martin, henry. 2010. "seven steps to heaven" _ a species approach to twentieth-century...

Seven Steps to Heaven: A Species Approach to Twentieth-Century Analysis and CompositionAuthor(s): Henry MartinSource: Perspectives of New Music, Vol. 38, No. 1 (Winter, 2000), pp. 129-168Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/833591 .

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SEVEN STEPS TO HEAVEN: A SPECIES APPROACH

TO TWENTIETH-CENTURY ANALYSIS AND COMPOSITION

HENRY MARTIN

PROPOSE TO OUTLINE a system of species counterpoint applicable to twentieth-century modal music. In so doing, I hope to provide this

repertory with the kind of broad-based analytical tools that have long been available for tonal music. It is well-known that traditional species counterpoint is useful both for analysis and as a step-by-step method for students of tonal composition. It is my hope that the twentieth-century update outlined here can function effectively in both roles as well. The first of two articles, this paper summarizes two- and three-part first species counterpoint as the basis of the method.

The original motivation behind this species counterpoint approach was the intractability of the analysis of much twentieth-century repertory,



Perspectives of New Music

particularly the music of Hindemith, Shostakovich, Bart6k, and Copland. Such music is often difficult to analyze from either a tonal or atonal perspective. Whereas clearly tonal or clearly atonal music have theories that are fairly exact in their application, much twentieth-century practice, influenced by the modality explored in this paper, is hazier; the musical grammar can shift on a sliding scale from common-practice tonality to outright atonality. This can be true not only among pieces by different composers, but also within the same piece or even the same passage. Hence, we need a general way of approaching those "in-between" works-those passages that are clearly not tonal, but clearly not atonal either and do not specifically depend on the practice of one composer. The sliding scale of the "seven steps" of diatonic modality to be presented here provides one means of getting a broad handle on much of this literature.

The presentation in this paper is theoretically focused, but I discuss several completed species counterpoint exercises in order better to show the method's advantages. A more pedagogically oriented presentation, based on material developed for use in class, is planned as well. I have found this material to be stimulating pedagogically-as step-by-step training for students of both composition and analysis, especially for those who have familiarity with traditional species counterpoint. Its greatest advantage is introducing students to compositional models that gradually become more and more dissonant until tonality has diffused into atonality. The method's efficiency is such that students are able to apply its principles immediately to their own work, whether it be composition or analysis.

Students of composition, in working through twentieth-century species exercises, begin with models that approximate the conservative Hin- demith or Shostakovich style. Later, by incorporating bimodal and bitonal two-part writing and more dissonant trichords, these students can experiment with the harmonic textures of Bartok or Stravinsky. At the limits of the theory, students write atonally, thinking in terms of atonal set-theoretical principles derived from experience with the prime trichords. In all cases, students learn the diatonic modes and the prime trichords thoroughly. As a result, they are subsequently able to work effectively with set-based atonal theories or jazz theories of modal design. Thus, students of jazz have also found it helpful in both composition and improvisation.

For students of analysis, the method is equally productive. Again, they will learn the diatonic modes and the twelve prime trichords thoroughly. Much twentieth-century literature can be tackled with this method to

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Seven Steps to Heaven

yield a "first-step" overview of the music. Of course, more particular theories should be brought to bear on specific composers and pieces as becomes necessary, but much as functional Roman-numeral analysis is largely applied as a first-step reading of a tonal piece, the seven-steps method and the trichordal theory has proven helpful for trying to unlock twentieth-century literature that reflects some tonal bias, i.e., music that traditional tonal theory might consider to have "wrong notes" and is unable to explain generally.

There is a long history of the pedagogical application of species counterpoint. Its fame as a tool for teaching composition spread through pub- lication of what is probably the most well-known compositional treatise in Western history, the Gradus ad Parnassum of Johann Joseph Fux.1 While Fux did not originate pedagogical use of the species, he consoli- dated the work of his predecessors. Many counterpoint treatises followed Fux's work,2 but the most thorough study of species counterpoint appeared in the twentieth century, Heinrich Schenker's extensive and definitive Counterpoint.3

In his Counterpoint, Schenker pioneered a species approach not intended as compositional pedagogy. Instead, Schenker showed that species counterpoint not only underlies tonal syntax in general, but also provides a superb point of departure for the analysis of tonal free composition. It was originally Schenker's inspiration that suggested the seven-steps method as a useful tool for analysis; from that idea the com- positionally oriented approach followed.

