harrison, daniel. 2002. dissonant tonics and post-tonality tonality

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Harrison, “Dissonant Tonics,” MTSNYS 4/28/02 1 Dissonant Tonics and Post-Tonal Tonality (MTSNYS Presentation Version) A red thread running through my work on harmonic theory is an interest in back-of- the-book topics, those issues that appear in the later chapters of our standard harmony texts. I’ve often felt that the typical plot of tonal-music instruction, as read in these texts, amounts to little more than this: the story of a brave beginning of theoretical rigor and explanatory power that becomes, by the end, an exercise of lever-pulling, wheel-whirling, huffing and puffing, all in the hopes of keeping Toto away from the curtain. This is to say that the final chapters of many of our standard undergraduate harmony texts offer a grab- bag of explanations of only apparently local use. When approaching techniques that have been associated with problems in the later history of tonality—equal division of the octave, extreme chromaticism and enharmonicism, not to mention questions of large- scale structure—their tone often becomes tentative and even apologetic. My contributions so far to harmonic theory try to convey a sense of how these topics may be presented less as signs of tonal dissolution than as signs of tonal development and enlargement. And it is in this spirit that I speak today. I want to deal with a matter of theory, frequently propounded early in instruction and with some authority, that is certainly at least partly responsible for the problems of back-of-the-book harmony, including the embarrassed agonies of any final chapters that attempt to treat post common-practice harmony. The topic is a large one, and I am able today only to outline the issues, do a little infilling, and point out connections with the work of other theorists as well as suggest directions for making those connections stronger. One of the fundamentals of tonal music, in both practice and theory, is that only a consonant triad can stand as a tonic chord of a key. This is why the keys of C major and C minor exist, for example, but not C diminished. For Rameau and, later, Riemann, consonance and tonic are related symmetrically, which is to say that “tonics are consonant” and “consonances are tonics” are equally true. Both theorists built out extensively from this relationship in their music theories. 1 Rameau, in comments about the authentic cadence that concludes a composition, noted that the consonance of the final chord—the tonic—distinguishes it from all other chords. Were the dominant chord also fundamentally consonant, “the mind, not desiring anything more after such a chord, would be uncertain upon which of the two sounds to rest. Dissonance seems needed here in order that its harshness should make the listener desire the rest which follows.” 2 (David Cohen glosses this and related passages in his recent article on musical perfections in MTS.) In this way, the tonic alone was fundamentally consonant (i.e., a

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Slightly modified version of a paper read at the Music Theory Society of New York State, April 28, 2002.

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Page 1: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

Harrison, “Dissonant Tonics,” MTSNYS 4/28/02 1

Dissonant Tonics and Post-Tonal Tonality (MTSNYS Presentation Version)

A red thread running through my work on harmonic theory is an interest in back-of-the-book topics, those issues that appear in the later chapters of our standard harmony texts. I’ve often felt that the typical plot of tonal-music instruction, as read in these texts, amounts to little more than this: the story of a brave beginning of theoretical rigor and explanatory power that becomes, by the end, an exercise of lever-pulling, wheel-whirling, huffing and puffing, all in the hopes of keeping Toto away from the curtain. This is to say that the final chapters of many of our standard undergraduate harmony texts offer a grab-bag of explanations of only apparently local use. When approaching techniques that have been associated with problems in the later history of tonality—equal division of the octave, extreme chromaticism and enharmonicism, not to mention questions of large-scale structure—their tone often becomes tentative and even apologetic.

My contributions so far to harmonic theory try to convey a sense of how these topics may be presented less as signs of tonal dissolution than as signs of tonal development and enlargement. And it is in this spirit that I speak today. I want to deal with a matter of theory, frequently propounded early in instruction and with some authority, that is certainly at least partly responsible for the problems of back-of-the-book harmony, including the embarrassed agonies of any final chapters that attempt to treat post common-practice harmony.

The topic is a large one, and I am able today only to outline the issues, do a little infilling, and point out connections with the work of other theorists as well as suggest directions for making those connections stronger.

One of the fundamentals of tonal music, in both practice and theory, is that only a consonant triad can stand as a tonic chord of a key. This is why the keys of C major and C minor exist, for example, but not C diminished. For Rameau and, later, Riemann, consonance and tonic are related symmetrically, which is to say that “tonics are consonant” and “consonances are tonics” are equally true. Both theorists built out extensively from this relationship in their music theories.1 Rameau, in comments about the authentic cadence that concludes a composition, noted that the consonance of the final chord—the tonic—distinguishes it from all other chords. Were the dominant chord also fundamentally consonant, “the mind, not desiring anything more after such a chord, would be uncertain upon which of the two sounds to rest. Dissonance seems needed here in order that its harshness should make the listener desire the rest which follows.”2 (David Cohen glosses this and related passages in his recent article on musical perfections in MTS.) In this way, the tonic alone was fundamentally consonant (i.e., a

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major or minor triad); the dominant, in its true and characteristic form, was a dissonant major minor-seventh chord. Riemann later identified the major added-sixth chord as the characteristic subdominant dissonance. For both Rameau and Riemann, dominants and subdominants, even those appearing in triadic forms, were understood theoretically as representatives of underlying dissonant, tetradic structures. The tonic triad, in contrast, was both consonant and complete.

