Riemann’s theory verged on a new

paradigm: the age of psychology was about to supersede the age of acoustics.” (Rehding 2003, 34).

iii

missing is an accurate theoretical differentiation between the method based on

symmetry and a specific dualistic interpretation of it. Symmetry is to dualism what

cogs and wheels to a machine: it could be a bicycle or a grandfather clock, but the

function of the purported mechanism is simply to turn. There is a circumstantial

connection and that is all it is: circumstantial. Obviously symmetry came first, as an

elegant design of things, and then the theories followed.

I cannot discern, yet, to what an extent my compositional method connects, or in

some manner complements the dualist theories of neo-Riemannian authors. First of all

it would be too premature to even suggest that possibility, and second there is still a

lot of work to do as regards the consistency of my own system and its applicability

outside my own practice. All it can be said with certainty is that, broadly, the system I

propose is used in this thesis as a theoretical tool (one that crystallizes my

compositional method) and that it appears to move among the same lines as those of

the neo-Riemannian authors, although our conclusions are different: for one, I am not

convinced that a geometrical account of the connections between chords is an

altogether satisfactory approach to “chromatic” music, as it is styled by the neo-

Riemannian authors mentioned above, and which comprises what it is commonly

known as extended common practice.

The piece to be analyzed, Molto Adagio, exists in no less than three different versions:

as the III movement of a double string quartet (original version), as an additional

ensemble arrangement of that movement, whose premiere took place at the

Hochschule für Musik und Theater Rostock (HMT)6, in Rostock Germany; and

finally as part of a string ensemble re-writing of three movements of the double

quartet with the title Music for a Mushroom Cloud.7 This last one is the version I

present here for analysis as it has the advantage of being available in audio recording.

***

I would like to thank the professors, composers, teachers, conductors, and staff at the

Estonian Academy of Music and Theater, especially Professor Margus Pärtlas for his

6 Named Rostock Molto Adagio, played by the Ensemble der HMT Rostock, Dir.: Konstantin Heuer. 7 Whose premiere took place at Mustpeademaja in Tallinn, Estonia, under the direction of Risto Joost,

where Molto Adagio stands as its last movement entitled And Silence Forever

iv

determination to see me through, Toivo Tulev for welcoming me back every occasion

I drifted, and for taking me hunting, Risto Joost for believing in my music, Professor

Mart Humal for supervising my thesis with a true touch of genius, Professor Kristel

Pappel for her unending generosity and sympathy (my academic guardian angel),

Professors Lippus and Siitan for welcoming an unorthodox student composer in the

musicology department (thank you), Helena Tulve for her timely advices, and, last but

not least, Professor Kerri Kotta, for positively bringing out, and seeing through, the

potential of a work that I myself could have hardly put into words. Thanks, finally, to

all the staff at EMTA (IT department guys, my cultural mentor Mr. Koit Pärna, and

also the sympathetic library staff) for making this possible. Margit Võsa, thank you

for making me a better person with your patience.

1

1 The System of Harmonic classes (SHC) as the Source of my

Compositional Method

No living theorist has had a more profound impact on the field of music theory than

Milton Babbitt. But it would be a mistake to separate Babbitt the theorist from

Babbitt the composer. Babbitt makes the shift from thinking "about music" to

thinking "in music" with integrity, not because his music is "theoretical" but because

his theoretical conceptions become aurally cogent. (Cherlin 1986, 75).8

Babbit’s example is illustrative of the musical aims of the author of this thesis. The

development of the System of Harmonic Classes is the outcome of the particular

manner in which I use tonal material in my compositions. What was a personal choice

became in time an inquiry into the tenets and propositions of the theory of harmony.

The highly idiosyncratic practice and use of harmony by diverse composers

throughout history9, at least those that influenced my music, made it inevitable to look

for a more comprehensive pattern that would include what seemed to me an identical

source: the common twelve tones used in Western music. The fact that, say, a C major

chord could be used by Wagner with more or less the exact harmonic consequences

(constituting a tonic resolution, for instance, or a deceptive cadence) that one could

find it in Monteverdi or Bach seemed to me encouraging enough to search the

connections; in other words, what connections, differences, or similarities, may be

involved. The recourse to music theory in my search for that blueprint, or harmonic

8 Michael Cherlin, Why We Got into Analysis and What to Get Out of It, Journal Of The Music Theory

Society of New York State vol.11); and it continues shortly after: “It might be correct to say that if

Babbitt were not such a powerful thinker in music it would be unlikely that he could be such a

powerful thinker about music.” Tymoczko (2011, 25) provides in his book a stern reminder of how far

“thinking” can sometimes push the boundaries of communicability by citing Milton Babbitt’s 1958

manifesto “Who Cares if You Listen?”, in which he cheerfully acknowledged that his music was not

enjoyed by laypersons, but only by a specialist musical community analogous to the specialist

community of professional mathematicians.” 9 For an illustrative account of the changing views and evolution of harmonic practice from the

Renaissance up to the Second Viennese School see Diether de la Motte, The Study of Harmony, An

Historical Perspective (Dubuque: William C Brown Pub 1991).

2

prototype, made it inevitable that part of the technical jargon and open questions I

found in theoretical treatises and scholarly research came to form part of the system

itself: in other words, at that point my compositional practice began to give way to a

“system” of composition, as I began to translate it into technical terms and reorient it

into a theoretical investigation running parallel with my composition work.

Both, compositional practice and theoretical inference go hand in hand to the point

where the music I present here seems to me sometimes to appeal to a reading

audience as much as to a listening one. The system I am about to present is the result

of the confluence of theory and practice, faithfully adopting the theoretical results in

my compositions.

1.1 What is Tonal?

Diatony therefore does not stem from the so-called Greek or Gregorian modes, but

rather from the composing-out process, which is governed by the principle of the fifth.

(Schenker 1979, 11-12)

I have always considered myself a tonal composer, with Tymoczko I strive for a

“music that brings people together, rather than dividing them”, and with him also I

think that “the traditional strategy—writing immediately attractive music that also

contains deeper levels of structure—is as potent as it ever was.” (Tymoczko 2011, 26).

This traditional strategy, if my reading of his book is accurate, is tonal music with its

accompanying musical structures and regularities. And yet, the path from the tonal

music I grew up with, the common-practice period, to the one I present in this paper,

is not straightforward. Schenker’s quote above stresses a very important point: the

synthetic character of musical practice.10 Whether our major diatonic scale is a

10 Brian Hyer’s article in the New Grove Dictionary of Music and Musicians is illustrative enough of

the difficulties we encounter when attempting to define tonality. He says, for instance, that “There have

also been some basic theoretical disagreements about whether its constituent musical elements are

melodies or harmonies: however narrow the definition, the domain of tonal music is so enormous,

diverse and complex that one can choose almost any combination of musical phenomena and

theoretical principles as a basis for discussion.”. Hyer, Brian. Tonality – The New Grove Dictionary of

3

historical product (something Schenker seems to implicitly deny) or a natural

phenomena (the overtone series), is not as essential as the results drawn by the

composer from the finished product, the major scale. Schenker’s final view of tonality

is of major relevance to this paper, if only to stand in contradistinction to that of the

present author. He maintains, shortly before the previous quote, that

The same relationship to a fundamental tone prevails also in the foreground: all the

foreground diminutions, including the apparent “keys” arising out of the voice-

leading transformations, ultimately emanate from the diatony in the background. I

have used the term tonality to include the various illusory effects in the foreground

[emphasis mine]; yet the tonal sparseness of diatony in the background and the

fullness of tonality in the foreground are one and the same.

To “flesh in”, as it were, the fundamental diatony of a musical work with “illusory

effects” seems to be the work of the “Schenkerian composer”. His view of tonality

seems to be marked by what Chua terms a “battle plan”11. Taking the opposite stance

one may come across a common criticism of Riemannian functional theory. Referring

to De la Motte’s functional views (and his silence over Schenker’s work) Jeffrey

Prater says:

The problem ...is essentially the same one found in Piston [,] namely the setting of

nearly all harmonic activity at a single fore-ground level. The central deficiency ... is

the over determination of most fore-ground (chord to chord) activity and a

corresponding inattention to the hierarchical relationships of the harmonies that direct

the foreground.12

Music and Musicians. Digital Edition. Edited by Stanley Sadie. Here I am only referring to salient

common-practice theorists that may offer an eloquent contrast with more recent views, such as those

held by Tymoczko, soon to be discussed, or Cohn. For Cohn see Audacious Euphony: Chromaticism

and the Triad’s Second Nature (Oxford University Press, 2012). 11 See Daniel Chua, On heroes, ‘Absolute Music and the Construction of Meaning’, (2003, 153). He

quotes Schenker’s ‘Beethovens dritte Sinfonie, in ihrem wahren Inhalt zum erstenmal dargestellt’, Das

Meisterwerk in der Musik (Munich: Drei Masken, 1930), 3: 29–101. 12 Quoted from (Motte 1991, xvii), taken from Martin Bresnick, review, p.321-22.

4

It seems intuitively clear that tonality and tonal may be found somewhere in between

these two extremes, which come down to a harmonic view of music (that which

primes chords and their relationships, and that includes functional harmony), or a

contrapuntal one (mainly the one outlined in the Schenker’s quotes above).

Schoenberg’s work comes a little closer to that middle way. He envisages function

and root movement as given premises, but he curiously opens the way for a more far-

reaching understanding of music structuring; in Theory of Harmony he writes:

We can assume that tonality is a function of the fundamental tone [tonic]: that is,

everything that makes up tonality emanates from that tone and refers back to it. But,

even though it does refer back, that which emanates from the tone has a life of its

own - within certain limits; it is dependent, but to a certain degree also independent.

What is closest to the fundamental has the most affinity with it, what is more remote,

less affinity. (1975, 150).

This conception eventually led him to the formulation of monotonality in which the

musical whole is organically tied up to a tonic of which everything else may be seen

as (using Schenker’s term) a prolongation:

Monotonality includes modulation-movement towards another mode and even

establishment of that mode. But it considers these deviations as regions of the tonality,

subordinate to the central power of a tonic. Thus comprehension of the harmonic

unity within a piece is achieved. (Structural Functions of Harmony, 1969, 19).

Despite Schoenberg’s mistrust of music theory, of which Theory of Harmony13 is

sufficient witness, and the open criticism of Schenker we can read in it, Schoenberg

does not, however, criticize the systematization of harmonic resources, of which he

himself was an advocate.14 My main contention, and this was Schonberg’s too, is that

13 Certainly a surprising title, as Schoenberg stresses the teaching quality of his book throughout,

beginning the preface to its first edition with “This book I have learned from my pupils.”, to its final

sentence: “In such a domain, who dares ask for theory!”. 14 Dr. Heinrich Schenker, for example, although he breaks through nowhere to complete clarity, is in

my opinion worthy of attention and respect just because he is one of the few who are really striving for

a system.” (1975, 408).

5

this systematization (the only one that can possibly define the limits of what is tonal,

hence Schoenberg’s recourse to it) tends to center on specific repertoires, which either

focus common practice or its extension.15 For Schoenberg:

A tonality is expressed by the exclusive use of all its tones. A scale (or part of one)

and a certain order of the harmonies affirm it more definitely. In classical and popular

music, a mere interchange of I and V is sufficient if not contradicted by extra-tonal

harmonies. In most cases, for sharper definition, a cadence is added at the end of an

entire piece or of its sections, segments, and e en smaller units (Schoenberg 1969, 11-

12).

Here Schoenberg seems to exclude music based on more indirect methods of handling

tonal materials, such as the presence of dominants, diminished, or half diminished

chords at significant structural moments. Reference to this will be made later in

section 1.2.2, where I discuss the possibility of establishing a tonality by these indirect

means. Analytical approaches to contemporary repertoire hinge occasionally on the

tonal qualification of the music. Thus, for Laufer, the decisive feature is that of

finding an appropriate analytical approach. Referring to chosen pieces by Debussy,

Bartók, Dallapiccola, and Tubin, he says of the music: “it is tonal, though not triadic,

not major or minor – tonal in the sense of being rooted in certain specific tones. There

are, for instance, no harmonic degrees such as tonic and dominant; no consonance and

dissonance, in the sense of consonance being triadic and dissonance being non-triadic;

no scales, as such; and so on.”16 (Laufer 2004, 89). This view, should it prove to be

applicable, associates tonality with centricity, and, at the very least, opens the way for

a more comprehensive view of tonality.

For Tymoczko the sense of tonality (regardless of historical period) is created mainly

by the following features, and I quote:

1. Conjunct melodic motion. Melodies tend to move by short distances from note

to note.

15 The so-called extended tonality.

(Harrison 1994, 5). 16 His emphasis throuought. Laufer offers a linear analytical approach that does not correspond here to

discuss, nor does the question of whether the application of it appears to be successful or not.

6

2. Acoustic consonance. Consonant harmonies are preferred to dissonant

harmonies, and tend to be used at points of musical stability.

3. Harmonic consistency. The harmonies in a passage of music, whatever they

may be, tend to be structurally similar to one another.

4. Limited macroharmony. I (Tymoczko writes) use the term “macroharmony” to

refer to the total

collection of notes heard over moderate spans of musical time. Tonal music

tends to use relatively small macroharmonies, often involving five to eight notes.

5. Centricity. Over moderate spans of musical time, one note is heard as being

more prominent than the others, appearing more frequently and serving as a

goal of musical motion.17

He rightly says, in my view, that “tonality constitutes a fairly unique solution to some

elementary compositional problems.” These problems are encountered by every

composer dealing with basic harmonic resources, such as the diatonic scale may be,

and their almost endless combinatory possibilities: major-minor and consonant-

dissonant being only the most salient.18 The idea behind it is that a continuity exists

between past and present compositional output that seems to be best analyzed when

these points of convergence (the five just mentioned although, as he observes, there

may be others) are taken into consideration. In the final analysis, I will try to show

that Molto Adagio can be described as tonal at least in the way described by

Tymoczko, exception made of the preference for explicit acoustic consonance, as

well as others (systematic use of tonal-diatonic materials derived from the

symmetrization of the major scale) that will be explained in Part 2.

Tymoczko opens the way for a more inclusive systematization of harmonic resources,

one that allows for a greater variety of combinations among its elements, and that do

not adhere strictly to the theoretical precepts held by previous theorists:

17 (D. Tymoczko 2011, 4), bolds mine. 18 “[I]t is perfectly possible, for example, to write music that is diatonic, or more generally

macroharmonically consonant, while also being serial, aleatoric, indeterminate, wildly polyrhythmic,

and so on.” (D. Tymoczko 2003, 26). This appears also to emphasize the distinction between what is

aurally tonal from that which is harmonically tonal; a crucial difference worth bearing in mind.

7

The idea that tonal music is hierarchically self-similar is central to the work of

Heinrich Schenker, who claimed that tonal pieces consisted of recursively embedded

patterns. […] Unlike Schenker, however, I view macroharmonies [identifiable

sequences/collection of notes] and scales (rather than chords or melodic lines) as the

primary vehicles of long-range harmonic progression […] (2011, 18-19).

He illustrates this with the following example:

Figure 1-1: Tymoczko's scalic approach

It is debatable whether Tymoczko’s views are fundamentally opposed to

Schenker’s,19 and a discussion of it would take us too far, but what is certain is that

tonal centricity appears to be losing ground to a more comprehensive view of tonality

as an organized collection of scales.

His analysis aims to show that tonal music “makes use of the same voice-leading

techniques on two different temporal levels: chord progressions use efficient voice

leading to link structurally similar chords, and modulations use efficient voice leading

to link structurally similar scales.” (17). The example above is certainly a simple one

but illustrates what Tymoczko sees as a technical feature further developed by

twentieth-century composers and up to the present day. The notion of linking

19 I owe this insight to Professor Kotta, at the Estonian Academy of Music and Theater.

8

“structurally similar scales” as a vehicle of “long range harmonic progressions”,

further extends the applicability of tonal principles to a significant part of the

contemporary repertoire, including my Molto Adagio.

Summing up, tonality, in Tymoczko’s words, remains poorly understood, lacking

“even the most rudimentary sense of the musical ingredients that contribute to the

sense of “tonalness.” The chromatic music of the late nineteenth century continues to

be shrouded in mystery. We have no systematic vocabulary for discussing Debussy’s

early 20th-century music or its relation to subsequent styles. […] As a result, many

young musicians are essentially […] rediscovering for themselves the basic

techniques of modern tonal composition.” (D. Tymoczko 2003, 19). The system I will

presently introduce is an attempt to situate my music within this, to some extent, still

speculative but promising area of contemporary tonality, which is necessarily still in a

descriptive phase, one that precludes a final definition of tonality.20

1.2 The System of Harmonic Classes

In this section I will discuss the conceptual framework associating the notion of

tonality, as discussed in 1.1, and my compositional system, which I term the System

of Harmonic Classes.