This paper begins with a brief general discussion of tonality versus modality. An important aspect of the tonal-modal distinction depends on a revised concept of intervallic consonance, which is then developed. From the revised concept of consonance, the second part of the paper- on two-part first-species counterpoint theory-follows. Its general idea is described first, followed by examples of analysis and sample exercises that illustrate the varying levels of the "seven steps."

In the third part of the paper, the three-part first-species counterpoint theory is summarized. First, the twelve prime trichords are analyzed; then follows a demonstration of how musical analysis proceeds from an under- standing of those sonorities. As with the two-part theory, sample exercises are provided to show the method in a pedagogical environment. The paper concludes with discussion of a fugal exposition of Hindemith in order to show how the two- and three-part theories can be united in the analysis of a twentieth-century work.

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TONALITY VERSUS MODALITY

The following practices, which can be called "cues," usually occur in music we call tonal. They are shown in roughly decreasing order of importance.

Tonal Cues

1. principal pitch-class collections usually reducible to major or minor scales;

2. normative dependence of dissonant melodic intervals on consonant intervals prolonged at a higher structural level;

3. functional harmonic succession based on triads; in two-part writing, on consonances that may imply functional harmonic succession;

4. harmonic rhythm arising from functional harmonic succession;

5. presence of Stufen arising from hierarchical, nested prolongations that ultimately give rise to tonal center and key;

6. norms of melodic writing in which conjunct intervals predominate;

7. half, full, and deceptive cadences;

8. meter;

9. phrase and section groupings that project two-, four-, and eight-bar symmetries.

In the twentieth-century repertory we are considering, these cues of tonal grammar vary in presence and degree of strength, often within the same piece or even the same passage. This variance contrasts with the repertory of common-practice tonality of the late eighteenth and early nineteenth centuries, in which all the cues tend to be present.4 This common-practice repertory is normatively "tonal." But because tonality varies according to the presence and strength of the cues, it is not always clearly defined: note that tonality first emerged from medieval and Renais- sance modal practice; then, beginning in the nineteenth century, tonality diverged into twentieth-century modal practice. From this perspective, it might be better to consider tonality a special case of the modality that has pervaded western music since the early middle ages.

Thus, tonality is fluid-its strength or lack of strength a dynamic judg- ment based on the salience of the cues enumerated above. When a suffi- cient number of cues are lacking, these weakened, quasi-tonal grammars

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may often be conceptualized as modal, i.e., relating to the diatonic modes of the major (Ionian) scale. Much of the twentieth-century repertory under consideration here is modal, although modality (like tonality) varies depending on the weight of the tonal cues. "Conservative" modal music will feature the cues shown above with considerable presence. "Less conservative" music will contain fewer of the cues. When the cues are lacking entirely or almost entirely, the music will probably be heard as atonal.

Of especial significance for the theory to follow are the first and second of the tonal cues. For now, we will take for granted the first cue, that the music under consideration be reducible to principal diatonic collections. The second cue is important because it is defined carefully in species counterpoint, which, though originally modal, also underlies common practice tonality. This second cue is also important, since the two- and three-part theory to follow is based on a revised concept of consonance that has proven very helpful for establishing a species environment remi- niscent of Fux and Schenker.

In order to motivate a suggested revision of consonance, let us examine the second cue in more detail. Traditional consonance and dissonance in species counterpoint can be summarized as in Example 1.

Pitch classes: (Upper voice)

C D E F G A B (Lower voice) --------------------------------------------

C: C D C D C C D D: D C D C D C C E: C D C D C D C F: C C D C D C D G: D C D C C D C D C A: C D C C D C D B: D C D D C D C

EXAMPLE 1: TRADITIONAL TWO-PART COUNTERPOINT

In Example 1, the pitch classes (pcs) of two voices in C major are repre- sented, across and down.5 Their intervals are either consonant (C) or dissonant (D). For example, E-C is either a third or sixth (consonant-C), while G-D is either a perfect fourth (dissonant-D) or perfect fifth (consonant-C). Counting this way, we find that two-part species counterpoint contains a balanced twenty-seven potential consonances and twenty-two potential dissonances.