We may no longer appeal in our harmony classes to things like “fundamental categories of chords” or “characteristic dissonances,” but we nonetheless speak about “tonic” and “key” in ways that are completely consonant with these old-fashioned ideas. In fact, from species counterpoint to Schenkerian analysis—the gamut of technical knowledge generally taught concerning tonal music—the tonic-consonance fundamental is an unquestioned given.

But we are also well aware of pieces and even entire repertories in which a traditionally dissonant chord projects tonic. And these tonic chords can and often do have the same phenomenological attributes possessed by consonant-triad tonics. Example 1 shows some easy-to-recall cases from semi-classical repertory, reminding us also that the technique is Gerswhin’s as well as Debussy’s. So here’s the problem: how can we square Debussy and Gerswhin with the still worthwhile concepts from Rameau and Riemann?

The starting point has to be discussion on the compositional discovery about chordal “statics” made towards the latter part of the nineteenth century: consonance could be relativized without damaging the tonal-system environment. In other words, tonics could be made from chords that were more consonant relative to non-tonics, but not necessarily triads. This accomplishment, of course, goes hand-in-glove with the well-known increase in dissonant chordal formations during the nineteenth century; increasing levels of dissonance allow a higher setting of the consonance band to maintain the same rough absolute distance between “imperfect,” non-tonic and perfect, “tonic” sonorities, the perfection of the latter, in essence, being defined down. Maintaining both the distinction between these two categories and the construction of progressions that highlight this distinction suggests an analytic method that Paul Hindemith, in the 1930s, identified as harmonic fluctuation.3

My interest in this paper is partly in the dynamic conditions of musical structure that allow for dissonant tonics, but mostly in the types of static structures (chords) that are used when the dynamic conditions allow. There is already some literature on tonal structure that focuses on dynamics almost exclusively—I’m thinking here of the 1960s work of Roy Travis and Saul Novack, as well as of Robert Morgan’s well-known article on dissonant prolongation. But the sonorities proffered as tonics in these studies invariably fail to give the same phenomenological experience as do traditional tonics—

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they don’t sound or feel like tonal-music tonics in the same was as those in Example 1. I’m interested primarily in those structures that do sound and feel like tonal-music tonics, which is a matter of chordal statics. But since there is no context- free musical situation in which tonics exist, there can be no satisfactory theoretical opposite to a purely dynamic, contextual view of tonic—that is, a pure statics of tonic. So, I will sketch a few dynamic influences.

Here is a familiar meaningful unit of chord grammar: I–V7, V7–I. Although only two chord types are used here, they are formed as antimetabole, as two phrases of the same two words but in reverse grammatical order. In the first phrase, tonic is subject and the dominant seventh predicate, if you will. But the dominant is dissonant and hence both imperfect and incapable of being an ending product. In the second phrase, tonic is consonant and predicated by imperfect-to-perfect succession from a chordal-dynamics point of view. In addition, the predication is underscored by the completion of chordal-succession and phrase symmetries. In this way, the wild, phrase-concluding C+7 chord of Example 1(a) receives a strong tonic accent and, therefore, sounds perfectly appropriate, even withstanding a change in the otherwise strict Tp4

relationship between antecedent and consequent. The same grammatical archetype is composed out in Example 1(b), although the dominant is expressed in this case through a ninth and not a seventh chord. Note that the Tp2 antecedent-consequent relationship is preserved by the dissonant tonic chord. In these two examples, at least, end position is a strong indicator of tonic and is indifferent to both absolute, triadic consonance and the products of motivic activity.

Examples 1(c) and (d) instance another grammatical unit: T–S–D–T, a single bilateral exploration of functional relations working out a Start–Depart–Return–Arrival paradigm. 4 The concluding C6 chord benefits from the same perfecting circumstances as in (a) and (b), but the starting chord benefits from another rhetorical convention—weaker than the first—that associates structural beginnings with tonics, allowing (c), the opening added-sixth chord, to act tonically. The tonic-beginnings convention is also at work in the antecedent phrase of Example 1(a), where little fuss ensues about the vocal D, a free-standing ninth against the C-chord accompaniment (perhaps it does “resolve” to C at the end of m. 2). The convention of tonic beginnings assimilates the D as a chord tone, rendering the acoustically consonant E a “dissonant” escape tone.5 At the start of Mackie Messer, the added sixth, A, slips in easily and over the course of the phrase will prove to be, essentially, the reciting tone of the tune. Harmonic activity—the T–S–D–T process—seems to take place underneath and without much notice from the A-obsessed tune, which only responds autonomically with a whole-step lowering of the motive’s pickup notes.