1.2.1 Content and Aim of the System of Harmonic Classes

Definition 1: symmetric pair

A symmetric pair is the combination of a major diatonic scale (say C major) with its

inversion in descending motion (in this case C Phrygian), and denoted as a binary pair

20 Especially if it can be proved to be a continuation of traditional tonality, in which case the latter

would have to be reviewed too. See Tymoczko 2011: 5, where he states that the point of his book is, no

other, than “to retell the history of Western music in such a way that the tonal styles of the last century

—including jazz, rock, and minimalism—emerge as vibrant and interesting successors to the tonal

music of earlier periods.”

9

separated by a forward slash Major scale/ Phrygian scale in its major scale equivalent

arrangement; thus C/A¨21 symmetric pair, as in

Example 1-1:

Example 1-1

By means of recombining the tones of both scales symmetrically around the

generating tone (c in this case) we obtain a number of seven-note synthetic scales (to

be defined in detail in section 1.2.4) which I will denominate harmonic classes. In

Example 1-2 a harmonic class constructed from c is shown in its extended melodic

form, as a scale, followed by an instance of its arrangement as a vertical configuration

(in this, I have chosen to arrange the tones by thirds for clarity, seconds or fourths

may have served us just as well, since there is no prioritization of intervals in the

SHC):

Example 1-2

We may notice how all the tones in these two arrangements can be rearranged

symmetrically around the generating tone c: b – d¨ constituting a diminished third

interval around it (a semitone on each side), a¨ – e an augmented fifth (a major third

on each side), and finally f – g constituting a major ninth (two perfect fifths on each

side) around c. Which is a consequence of these tones (belonging to the harmonic

class) having been derived from the symmetric arrangement of the C/A¨ symmetric

pair, since all the tones present in that pair can be arranged in pairs that are symmetric

around c.

21 C-Phrygian and A-major are identical scales (not modes!) possessing identical tone content.

10

Even at this early stage, the question may arise as to why the composer feels

compelled to effect such an arrangement of traditional harmonic resources (a major

scale and a minor mode); although a context will be furnished in the following

sections, I promptly beseech the reader to allow for a non-theoretical explanation as

the ultimate basis of the system. At the heart of this arrangement we may find, not

only the quest for larger and more intimately felt means of musical expression

(something which by itself would warrant a great deal of spontaneous

experimentation), but the problematization of a central tenet of western musical

practice: the ascending character of scales, which is primarily a harmonic property.

Take for instance the definition of scale: “[A] sequence of notes in ascending or

descending order of pitch.”22 In Western music, for all practical purposes,

“descending order” amounts to the same notes being read or played in a downward

fashion, c – b – a – g, being the descending order of g – a – b – c: that is, a backward

spelling of the same musical object, a directional choice. The fact that there is a

noticeable aural difference when approaching the tonic c (c = 1 in our examples) from

7 or approaching it from 2 does not enter into the ascent-descent structure of the

definition. Yet, from and analytical point of view the semitone step moving from the

degree VII of the scale to the octave (b to c in our example) defines a harmonic event

considerably more significant than the movement from d to c, which is according to

the definition its descending counterpart. The presence of the semitone step 3 – 4 is

also polemical, in the sense that it constitutes a kind of mirror image downward

resolution of the ascending 7 – 8 leading tone paradigm. Doubtless the tonic tone

should be consider hierarchically higher to the third of the triad built above it, but it is

22 “(Fr. gamme; Ger. Tonleiter; It. gamma). A sequence of notes in ascending or descending order of

pitch. As a musicological concept, a scale is a sequence long enough to define unambiguously a mode,

tonality, or some special linear construction, and that begins and ends (where appropriate) on the

fundamental note of the tonality or mode; a scale, therefore, is usually thought of as having the

compass of one or more octaves.” And then it goes on to say that the rest of the article will be limited

to the scales of European musical theory, that is, limited to ascending scales. William Drabkin/R. Scale.

– The New Grove Dictionary of Music and Musicians. Digital Edition.

11

precisely this third that gives the chord a major character and thus, arguably, an

essential feature of this sonority.23

As I will show in the next section, a given major third is at the center of the major

scale by means of being the primary resolution to the tritone. It is my view that this

specifically highlights the tritone as the center of the harmonic structure of the major

scale, and not the major consonant triad associated with the major third that

constitutes, as noted, the tritone’s primary resolution. The major and minor consonant

triads that incorporate this major third (1 and 3 in the major scale, contained in chord I

and chord VI) can be, thus, conceived hierarchically equivalent as far as the resolution

of the tritone dissonance is concerned.

The major scale itself may, consequently, already possess “descending” traits that

would admit a different reading of its harmonic properties, and I am referring to a

harmonic progression that may resolve to the tonic triad in a downward motion in the

subdominant sonority (the subdominant sonorities including the 4 – 3 downward

motion). And yet its character is somehow weakened by the harmonic properties of

that progression: the lack of a suitable dissonance (within the scale), and/or the

presence of the tonic, whichever the choice of subdominant sonority may be: chord IV

or II7. This path, therefore, does not provide us with an answer as regards the

possibility of a hypothetical western music practice favoring the descending motion of

scales and its corresponding “descending harmonies”; a premise, too, that constitutes

the foundation of my initial inquiry. The major scale (also the minor scale) is too

strongly determined harmonically to allow much room for anything else besides what

we already have. But should that possibility exist (and my compositional practice is

greatly involved in experimenting with this possibility), it must lie outside the strict

confines of the major scale itself.

Ultimately, it may be that a hypothetical musical practice making use of “descending”

harmonies is not plausible. It could be, as Tymoczko observes, that tonality as we

know it is simply the best solution, or at least the most appealing for the cultural area

23 If not its character of tonic chord altogether, the dominant seventh being indisputably a chord

borrowed from the major scale or, at least, from the descending minor scale – this one, a descending

scale sharing some of the harmonic characteristics belonging to the major scale.

12

where that music developed. What we are facing, then, is an inquiry into a system

encompassing the major scale, with its characteristic melodic and harmonic elements,

but supplemented with a set of, predictably, similar fundamentals that may re-

contextualize its aural and harmonic characteristics. This system may initially be

researched by studying the consequences of symmetrically inverting the scale

(downwards) and recombining it with its original ascending form. And this is the

main object of this thesis.

Before returning to this question, in sections 1.3 and 1.4, I will proceed to define a

number of concepts that may facilitate its comprehension.

1.2.2 Primary Dominant Factor and Harmonic Domain

For strictly analytical purposes concerning the piece here presented, I will call the

tritone the primary dominant factor (or simply dominant factor, as other possible

“dominant” intervals exist, such as the augmented fifth, that will not be discussed in

this paper), and designated by the Greek letter “ τ ”.24 The tritone’s harmonic

resolution to the major third (minor 6th) will be called its τ-resolution. Any scale with

only one tritone would therefore possess a unique τ-resolution to the major third of

degrees I or VI of the major scale. These, in conjunction with V7, define one unique

major diatonic scale. Thus:

Definition 2: Primary Dominant Factor

Primary dominant factor is the potential of the tritone to resolve to the 1 - 3 major

third and, therefore, to imply or establish a major diatonic scale.

Needless to say that the tritone can resolve to several other intervals included in scales

other than the major diatonic scale, and also to another scale (major or not) by

respelling its tones enharmonically. In the latter case we are simply implying a

24 The importance of this designation is that it emphasizes the tritone’s structural role, traditionally

represented by the dominant V7.

13

different scale than the one where the tritone has been defined as 4 – 7 (or 7 – 4, since

the order is irrelevant in this case). And then, in both cases, there is the fact is that any

other 7-note scale (the ones relevant in the construction of the SHC) will include at

least another tritone, and thus possess greater complexity than the major diatonic scale,

which, for this reason, can be said to be the simplest scale that a tritone can establish.

In practice, though, any number of tritones (of the possible six in the chromatic scale)

may be present in a composition and it may hardly be the case that they all adhere

strictly to a τ-resolution. And yet, it is crucial to notice that the introduction of the

tritone allows us to identify the potential presence of the major scale.

Definition 3: Harmonic Domain

I define harmonic domain as the tones of a major diatonic scale associated with a

specific tritone by the resolution of this tritone to the major third 1 - 3.

Thus, a harmonic domain will include the major and natural minor modes possible

within the major diatonic scale as they all contain the same dominant factor and are,

therefore, harmonically equivalent. It is important to note that, similarly to the manner

in which the tritone may stand for a V7 dominant sonority, the harmonic domain

stands for the major scale regardless of the number of tones (belonging to that major

scale) present. As I will show (see 2.3 below), a harmonic domain helps us to chart

the tonal areas through which the music moves.

A further stage in defining the implicit harmonic functionality within a major diatonic

scale is to recall its association with the triad. In the context of the SHC the triad can

be understood as the direct product of a τ-resolution; the perfect fifth above the root (1)

and below the third (3) being a derivative of possessing a step-wise arrangement of

the scale. Analytically, also, the consonant tonic triad, its relative minor, and the

major diatonic scale to which they belong are established as a logical necessity.

From this standpoint the most relevant (harmonic) event in the unfolding of the scale,

through chords and their connections, is not so much the existence of a tonic or

dominant chord but their relation: only when the τ is present an intrinsic harmonic

relation (a progression of the tritone to a major third or minor sixth) is to be found in

the scale.

14

The overall idea is that of centering the music on the dominant factors (the tritones)

present and their functional character. Here I am reminded of A. B. Marx, who, in

Kopp’s words (2002, 48), held that:

[…] the dominant seventh, which he calls the “dominant chord,” is unambiguous,

implying only its associated tonic; it is the “origin of harmonic motion.”25 […] Marx

finds V7 to be the strongest indicator of key, while his dominant triad [the V chord]

does not share at all in this property.

For Yavorsky, too, the tritone is the prime engine of harmonic motion, as he puts it:

“From my studies… I have come to the conclusion that the basic cell in musical

language is the tritone and its resolution.”26 In Rethinking Octatonicism, P. A. Ewell

draws a succinct account of Yavorsky’s theoretical tenets, and clarifies the role that

the tritone had in Yavorsky’s system:

Gordon McQuere argues that the paramount element in Yavorsky’s system is the

tritone and the motion that is required to resolve it (2009, 113). Though this is

certainly important, it is not at the foundation of the system [emphasis mine]. The

reason that the tritone became so important to Yavorsky was because the functional

tonal system had, by the late nineteenth century, been weakened, and he felt that

something other than the T–S–D–T formula was necessary to account for structure in

music. It was clear to Yavorsky that the seven-step system of functional tonality had

been replaced by a twelve-step system. This notion he learned from his teacher at the

Moscow Conservatory, Sergei Taneev (1856–1915). (Ewell 2012, 2).

Contrary to Yavorsky (if Ewell is correct in his interpretation) the potential resolution

of a tritone, which may be called a dominant event, is by itself sufficient to map the

major scales (and thus “tonalities”) potentially present in the work as determining

25 Is not this what Yavorsky is saying? 26 Letter to Taneev, April 1906, quoted in Philip A. Ewell, Rethinking Octatonicism: Views from

Stravinsky’s Homeland, Society for Music Theory, December 2012. I will have occasion below to

dwell in more detail in Yavorsky’s musical thought.

15

compositional factors.27 To illustrate this in a tonal context we may take at random a

chorale by Bach. In Example 1-3 below, taken from the cantata Ich ruf zu dir Herr

Jesu Christ, BWV 177, the tritone components have been marked with diamond

shaped note-heads. We notice how Bach uses three tritones in total: a-e¨, f© -c, and

finally b§ - f§, whose harmonic domains are B¨ (and therefore G-minor), G-major, and

C-major diatonic scales respectively.

Example 1-3: Ich ruf zu dir Herr Jesu Christ, BWV 177

According to the definition of primary dominant factor (tritone resolves to major third

by semitone step), this fragment would represent the confluence of three different

27 This idea is not new, Bailey says: “Still another innovation in Tristan is what we might call the

indirect method of exposition, wherein certain fundamental and motivic elements first appear by

implication rather than by explicit statement.” (Bailey 1985, 125). He ends the paragraph by saying that

“The dominant is in fact so intimately bound up with its tonic that it can suggest or present it by

implication and thereby substitute for it.” Also, in an interesting footnote in ‘The Love of

Fundamentals…’, Vol. V of Composition as a Problem, Charles J. Smith, in discussing Rachmaninov

Piano Concerto N. 2 in C minor (Op. 18), I, mm. 83-87, mentions the fact that a borrowed G7 chord (m.

84) in an E¨ major passage may have “tonicised in absentia […] the global Tonic of the whole

movement. In other words, Rachmaninov might be thought of as having worked out a way for this

passage to swap Dominants – borrowing the usual Dominant of the overall C minor as a special kind of

enharmonic Dominant for the E¨-major second key area.” (Smith 2008, 11). This, in my opinion, would

deserve more than a passing footnote as it opens an analytical ‘can of worms’ (perhaps that is why is

said only in passing!) by suggesting that there are possible structural references not written out in the

score itself, thus “in absentia”. An example of this will be discussed in Section 2.2.4.2 (Module 2: mm.

26-35) below. More recently, quoting Harrison (1994) “[…] in the context of late nineteenth-century

music, like jazz pedagogues, Harrison argues that a chord with a prominent leading tone and fifth scale

degree can often be interpreted as a dominant chord, no matter what other notes it contains.”

(Tymoczko 2011, 356).

16

harmonic domains, those corresponding to the major diatonic scales just mentioned.

A resolution to G major is certainly possible (and as a Tierce de Picardie common)

and so is a resolution to C major. We know that the harmonized f© is in itself an

alteration of the B¨ major scale, and this results in the neutralization of the possible C-

major diatonic area in favor of that of G-major28, although in the first beat of the last

bar a C-major triad is by no means a remote possibility. The presence of an a¨, in the

second to last measure, hints to an E¨-harmonic area, although the notes of its primary

dominant factor29 do not sound simultaneously. According to our definition of

primary dominant factor, the tritones present imply major diatonic scales that could be

present, and therefore may have tonal implications in what follows or what is

immediately juxtaposed around it.

In the chosen excerpt the composer could have decided at any moment to make use of

a different tritone resolution to lead the music somewhere else. What is to the

composer a question of choice is, from the overall perspective of this paper, a

significant analytical element since it would introduce a different harmonic domain

that may, or may not, correspond to the same symmetric pair. What it means is that,

harmonically, those areas of choice run parallel to the music as it is written on the

score. Although we do not see it (or hear it!) they belong to a more comprehensive

system which I term the System of Harmonic Classes. Its fundaments are somewhat

similar to Schoenberg’s conception of monotonality,30 which is a comprehensive and,

crucially, hierarchically organized conception of the 12-tone system. The details and

development of my system differ greatly from those of the Viennese composer, but its

fundamental tenet, that of a systematic overarching conception of musical harmonic

resources, is germane to his.

28 Crucially avoided by the resolution of the c-f© tritone to a tonic octave in the last measure of the

example. 29 The d-a¨ tritone. 30 The fundamental contribution of his Structural Functions of Harmony, and a result of his views

about regions, which correspond greatly with my harmonic domains. In a broad sense I consider my

harmonic thinking as a continuation of , or a reflection upon, Schoenberg’s ideas, to which a good deal

of Riemannian and, unavoidably, Schenkerian contributions may be added, as I will attempt to show in

this paper.

17

1.2.3 Harmonic Area and Tonal Modules

Molto Adagio is harmonically structured around a small number of relatively simple

harmonic relations, such as C major relations to G (V) and D major (V of V), or to F

(IV) and D¨ major (¨II); and yet, the highly complex texture of the work, and the

manner in which tonal materials are weaved in this harmonic texture, resist a

straightforward tonal approach. The simultaneous presence of tones belonging to

different scales (harmonic classes strictly speaking), and the difficulty in establishing

the nature and function of the sonorities these tones constitute, represent an additional

analytical difficulty, as it does the existence of different tonal centers throughout the

piece.

Thus, in order to organize the tonal elements and tendencies present in the work, we

must develop a system, centered on the major scale, and that may possibly inherit its

properties: progression, symmetry, and possibly function. We may begin by defining

the elements of this system.

Definition 4: Harmonic Area

I will call harmonic area the group of harmonic domains (see Definition 3)

associated with all the tritones present in a symmetric pair (see Definition 1).

As it may be deduced from the definitions involved, the harmonic area generated by,

say, the C/A¨ symmetric pair, will contain five possible harmonic domains,

corresponding to the five tritones present in that symmetric pair; starting with the

tones of the C-major scale they are: d – a¨, e – b¨, f – b, g – d¨, and a – e¨. They define,

respectively, the E¨, F, C, A¨, and B¨ harmonic domains. Or, in descending order of

flats: A¨, E¨, B¨, F, and C.