133




The balance between consonance and dissonance is described as "potential" for an important reason. Composers determine the levels of consonance versus dissonance in each specific work within the context of its general system. This can be seen in traditional species counterpoint itself: a second-species example might feature a large number of dissonances on the upbeats or avoid them entirely. The same principle is true of twentieth-century writing: the method to be developed here provides a measurable potential for consonance and dissonance, but the actual ratio of consonance to dissonance in a composition varies according to the piece.

In traditional species counterpoint, any dissonant interval must be treated precisely; it must function as (1) a passing tone (almost always between two consonances), (2) a suspension (properly prepared as a consonance and resolving downward by step to a consonance), or (3) a neighbor tone or adjacency (between two consonances). The passing tones and suspensions may be considered as prototypes for, respectively, the rhythmically weak dissonances and the rhythmically strong. Thus, the wide variety of dissonances permissible in two-part writing more generally-appoggiaturas, anticipations, escape tones, incomplete neighbors, and so on6-are based on either the "off beat" dissonance (the passing tone) or the "on beat" dissonance (the suspension), as developed in species counterpoint.

Using traditional species counterpoint as a model, we can also argue that dissonances are dependent on consonances. First, as suggested above, there is the simple definition of the dissonance, as in the key rule of the second species: any dissonance must occur on the upbeat of the (2/2) bar and resolve stepwise (in the same direction) to a downbeat consonance. Similarly, a dissonant suspension must be prepared as a consonance on the upbeat of the previous bar and be resolved downward by step to a consonance. The definitions both hinge on the presence of consonances that give meaning and impart function to the nonchord tones; that is, the conceptually prior consonances give the dissonances a raison d'etre.

A second illustration of the dependence of dissonances on consonances follows from the above and involves structural levels. Virtually without exception, a more background structural level must absorb and convinc- ingly account for the dissonances of the more foreground level. For example, a dissonant passing tone prolongs the third created between the initial tone of the passing motion and its goal tone. The suspension creates a temporal displacement whereby motion to the resolution is delayed through the dissonance of the downbeat; that dissonance would be sub- sumed and the temporal shift "corrected" at a higher background level.

134




Thus, dissonances are nearly always shown as more foreground elabora- tions of the consonances they depend upon.7 Yet much two-part twentieth-century writing does not always project dissonant intervals as functionally dependent on consonances. Two illustrations follow.

"Fugue 3" of Shostakovich's Twenty-four Preludes and Fugues, op. 87 (Example 2) begins by projecting a modal environment in its very opening gesture, which outlines the dissonant major seventh from G4 to F#5. After the wedge-like convergence of the compound melody in measure 2, the major seventh is further emphasized by the C5-B5 leap in measure 3. Hence, while the pitches from G Ionian are used exclusively in the subject (satisfying cue 1), these standing, unresolved dissonances alert us that the syntax is not classically tonal (cue 2 unsatisfied).

At the entrance of the answer in measure 5, the counterpoint proceeds momentarily in classical fashion. The C#s atop the run in the alto (measure 5) can be heard as prepared by the F#5-E5-D5 eighth notes in the soprano; further, note the conventional 4-3 suspension at measure 6. Yet, in measure 7 the leap of G4-F 5 in the alto brings us back to a non- traditional orientation: both the G4 and F#5 are consonant with the D6 of the soprano, but the harmony is vague: is it a G major seventh? Or is there a change of harmony: D major to G major to D major-all in the span of the first half of measure 7? It seems preferable to infer a D Ionian mode through the passage rather than seek specific functional usages.

Once the less harmonically specific D Ionian is conceded, then the sequence of dissonances in measures 7-9 becomes more easily compre- hensible: while the E5-A5 fourth at beat 5 of measure 7 can be understood as a passing sonority, the sequences of fourths and sevenths in measures 8-9 are not treated as functionally dependent on any consonant intervals. The same E5-A5 fourth at beat 3 of measure 8, for example, proceeds to the B4-A5 seventh. Then follow the parallel fourths Ds-G5 to C#5-F#5 from measures 8-9. Since the fourths and sevenths in measures 7-9 of Example 2 are not functionally dependent on more consonant intervals (as passing sonorities, suspensions, or other nonchord tones), it seems preferable to group them as stand-alone intervals. These usages, in which intervals that are dissonant in tonal practice do not seem to have foreground consonances to depend on, are labeled M.