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Example 2 presents a more substantial situation involving concluding-tonic rhetoric. Excerpted from the closing measures of Maurice Duruflé’s Requiem, op. 9 (1947), the passage is based on a cantus firmus, In Paradisum, that appears in the top staff. This

antiphon is in Fƒ mixolydian, with a key signature of five sharps; that this signature is the same as that of B major will prove to be significant in the following discussion. PLAY

EXAMPLE Underlaying the chant are a pedal on Fƒ and a haze of chords—mostly conventional seventh- and ninth-chords—that, after reaching a nadir in reh. 101+2,

gradually ascend until the uppermost note reaches Fƒ5 at reh. 102+2. At this very point the haze and the pedal clear away, revealing a simple B-minor triad, the first triad heard in the movement. It is not immediately certain what kind of tonal cue this triad gives. A diminuendo having started two measures before and the chant having reached its final, a conclusion seems imminent. But upon what tonic? It is not impossible or even unexpected that the piece should end on the B-major chord shown at the asterisked measure of the example. The key signature and position-finding data suggest as much,

and the Fƒ pedal heard since the beginning of the excerpt can be heard as creating a lengthy dominant area preceding the final tonic arrival. Duruflé, however, famously

concludes on an Fƒ dominant-seventh chord, upon which he subsequently heaps a major ninth. Attempting to reconcile “key-signature hearing” with this fact, we might interpret the final chord as an unresolved dominant, perhaps yearning for some resolution in the

afterlife. But key-signature hearing seems to miss the point here. The concluding Fƒ sonority asks to be heard as the tonic section of a plagal cadence, with the B-minor triad acting as subdominant. This Kirchenschluß is appropriate both to the sacred genre of the work and to the mode of the cantus-firmus chant. As the molto rit. comes to bear during the cadential motion (joining the diminuendo begun some measures before), the apprehension of conclusion changes from “imminent” to “in-process”; dynamics, tempo,

and chord voicing all converge upon the Fƒ goal. All then penetrate and bury themselves in this target, for the (sempre) diminuendo and ritardando (très long) continue past the initial arrival, and the major ninth appears afterwards as well. The chord then simply fades away, not in unrequited yearning for a B-major tonic but apparently in the eternal

peace and perfection of an Fƒ tonic.6 So far, I have proposed that tonics like to inhabit environments at the beginning and

ending of formal processes, the former in the guise of a perfected structural subject awaiting a verb and the latter in the guise of perfected predicate. While concluding tonics are most frequently the final sonorities of processes, initial tonics need not be the first sonorities; they can and do take various kinds of accenting non-tonic prefixes, of which the dominant- functioned anacrucsis is the most common. Nevertheless—and despite the fact that there are other rhetorics of tonic, some of which I have discussed elsewhere7—

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the general proposal about structural placement of tonics can serve well for the time being for a number of musical situations I want to discuss in this paper.

The situation with chordal statics is, as I earlier made plain, a touchy subject. Not only are deep values about consonance and dissonance implicated, but there is also a problem that atonal compositional procedures first brought to light: in any piece where exercise of pure process is paramount, it is possible to claim as tonics things like hexachordal regions, referential sets, states of certain tonal arrangements, etc. Under this regime, as Roy Travis noted in 1959, "[there is] no reason why a major or minor triad, a seventh chord, a fourth chord, a polychord, or any other conceivable combination of tones ... cannot become the tonic sonority of a tonal music."8 In other words, a tonic can sound like anything as long as it plays the appropriate role in the piece, even if it does not have the natural resonance of traditional tonal tonics.

In what follows, I am interested in dealing with sonorities that do have such natural resonance. For that reason, the focus on what follows is upon rooted chords. In a longer paper on this topic, I go on at some length about perception, psychoacoustics, and phenomenology of chord roots in general, and I end up proposing that chord roots are like virtual pitches—not pitch classes primarily—that are presented to consciousness as a byproduct of the auditory system. The are as illusory yet as sensible as difference tones, as technologically useful as superheterodyne radio tuning—the stuff of AM radio. In what follows, I’ll take rooted chords as those harmonic structures designed to create virtual pitches by exploiting peculiarities of the human auditory system.