The significance of the harmonic areas is paramount in the SHC. They correspond

loosely to the notion of key or key area in traditional tonality, and so our analysis of

Molto Adagio will center in defining these areas in order to offer the reader a clear

picture of the overall harmonic structure behind the music. Due to the complex

character of the relations between the two diatonic scales, forming a symmetric pair,

the concept of key is severely limited and perhaps also misleading. In this sense

18

Schoenberg’s notion of region, introduced in Theory of Harmony and charted in

Structural Functions of Harmony, is more appropriate for the task. Regardless of the

difficulties surrounding this concept,31 intuitively is sufficiently defined to help

envision my notion of harmonic classes within a harmonic area; harmonic class

being the equivalent of region or segment (in Schoenberg’s sense), and harmonic area

the equivalent of tonality as a whole, including all the incursions into different regions.

As I will explain in the next sections two symmetric pairs, say C/A¨, can combine

their tones in an organized manner to generate a number of scales I term harmonic

classes. These harmonic classes in turn provide us with the sonorities that constitute

the harmonic make up of the music I write.

What is important now is that Schoenberg relates region to the introduction of

substitute tones foreign to a given scale. Developing on this the Austrian composer

identifies these substitute tones as belonging to what he calls segments of the tonality

they are inserted in; as he says “Intermixing of substitute tones and chords with

otherwise diatonic progressions, even in non-cadential segments, was considered by

former theorists as modulation. This is a narrow and, therefore, obsolete concept of

tonality.” (1969,19). This amounts to an extension of tonality achieved by grafting

diversely related fragments of other scales, a procedure that resembles, even if

superficially, the origins of the SHC and the conception of harmonic area.

This concept, the harmonic area, allows us to adopt a provisional “tonal” view of the

music as it involves sonorities drawn from tone-centered scales. At least while we

remain within a given collection of harmonic classes sharing the same generating tone

the harmonic area they define will allow us to orient ourselves harmonically within

the piece. There are, though, important reservations that may be taken into account at

this point. Tone-centered harmonic classes constitute the main harmonic resources of

31 In the preface to the English revised edition L. Stein (editor) warns us that the theory of regions is

not complete, and that “Schoenberg, as was his usual custom, postulated certain hypotheses regarding

the main problem of key relationships within a composition - how, in fact, harmony functions in

determining the structure of a piece. Of the relationships between the regions many. but not all, are

explored, at first in four-part harmony. Moreover, when the regions are later applied to the analysis of

examples from literature, many instances occur where other explanations of regions could be given. An

examination of the classification of region”

19

G/E¨

the piece to be analyzed: they provide the music with all the melodic lines and

sonorities to be found (taken, say from the C, F, B¨, E¨, A¨ harmonic domains, either

in isolation or combined), interacting between each other in various ways (sonorities

of a harmonic domain may support melodic lines from another, or different harmonic

domains may combine into one single sonority at some point, and so on).

It is the connection between various harmonic areas that is problematic. When we

come to analyze the piece the strategy to follow will have to take into account the

heterogeneous nature, or at least the diverse origin of the harmonic classes

(“scales”)32 involved, which ultimately derive from different harmonic areas

(“tonalities” in the wide sense envisioned by Schoenberg). We may visualize this by

thinking of the different harmonic areas as the different layers in

Figure 1-2; here, every flag-shaped layer corresponds to the appearance of a harmonic

area within a specific length of time in the music (time being indicated by the arrow

at the bottom of figure, and the appearance and ending of the harmonic area by the

vertical lines that intersect it):

Figure 1-2

It can be observed that the different harmonic areas juxtapose at different intervals in

time (i.e., they sound simultaneously). The now familiar C/A¨ symmetric pair (here

standing for the harmonic area it generates) may have no direct harmonic

connection33 with the material present in other layers (those representing the D/B¨ and

G©/E harmonic areas). Every one of these layers may be represented by a chord, a

32 I remind the reader that scale and tonality may not be equated in the context of this paper, since most

of the scales used, or strictly speaking harmonic classes, generally posses more than one tritone and

thus, potentially at least, more than one possible dominant-tonic resolution. 33 Or indeed it may.

C/A¨ G©/E

20

melodic fragment of a scale, a harmonic domain or group of them,34 or in fact any

other harmonic or melodic resource provided by the harmonic area in question. For

example, a chord belonging to C/A¨ may be harmonizing a melodic fragment

belonging to G/E¨, and both may be joined by harmonic intervals belonging to G©/E

(perhaps extending the C/A¨ chords, or punctuating the G/E¨ melody with a

contrasting melody, just to put an example).

It is here where the tonal relation between the layers may become significant and, so

to say, glue the group of layers harmonically. In Figure 1-2 this could correspond to a

potential dominant-tonic relation between the G/E¨ and the C/A¨ harmonic areas. On

the other hand, it may be possible that no harmonic relation is found between the

harmonic areas present simultaneously at a given time. It may be, in this case, that the

harmonic relation may be found with a subsequent harmonic area, or group of them,

as in Figure 1-2b, where the confluence of harmonic areas have been simplified and

depicted as circles:

Figure 1-2b

Each one of this circles, or spheres, can be thought as a unit carrying its own specific

harmonic configuration, and in possible harmonic relation with the harmonic

resources within it, preceding it (another circle), or following (yet another circle).

This “circles”, or self contained definable harmonic units I call tonal modules.

Definition 5: Tonal Module

I will define tonal module as a musical passage containing, and delimiting, an

identifiable harmonic area (or areas) and/or tonal center(s).

34 Defining, thus, the harmonic area beyond doubt.

Third Confluence:

G/E¨, G©/E, C©/A?

Second confluence:

C/A,¨ G/E¨, G©/E

First confluence:

C/A¨, G/E¨

21

As I will show (see Section 2), it may be possible to use linear analytical charts to

employ a tentative tonal approach for the understanding of these tonal modules. Even

though the harmonic materials may be dissimilar, an overall tonal tendency may be

identifiable either within itself or in relation to other tonal modules. It may be, as I

said earlier, that the fragment of a scale is harmonized by heterogeneous harmonies

(composite chords), but carries an important memorable motif, or it may be, for

instance, a dynamic build up to a chord that is set to clash with, or resolve to, another

chord made of different harmonic materials, thus beginning a new tonal module in

what could be arguably called a cadence.

Whichever the case may be, what interests us now is not the way in which the piece

may be said to be tonal, that will be the content of section 2; our main pursuit in this

section is lay out the way in which the SHC informs the scales and sonorities found in

the music.

1.2.4 Harmonic Areas and Vertical Sonorities

As it has just been mentioned, and it is commonly found in the music to be later

analyzed, several harmonic areas may be present simultaneously in a given passage:

in Molto Adagio this usually occurs when different voices are given different

harmonic classes. Let me illustrate this with an example taken from J. S. Bach:

Example 1-4: Herzlich lieb hab Ich dich, o Herr, BWV 340

No analyst would entertain the idea of evaluating the first beat of the first bar other

than as it has been indicated in the example, that is, a dominant of the dominant. As

far as music flow is concerned the temporary borrowing of the a§ would not be seen

as affecting the voice leading linear succession or, even less, disturbing the

22

progression I – V – I. Let us now make an experiment and modify the tones in the

cadence above to include several more “borrowed” tones:

Example 1-5

In the example I have kept the outer voices as Bach conceived them, coinciding in a

E¨-major scale, but modify the two inner voices sounding a d-e¨ (absent)-f© -g-a-b¨ -c©

harmonic class, which belongs to the D/B¨ symmetric pair. Chords are turned into a

vertical assemblage of different scales, which is contrary to the traditional practice of

using one scale to inform the vertical sonorities as well as the melodic lines. With

some effort we may still be able to convincingly attempt interpreting the chords in the

second bar of the A) excerpt as somehow altered degrees. The fragment would not be

tonal anymore, at least in its Baroque-Classical context, but the modified pitches are

still there, chromatically altered. Now if we take a look at the B) excerpt that follows

it we perceive a different picture altogether. The notes with diamond note-heads

inside the curve correspond, potentially, to either a modified form of the E¨ scale or to

a different scale altogether, regardless of the fact that some of the tones, as in the case

of g and b¨, may be shared; outside the curve we still identify notes that can easily

correspond to E¨.

The principle involved here35 is not entirely compatible with linear thinking, as it

dramatically disrupts the vertical tonal coherence and ultimately affects the choice of

suitable notes to connect in a leading tone linear chart, in other words, the notes

present in a linear representation may belong to different harmonic layers in the music,

or simply to one of them while the others reveal a different layer/harmonic area:

35 Maximum harmonic independence of voices.

23

harmony reveals itself as composed of different elements (scales and chords

belonging to different harmonic layers within the music) organized by strict tonal

principles that do not correspond to traditional tonality. This is more acutely certain

when the tones/scales constituting a specific passage belong to different symmetric

pairs, as if in the example above the harmonic class (that’s what it is) inside the curve

would belong to “D” harmonic area.

Chords then, in a rough simile, could be seen as an “intertwining” line (our vertical

sonority) through diverse threadbare “nets” (our different scales): the resulting knots

(tied to different layers) being the chord’s tones – Figure 1-3 a), as opposed to the

customary view of a chord “borrowing” tones from a different scale – Figure 1-3 b):

Figure 1-3

a) Modal “net” mixture b) Traditional tonal “net”

This seems to fly in the face of our common intuition of a homogeneous tonal texture

in which (like the stitches in Figure 1-3 b), in the example above) substitute notes and

borrowed chords are somehow “inserted”36 into a given musical passage.37

Nevertheless this kind of modal mixture is common in the music that I present in this

paper. The sonorities can indeed be taken vertically as a sound object (as I did in

many cases), but that is the limit of it: if we do not take into account the diverse scales

present in the various voices that compose a “chord” we will inevitably distort our

understanding of the compositional process.

36 Chromatically or quasi-diatonically as Schoenberg points out (1969, 18). 37 Think of augmented sixth chords, Neapolitan sixth chords, or fully diminished seventh chords which

contain a non-diatonic tone as regards the major scale that borrows them.

24

1.3 Construction of the SHC

1.3.1 Left-Right Symmetry in the Major Diatonic Scale

In this section I will explain the process by which the System of Harmonic Classes is

constructed and how the definitions given in the previous sections constitute an

integral part of this system.

The first relevant concept is that of the primary dominant factor, or resolution of the

tritone (by contrary semitone motion) to the 1-3 (or 3-1) major third. As it was

mentioned previously38 a semitone descending motion to the tonic is not possible in

the major diatonic scale, which, having just one tritone privileges so to say the

ascending semitone motion of the leading note to the tonic in a 7 – 1 motion, whether

the tritone is taken as an augmented fourth (say f – b in a C-major scale) or a

diminished fifth (b – f). According to our definition of primary dominant factor an

authentic “descending” resolution entail the presence of an equivalent major third;

that is, a f2 – 1 descending semitone motion to the tonic should be accompanied by a

5 – f6 semitone motion, as shown in Example 1-6:

Example 1-6

What this means is that the tonic of a major diatonic scale may possesses a

descending configuration that connects it not only to the customary major tonic triad

but to another triad, also possessing a major third, to which the tonic c belongs. The

configuration we are looking for must not only conform to the definition of primary

dominant factor but also conform to that of harmonic domain, as anything other than

a major diatonic scale (all its modes included) would contain another tritone;39 thus

our only possibility is a descending C-Phrygian scale, which conforms to all our

38 See section 1.2.1, Content and Aim of the System of Harmonic Classes. 39 And therefore another potential “tonic”.

25

definitions (now viewed in the new context of a harmonically generated symmetric

pair) and establishes c as the ultimate generating and resolving tone of the system

(Example 1-7):

Example 1-7

In principle the new harmonic domain appears to challenge the hierarchy of the C-

major tonic triad, since the introduction of the new tritone does not provide a τ-

resolution to it. Besides, C-Phrygian belongs by definition to the A¨ major diatonic

scale, or, in our terms, the a¨-d harmonic domain. We should therefore re-define tonic

note as the tonal center of an extended harmonic motion; not only are we now in

presence of an ascending sequence of tones as a harmonic property of the major

diatonic scale40 but, also, an equivalent descending arrangement necessarily

symmetric to it which complements the harmonic tendencies of all possible harmonic

motion to the tonic41. We have moved from the major diatonic scale, defined by a

single dominant factor τ, to a more complex system where that scale is contained.42

1.3.2 Modal Layers

We may call the generating tone of a given symmetric pair of scales (see Example 1-7)

To, and thus in the case of C-major diatonic ascending C-Phrygian descending

symmetric pair we may denote it by To (c). In order to explore the harmonic

possibilities of the system we need to construct the vertical sonorities possible within

40 Harmonically characterized as a major diatonic scale. 41 This amounts to dissociating the tonic from the major tonic triad, C major in our examples. 42 I have yet to define how these scales interact harmonically. The introduction of new tones evidently

suggests the possibility of an exchange between both scales. This transcends the traditional notion of

“borrowed” tones as it can be argued that both scales are directly related through a tonal function:

progression to the tonic.

26

it. We begin by stacking our thirds (any other interval would give us, eventually, the

same results when the process is complete) above and below the main generating tone

To (c), which for convenience will be denoted by a diamond shape note-head:

Figure 1-4

We need now to identify the different tones that constitute this vertical configuration.

I will designate them preliminarily by the letters A (if it belongs to the ascending C-

major diatonic scale) and B (if it belongs to the descending A¨-major diatonic scale):

Figure 1-5

If we now go back to Example 1-7 we may see the possibility of supplementing the

given symmetric pair of scales with a corresponding number of tones above and

below in order to obtain their full collection of notes; that is, constructing symmetric

chords in parallel motion step-wise upwards and, correspondingly, downwards too. In

this case we would obtain a familiar collection of parallel 7th-chords of C-major and

C-Phrygian diatonic scales (Figure 1-6)43:

43 This method of opposing triads appears in Persichetti’s Twentieth Century Harmony, ‘Mirror

Writing’, Ch.8, 172. Persichetti’s work, apart from being a most inspirational and comprehensive

source of compositional techniques and possibilities, is also one of the works that have directly

contributed to the shaping of the SHC here presented. Persichetti does not further elaborate, in a

systematic or any other manner, the results of this process, but proposes them as they stand as a

compositional choice.

27

Figure 1-6

No harmonic consequences seem to derive from Figure 1-6, apart from the fact that

some of the vertical configurations constitute familiar collections of tones such as

some heptatonic and octatonic formations. In any case they would remain unrelated if

we do not find a principle that unites them in a logical, if not functional manner. What

we may conclude from this arrangement, though, is that it can be viewed as a double-

layered arrangement of tones, each layer in turn constituted by A-elements

(originating in C-major) or B-elements (originating ultimately in A¨-major). This

takes us to the next step in the process. Thus the A-elements in Figure 1-6 are c-d-e-f-

g-a-b (no need to include the octave), above it e-f-g-a-b-c-d, similarly above it g-a-b-

c-d-e-f, and then b-c-d-e-f-g-a. The B-elements being c-b¨-a¨-g-f-e¨-d¨, below it a¨-g-

f-e¨-d¨-c- b¨, then f- e¨- d¨-c- b¨- a¨-g, and finally d¨-c-b¨- a¨-g-f-e¨; these are the

modal layers present in the harmonic area represented by the C/A¨ symmetric pair.

1.3.3 Modal Interchange

Whichever consequences may be derived from our system so far, one thing is certain:

every modal layer, or horizontal sequence of notes in Figure 1-6, corresponds to a

specific mode or scale, which, for the purposes of the SHC, are equivalent as they

possess in each case one tritone only, the primary dominant factor. This means that

we are potentially making a different use of the familiar tonal materials, i.e. the major

diatonic scale and its derivate modal formations, since we are isolating sequences of

notes (say, for instance, the sequence of A-elements known as the Myxolydian mode,

starting in g in Figure 1-6) from the context of the chain of chords where it is found,

An example may illustrate this (Figure 1-6b):

28

Figure 1-7

I have omitted all the pitches from Figure 1-6 except for those belonging to ascending

C-major scale (the Ionian modal layer, if the reader prefers), and the vertical

configuration in the third place from the left. The modal layer consisting of c-d-e-f-g-

a-b (any other modal layer would have done just as well) is harmonized by a vertical

configuration from a different scale; a harmonic class, strictly speaking, acting as a

chord: b¨ -d-f- a¨-e-g-b-d. That is, an A-element sequence of tones can be put in

relation to an array of B-elements vertical configurations.