My work on dissonant tonics has so far involved a gross classification system for the kinds of tonic effects rooted chords can create, with a particular interest developed for one of the categories. The dissonant tonics I have listened for and analyzed are found in a repertory that begins at the turn of the twentieth century and that variously continues, in both art and popular styles throughout the century. It has not yet comprehended, I think, the complete range of possibilities in this vast repertory of post-common-practice tonal music. Still, the structures here are found in the literature and can, compositionally speaking, be made fairly easily to produce the desired tonal effects. I’m still gathering more examples though, so please offer me effects I haven’t considered afterwards. The first type of dissonant tonic I’ve considered is the colored triad. Example 3 shows the opening two phrases from Bartók’s Bagatelle, op. 6, no. 4. The first, mm. 1–2, is composed entirely of root-position triads and has a D-minor tonic. The second phrase repeats the outer-voice counterpoint of the first but thickens the texture in two ways. First, the bass-clef open fifths of the first phrase are filled with thirds. This gives the chord mass greater perceived depth as well as thickness. Second, sevenths are placed in the individual chord stacks, one in each hand. Since the second phrase is heard solely as a

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textural variant of the first, this thickening of texture does not affect the sense of D-minor tonic, even though that tonic is expressed in the second phrase as a minor seventh chord and not a triad.

The ease with which the tonic can be described as D minor even though the second phrase has a seventh chord as a tonic points out the inadequacy of the term “seventh chord” to denote the simultaneity in question for this particular analytic circumstance. For the evidence of the passage suggests that the sevenths are not organically part of the chord but supplementary or added tones. In this sense, one of the naming conventions for seventh chords accurately reflects the nature of the chordal association. Calling the tonic chord a “minor minor-seventh” shows it to contain, first, a minor triad and, second, an appended minor seventh. 9 In this way, the chord is still heard primarily as a triad, and the appended note produces what has traditionally if uncritically been called a “coloristic” effect—something that pertains to neither structure nor function, but rather operates somewhere within or upon the structure as a timbral effect.

Colored tonics depend upon what Neil Minturn has termed tonal interpreters, “a harmonic triad, a diatonic scale segment, or a functional bass segment or progression whenever such … is heard to organize pitches around it into a locally tonal scheme.”10 In other words, tonal interpreters are creatures of dynamics because of their power of organizing, which can only result from context and placement. In the Bartók example, these are easy to recognize. From small scale to large, we note the preponderance of outer-voice octaves in the first, second, fifth, and eighth chords, all of which emphasize the functionally important notes D and A. (The same applies to the chords of the second phrase.) Moreover, D opens and closes each phrase, occupying thereby traditional tonic spots. We also note that these octaves come about through intensive contrary motion, making them appear to be more subtle and secondary rather than obvious and primary tonal effects. Further, both outer-voice lines present diatonic scale segments anchored, as just mentioned, upon D. The passage as a whole is thus heard to be constructed of pitches from the D-minor natural scale And finally, to return to our initial observation, the first, triadically-based phrase vouchsafes all these tonal features to the second, which repeats its essential counterpoint and scale content. In a sense, the first phrase is, as an entity, the tonal interpreter for the second.

Another example, suggested by studying Minturn’s work on Prokofiev, is offered now: the conclusion of the last movement of Sonata No. 4. Here, two coloring agents are applied to the final, tonic chord. As the functional analysis in Example 4 shows, the dynamics involved here are clearly the same as in the authentic cadence. This allows the static structure of the tonic chord to bear greater dissonance, and the chord indeed has two coloring agents. The “left-hand” agent cleaves to the major third of the chord. The

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“right-hand” agent is attached to the pitch-class root at the upper boundaries of useful pitch space for chord-building, safely far away from the C1 root sounded at the beginning of the measure.

The second type of dissonant tonic are constructed as polychords. The disposition of notes in the tonic chord of Example 4 is likely a factor in its ability to bear two coloring agents. The right- and left-hand portions of the chord are separated by an octave, with the right-hand part close to the border of useful pitch space. It is not difficult to hear the chord as having two distinct sites of activity, one in the fourth octave and the other in the sixth, with the empty fifth forming a buffer zone between them. Buffer zones such as this suggest that the (singular) tonic chord at the end of the Prokofiev example could also, perhaps more profitably, be called the (plural) tonic chords. For the passage at hand, this description does not offer much advantage. In the passage shown in Example 5, however, the utility of such a description is clearer. This is a case where what might be described as multiple coloring agents of a single triad actually affiliate to form a separate triad—bitonality, manifested in polychords.11 I could obviously go further in the analysis of this passage, discussing, for example, how the tona l stream separation is accomplished compositionally. But let’s move on.

The next type of dissonant tonic comprises many individual techniques and exemplars. They have in common what Leonard Meyer might call a “secondary-parameter” root, or Paul Hindemith a “root representative.” This is a chord tone promoted to the office of root by some contextual power—it gains the office extra-constitutionally, as it were. This tone is almost always a marked bass note of some kind: the loudest, longest, deepest, prettiest, biggest, and baddest. The environments that support this kind of chord-tone promotion vary, but one is the presence of intervallic tracery or patterning in the chord stack. A simple case of what I mean is the stack of perfect fifths shown in Example 6. (Any other cyclic pattern, such as by perfect fourth or diatonic third, can also work.) The chord tone marked as root representative (A2) is the beginning as well as the lowest and longest of the entire chord construction. Even if the chord stack were to go on indefinitely from here up to limits of hearing and fill up with all manner of chord tones, the sense of A as root would not be shaken thereby. 12 In addition, the passage could begin with, say, a D3, with the A3 coming in as written in m. 2 and the rest of the chord unchanged; D would then be offered up as root and not A.

In Hindemith’s chord group system, aspects of which inform my own classification of dissonant tonics, a stack of two fourths, which can sound both euphonious and rootless, is explicitly singled out for having an indeterminate root; the other chord singled out in this category is the augmented triad. Hindemith called the proffered roots of these chords root representatives, suggesting that they were a circumstantial first-among-

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generational equal rather than a source of descendent tones. I adopt the term to cover other structures such as those described above.

The three types discussed so far, colored triads, polychords, and chords with root representatives, are all somewhat inorganic structures. Example 7 shows a rough schematic of how this assertion is framed, with the relative sizes of the elements reflecting the strength of their contribution to the simultaneity. The last category of dissonant tonic, not shown in Example 7, is what I call chord with amplified root, which seems to me to be different from the others in this regard. A chord with amplified root offers the same sense of unity through harmonious proportion that marks the major and minor consonant triads. That is, the arrangement of chord tones is such that it is difficult and even undesirable to pick out one ore more as superfluous to the structure of the chord. As a result, it is not too far a stretch to say that this particular type of dissonant tonic actually doesn’t very sound dissonant at all. In some respects, these chords sound like colored triads, but what would be considered coloring agents participate actively in both voice- leading and harmonic structure in ways that I can only discuss briefly today.

Example 8 shows a schematic for an amplified-root stack. The left-hand column begins at the bottom with the essential rooting interval of a perfect fifth, indicated as R [root] and F [fifth]. Octave equivalents of Rs and Fs are shown in brackets. The right-hand column shows available “fills” within various bands, notated with the usual diatonic interval abbreviations. These fills are, usually, slightly asymmetrical bisections (equals maximally even) of a given bandwidth. This kind of division locates two points that are as far apart as possible from the bandwidth boundaries and thus avoid the dangers of proximity dissonance that produces intense beating and harshness. Once it is used, a fill then marks out a new and narrower bandwidth in the adjacent upper octave, which is why bracketed intervals appear in the left-hand stack above their occurrence in the right-hand one.Before I unpack the example and the chords it describes further, let’s note that it basically produces chords with decreasing distances between chord tones in the upper bands. In this way, the chords mimic the structure of the harmonic partial series. This coincidence has been noted for about a century now, and terms like “extended chords,” “higher consonances,” and the like all reflect this observation. Schoenberg, who was among the first to explore this issue seriously, 13 also noted that this conception of chord structure is not fully compatible with the principle of tertian construction, to which it has often been yoked. But that’s another paper.

Example 8 reveals some tension between pitch and pitch-class in the construction of chords, a tension first discussed theoretically by Rameau with respect to ninth chords, but recognized even before that in counterpoint theory and thoroughbass, where the differences between the figures “2” and “9” are profound. In general, we can see that the

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marking off of bandwid ths is influenced by pitch-class considerations, but the actual deployment of a chord tone is a decision about pitch and resonance. The basic criterion here is Hippocratical—that is, “do no damage” to the sense of concord. In practice, this means avoiding ic1s as much as possible, and when not, to deploy them as 9ths instead of 2nds. (The avoidance of ic1 discourages a m9 fill in the first upper band, shown in parentheses; in the next octave, however, it is the only new chord tone available [in 12-tone ET] for filling the band there.) Another, more powerful discouragement concerns the P11, which ambiguates the R–F root relationship by attempting to arrogate R for itself. In other words, the presence of a P11 in an amplified-root chord generally signals a kind of chord inversion, not root position. Hence, the “no-entry” symbol for the interval. (Interestingly, vague distrust of P4/11 as a chord tone has infiltrated various theories of harmony that seriously consider “extended chords,” especially in jazz harmony; George Russell’s Lydian Chromatic Concept of Tonal Organization for Improvisation is perhaps the best known.)

In the interests of time, I want to concentrate upon consequences of filling the second band. Fills in the upper band are a fascinating study that leads to many intersections with jazz theory, which I have not yet explored to my satisfaction. Fills in the second band, on the other hand, lead to an intersection with a theory of tonicity neither recently discussed nor developed in the music-theory community, which I will come to presently.