As I mentioned earlier, triadic configuration is a question of choice: in the SHC any

other interval would do just as well to construct its basic design, such as the interval

of a second around To (c), as in Figure 1-8 below:

Figure 1-8

The configuration in this arrangement corresponds to those in Figure 1-4 and Figure

1-5; as previously stated, the choice of thirds does not possesses any priority over

other arrangements, as long as the pairs of tones (g-f, a¨- e¨, b- d¨ in this case, Figures

1-4, 1-5, and 1-7) are symmetric in respect to To (c); any interval will do. We will see

the importance of this as we progress further. For now it is worth noticing that the

cluster sonority above c in Figure 1-8 is also by A- and B-elements, although in

different order above and below c. Thus, for instance, f is a B-element because in the

tones generated by To (c) in this examples (Figures 1-4, 1-5. 1-6, 1-7, and 1-8) the f

has been chosen consistently to correspond to a B-element. In fact, what has happened

is that by moving f above the generating To (c) we have shifted the entire modal layer

too! The pitch f, in our chose configuration, will always generate an f- e¨- d¨-c- b¨- a¨-

g mode (Figure 1-8b):

29

Figure 1-8b

In Figure 1-8b I have expanded the context of Figure 1-8, to illustrate the f-generated

modal layer including the arrangement by seconds and by thirds (Figures 1-4, 1-5, and

1-6). Both arrangements are equivalent and, in fact, harmonically the same within the

context of the SHC.44

Going back to Figure 1-6 we notice that the AAA/BBB arrangement may seem also

arbitrary from this new perspective of modal layers, that is, any other symmetrical

arrangement of A’s and B’s would do just as well. Let’s choose, for our remaining

exposition, AAB above To (c):

Figure 1-9

We notice immediately that the outside voices have swapped their modal origins, b¨

belonging to A¨-major and d§ to C-major. What this means among other things is that

the aural quality of the vertical configuration has changed regardless of how we

decide to understand it. One way of understanding it, of course, is to see c as the root

of a dominant seventh chord symmetric to a half diminished chord. But, as the

following pages will suggest, the tonal implications of the SHC configuration are only

44 They may, of course, sound different, but this is not a decisive harmonic characteristic, at least

within the context of this thesis.

30

one characteristic among others, although, to be certain, one that the composer feels a

special inclination to.

Applying the parallel chord construction (similar to that of Figure 1-6) to the vertical

configuration of Figure 1-9, we obtain (Figure 1-10):

Figure 1-10

Note that in all these examples accidentals only affect the note they modify, and should not be

read as affecting other notes.

Here the arrows denote the fact that all the notes originating from any given tone

above/below To (c) will maintain their modal quality as an A- or B-element modal

layer (as do, obviously, the ascending C-major and descending A¨-major diatonic

scales). What is more, should we favor a different arrangement of notes in the vertical

configuration, say by seconds as in Figure 1-8 above, the tones above and below To (c)

would still preserve their modal origins. For instance the bottom pitch d in the first

vertical configuration (the one built around To (c)) would be promoted together with

its Dorian modal character, the f (two pitches below To (c)) would now prompt an

ascending modal structure, but one that retains its Aeolian character, and so with the

rest. This is equivalent to say that we have operated an exchange between the upper

and lower layers of this particular SHC configuration: A¨-major lends its B-element

sequence of notes to C-major above and this in turn sheds an A-element sequence that

now appears below. This mechanism I label modal interchange. There are symmetric

modal interchanges, such as the one in Error! Reference source not found., in

which both the ascending and descending groups exchange the same number of modal

31

layers (these are the object of the present paper), but there are also cases in which a

different number of layers is exchanged and asymmetric SHC configurations result.45

The different horizontal configurations, what I have termed modal layers) constitute

the modal layers that combine, in different intervallic arrangements, around the

symmetric C/A¨ (in this case) pair of scales (see 1.3.2 Modal Layers). They are in fact

as important as the symmetric pair itself despite the fact that they were originated by

it.

1.3.4 Harmonic Classes

This section will deal with the specific combinatory process by which we obtain the

synthetic modes we termed harmonic classes (see 1.2.1). I will turn to a C/A¨

symmetric pair which constitutes the main harmonic area to be found in the analysis

of the piece.

1.3.4.1 Interval Equivalence and Virtual Configurations

As it has been stated above, the same procedure of designating the symmetric tones

around To (c) and constructing the modal layers they generate can be applied should

we use the intervals of a second, a third, or a fourth (and, needless to say, their

inversions). By now it should be noticeable that some of the vertical configurations in

intervallic construction by thirds or seconds do not have all the representative tones

(with or without accidentals) of a 7-note scale (Figure 1-11):

Figure 1-11

Or, what is the same (Figure 1-12):

45 These are the most common and present many possibilities as regards the synthetic derivation of

scales such as the one to be seen later for the major diatonic scale, Section 1.4.

32

Figure 1-12: Modal Layers by Seconds

Where I have separated the modal layers into a 3-stave system for clearer reference.

These modal layers are AAB above the C/A¨ symmetric pair and ABB below (starting

from the bottom pitch, bottom staff, as we naturally read them). By reorganizing the

modal layers around To (c) to conform to an SHC configuration by thirds we obtain

Figure 1-13:46

Figure 1-13: Modal Layers by Thirds

Again, the modal layers present in Figure 1-12 have been moved around the pivot c,

or more accurately around the C/A¨ symmetric pair, while maintaining their

characteristic configuration (A-element in a configuration by seconds will remain an

A-element in a configuration by thirds, regardless of their position above or below To

(c), and so will B-elements). Thus, the modal layer configuration will be now AAB

above the C/A¨ symmetric pair and ABB (bottom pitch, bottom staff) below. They

46 Notice that we could have started our exposition with an SHC configuration by thirds (or fourths as

in Figure 1-14) and rearrange the layers around To (c) according to whichever interval may be needed

to complete the full intervallic arrangement by seconds, thirds, and fourths. We simply had to start

somewhere.

33

(accidentally we may say) coincide, the modal layers generated by g and b¨ having

move downwards, and those generated by f and d¨ upwards.

Let us now take, for instance, the fourth verticality in the SHC configuration by thirds

(Figure 1-13). It relates to the vertical configuration corresponding to the IV degree in

the ascending C-major scale. We notice how in Figure 1-13 its tones are a-c-e¨-g

below and f-a-c-e¨ above; whereas in Figure 1-12 (arranged by seconds) the

corresponding notes are d-e¨-f-g and f-g-a-b¨: in the former b¨ and d are missing, and

in the latter c is missing. To configure a full 7-note scale in each verticality we must

add the tones present in all possible SHC intervallic configurations: SHC

configurations by thirds (Figure 1-12), by seconds (Figure 1-13), and by fourths

(Figure 1-14 below). This is equivalent to adding extra modal layers around To (c)

generated scales (the basic C/A¨ symmetric pair) until all the notes are replicated

above and below them. We may conclude that the missing notes are virtually there,

although not shown in the examples (Figure 1-13b):

Figure 1-13b: adding an extra layer

Where the diamond notehead in Figure 1-13b corresponds to the added modal layer

above and below the C/A¨ symmetric pair. Evidently, in the first vertical

configuration, the one generated by To (c), the d at the top must replicate the d below,

and so with the b¨ to avoid introducing a chromatic note into the 7-note scale. The rest

of the modal layer will be determined by this, and as a result, an additional tone will

be added to the rest of the vertical configurations. If we continue this process of

adding modal layers we would eventually obtain all the tones we need: a complete set

of 7-note vertical configurations. Doubtlessly it is easier to consider the different SHC

34

configurations (by seconds, thirds, and fourths) equivalent, and borrow the notes we

need from any of them at any given time.

In compositions using the SHC, such as the one I present in this paper, there is no

normative criteria that indicates what should come after or before any sonority.

Sometimes smooth voice leading is a factor in the choice of tones, sometimes it is

color or the anticipation of another scale/sonority. The same way the composer could

build a complex sonority over almost any note just by adding the suitable tones from

the harmonic class to which that given note may belong: they are all virtually (that is

to say, potentially, at the discretion of the author) there.47

Finally the SHC configuration can also be built in fourths (Figure 1-14), with a BBA

modal layer configuration above, and a BAA configuration below:

Figure 1-14: Modal Layers by Fourths

We may notice that the SHC configuration by fourths does not supply any of the

verticalities present with any additional tones that were not there in the previous SHC

configurations (by seconds and by thirds), with the exception of the first verticality

(something obvious, since it simply rearranges the modal layers around To (c)).

1.3.4.2 Harmonic Classes as SHC degrees

Once we have all the basic configurations by thirds, seconds, and fourths, and

following the principle that they are all interchangeable (any number of modal layers

47 This cannot be said of tonal music since adding further notes to a triad can dramatically change its

function. In the SHC the function of a note/sonority is primarily to be representative of the harmonic

class to which they belong.

35

can be constructed around To (c) and re-arranged in any intervallic configuration), we

take those vertical configuration as independent scales or harmonic classes. In the

specific SHC configuration shown in Figure 1-12, Figure 1-13, and Figure 1-14 the

harmonic classes are:

SHC-I (verticality corresponding to degree I of the ascending C-major scale, let us

call it verticality-I): c-d-e-f-g-a¨-b¨

SHC-II (verticality-II): c-d-e§ -e¨-f-g-a§ -a¨ -b

SHC-III (verticality-III): c-d¨ -e-f-g-a¨-b (Double Harmonic)

SHC-IV (verticality-IV): c-d-e¨ -f-g-a-b¨ (B¨-major scale!)

SHC-V (verticality-V): c-d§ -d¨ -e¨-f-g-a-b¨ -b§

SHC-VI (verticality-VI): c-d¨-e§ -e¨ -f-g-a¨ -a§ -b

SHC-VII (verticality-VII): c-d§ -d¨-e§-e¨-f-g-a§ -a¨ -b§ -b¨

As we see some of the harmonic classes are clearly synthetic, such as SHC-I which

presents a major diatonic tetrachord plus a melodic minor tetrachord in the same key.

Four SHC-degrees (II, V, VI, and VII) are chromatic, meaning the same tone is

simultaneously present with different accidentals, and two are familiar scales such as

the Double Harmonic48 (SHC-III) and the B¨-major scale (SHC-IV).

1.4 The Major Diatonic Scale as a Harmonic Class

Diatonicism, the bedrock of Western theories of scale for both modal and major-

minor tonalities, is largely responsible for our ability to perceive harmonic function.

Because of the special intervallic properties of the diatonic scale, which allows us to

deduce the entire structure from just a few notes, we are able easily to experience that

tonal centricity on which harmonic function depends. Diatonicism should thus rightly

be involved in any theory of harmonic function, even one that deals with highly

chromatic music. (Harrison 1994, 25)

48 Typically a minor scale with two augmented seconds (Persichetti 1961).

36

As we saw the SHC can be viewed as an organization of diatonic scales and modes

(horizontal configurations) and synthetic symmetric modes (vertical configurations);

the latter I labeled as harmonic classes. From a purely harmonic perspective there is

no reason why the horizontal and vertical configurations should be deemed

qualitatively different: they all define specific harmonic domains which in the case of

the major diatonic scale is customarily called key area and more generally tonality.

The previous section also showed that a harmonic class may also constitute a major

diatonic scale; this is the case with c-d-e¨ -f-g-a-b¨ , which is symmetric around c and

it contains the tones of the B¨-major scale.

What this means in practice is that the major diatonic scale can be seen, within the

frame of the SHC, as just another possible harmonic resource: traditional tonality,

then, may be viewed as a special case of a system of harmonic classes such as the one

presented in this paper. If this is the case, we should find a term that describes the

equivalent of tonality for any other of the harmonic classes, but for our purposes tonal

or tone centered may be sufficient.49

What the analysis must take into consideration is that the introduction of this major

diatonic scale does not represent a change in harmonic medium as, say, shifting from

diatonic to pentatonic would be in the common practice period, or from tonal to serial

in the same piece of music: in other words it is not a case of harmonic “collage”. The

major diatonic scale, with only one tritone, maximizes the number of consonant

intervals and it is therefore highly stable harmonically; and this, the specific aural

character of the scale, is what should be taken into account when faced with the

inclusion of that scale in the context of the SHC.50

49 As pars pro toto, that is, the attribute (tonality) of a part (the diatonic scale within the SHC) used to

represent the whole (whatever is being established when a given number of harmonic classes,

including the major scale, are conjunctly used constituting an organic whole is supposed to do when

being established). 50 Let us remind ourselves that in the major scale, for instance, a highly dissonant sonority such as the

diminished-7th chord is not possible, as it is composed of two tritones.

37

2 Analysis of Molto Adagio

Because some species of classical diatonic theory is the first, and in many cases the

only, music theory that we are exposed to, its status as theory, a way of looking,

easily hardens into nature: the way that music is. In response, I will occasionally

invite readers to “forget” that major and minor triads are acoustic consonances

generated from roots that occupy a position in a diatonic scale with respect to some

tonic. (Cohn 2012, Introduction, x).

The analysis of this piece needs to take into consideration the composer’s approach to

tonal materials. As Cohn suggests in the preceding quote, familiar harmonic resources

such as triads may be occasionally used in non-diatonic contexts. This does not mean

that these sonorities (triads) are deprived of their potential functional character, but,

on the contrary, that this functional character is a possibility among others. In Molto

Adagio the interaction of the diatonic scale with other scales, together with the

presence of triadic sonorities that can be potentially understood as triads (again,

because of their potential functionality), makes it necessary to integrate a tonal

perspective within the framework of the SHC. In order to do this a linear analytical

approach, emphasizing the relevant voices and sonorities, and in particular those that

may be structurally significant, will be offered (see 2.2.4).

As I will show, the scale, not the chord, is the focus of harmonic action. This

translates into an overwhelming emphasis on the horizontal movement of the voices

that achieve an almost complete independence from their vertical nexus. The different

voices represent often different scales (harmonic classes to be sure) carrying their own

central tone (To (c), To (f), and others) which sometimes coincide and sometimes not

with that of other voices. When they do coincide we are presented with a recognizable

harmonic area where different harmonic classes belonging to the same central tone

can be found. Thus, at the beginning of the piece (say the first eight bars) two

symmetric harmonic classes, c-d-e-f-g-a¨-b¨ and c-d¨-e¨-f-g-a-b define a C/A¨

harmonic area that may be considered by the composer a tonal unit: that is, within

that specific passage the composer may have chosen to use the tonal tendencies

present to structure the music.

38

The question then arises as to what happens next, that is, how the transition is

accomplished from this (or any other within the piece) well defined harmonic area to

the next. In common practice the favored technique of effecting a harmonic transition

to a different tonal area is that of modulation. Usually a pivot chord, or note, is

introduced at the required moment, such as the end of a section, and a new tonic chord

is presented and established soon after. In Molto Adagio, too, a similar technique is

present. Although we cannot rigorously speak of modulation within the framework of

the SHC, the transition to a new area of harmonic activity, introduced via a smooth

handling of chordal voices, or a non-harmonic tone in a tritone relation to the current

tonal center, such as for instance f© in the context of C/A¨ symmetric pair, can be said

to be cadential or modulatory. These areas are, as a rule centered upon one,

sometimes two, tonal centers. These tonal centers are established by means of the

harmonic classes generated by that specific tone. To define these areas, where a

specific harmonic activity can be identified, I will speak of a tonal module, which was

defined as a musical passage containing, and delimiting, an identifiable harmonic area

(or areas) and/or tonal center(s). We may clarify this with two examples.

The tonal center, for instance, may be in the form of a diatonic scale unrelated to the

harmonic areas present in the module. An instance of this is the B¨-major scale near

the end of Molto Adagio (Vln. I, mm. 66-79, see appendix) that stands in contrast with

the main F-, D¨-, and D-centered harmonic classes (see 2.2.3).

Occasionally, though, the presence of harmonic classes from different tonal centers

can make it more difficult to identify the overall harmonic tendency of the music,

such as in mm. 55-56 (Example 2-1):

39

Example 2-1

Here we can see a confluence of a D-centered harmonic class in Violin I (1 and 2): d-

e¨-f-g- a¨-b-c© 51, and G-centered harmonic class: g-a¨- b¨-c-d-e-f©52, in Violin II (1and

2)53, and Violoncello The violas carry fragments of different harmonic classes, which,

fully identifiable or not, certainly reinforce the D-centered harmonic configurations

with the presence of the c© and f©. The occurrence in these measures of eª-e¨, bª-b¨,

and aª-a¨54, clearly point to the G/E¨ and D/B¨ symmetric pairs. Taking into account

the presence of c© in most instruments we may rightly conclude that the music is

moving away from the G-centered harmonic area. The c©-g dominant factor (Vc. and

Vla., m.56; notice how the Vc. plays now a D-centered harmonic class) resolves, if

not by smooth voice leading at least definitely so harmonically, to a harmonic

configuration in Vl. I and II, that clearly defines a D-centered harmonic area (in fact

D major itself, if only in this measure). Thus, overall, we see that within a tonal

51 See the full context in the appendix, in particular mm. 54-57. The symmetric pairs around d are e¨-c©,

f-b, and a-g. 52 Symmetric pairs around g: a¨- f©, b¨-e, d-c. 53 I will be referring to the instruments simply as Violin I and Violin II when the whole section carries

the same harmonic material, even when they are marked divisi. 54 The pitch a appears in m. 54, and then again in m. 57; see appendix.