The two fills available in the second band—which, to avoid proximity dissonance, cannot be used simultaneously—participate in two types of relationships with the fills in the first band. The solid- line relationships mark out collections belonging to set-class 4-26[0358], the minor minor-seventh chord. One, the version with m3 and m7, is in the

classical “root position” of the chord; the other, with M3 and M6, is in # position, which Rameau (sometimes) and Hindemith (always) considered a root position, as well

The dotted line relationships mark out collections belonging to set-class 4-27[0258], the two forms of which found here are in an inversional relationship; this is reflected by the difference in dotting patterns. The presence of the tritone in the 4-27s makes them somewhat difficult to stabilize as tonics. Even so, we have already seen, in Example 2 (the ending of the Duruflé Requiem), a fine illustration, with the addition of the M9 from the upper color band to boot. The 4-26s have proved to be easier to work with because it is possible to extract consonant-triadic subsets to be used compositionally as apparent consonant tonics.What I am getting to here is the crossroads with Robert Bailey’s notion of double-tonic complex, which he first developed in connection with study of Tristan und Isolde.14 (There is also resonance with structures that Joseph Straus identified as tonal axes in the neo-classic works of Stravinsky. 15) The original form of Bailey’s idea involved a very tight tonal pairing of two tonic triads

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…linked together in such a way that either triad can serve as the local representative of the tonic complex. Within that complex itself, however, one of the two elements is at any moment in the primary position while the other remains subordinate to it.16

This is how Bailey must explain the situation in Tristan—which is to say that Wagner’s art did not recognize a fusion of the two triads as a possible tonic sonority. Such fusion began to occur in the works of Mahler (Rückertlieder, Das Lied von der Erde) and Debussy. By the time of Stravinsky’s neo-classic works, Bailey’s primary/subordinate condition gives way to one in which fusion is the norm, as Straus asserts:

[The tonal axis] must function in the piece as a referential sonority. It must occur prominently as a discrete harmony within the piece, particularly in cadential situations. It must be the essential harmonic generator of the piece; other harmonies derive from and relate to it.17

Between Bailey’s analytic method—which uses traditional harmonic analysis while on the lookout for manifestations of one or the other of the two tonics—and Straus’s—which primarily uses motivic and set-class techniques—there is room for other approaches. For symmetry’s sake, I’d like to conclude today by trying one out with the opening measures of Duruflé’s Requiem. Example 9 shows three basic harmonic and voice-leading elements of the section. At (a) is the tonic sonority, shown in whole notes and analyzed as T(d/F), the “d” and the “F” referring to the embedded consonant triads shown individually as half notes. Three closely related non-tonic sonorities are used in the passage, shown at (b). These sonorities barely disturb the pitch-class content of the tonic, as the whole notes show. From a voice- leading standpoint, they come about through neighboring motion. They are thus labeled as N(1)through (3). The boxed text underneath explains how the disturbances in N(1) and N(2) affect harmonic function. N(1) is mildly subdominant, and N(2) is more strongly subdominant. N(3) is a somewhat different creature that occurs once in the passage to perform a special task, which I’ll discuss shortly. Example 9(c) shows how a certain text-painting effect in the passage will

be set up. The essential move here is activating and subsequently deactivating Fƒ from F½. The example shows this move to be accompanied by semitonal contrary motion, using

notes that are locally chromatic, Eß and B½. Given the text, which I’ll discuss in the next

example, I suggest that the most appropriate way to analyze the introduction of the Fƒ is as a “shining” version of the local d-minor triad.

Example 10 shows a slightly reduced score of the passage. The accompaniment is missing a sixteenth-note figuration pattern that largely arpeggiates the harmonies shown, but otherwise it is generally accurate as to register and voice-leading. As in the other example from Duruflé (Example 2), the passage is based on a cantus firmus, the introit

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Harrison, “Dissonant Tonics,” MTSNYS 4/28/02 11

Requiem aeternam, a simple hypolydian chant on F, sung by the tenors and basses; the sopranos and altos sing “aah.” PLAY PASSAGE The passage partly reconciles both Bailey’s and Straus’s generalized conditions for recognizing double-tonic/tonal-axis structures—to the extent that these structures are related to an amplified-root tonic. The controlling tonic sonority is heard as a simultaneity in the opening measures, at a point of cadential articulation in m. 11, and at the return from “light” to the “dark” shade in m. 17. The consonant-triad subsets are used at the beginnings of phrases in mm. 7 and 13—D minor in the former and F major in the latter. And the F-major subset is used as the concluding tonic of the passage at m. 19.

The passage as a whole is not highly differentiated as to harmony. The restricted range of the chant, the leisurely tempo, and the use of the N(1) and N(2) chords as subtle non-tonics imbue the passage with a cloud- like, floating quality. In this environment, it seems inappropriate to claim that the concluding F-major chord represents the modulatory goal of the passage from its initial F/d state. This chord comes off instead as a less shaded version of the overall tonic, just as the d-minor chord in m. 7 is a more shaded version. Two related points of harmonic articulation in the interior of the passage—perhaps the closest thing to modulations—stand out in particular and deserve

comment. The introduction of N(3) and its B½ in m. 12 marks the F-major chord in m. 12. This chord, in turn, initiates the move, sketched in Example 9(c), to the “shining” tonic of

m. 16. The cloud cover returns, along with Bß, in m. 17. This coordinated series of harmonic events is in service of the text here—“may light perpetual shine upon them”— with the moment of maximum brightness occurring at the verb luceat. Thereafter, as I mentioned, the passage and its text conclude with the less-shaded subset of the tonic sonority.

The point of the preceding discussion has been to show that the effects of an amplified-root tonic can extend deeply into structure, unlike a colored triad, which is a more local effect. The observations of Bailey, made in connection with late nineteenth-century music, and those of Straus, made in connection with Stravinsky’s neoclassic work, both relate, I believe, to amplified-root tonics as I’ve discussed them today.

In closing, let me recapitulate not the dissonant-tonic types I’ve discussed today, but rather the list of theorists whose work I’ve explicitly referenced today. Bailey’s ideas about double-tonic complexes, which have been commented upon by Deborah Stein, Christopher Lewis, Harald Krebs, William Kinderman, and others, are clearly germane to amplified-root tonics, especially those without upper-band fills. Joe Straus’s tonal axis is, as I’ve stated, a related manifestation of the same structural opportunity. Ernst Kurth, whose name hasn’t been mentioned until now, has advanced related ideas, I believe, in his notion of underthirds; and I explicitly employed his vocabulary when discussing

Page 12: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

Harrison, “Dissonant Tonics,” MTSNYS 4/28/02 12

shading effects in the last example. For amplified-root tonics with upper-band fills, pertinent work has been done in jazz theory. To my knowledge, the voice- leading and deep-structural consequences of these chords has not yet been studied in either the jazz or legit domains.

For colored triads, or added-note chords in general, there is a considerable body of rather uncoordinated literature running back to the first part of the twentieth century, although almost all of it avoids dealing with the consonance = perfection = tonic principle. I’ve advanced the idea today that reconciliation lies in recognizing added notes as timbre tones instead of chord tones, and that Neil Minturn’s idea of tonal interpreter can help in distinguishing the two. I’ve written previously about bitonality and the structural environment that can support it, specifically in a chamber symphony movement of Milhaud. More work awaits in that area. Do look for an article by Dmitri Tymoczko in your next issue of Spectrum for a pertinent contribution using Stravinsky’s music.

Finally, I’ve been explicit in making connections to Hindemith’s theories. The more I consider these matters and the music that brings them up, the more I think that Hindemith actually got a lot of it right. This is not the place to launch an examination and critique of his theories, but I have one currently in drydock that, when completed, I hope will substantiate my hunch. But that’s yet another paper.

Page 13: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

Harrison, “Dissonant Tonics,” MTSNYS 4/28/02 13

NOTES 1 Both, but especially Riemann, devoted theoretical capital the equivalance class of “consonant triad” for

the support of the equivalence class of “key,” which lead ultimately to harmonic dualism. Major and

minor keys, representing the two possible “moods” of tonal pitch materials, both aesthetically equal,

needed to be supported by major and minor tonic chords, the static structure of which could be

theorized as transformationally equal. 2 Rameau ref. 3 Unterweisung ref. 4 Note Enharm comments on this matter. 5 Ref Larson Problem of Prolongation in Tonal Music 6 Interestingly, were the asterisked B-major chord sounded after the Fƒ chord, the effect would be strangely

anodyne, even if the ritardando and diminuendo were stripped out of the passage in an attempt to

prevent Fƒ from setting up. 7 HFUNC ref. Cf. Clevenger dissertation, 51. 8 Roy Travis, "Towards a New Concept of Tonality," Journal of Music Theory 3 (1959): 263. Among the

various other articles advancing this idea, the most frequently cited is Robert Morgan, "Dissonant

Prolongations: Theoretical and Compositional Precedents," Journal of Music Theory 20 (1976): 49-91. 9 There is some confusion about the hyphenation in this chord labeling system. Some (e.g., Aldwell &

Schachter) would hyphenate between the two “minor” terms: “minor-minor seventh chord.” While the

underlying understanding of the chord is still a triad with a seventh, this hyphenation scheme seems to

emphasize the unification of these elements into a single entity: a seventh chord. Hyphenating minor

and seventh, in contrast, pulls the triad and added interval aprt, emphasizing thereby the supplementary

nature of the added note. 10 {Minturn, 1997 #45}, 61. 11 The discussion in the present section relies to some extent upon earlier observations about bitonality

made in {Harrison, 1997 #55}. 12 See discussion in EKATA. 13 See Schoenberg, Theory of Harmony, 320 ff. 14 Bailey, Norton Score of Tristan Prelude, 121–3. 15 Straus, JMT 26.2 (1982). 16 Bailey, 122. 17 Straus, 265.