40

module different harmonic configurations can coexist, and that, however complex the

music may appear, a tonal direction can be discerned. In the analysis I will use linear

analytical charts (see section 2.2.4, Linear Approach) to highlight tonal centers (or at

least tonal tendencies), cadential moments, and structurally significant harmonic

events. Let us remind ourselves that, whichever the harmonic makeup of Molto

Adagio, ultimately the successive harmonic areas that support the music, the tonal

quality of the piece resides in the harmonic relations between those harmonic areas. It

is not the aim of this thesis to strip Molto Adagio measure by measure to show the

countless connections between the possible harmonic domains and harmonic classes

of all kinds found in it. Quite the opposite, this paper takes a “holistic” view, so to say,

of how tonality (in the sense I have endeavor to define above) is constructed using the

material provided by the SHC, which is, whichever its implications, an organized

collection of harmonic resources.

2.1 Compositional Background

Molto Adagio was originally the second movement of my Double String Quartet. It

was written in the winter of 2009, while the composer was on a sabbatical break from

his studies in Estonia. The aim of the work was in principle to try out the composer’s

own system of harmony in an extended work and within a suitable genre: that of a

string quartet, which is the medium par excellence to experiment and display at least a

composer’s harmonic thinking. It soon became clear that a four-part setting (which is

what makes a quartet suitable for expressing and develop ideas related to harmony)

would not suffice, because the chordal makeup of the piece constantly calls for

sonorities that average 5 to 6 voices (not rarely 7) treated independently. Also, the

characteristic double-mode approach to harmony (explained above as the symmetric

pair, the basis of the SHC) would benefit from an antiphonal setting between two full

distinct groups of instruments. This is clear, for instance, in the first movement of the

piece, which features a double exposition dealing with the same motivic material in

two different but related harmonic classes, the aim being: how would a different scale

“see” or change the character of a specific motif? The SHC does not favor nor

exclude any specific scale or mode (traditional or not), key or chord.

41

The difficulties of having such a piece performed soon obliged the composer to seek

alternative settings, and so a mixed chamber version of the piece55, and then a string

chamber version (the one analyzed here) were eventually written.

2.2 Musical form, harmonic and motivic material

The work is written in an ABA’B’A’’ form, the letters A and B referring to two

distinct motif complexes to be defined shortly (see 2.2.2). Section A is the exposition

part, section B constitutes a contrasting section, A’ is a very short restatement of

section A, and finally section B’ serves as a “retransition” to the closing recapitulation

in section A’’. The work progresses harmonically from a C-centered harmonic area to

a F-centered one, and thus cannot be said to have a definable key in a traditional tonal

sense; it nevertheless displays clear tonal tendencies that a detailed analysis can reveal,

as I will attempt to show in section 2.1.2.

Traditional tonalities are signed as usual, such as for example D¨, meaning D¨ major,

while harmonic areas with identifiable tonal centers are placed in inverted commas,

thus “F” means an F-centered harmonic class or group of harmonic classes. The

harmonic areas articulating the main formal sections of the work are as follows:

55 Featuring Flute, Oboe, Bass Clarinet, Trumpet, Trombone, and the usual string section with double

violins and including Double Bass.

42

Table 1

Section A

mm. 1-35

Section B

mm. 36-44

Section A’

mm. 40-50

Section B’

mm. 51-57

Section A’’

mm. 58-79

Module 1

(mm. 1-

26)

Module 2

(mm. 26-

35)

Module 3

(mm. 36-

39)

Mod. 4

(mm.

40-44)

Module 5

(mm. 45-57)

Module 6

(mm. 58-79)

Symmetric Pairs / Main Tone Centers

C/A¨

D¨

(“C” still

present)

F/D¨

C/A¨

D/B¨

G/E¨

F/D¨

(B¨ major, “D” harmonic

class also present)

Instructions in the score

Molto Adagio (mm. 1-35)

Abiertamente

(mm. 36-39)

Expresivamente (mm. 40-

41), Con Extrema…

(mm. 42-50),

Súbitamente (mm. 51-52)

Ampliamente

(mm. 53-63)

Intensamente

(mm. 64-79)

Modules refer to tonal modules, as defined above (see Definition 5). As Table 1

shows, the formal areas may contain more than one tonal module. Sometimes, as in

the case sections A’ and B’, a tonal module may overlap the formal sections. Before

we embark on a detail analysis of Molto Adagio we need to take some considerations

into account; especially what I have termed the triadic grid.

2.2.1 Triadic Grid

The composer has strictly followed a triadic pattern to facilitate the writing of this

piece. It was a question of choice: any sonority pattern could have done it just as well,

the analysis probably being a little more complex but functionally (in the sense of a

given sonority representing the particular scale from which its tones are drawn) the

same. Triadic sonorities afforded me a familiar terrain and I moved through them

comfortably, always with an eye on “what could they mean?” if they were the

traditional triads they resemble. Often in the piece this resemblance between the

43

triadic units I used and the triads (as they appear in traditional tonality) they recalled

had a defining effect on the music that was being written. If it is certain, therefore,

that tonal associations and familiar sonorities should not be taken at face value when

analyzing the piece, the consistent triadic nature of vertical configurations must be

established from the very beginning, as it was the author’s purpose. The steady use of

“stacked up” thirds that compose most of the harmonic texture of the work constitute,

in this sense, a network , or “grid”56 of vertical sonorities structuring the harmony of

the piece throughout. These vertical triadic sonorities carry intervals of a third up to

the ninth of the chord, although vertical sonorities with more than 5 notes can also be

found (see for example mm. 40-41). An example of a 5-note triadic sonority may be

found in bar 8, second beat, constituting a possible “Cm§7¨9” (see Example 2-2):

Example 2-2: m. 8, second beat

The fifth of the chord, as it is sometimes the case when it constitutes a perfect fifth

above the tonic, has been omitted. The following example may be taken as a sample

of the way sonorities are commonly built in the piece (Example 2-3):

56 I used the word grid to indicate that the harmonies used in this composition mostly are built up by

thirds.

44

Example 2-3: “C chord”

Tonally speaking, the circled chord in the example is an inversion of the chord which,

in its prime form, could be spelled: c – e – g§ /g¨ – b§ – d, that is, a C9 chord with a

double (natural and flattened) fifth added.

By mapping the most salient of these sonorities we will be able to follow, later on, the

more complex crisscrossing of symmetric pairs and their wealth of harmonic classes

that, at a deeper level, are the fundamental pillars above which the music is

constructed. They move in different and often contradictory directions; approaching

them without the safeguard of pivot sonorities may render the analysis unmanageable.

We must bear in mind then, that when the author speaks about progressions and voice

leading (referring to the piece in hand) he ascribes the concepts a family resemblance

with those of traditional tonality.

2.2.2 Motivic Material

A two-note descending motif (motif 1, circled in Example 2-4 below), is a descending

major second with which the piece starts (recalled in m.42, at the start of the brief

recapitulatory A’ section):

45

Example 2-4: motif 1

It is usually found in melodic juxtaposition to itself; either repeated such as in mm. 1-

2 (Vln. I and Vln. II, Example 2-4 above), or in dissonant pairs such as in mm. 3-4

(same instruments, see appendix) and mm. 45-46 (also), among other occurrences. It

seldom ascends, and when it does then it is for specific expressive purposes, such as

the predominantly dissonant counterpoint between Violoncello and Double Bass in

mm. 12-15, or “leading tone” resolutions,57 such as mm. 44 and 48 (both in the Vc.).

In the exposition part (section A) an ascending following motif 1 stands usually in

high contrast to it. This interval varies in size: a tritone (m.2, Vc.), a perfect 4th (m.25,

Violin I), a perfect 5th (m.43, Vc.) a minor 3rd (m.5, Vln. II), a minor 6th (m.7, Vln. I),

and so on (see appendix) – a series of them in succession having a clear dramatic

content.58

Another descending motif (motif 2), is a stepwise descending sequence of three notes

that should be considered a scale fragment (see the encircles notes in Example 2-5):

57 Meaning tones that have a leading ttone quality, but that do not represent the 7 or 2 degrees of the

scale. 58 Such as in mm.7-10 and 24-29. This is “echoed” in the B sections by rising scale-wise fragments in

melodic contrast to motif 2. I am tempted to call this rising interval a motif too, maybe the “implacable

answer”, but I did not write it as such, as a motif, but only as a gesture that stands in emotional contrast

to motif 1.

46

Example 2-5: motif 2

It suggests different tonal centers at different times, and usually it is followed by a rest,

a sequence, or a leap. Motif 2 is most frequently displayed in sections B and B’,

although the expositional first bars of the piece already display it in both the

Violoncello (m. 2) and Violin II (m. 4). In sections B and B’ motif 2 is usually

accompanied by stepwise ascending scale formations, as can also be seen in Example

2-5, just above the circled instances of motif 2 (see appendix for an overall view).

47

2.2.3 Harmonic Material

At first sight, a tonal approach to the analysis of this piece may render an overall

harmonic motion from a main C-centered harmonic area (harmonic classes generated

by the C/A¨ symmetric pair), which can be seen as a “dominant”, finally resolving to

F-centered harmonic area, representing a “tonic”. More precisely, we could possibly

observe a motion from “C” (mm. 1-26) to “D¨” (mm. 26-35, related to the

“subdominant” “F” through the F/D¨ symmetric pair), moving back to “C” (mm. 40-

44), and quickly shifting the tonal emphasis to “D” and “G” (mm. 45-57), which

merge into the final F-major area (mm. 58-79); harmonic areas and expressive

sections may overlap (See Table 1, section 2.2 for reference):

A Section: Molto Adagio

Module 1, mm. 1 – 26; C-centered harmonic area

(“C” harmonic classes) in all instruments.

Module 2, mm. 26 – 35; D¨ major introduced (with a preceding c – g¨ tritone in

Violin I and Viola, m. 26, signaling a “A¨7” dominant of D¨) in m.32; all instruments.

B Section Abiertamente, danzante (“Open[ly], dance-like”) and Expresivamente.

Module 3, mm. 36 – 39:

D¨ major is joined by its symmetric F pair by means of the f-g-a-b¨-c-d¨-e¨ harmonic

class in the upper strings. This section ends with a possible “(I-) V-VI¨” deceptive

cadence in “F”59 (m. 39-41, see Example 2-11: “Deceptive Cadence”).

A’ Section:

Module 4, mm. 40 – 44; C-centered harmonic classes in all instruments.

Expresivamente; this marking may be considered the harmonic and expressive center

of the entire piece. It comprises two measures (40-41) and one C-centered harmonic

class: c-d¨-e¨-f-g-a-b.

59 “F”, to be sure, constituting chord I.

48

Con extrema delicadeza y dulce añoranza; mm. 42-44 introduce the c-d-e¨-f-g-a-b¨ C-

centered harmonic class. Violin II retakes the main descending major second motif a

minor third higher (g-f), over a 6-note chord whose prime form is that of an E¨9

(whose tone content is e¨-g-b¨-d-f), with an added sharp 11th (a§) in the bass. The tone

f© is introduced in m. 44 (Vc.), opening the G (V) and D (V of V) harmonic areas.

Module 5, mm. 45-57; This module overlaps the A’ and B’ sections. Here we find G-

and D-centered harmonic classes: g-a¨-b¨-c-d-e-f©60 (Vln. II), and d-e¨-f©-g-a-b¨-c©

introduced in the bass61 (Vc., although this voice also reinforces the tones other

harmonic classes in the upper voices, such as a¨ or b§ ) and take over by Violin I. The

harmonic materials that make up Module 5 overlap the next section (B’, see next

bullet point), thus avoiding to conclude this section (A’) with the customary cadence.

The presence of “G” and “D”, with their obvious tonal implications of dominant, and

dominant of the dominant, possess a profound (if I may say so) significance as

regards the overall orientation of the SHC. They highlight, by their previous absence

and eventual dissolution, the “subdominant” tendencies of the harmony noted above

in reference to the first three modules of the piece. We will have occasion later on

(2.2.4.4) to return to this important point.

B’ Section (mm. 51-57) Súbitamente suplicante, creciente intensidad (“Suddenly

imploring, increasing intensity”) and Ampliamente; resolute & opening up (mm. 53-

63). The first part of this section, Súbitamente, consists of two bars (mm. 51-52) that

state briefly the syncopated motif of section B (introduced for the first time in Vlns.I

60 Although an e¨ (m. 45) and a c© (m. 56) appear in this harmonic class, they can be considered to be

reinforcing the d-e¨-f©-g-a-b¨-c© harmonic class in Violin I. 61 Mm. 43-44 in the Violoncello sums up, as it were, all the motivic materials found so far: a falling

major second followed, or contrasted, by a rising larger interval (a 5th in this case), and a three-note

falling motif resting, surely enough, on a dissonant tone (f©); this tone retakes motif 1, now rising (f© -g,

structurally important as it will be restated in this form at the concluding measures of the piece as a-b¨):

the vertical sonority above f© is clearly a D7(¨9), signaling a shift to a D-centered harmonic area.

49

and II and Vla., m. 36), characteristically a descending interval of a second. The next

part, Ampliamente, elaborates this, together with motif 2, reaching a climax at m. 58,

in which a change of harmonic area occurs.

A’’ Section

Module 6, mm. 58 to the end (m. 79): a fff brings about62 the “subdominant” “F” area,

that is, the F/D¨ symmetric pair (Vln. II, Va., Vc.1), which is the generator of the F-

centered harmonic classes.

The following bars (mm. 58-63) serve as a transition to the final A’’ section. The

harmonic significance of this transition is that of a bridge between Module 5 (G- and

D-harmonic areas) and Module 6 (F-harmonic area). In mm. 59-61 Violin I plays an

unmixed F-major scale that takes the piece to its highest pitch, f§, over a descending

e¨-d¨-c, unmistakably establishing the F/D¨ symmetric pair.

Intensamente (“Intensely”, mm. 64 to the end); the sixtuplet that follows, continued in

a stepwise wavy line to reach the high c, possesses an expressive character; the

implied F resolution on the third beat of m. 64 (f-e-f in the crotchet triplet) is thwarted

by the presence of b§ in the Violoncello1; the cellos carry a dissatisfied D-major line

until the end, although the sharps are gradually dropped. These two bars are best

looked at as an extension of the transition to the next section.

Characteristic of this section is the reversal of the descending major second of motif 1

(that constitutes the main motif of the piece): this reversal takes us to a motif of major

structural importance (m. 74, Vln. I, and m. 78, Vc.), since it implies a 3-4 (-3)

neighbor-note figure that serves to signal the primary dominant factor (or potential

tritone resolution, see 1.2.2). All harmonic classes present coalesce into F major after

this implied resolution in m.74.

Final cadence (mm.73 to the end) features a clean F major scale in pandiatonic63

writing.

62 Abruptly, although Violin I –m.57– uses f§ to bridge the sharps/flats harmonic areas 63 “Pandiatonic writing is a specific kind of “static harmony” in which “[T]he vertical structures are

combination of any number of tones from the prevailing scale, placed in variable spacings.”

(Persichetti 1961, 223).

50

As we can see, it is possible, at least temporarily, to entertain the possibility of

analyzing the piece borrowing notions from harmonic analysis: mainly those of chord

and function.

2.2.4 Linear Approach

The problem with drawing a full voice-leading graph representation of my piece is

that, despite the fact that the melody shows clear scalic tendencies, with structural

“peaks” in c64 (m. 25-28, the proposed “dominant”) and f (m. 61, f being the proposed

tonal center of the piece), there is a discontinuity, or even incongruence, between the

contrapuntal and harmonic layers of the movement. As Humal notes65:

The contrapuntal structure of tonal music is based on a continuous process of voice

leading. Form, on the contrary, is based on the articulation of the whole into discrete

units, usually by means of harmony.