Page 14: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

& C ˙ œ œC

I'm just wild

œ œ ˙a - bout

˙ ˙G7

Har - ry,

˙ ˙and

˙ œ œHar - ry's wild

œ œ ˙a - bout

wC+7

me!

˙ Ó

&?

###

# # #43

43.˙œ œœ œœ

˙ œœ œœ œœ

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˙ œ

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.˙œ œœ œœ

.˙œ œœ œœ

& C Ó .œ jœOh, the

˙ ˙C6

shark has

˙ .œ jœpret - ty

˙ ˙Dm7

teeth, dear,

˙ .œ jœand he

˙ ˙G11

9

shows them

˙ .œ jœpearl - y

wC6

white.

˙

I'm Just Wild About Harry Eubie Blake

Waltz #1, from The Skaters Emil Waldteufel

Mackie Messer Kurt Weill

Example 1

(a)

(b)

(c) (d)

Harrison, "Dissonant Tonics," MTSNYS 4/28/02

Page 15: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

&

???

#####

###############

44

44

4444

Chant

Accompaniment

p

ppp

Ó. œ œ

Ó ˙̇̇n

Ó ˙˙˙

w

Reh 101œ œ œ œ œ œ˙̇̇n ˙̇̇ww

w

.˙ œ œ

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w

œ œ œ œ œ œ

˙ Œ̇ œww

w

˙ Œ œ œn

˙̇̇# ˙̇̇

w

w

3œ œ œ .œ Jœ

www˙ &̇

w

&

?

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#####

###############

23

23

2323

44

44

4444

F

F

œ œ 3œ œ œ

˙̇ ˙̇n&wwww

dim. poco a poco

dim. poco a poco

Reh 102œ ‰ Jœ œ œ

˙ ˙˙ ˙wwwnw

Jœ œ Jœ3œ œ œ œ œ œ œ

˙ ˙n ˙˙̇ ˙̇ ˙̇..wwn.w˙n ˙ ˙

sempre dim.

sempre dim.

sempre dim.

sempre dim.

w

wwnw

ww ?w∑

Molto rit.

˙̇nn w ˙̇̇nn˙̇̇̇̇nnn ˙̇̇̇̇nnn˙n ˙n∑

∏∏∏

wwww# wUŒ trés long.˙#uwwwwuwwwu

wwwwwwwwwww*

Example 2

Harrison, "Dissonant Tonics," MTSNYS 4/28/02

Duruflé, Requiem, In Paradisum, end.

Page 16: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

&

?

4

3

4

3

ƒ

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Example 3

Harrison, "Dissonant Tonics," MTSNYS 4/28/02

Example 4

Bartok, Fourteen Bagatelles, op. 6, no. 4

Prokofiev, Piano Sonata No. 4, conclusion

S— D———— (antic.) T—————

Page 17: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

&

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4

2

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Example 5

Example 6Ravel, Daphnis et Chloe, opening

Harrison, "Dissonant Tonics," MTSNYS 4/28/02

Milhaud, Saudades do Brasil, Corvocado, end.

Page 18: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

Harrison, “Dissonant Tonics,” MTSNYS 4/28/02

Example 7

I. Colored Triads

Triad Color agent

II. Polychords

Triad Triad

III. Chords with Root Representatives

Root rep. Chord

Page 19: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

Harrison, “Dissonant Tonics,” MTSNYS 4/28/02

Example 8 [R] | [m7] [M6] | [F] | A11 | P11 [M3] [m3] | M9 | (m9) [R] | | m7 | M6 | F | | | M3 | m3

Upper bands

pitch space pitch-class space

Second band; two fills

First band (Mode); two fills

| | R

Page 20: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

&

(a)

T(d/F)

wwww

T(d)

˙˙˙

T(F)

˙˙˙

(b)

w

ww

N(1)

œœb w

ww

N(2)

œb

N(3)

www

œn

œ œ

T(d) shining

(c)

T(F)

˙ ˙# ˙#

T(d/F)

˙n˙b ˙˙ ˙

Example 9

Bß is ß^6 of T(d) and ^4 of T(F). Bß thus sug- gests a general S(d/F) . G as ^4of T(d) pro- motes the gen-eral S function.

The "casting out" of G to the bass increases the sali- ence of the gen- eral S function.

Harrison, "Dissonant Tonics," MTSNYS 4/28/02

Page 21: HARRISON, Daniel. 2002. Dissonant Tonics and Post-Tonality Tonality

&

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b

b

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4

3

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3

4

3

4

3

4

2

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SA

TB (Chant)

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9

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11

Œ ˙˙

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ne,

.

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T(d/F)

.

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.

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œn œœ œ

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et

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lux per -

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˙

&

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3

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pe - tu - a

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.

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17

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at

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N(1)

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e - -

.

.

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19

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is.

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Example 10Duruflé, Requiem, Introit, opening

Harrison, "Dissonant Tonics," MTSNYS 4/28/02