There are certainly leading-tone motions (fragments of them exist at one point or

other in all the voices), or chord progressions within this piece, but they cannot

account for the overall harmonic structure, which is organized around connecting

different symmetric pairs (see Ex. 1-1): that is, binding harmonic classes together

which have no traditional tonal relation to each other, apart from a given tone (the

tonal center of the symmetric pair), which generates them. This does not mean that the

piece lacks “tonal” continuity as such. As I explained in Part I, the principles that

govern the SHC are essentially tonal: tritone dissonance, hierarchy of the diatonic

scale, functional (at least in semblance) chord connections, and so on. Besides, the

triadic grid (see 2.2.1) that makes up the present piece makes it almost unavoidable to

draw a parallelism between tonal progressions and those present in this piece, and so

the harmony used to articulate the different “discrete units”66 (which we may identify

64 With an expressive, non structural, lean on its upper neighbor, d§. 65 (Humal, Studies on Tonal Structures; Introduction and Fourteen Analytical Studies (Concise English

Version): Contrapuntal Analysis 2007, 136). Emphasis mine. 66 This is somewhat problematic. As Professor Kotta points out there can be discrete units of texture

and discrete units of form. The presence of various harmonic materials within a given tonal module

may, according to him, constitute textural shift, and therefore a discrete unit of texture. I am inclined,

51

as tonal modules, see 1.2.3, or in other words), can be presented as following a tonal

plan,67 employed here as a heuristic tool, and therefore justifying, at least, a modular

linear approach (see Definition 5) in the analysis of Molto Adagio.

2.2.4.1 Module 1: mm.1 - 26

The beginning of the piece may be presented as in Example 2-6.

Example 2-6

For quick visual reference, and clarity, white noteheads represent either minims or

semibreves, just as in the score; the shorter values (crotchet, quavers, semiquavers)

have been represented simply by black noteheads (dotted noteheads represent dotted

values in the score). The solid stems simply connect significant tones, either because

they form part of a progression, an important melodic line (in which different

instruments may contribute to the melodic shape, such as the one in the upper staff of

Example 2-6, which is shared by Vln. I and Vln. II). As it can be observed in the same

example, a bracket, or joined stems, can also represent a primary dominant factor

defining a specific harmonic domain. The broken-line stems only appear in this graph,

and they will be discussed shortly.

however, to see the tonal module as a harmonic confluence in a given, identifiable length of time,

regardless of how conflicting the harmonic configuration may be. Also, in the analysis the tonal

modules have been used consistently as units that provide the music of Molto Adagio with an

alternative musical form (that construct its tonality, strictly speaking). It would be perhaps ideal to find

a concept that isolates those textural shifts as formally significant (besides their description as

harmonic classes, etc.), and this may be a task to tackle in a future work. 67 Although as noted the harmonic areas at some point divorce their content from the musical form that they commonly support. See Figure 2-4 and Figure 2-5 below for an intuitive graphic representation of this.

52

The overall harmonic tendency of this opening passage is the establishment of a C-

centered area, first in what could be defined as a “dominant” tendency (all the notes

present in mm. 1-2, except the initial f in the bass, spell a C9 chord), and then as a

“tonic” tendency, which is announced by the f-b tritone, as I proceed to explain.

The high pitches, represented in the upper staff, describe an e-b¨ (Example 2-6)

melodic line in which the high f is reached, ultimately, by an upward leap of a minor

sixth (a – f). In m. 19 (see appendix), the Violin I, 1 prepares a change of module by

taking up the C-major scale (m. 19), and abandons it (m. 30) after the appearance of

the chromatic g¨ tone (m. 26, see appendix for details). The fact that the highest pitch

in the music so far, c (Vln. I, 1, see appendix), is reached without smooth motion or

due preparation, after a consistent step-wise ascent, is significant (broken-line stems

in Example 2-6). The isolated high c in m. 6 is carried on to m. 7 (see appendix),

where it is joined by e and b¨ (Vc. and Vln. II, see appendix), thus constituting the

primary dominant factor of the F harmonic domain. In other words, the high c in

Violin I, 1 signals unambiguously the establishment of the F harmonic domain (i.e.,

“dominant” C). The rupture of “linearity”, (broken-line stems), is in contrast with the

fact that from now on the music’s tendency is to establish c as a tonal center

(specifically in Vln. I, 1, as I just mentioned).

The bass in this opening passage (see Example 2-6) has a downward motion that

emphasizes the dominant harmony of c. It is possible to see the e in m. 2 as an

embellishment, no doubt, but its inclusion in the primary dominant factor defined by

the e-b¨ tritone (F harmonic domain) invests this pitch with a higher harmonic

significance. Extending the melodic progression in the bass further to m.12 (refer to

appendix for details) allows to span the initial f an octave lower (Example 2-7):

Example 2-7

Where the graph should be interpreted as in Example 2-6. The linear descent in the

bass masks the fact that the harmonies it supports are drawn from a heterogeneous

collection of scales (although still in Module 1, where the harmonic classes are

53

always related to a c center). By the time we reach m.12, the different instruments

have taken various fragments belonging to either one of the C-centered harmonic

classes already described, and built non-tonal chords whose semblance of

functionality resides in the similarity they may possess with tonal chords.68

Tonally, the harmonic continuity of this passage (mm. 1-8) is guaranteed by the

presence of two C-centered harmonic classes: c-d-e-f-g-a¨-b¨ (Vc., m. 1, taken up by

Vln. II in m.5, as can be seen in the appendix), and proceeding to c-d¨-e¨-f-g-a-b C-

centered harmonic class (which first appears in Vln. I and Vla. in m. 4). Together

with the C-major scale played by Violin I, 1 in mm. 19-30, the hierarchic relation of

these harmonic classes (C major can be considered a harmonic class, as viewed in 1.4)

is reflected in Figure 2-1:

Figure 2-1

The C/A¨ symmetric pair has been isolated in the middle staff for clarity; further, the

harmonic configuration of SHC-I (configuring now a C-centered c-d-e-f-g-a¨-b¨

harmonic class, see section 1.3.4.2) corresponds now to an ABB-element

configuration of modal layers (symmetrically reflected below To (c) as AAB-element

configuration, starting from the bottom note). Additional modal layers, denoted by

diamond noteheads, have been added above SHC-I to obtain the necessary number of

pitches to complete SHC-IV and SHC-V, which are the only degrees that correspond

68 As in mm. 19-23, where the vertical sonorities describe a “II – I – IV” progression in a C-centered

context of harmonic classes.

54

to the c-d¨-e¨-f-g-a-b harmonic configuration; all other SHC degrees will posses either

b¨ or a¨ (or both). These additional modal layers, it may be remembered, preserve

their modal (A- or B-element) character whether above or below To (c).

Among the various consequences that can be derived from Example 2-8, is the fact

that the progression between the two C-centered harmonic classes present in Module

1 of Molto Adagio corresponds to a “I – IV” or “I – V” progression indistinctly, since

the pitch content of SHC-IV and SHC-V is identical. Going back to Table 1 we may

observe that, most of the tonal modules present correspond to the hierarchic relations

established in Example 2-8: “subdominant” (F/D¨) and “dominant” (G/E¨); D/B¨

being easily understood as “V of V”. These harmonic relations work at a deeper level

than the progression between SHC degrees,69 but nevertheless reflect them. Once the

C-centered Module 1 has been established, the descent to the lower part of the

symmetric pair begins.

2.2.4.2 Module 2: mm. 26-35

This section has a clear D¨-major aural feeling. The appearance of c-g¨ tritone in m.

26 announces the proximity of a more or less orthodox modulation to D¨, which is

echoed in all voices except for an errant aª in Violin I (m. 35, see appendix). One may

dispute whether this aª is structurally significant enough. The inclusion of an “odd”

note should remind us that the texture of the piece is multilayered: at any point in time

may one of these layers come to the surface and affect the harmonic course of the

music. The tritone e¨-aª constitutes the primary dominant factor of the B¨ harmonic

domain; and soon enough (mm. 36-38, Module 3, see appendix for details) a B¨

ascending melodic minor (b¨-c-d¨-e¨-f-g-a) in Violin I appears as if to confirm the

possible structural character of the tritone.70

69 A progression that, not without nuance, could be said to be “nested” in the tonal modules present 70 Here, although we are not presented with an immediate τ-resolution, we can nevertheless deduce the

presence of a dominant by the delayed appearance of its resolution to the minor mode, a common tonal

occurrence.

55

Were we to take a linear view of the movement in the bass it may look like this

(Example 2-8):

Example 2-8

The bass line is clearly ascending from c to e¨ never quite reaching f, supposedly the

tonal center of the piece. This nevertheless will occur in the final cadence of next

harmonic area (mm. 39-41, see Example 2-9). Should we hypothesize that the

semblance of the vertical sonorities with triads is justified, the overall bass movement

c to f (reached in the next section, not this) encloses an A¨,71 which indicates a F/D¨

harmonic area.

Most importantly, A¨ is the symmetric companion of C, and this seems to aid the

establishment of the C/A¨ harmonic area, constituting a double dominant complex: C-

F, A¨-D¨. On the one hand, the e-b¨ stated at the beginning of the piece, points to the

establishment of an F harmonic area (that is, a symmetric pair with an F scale

component), and, on the other hand, the inclusion of a g¨ indicates that we have

moved away from the C/A¨ symmetric pair. This is of the utmost harmonic

importance: the inclusion of a dominant factor over the tonal center of a symmetric

pair amounts to the removal from that center.72 Thus Harmonic Area 2, by using A¨

bass sonorities (see Example 2-8 above) in conjunction with g¨ both, explores the

descending area of the C/A¨ symmetric pair and signals its departure.

71 With the presence of a¨, c, e¨, g¨, and b¨, spell the now usual ninth chord, with the d¨ as a non-

essential added eleventh. See Example 2-9. 72 The tone c combines with g¨/f© into a tritone, demarcating the D¨ and G harmonic domains, neither

of which belongs to the C/A¨ symmetric pair.

56

In the context of the SHC the predominant harmonic class in this Module is the D¨-

major scale with a predominance of the f pitch.

2.2.4.3 Module 3; mm. 36-39

This fragment not only establishes a new harmonic area but also an entirely new

formal area, that of the B section. In Example 2-9 I have also included (within the

square) the cadence that closes the first half of the piece (mm. 39-41).

Example 2-9

In this passage we observe the presence of one main harmonic area: that of F/D¨

“subdominant”; that is, g-e¨/g¨-e, a-d¨/a¨-d, and b¨-c symmetric pairs around an f

center (see Example 2-10).

The generating To (f) tone of the F/D¨ symmetric pair can constitute the bass tone, or

“root”,73 of a V7 chord, which is by implication a dissonant chord constructed over the

V degree of a given scale. In the case at hand the tritone e¨-aª (m. 39, see Example 2-

10) is the dominant factor of the F7 chord. This chord does not have to appear in its

73 I am aware of Smith’s warning against the assumption that “chromatic chords are analogous to

diatonic and can be likewise characterized [in terms of roots and inversions]”, which, he continues, “is

falsified by almost everything we find in complex chromatic music.” Smith, Charles J. "The Love of

Fundamentals Is the Root of All Evil." In A composition as Problem V (2008, 9), page 23. This,

nevertheless, would not exclude that highly “chromatic chords” could possess tonal tendencies and be

exploited as such, and certainly Smith does not claim this at any rate. Elsewhere in the same article he

speaks of how the bass-line, and not the roots, may occasionally take control of harmonic progressions.

57

entirety, according to the definition of harmonic domain. The preponderance of the

note f in the bass, then, indicates a B¨ harmony. The descending fourth in the bass,

that stretches the overall bass motion in the passage, points to a return of the V of V,

or “C”.

In the context of the SHC this Module can be understood as a transposition of Module

1 down a semitone (Figure 2-2):

Figure 2-2

In contrast with Module 1, the Viola (and the majority of tones present, see Example

2-9) favor the descending mode of the F/D¨ symmetric pair (f Phrygian), while Violin

I, 1 plays an ascending f-g-a-b¨-c-d¨-e¨ harmonic class (SHC-I, mm. 36-40 in the

score, see appendix); whereas Violin II centers majorly on a f-g¨-a¨-b¨-c-d-e

harmonic class (SHC-IV and V). In other words, not only has the “tonic” c of Module

1 given way to an F-centered passage ( C to F “subdominant” shift, see Table 1), but

also the harmonic weight, so to say, has swung down to the lower (flat) part of the

symmetric pair.

Figure 2-2 illustrates the difficulties involved in characterizing the tonal traits of

Molto Adagio. At a deep level (al level not aurally evident, perhaps), the different

tonal modules are linked together by what we could safely qualify as traditional tonal

functions (tonic, dominant, and subdominant). And yet, from the perspective of the

SHC, a different picture emerges in which a “tonic” – “subdominant” C to F

progression (effected from Modules 1 to 3, see Table 1), represents a semitone

transposition of the harmonic classes in Module 1, C to D¨.

58

Tonally (in more restricted, traditional terms) we may notice a significant harmonic

event taking place in mm. 39-40 (see Example 2-10):

Example 2-10: “Deceptive Cadence”

The three chords in mm. 40 (continued from 39) to 41 are problematic in terms of

linear analysis. There is a clear caesura in the music, dramatically tied to the lonely b¨

in 1 Violoncello (m. 40), alienated now by the rest of the ensemble (as the rest given

to the instrument clearly indicates). Possible ways of viewing these two bars is that of

counting the notes and assimilate them into a single chord, in which case a possible

chord of the ninth (second beat of 39 to first beat of m. 40), whose values all

correspond to a triadic F9 sonority, would be the logical choice. Also, a close look at

these two bars reveals the use of the c-d¨-e¨-f-g-a-b harmonic class, whose center is

undoubtedly c, with symmetrical pairs d-b¨, and e¨-a around it. That is, a full

statement of a C/A¨ harmonic class, which, in the case of m. 41, amounts to a

complete sounding of all its tones. Thus, the sudden return of a harmonic area alien to

the newly established F/D¨, points clearly to at least a meaningful harmonic event: a

cadence from one harmonic area, generated by To (f) to another generated by To (c). If

I may be permitted to use a graphic illustration of this it may look like this (Figure 2-3,

each successive circle being a consecutive harmonic area):

59

Figure 2-3

We are in the presence of a gradual layering of harmonic areas, which gathering

harmonic “sediments” from previous areas, construct a very complex (but highly

suggestive for the compositional process) network of possible tonal connections. The

tone centers, c – f – c are positively suggestive of an overall progressive motion. We

owe this “subdominant” inclination to the inevitable association of the central

generating tone with its minor 7th, which in the course of the music becomes almost

inevitable, and converts it into the dominant sonority of its lower fifth triad.

2.2.4.4 Module 4: mm. 40-44, and module 5: mm. 45 – 57

The five bars going from m. 40 to m. 44 (see appendix) should be considered a

transition to the next harmonic area, although strictly speaking they constitute an

independent harmonic module. They introduced motif 1 once more in what appears to

be the key of B¨. The Violoncello in m. 44 introduces what seems to be a D7(¨9), and in

fact we could perhaps discern the symmetric d-e¨-f©-g-a-b¨-c© harmonic class

belonging to a D-centered harmonic area. The almost immediate inclusion in m. 45 of

bª (Vln. I), cª (Vln. II), and then fª (Vln. I, m. 48), now in module 5, would appear to

confirm the D/B¨ symmetric pair (Figure 2-4):

Figure 2-4: Symmetric D/B¨ pair

3) To (c)

C/A¨

harmonic area returns

Possible "C" in m. 40

2) To (f)

F/D¨

harmonic area Possible "F" in

m. 39

1) To (c)

C/A¨

harmonic area (mm.1-25)

"A¨7" closure

60

The appearance of a¨ (tritone over d) in Violin II (m. 47) leads the music to the

possible establishment of the G/E¨ symmetric pair, investing the music with a

considerable degree of complexity but, nonetheless, also of direction, should we take

G and E¨ as natural resolutions of D/B¨; something that provides this passage with a

contrasting ascending pull, in contrast to the rest of the previous descending tendency

of Modules 1 to 3 (C to F “subdominant” pull, Figure 2-5):

Figure 2-5: Symmetric G/E¨ pair

The Violoncello conflates these two symmetric pairs in an almost non-portrayable

manner, and the more significant vertical configurations (see example Example 2-11)

hardly appear to make sense at all, if we except the possible connection between m.

44 and m. 50, joined in example 2-11:

Example 2-11: mm. 44-50

The impression of this passage is that the vertical configurations have given way to

another principle, which may be postulated as the (almost) total independence of

voices. If we look at Violin I (mm. 45-57), we find a consistent use of the D-centered

(To (d)) harmonic class: d-e¨-fª-g-a-bª-c©. In contrast Violin II (also mm. 45-57)

shows the G-centered (To (g)) g-a¨-b¨-c-d-e§ -f© preponderant harmonic class. The

Viola presents us with a purposely inconsistent (because reinforcing other voices

harmonically) use of either D- or G-centered harmonic classes, and, as a matter of

fact, what we see is a mixture of both: c§ and c© in m. 47-48, respectively, exclude

61

G/E¨, and a¨ in m. 52 seems to exclude D/B¨. All bass instruments play a mixture of

the two, sometimes tending towards one area and sometimes the other.

In the context of the SHC it is clear now the arrangement of A- and B- elements has

changed; it is not a case anymore of transposition (such as the one effected between

Modules 1 and 3 harmonic classes, see Figures 2-1, and 2-2), but a complete new

harmonic configuration around the generating tone d (Figure 2-6):

Figure 2-6

SHC-I, surely, still constitutes a transposition of the original C-centered SHC-I

configuration (Module 1, see Table 1) around To (c). And yet, the SHC-degree

progression does not fit anymore a I-IV, or I-V SHC-progression but, as Figure 2-6

shows, it now constitutes a I – III SHC-degree progression. The descending

subdominant link has been severed, and it will be down to the final Module to re-

establish the balance, fulfilling the “subdominant” C to F descending movement.

(relinquishing the establishment of the upper members of these two symmetric classes

(G and D major scales), the ascending harmonies strictly speaking, will have the

general effect of a downward or “flattened” overall course of the music). The

composer could have chosen the upper path and push the music forward on a more

traditional circle of fifths: D to G, and then G to C, while their symmetric pairs would,

likewise, move B¨ to E¨, and E¨ to A¨.74 And yet, the introduction of a SHC-degree I

– III progression, if tonally significant at all, would seem somehow to introduce an

element of non-predictability in the course of the music.

74 Notice how the piece, up to this point, seems to move on a characteristic I – IV – I – (V of V) – V

harmonic progression making the V – I almost inevitable.

62

2.2.4.5 Module 6: mm. 58 – 79

The following bars (mm. 58-63) connect together two different sections, B’ and A’’,

by means of juxtaposing their respective motifs. Here we find the dynamic climax of

the piece, mm. 58-61, with all violins (except the highest division of Vln. I, which

play an almost complete F-major scale) playing the F-centered f-g¨-a¨-b¨-c-d-e

symmetric harmonic class in fff. The climax is reached in m. 61 (high f in the top

violins, playing an unmixed F-major scale, as mentioned). The mf in m. 62 signals the

approach of a new passage with a gradual diminuendo to ppp through the following

bars. A dissonant chord (f over a c©-d¨-f-g¨-a in the last beat of m. 63 (p) continued in

m. 64 (ppp)), represents the characteristic mixture of the D- and F-centered harmonic

classes in this passage. Here it is at a glance (Example 2-12):

Example 2-12: mm. 58-63

The Viola remains indecisive until m. 64 (with a repeated bª that seems to contradict

the preeminently F-centered fragment75). Apart from a fleeting episode in which

Violoncello, superimposes a D¨-centered harmonic class (inside the curve in Example

2-13) over a D-centered harmonic class, the cellos will remain inside the D/B¨

harmonic area until the end of this passage with specific emphasis on the notes c©

and f©. That is, subdominant harmonies over dominant harmonies, in a C-centered

context.

75 To (c), as a matter of fact, would be the likely candidate for the violas if we exclude the g¨ in m. 62.

63

Example 2-13: D¨ over D harmonic classes (mm. 59-60)

The strong dissonances present in this passage are enclosed within an overall

progression that seems to insist on an “F” center, as the following bars show

Example 2-14):

Example 2-14: mm. 64-67

A look at the vertical sonorities will be enough to convince us of the final dominance

of the F/D¨ related sonorities (Example 2-15, see appendix for details):

Example 2-15

The sharps, although still there, may sound to the ear as dissonant ornaments or, once

the chords are isolated, as enharmonic tones. D is fading fast and the pull now is

towards the flats; the f§ – a§ – f§ , together with the reiterated c natural above leave no

doubt. Aurally the e¨ adds a strong dominant sound, to the point where one might

expect the arrival of a B¨ sonority (b¨ arrives finally in m. 72, almost as a premature

cadence). The dissonant layers, specifically the D-centered harmonic class in

64

Violoncello are gradually “peeled off”, as it were, revealing the increasingly diatonic

sound of what, ultimately, we identify as F major (mm. 74-79). It is this fact that

allows us to reassess the overall piece as juxtaposition of harmonic layers with one of

them, the preponderant “F”, being finally established (Example 2-16).

Example 2-16: mm. 73-76

The upward a-b¨ semitone is in fact motif 1 (signifying a new expressive content),

which in the final bars closes the piece with a mild, and somewhat questioning,

upward leaning appoggiatura. The music chose to reflect the final resolution of the

conflicting harmony present in the piece with the statement of the F-major scale, a

most evocative, peaceful, and hopeful scale in the mind of the composer.

65

2.3 Tonal layers in Molto Adagio and their analytical representation

What connects the different formal areas of the piece is, in a sense, “outside” the

piece in a strictly harmonic sense; that is, not reconcilable with traditional tonal

functionality. The overall motion V to I that, I argue, connects the beginning and end

of the piece, cannot be accommodated in a linear graphic representation. The tonal

plan of the piece is best looked at as a “harmonic landscape”. If we look at the chart

below (Figure 2-7) we may imagine that we are visually contemplating the music as if

from a high top, and that all that we can see is a range of bulky mountains (the

different harmonic areas) rising above cloud level (the square bottom of the figure):

that which is in our visual field is what we hear, and so it is that sometimes some of

the bulk disappears from sight, but we know for certain that it may reappear at any

time; virtually out of sight but present in the harmonic plan.

Figure 2-7: “simultaneous” presence of different harmonic areas (front to back)

As I have repeatedly attempted to show, there exists, also, a clear hierarchical

potential in the use of the SHC, of which this piece is an example. The question

whether that hierarchy can be accommodated to the main principles of traditional

tonal function is one of the aims of our discussion (see 1.1 What is Tonal?).

The extenuating difficulty of representing at a glance the overall shape of a piece of

these characteristics has been accompanying our analysis from the start. Linear

"C" Area "F/Db" Area "G" Area "D" Area (D ninth chord)

66

sketches describe only a part of the structure, revealing areas where the tonal

analytical approach may be employed, at least heuristically; these areas are the tonal

modules. But underlying the whole structure there are a number of generating tones

(To (c), To (f), and others) that act as gravity points attracting towards themselves a

considerable number of complex sonorities and scales immersed in what I term

harmonic areas (the globes in Figure 2-8 below).

Figure 2-8: Graphic representation of tonal modules with their associated harmonic areas76

The circles (globes or ellipses) refer to the tonal modules present in Molto Adagio, as

they appear in succession, the last ellipse referring to the goal harmony “F major”.

The arrows in the example signify possible tonal connections between harmonic

areas. These harmonic areas transcend (mainly by overlapping or creating “belated”

transitions) the sectional division of the piece, providing it with an alternative

“program”; a purely harmonic one at that. Figure 2-5 illustrates what we may term

modular tonality, that is, the specific combination of areas and passages that can be

conceived as tonal in substance. This organization of the music is unthinkable without

the notion of symmetric pairs, which expand the music flow into simultaneous

(virtually simultaneous, at times) diverging strands of ascending and descending

scales; the harmonic classes. The tonal modules, in this sense (those “globes”) can be

seen as surfacing “bubbles” of continuous simmering harmonic play of opposites.

76 Harmonic areas and events not proportional to the sizes depicted here.

"C9"(m. 2)

C/A¨ F/D¨

C/A¨ D/B¨ G/E¨

F major

67

3 Conclusion

As we know (see 1.3.4) the tonal content of symmetric harmonic classes can be

further simplified by dividing it into the number of harmonic domains present in it,

disassembling it, as it were, into its basic units, which are no other than the diatonic

scales that constitute the symmetric pair to which these harmonic classes belong (see

Example 2-18):

Example 2-1

Above and below the c-d-e-f-g-a¨-b¨ harmonic class we have the diatonic components

of the SHC, the C/A¨ symmetric pair in this case, the tones inside the rectangles being

two pairs of non-common tones that (through the combinatory process explained in

1.3) go to build a number of harmonic classes. In this particular case, we may say that

A¨-major is “included” in the given harmonic class through the presence of its

dominant factor, the tritone g-d¨. This dominant factor defines what I termed a

harmonic domain, which, in the absence of a full statement of the major A¨-scale (to

which it refers), acts as a tonal reference to it (a kind of pars pro toto substitution).

Let me illustrate this with a previous example, that of Ich Ruf zu Dir (Example 1-3).

In Example 2-19: Chart of harmonic domains in Ich Ruf zu Dir, I have looked at

Bach’s music (middle system) from the perspective on two different harmonic

domains: the c-f© harmonic domain (corresponding to a potential G major, top

system), and the d-a¨ harmonic domain (corresponding to a potential E¨ major,

bottom system). This time the process of “simplification” entails a disassembling of

the G-minor scale into its hypothetical constituent units, which are no other than the

68

two major scales contributing to its formation: that of G major, providing f© , a§, and

b§, and the scale of E¨ major, that provides e¨, b¨, and a¨. The tone e§ is not present

and the tones of c, d, and g are common to both (see Example 2-19).

Example 2-2: Chart of harmonic domains in Ich Ruf zu Dir

G major and E¨ represent a symmetric pair (such as the one in Example 1-1). G/E¨

appears to be an appropriate choice, since it accommodates all the tones present. More

importantly, this symmetric pair also has g as the generating tone, To (g), since the c©

/d¨ tone, that would constitute a tritone above that tonic, is excluded.

The chart above (Example 2-19) does not imply that Bach is thinking in symmetric

(or dualistic) terms. A chart of harmonic domains only represents the potential totality

of sonorities that a given musical sample may contain, emphasizing, at the same time,

the main areas where the harmonic activity (the one present in the actual music) take

place. Further, it helps us to map the most likely tonal areas that the music may follow,

and, thus, the analysis does not consist so much in describing what it is actually

written in the passage, but revealing the harmonic resources employed, and their

unlimited potential. This, I emphasize, implies that harmonic areas, in the context

offered in this thesis, describe the likely tonal paths that a given major scale, and its

associated minor, may follow. A G-minor scale has the tendency to ascend into the G-

69

major area (“sharpen”, in a musical sense), or descend into B¨ or E¨ (“flatten”); and

these are, by far, the most common modulations one may find in tonal music: all of

them are present, either as passing sonorities or possible resolutions, in this chorale.

Symmetry, I emphasize, merely relates these areas, and thus it is used as a convenient

tool. More than an analytical rudiment, we may more appropriately say that symmetry

allows us to focus the analysis in the possible relations between relatively distant (or,

if preferred, non-diatonically, or chromatically related77) chords and scales.

We may observe, in Example 2-19, that not only the tone c© /d¨ (equivalent as it

constitutes, in either case, the tritone over the tonic) is missing, but that e§, as

mentioned, is missing too. This is particularly obvious in the upper part of the graph,

corresponding to the c-f© harmonic domain, where we lack a full statement of the G-

major scale. The definition of harmonic domain does not say that it should be stated;

only that it could be stated. And so we may conclude that the e§ tone is virtually78

there, and that a subsequent appearance of it (as part of a consonant or dissonant

sonority, in the course of the music) can be expected. Equally, as it has been

mentioned previously in the context of this example (Example 1-3), b§ could have

equally appeared as in a more prominent role than the one present.

We would have, merely, shifted the emphasis from one harmonic domain (d-a¨) to

another (c-f©), but without leaving the G/E¨ harmonic area, which also has other

possibilities. What this shows is that the, generally speaking, harmonic area that Bach

chose, accounts for a number of tonal (in his case) choices that the composer may

have made at any time, and that can be coherently structured using the SHC. These

choices correspond to the composer’s harmonic resources, and they are certainly

always present in his mind. In more extended pieces (such as a first movement sonata

form may be), these possibilities are always almost invariably explored, thus opening

77 See Kopp, Chromatic Transformation in Nineteenth-Century Music (2002), where he uses

“chromatic” in the discussion of mediants with a non-diatonic relation to a given tonic major chord. 78 Virtual refers to the possibility of a tone (as in this case), sonority, or scale, appearing at a given

moment and being harmonically congruent with what has been previously stated. This may count as the

essential function of the notion of harmonic domain, that of providing the analysis with potential areas

of harmonic activity.

70

the possibility of understanding tonal connections as part of a more comprehensive

structure transcending the scale.79

The question immediately arises as to what would be the choice of symmetric pair,

should the music (that of Bach in the example, or any other) reach the point where the

tritone above the tonic is stated. Or, put in a different manner, what choice of

symmetric pair would account for a passage that eventually uses the twelve tones of

the chromatic scale, including a possible enharmonic spelling of a tone already

present. The answer to this is that a further symmetrization of the given symmetric

pair is needed, one that accounts for all the possible harmonic domains that the

twelve tones of the chromatic scale, including their enharmonic equivalents, contain.

If we were to further supply symmetric scales above and below, say G/E¨, for

convenience, we would obtain a closed harmonic sequence, or cycle, of four different

scales: C¨/G/E¨/B, being C¨ and B the symmetric “outer” scales of E¨ and G

respectively (Figure 2-9), and at the same time enharmonically equivalent scales80:

Figure 2-1

79 A claim that may be, arguably, found in the writings of Schenker and Schoenberg, to name two of

the most influential theorists of tonal music. This was supposedly achieved by reducing the harmonic

resources to the expression of a fundamental triad (Schenker, see Free Composition, Longman, New

York, 1979), or enlarging the tonal field to embrace a harmonic whole (Schoenberg’s notion of

monotonality, in Structural Functions of Harmony). 80 Note that any further symmetrization of B or C¨ would take us back to the original To (g). The

generating tones b and e, can be seen, respectively, as the sharpening or flattening tendencies that may

be present in the music.

71

In Figure 2-9 we observe how the symmetrization of the G- and E¨-major scales

(generating B major above G, and C¨ major below E¨) take the tonal center away from

the original To (g) over to a different tonal center, in this case either To (b) or To (e¨).81

This is a consequence of introducing the tritone above the original g generating tone

which, as a consequence of this, does not constitute anymore the center of harmonic

activity. This activity, inevitably, may be centered in scales with more flats or more

sharps, and consequently easier to relate (and possibly contained) to the new tonal

center and the scales it generates.

What symmetry allows us to do is to relate the harmonic domains present in the

music by integrating them into a higher structural unit, the harmonic area

(corresponding to a given symmetric pair). These harmonic domains are usually only

partially stated and serve, at the very least, to identify possible tonal tendencies

present in the music. If we were to compare the tonal relations established by the

symmetrization of scales, on which the SHC is based, and a traditional system of tonal

relations such the cycle of fifths, we may find that the SHC provides us with a tonal

“shortcut” to chords and scales distantly related to the initial tonic, which in the SHC

represents the generating tone; in Ich Ruf zu Dir this tone corresponds, as it could be

expected in any case, to g.82

As Tymoczko writes, “[T]he word “tonal” is contested territory”; the notion of

tonality can be used historically to determine what we know as “common practice”,

but it can also be used antithetically, to describe what is not “atonal”, in a wide sense,

“understood in contrast to music that was deliberately written to contrast with it.”

What organizes Molto Adagio is certainly a tonal (perhaps modal) principle, that of

relating different keys; only these keys are organized within harmonic areas

81 The possible harmonic classes (which are synthetic scales, by definition) that could be deduced from

a combinatory use of the major scales present in the Figure 2-6 would take us too far. But, as it may be

observed, a potential great number of known scales, and others not so common, may be constructed

with the tones present. 82 Notice that the G/E¨ harmonic area we can construct not only the dominant of the G-minor tonic

chord, but also the dominants of E¨ and B¨ (and of course G major), which correspond to the most

likely modulations that may be introduced in a minor key.

72

possessing one tone, the generating, or To tone, which does not combine into a tritone

with any of the other tones present in that specific harmonic area. These areas are not

represented by specific sonorities such as C7 or F but by a whole network of harmonic

classes; these harmonic classes take the place of chords since they can, at any time,

be used in the form of vertical sonorities, even of a triadic nature, and

indistinguishable from traditional triads. Thus, C7 can be understood to be a vertical

sonority representative of the c-d-e-f-g-a¨-b¨ harmonic class, and F major, its possible

resolution in a traditional tonal context, a representative of another related harmonic

class, say c-d¨-e¨-f-g-a-b (also generated by To (c)). In this manner, the progression V7

– I could be hypothetically seen as representing two different harmonic classes related,

ultimately, by the symmetrical relation of two diatonic scales generated by a common

tone To: that is, V7 – I can be said to functionally relate, by the agency of symmetry,

two different scales (harmonic classes) with contrasting ascending (“sharpening”) and

descending (“flattening”) harmonic tendencies; this is what has made possible the

music of Molto Adagio, the ubiquitous use of these tendencies in the shaping of the

musical course. They bear witness, as it were, to the underlying symmetricity of the

overall musical plan. And thus, if dualism may be applicable at all, it may need to

refer to the process, not only of opposing triads (major and minor), but also of a

number of different scales that result from the re-combination of diverging (ascending

and descending) harmonic tendencies: the harmonic classes. Another way of looking

at this is to view the SHC as an ulterior “harmonization”83, so to say, of traditional

tonal materials: any chord (for instance the ones present in Ich Ruf zu Dir) can be

further extended to include any of the tones present in the harmonic domains that

these chords belong to. This, which may be deemed a trivial exercise, points instead

to the theoretical possibility that, not only chords, but scales too, may be related in a

strictly functional manner, that is, connected essentially by prescribed tonal paths.

Thus, the compositional work of contemporary tonal composers (in the sense adopted

in this thesis), such as myself may be, could be seen as a continuation of the

traditional area of common practice and extended common practice,84 and provide, at

83 In the sense of increasing the possible choices that, not only the composer has, but that the analyst

may contemplate and anticipate. 84 See Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common

Practice (2011)

73

least potentially, some heuristic tools for a more comprehensive approach to tonality

and the theory of harmony.

Whether other consequences, especially of a theoretical nature, may be derived from

this system, the author is not yet in a position to decide. There are suggestive

possibilities, such as that of integrating non-diatonic pitch content chords into a given

major scale without recourse to notions such as “substitution” or “artificial”, applied

commonly to certain tones and chords; another possibility that the system points to is

that of establishing theoretically the existence of a cycle of thirds, which seems to be a

recurrent theme in the analysis of the extended tonal practice period.85 Further

research may be needed to either establish, or discard these possibilities, and all that

may be said at present is that the system provides the composer with a tool to widen,

organically, the extent and expressivity of tonal practice.

85 For a detailed discussion of this see Kopp, Chromatic Transformation in Nineteenth-Century Music

(2002).

74

4 Bibliography

Bailey, Robert. Richard Wagner, Prelude and Transfiguration from Tristan and

Isolde. Edited by Robert Bailey. New York: Norton Authoritative Scores;

W.W.Norton & Company, 1985.

Cherlin, Michael. "Why We Got into Analysis and What to Get Out of It." Journal Of

The Music Theory Society of New York State , 1986: 53-75.

Chua, Daniel K. L. Absolute Music and the Construction of Meaning. Cambridge:

Cambridge University Press, 2003.

Cohn, Richard. Audacious Euphony: Chromaticism and the Consonant Triad's

Second Nature. New York: Oxford University Press, 2012.

Ewell, Philip A. "Rethinking Octatonicism: Views from Stravinskys Homeland."

Society for Music Theory, December 2012.

Harrison, Daniel. Harmonic Function in Chromatic Music: A Renewed Dualist

Theory and an Account of its Precedents. Chicago and London: University of Chicago

Press, 1994.

Humal, Mart. “Studies on Tonal Structures; Introduction and Fourteen Analytical

Studies (Concise English Version)”: Contrapunctual Analysis. Tallinn: Estonian

Academy of Music and Theatre, 2007.

Klumpenhouwer, Henry. "Dualist Tonal Space and Transformation in Nineteenth-

Century Musical Thought." The Cambridge History of Western Music Theory, by ed.

Thomas Christensen, 456-76. Cambridge: Cambridge University Press, 2002.

Kopp, David. Chromatic Transformation in Nineteenth-Century Music. Cambridge:

Cambridge University Press, 2002.

Laufer, Edward. "An Approach to Linear Analysis of Some Early Twentieth-Century

Compositions." Composition as a Problem, IV. Tallinn: Estonian Academy of Music,

2004.

Motte, Diether de la. The Study of Harmony, An Historical Perspective. Dubuque:

William C Brown Pub, 1991.

Oxford University Press. A Dictionary of Critical Theory. Ed. Ian Buchanan. Oxford:

Oxford University Press, 2010.

Persichetti, Vincent. Twentieth Century Harmony. New York, London: W. W. Norton

& Company, Inc., 1961.

75

Riemann, Hugo. Harmony Simplified or the Theory of Tonal Function of Chords.

London/New York: Augener Publishers Ltd., 1893, 1900.

Rehding, Alexander. Hugo Riemann and the Birth of Modern Musical Thought.

Cambridge: Cambridge University Press, 2003.

Schoenberg, Arnold. Structural Functions of Harmony. London and Boston: Faber

and Faber, 1969.

Schoenberg, Arnold. Theory of Harmony. Translated by Roy E. Carter. Los Angeles:

University of California Press, 1975.

Smith, Charles J. "The Love of Fundamentals Is the Root of All Evil." A Composition

as a Problem, V. Tallinn: Estonian Academy of Music and Theatre, 2008.

The New Grove Dictionary of Music and Musicians, Second Edition. Ed. Stanley

Sadie. London: Macmillan, 2001.

Tymoczko, Dimitri. "Progressions fondamentales, functions, degrés, une grammaire

de l’harmonie tonale élémentaire." Musurgia, 10 3-4, 2003: 35-64.

Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the

Extended Common Practice. New York: Oxford University Press, 2011.

76

5 Töö lühikokkuvõte

Helistik minu Molto Adagio’s

Käesoleva väitekirja eesmärk on analüüsida minu keelpilliansamblile kirjutatud teost

And Silence Eternal – Molto Adagio (lühidalt: Molto Adagio). Teost iseloomustab üldiselt

tonaalne suunitlus ja see juhindub sellistest tonaalse muusika põhimõtetest nagu

funktsioon (printsiip, mis määrab akordide omavahelised suhted) ja hierarhia

(printsiip, mis näitab, millised kooskõlad on esmatähtsad). Analüüsimaks Molto

Adagio’t, on väitekirjas kirjeldatud selle aluseks olevat kompositsioonimeetodit –

harmooniaklasside süsteemi (System of Harmonic Classes). Tingituna muidu

traditsiooniliste harmooniavahendite suhteliselt ebatavalisest kasutamisest selles

süsteemis, on töö esimeses osas (“Harmooniaklasside süsteem”) püütud kujundada

teoreetiline raamistik, selgitamaks analüüsi iseärasusi.

Lühidalt öeldes tekivad harmoniaklassid tõusva diatoonilise mažoorhelirea

kombineerimisel selle peegelkuju, sümmeetriliselt paikneva laskuva früügia helireaga

(vt. näide 1-1 töö põhiosast). Nende kahe samast helist lähtuva ja vastassuunaliselt

liikuva helirea ühendust võib nimetada sümmeetriapaariks (symmetric pair).

Kui kummagi nimetatud helirea helid põhiheli (antud juhul c) suhtes

sümmeetriliselt ümber asetada, siis tekib hulk seitsmehelilisi sünteetilisi heliridu,

mida ma nimetan harmooniaklassideks. Näites 1-2 on näidatud helist c moodustatud

harmooniaklassi nii selle meloodiliselt väljaarendatud kujul kui ka vertikaalse

konfiguratsioonina.

Analüütilise järjekindluse huvides nimetan tritooni esmaseks dominantfaktoriks

(primary dominant factor). See tähendab tritooni võimet laheneda suurde tertsi

helistikuastmetel 1^–3^ ja seega suhestuda diatoonilise mažoorhelireaga. Diatoonilise

mažoorhelirea helisid, mis on seotud teatud kindla tritooniga selle lahenemisel suurde

tertsi, nimetan harmooniadomeeniks (harmonic domain). Harmooniadomeen hõlmab

diatoonilise mažoorhelirea piires võimalikke mažoori ja loomuliku minoori heliridu,

sest mõlemad laadid sisaldavad sama dominantfaktorit ja on seega harmooniliselt

ekvivalentsed. On oluline märkida, et nii nagu tritoon võib esindada dominantseptakordi,

esindab harmooniadomeen mažoorhelirida, sõltumata selle olemasolevate helide

arvust.

77

Molto Adagio on struktureeritud väheste ja suhteliselt lihtsate harmooniasuhete

põhjal, nagu näiteks C duuri seos G duuriga (V aste), D duuriga (V/V), F duuriga (IV) või

Des duuriga (II). Siiski ei võimalda teose keerukas harmooniline faktuur ja viis, kuidas

elimaterjal on sellesse põimitud, sirgjoonelist tonaalset lähenemist.

Organiseerimaks teoses leiduvaid tonaalseid elemente ja tendentse, on kasutatud

süsteemi, mis keskendub mažoorhelireale ja lähtub selle iseärasustest (järgnevus,

sümmeetria ja võimalikult ka funktsioon), s.t. süsteemi, mis juhindub mažoorhelirea

loogikast, kuid on sellest tunduvalt keerulisem.

Sümmeetriapaaris leiduvate tritoonidega seotud harmooniadomeenide rühma

tähistamiseks kasutan nimetust harmooniaala (harmonic area). Harmooniaaladel on

harmooniaklasside süsteemis väga oluline osa. Nad on võrreldavad helistiku või

helistikuala mõistega traditsioonilises tonaalsuses, mistõttu Molto Adagio analüüs

keskendub nende alade määratlemisele, pakkumaks lugejale ettekujutust teose üldisest

harmooniastruktuurist. Teoses mingis lõigus võib olla korraga mitu harmooniaala;

Molto Adagio’s juhtub see enamasti siis, kui eri hääled kuuluvad erinevasse

harmooniaklassi.

Seega võib harmooniaklasside süsteemi vaadelda nii diatoonilste heliridade ja

laadide (horisontaalsete konfiguratsioonide) kui ka sünteetiliste sümmeetriliste laadide

(vertikaalsete konfiguratsioonide) organiseerijana, viimaseid ma nimetangi

harmooniaklassideks. Puht-harmoonilisest seisukohast pole põhjust käsitleda

horisontaalseid ja vertikaalse konfiguratsioone kvalitatiivselt erineval viisil – nad

mõlemad määravad teatud harmooniadomeeni, mida diatoonilise mažoorhelirea puhul

nimetatakse tavaliselt helistikualaks või üldisemalt helistikuks. Näeme samuti, et

harmooniaklass võib koosneda diatoonilisest mažoorhelireast, näiteks helireast c–d–

es–f–g–a–b, mis on c suhtes sümmeetriline ja koosneb B-duur-helirea helidest.

Praktikas tähendab see, et diatoonilist mažoorhelirida võib käsitleda kui üht

võimalikku ressurssi harmooniaklasside süsteemis; seega kujutab traditsiooniline helistik

endast üht harmooniaklasside süsteemi erijuhtu.

Väitekirja teine osa (“Molto Adagio analüüs”) sisaldab vaadeldava teose analüüsi,

lähtuvalt töö esimeses osas kirjeldatud põhimõtetest.

Analüüsitav pala Molto Adagio eksisteerib kolmes erinevas versioonis: III osana

topeltkvartetist keelpillidele (originaalversioon), antud osa seadena ansamblile, ette

78

kantud Saksamaal Rostockis (Hochschule für Musik und Theater Rostock), ja lõpuks

osana topeltkvarteti kõigi kolme osa ümbertöötlusest keelpilliansamblile pealkirjaga

Music for a Mushroom Cloud. Viimati nimetatud variant (millest on olemas

heliülesvõte) on ka käesoleva analüüsi aluseks.

Diatoonilise ja mittediatoonilise helirea vastastikune mõju koos kolmkõladena

tõlgendatavaate harmooniate olemasoluga Molto Adagio’s tingib vajaduse vaadelda teose

helistikku harmooniaklasside süsteemi raamstikus. Selleks on kasutatud lineaaranalüüsi,

mis rõhutab häälte ja kooskõlade strukturaalset tähtsust.

Kuigi rangelt võttes ei saa harmooniaklasside süsteemi puhul rääkida

modulatsioonidest, võib üleminekut uude harmooniaalasse, mis toimub akordihäälte

sujuva liikumise või antud tonaalsest tsentrumist tritooni kauguse mitteakordiheli

käsitluse teel, vaadelda kas kadentsilise või modulatsioonilisena. Määratlemaks alasid,

kus niisugust spetsiifilist harmoonilist aktiivsust saab tuvastada, kasutan

helistikumooduli (tonal module) mõistet. Viimast võib defineerida kui muusikalõiku,

mis sisaldab ja määratleb mingi tuvastatava harmooniaala (või alad) ja/või tonaalse(d)

tsentrumi(d). Helistikumooduli analüütiline tähtsus seisneb selles, et see võib

sisaldada nii tuvastatavat lineaarset liikumist kui ka terve teose seisukohast olulisi

tonaalseid implikatsioone. Neid võimalikke tonaalseid tendentse ja kadentsimomente

on näitlikustamatud lineaaranalüütiliste skeemide abil.

Teos on kirjutatud vormis ABA’B’A’’, kus tähed A ja B tähistavad erinevaid

motiivikomplekse. Lõik A on ekspositsiooniline, lõik B moodustab kontrastse osa, lõik A’

on A-osa tugevasti lühendatud kordus ja lõik B’ on üleminek repriisile (osale A’’).

Harmooniliselt areneb teos C-kesksest harmooniaalast F-kesksesse, mistõttu selles

puudub traditsioonilises mõttes peahelistik. Siiski võib detailse analüüsi tulemusena leida

teoses selgeid tonaalseid tendentse.

Nagu öeldud, võib tonaalsel lähenemisel teose analüüsile märgata esmapilgul

üldist harmoonilist liikumist peamisest C-kesksest harmooniaalast (sümmeetriapaarist

C/As lähtuvaist harmooniaklassidest) “subdominantsesse” F-kesksesse harmooniaalasse

pala lõpus. Täpsemalt võib täheldada liikumist tsentrumist C (taktid 1–26)

tsentrumisse Des (taktid 26–35, seotud “subdominandiga” F sümmeetriapaari F/Des

kaudu), seejärel tagasi tsentrumisse C (taktid 40–44) ja siis kiiresti tsentrumitesse D ja

G (taktid 45–57), mis haakuvad lõpetava alaga F-duuris (taktid 58–79).

Nagu näha, on vähemalt esialgu võimalik teost analüüsida harmooniaanalüüsi

mõistete – põhiliselt akordide ja funktsioonide – abil. Täielike häältejuhtimisskeemide

79

puhul on aga probleemiks asjaolu, et kuigi meloodias ilmneb selgeid lineaarseid

tendentse (koos strukturaalsete haripunktidega “dominanthelil” c taktides 25–28 ja

oletataval tonaalsel tsentrumil f taktis 61), valitseb teose kontrapunktiliste ja

harmooniliste tasandite vahel diskontinuiteet või isegi ühtimatus.

Nagu antud teoses ilmneb, eksisteerib harmooniaklasside süsteemis selge

hierarhiline potentsiaal. Küsimus, kas seda hierarhiat saab kohandada traditsiooniliste

tonaalfunktsioonidega, ongi üks arutluse eesmärke käesolevas töös. Püüe kujutada

ülevaatlikult teose üldist kuju oli analüüsi algusest peale seotud suurte raskustega.

Lineaaranalüütilised skeemid kirjeldavad ainult osa struktuurist, näidates alasid, kus

saab (vähemalt heuristiliselt) rakendada tonaalanalüütilist lähenemist; need alad ongi

helistikumoodulid. Kuid kogu struktuuri aluseks on teatud generatiivsed helid (c, f jt.).

Viimased toimivad raskuskeskmetena, mis tõmbavad enda poole suure hulga vastavas

harmooniaalas sisalduvaid keerukaid kooskõlasid ja heliridu.

Praegu veel vara öelda, kas harmooniaklasside süsteemist saab teha kaugemale

ulatuvaid, eriti teoreetilist laadi järeldusi. On lootustandvaid võimalusi, nagu näiteks

mittediatoonilise helikoostisega akordide integreerimine antud mažooriheliritta,

kasutamata seejuures mõisteid nagu “asendus” või “kunstlik”, mida üldiselt

seostatakse teatud helide või akordidega. Teine süsteemist tulenev võimalus (mis näib

olevat sagedane teema laiendatud tonaalsuse ajastu teoste analüüsimisel), oleks

tertsiringi teoreetiline põhjendamine. Nende võimaluste kinnitamiseks või

kummutamiseks oleks vaja edasisi uuringuid. Praegu võib vaid öelda, et vaadeldav

süsteem pakub heliloojale uusi vahendeid, laiendamaks orgaaniliselt tonaalse praktika

ulatust ja väljendusvõimalusi.

80

6 Appendix: And Silence Eternal

Score

IV Movement of Music for a Mushroom Cloud

Being an arrangement for Strings of Molto Adagio86

86 See Introduction.

81

82

83

84

85

86

87

88

89

90

91