Charlie Parker's "Ah-Leu-Cha" (Example 3) also shows fourths, major seconds, and minor sevenths as functionally independent. In measure 1, the consecutive seconds followed by a fourth cannot be explained as deriving from some prior conception of consonance.8 In measure 5, the third and fourth beats parallel measure 1, with three consecutive functionally independent intervals: note that the M2-M2-P4 interval succession in measure 1 is answered in measure 5 with P4-M2-M2 (the major

135




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seconds expanded as ninths). The consecutive perfect fourths of measure 4 are paralleled by the concluding perfect fourth in measure 8 (as an elev- enth). The two parallel events (measure 1 to measure 5 and measure 4 to measure 8) further exemplify the independence of these intervals, help unify the passage, and show a balance in construction that argues against the haphazard. Again, these independent dissonances are labeled M.

Examples such as the Shostakovich and the Parker suggest that in modal music, perfect fourths, and minor sevenths/major seconds (interval class [ic] 2s) could be thought of as "virtually" consonant. Given the resonance of history, where the traditional consonances (ics 0, 3, 4, and perfect fifths) are strongly established, it also would seem that any functional independence of perfect fourths and ic 2s would have to take into account traditional tonal practice, which lingers tellingly in cultural mem- ory and is supported (at least to some extent) acoustically. I propose that these perfect fourths and ic 2 intervals be called modal consonances: they are not "as consonant" as ics 0, 3, 4, and perfect fifths, but neither are they treated as dependent on tonal consonances at the foreground.

Some theorists might suggest it simpler to abandon the dissonance- consonance principle entirely in twentieth-century two-part writing. An argument can be made that in many similar contexts, the remaining intervals (ics 1 and 6) are also treated as independent, i.e., as consonant. But a graduated approach with the modal consonance as intermediary between tonal consonance and dissonance seems preferable, since the degree of tonal projection varies widely from piece to piece. Some pieces, such as the Shostakovich and Parker excerpts, are tonal except for their atypical (relative to traditional practice) use of fourths and ic 2s. Other pieces, as will be demonstrated below, vary more dramatically from tonal norms. The modal consonance, in fact, seems better suited as a (fairly consonant) independent interval in those pieces "closer" to the tonal benchmark. As practice veers closer to nontonal usages, the modal consonance loses its particularity. Again, this will be evident in examples to follow.

TWO-PART FIRST-SPECIES COUNTERPOINT

As based on the practices of the Shostakovich and Parker examples, a consonance-dissonance grid that includes modal consonances is given in Example 4. Intervals are categorized as tonal consonances (T), modal consonances (M), or dissonances (D).

The model shown in Example 4 is called "ic 0" (or "Type A") modal counterpoint for reasons that will be explained shortly. The only disso-

138




Pitch classes:

(Lower voice)

(Upper voice)

C D E F G A B

C: T M T M T T D D: M T M T M T T E: T M T D T M T F: T T D T M T D G: M T T M T M T A: T M T T M T M B: D T M D T M T

EXAMPLE 4: IC 0 DIATONIC MODAL COUNTERPOINT (TYPE A)

nances are major sevenths, minor seconds, diminished fifths, and tri- tones-that is ics 1 and 6. In writing two-part first species counterpoint, only tonal and modal consonances are permissible.

The interaction of tonal and modal consonances may be examined further in Example 5, which shows a first-species realization in F Lydian. The cantus firmus (C. F.) in the bass projects intervals that emphasize the lydian quality-a prominent tritone between the F3 in measure 1 versus the B3 in measure 3, for example. Most of the motion of the C. F. is stepwise, but after leaps of a fifth (measures 1-2) and a fourth (measures 4-5), the melody returns by step in the opposite direction.9

1 2 3 4 5 6 7 8 9 10 11

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C.F.

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T T M M M M M T M T T

EXAMPLE 5: F LYDIAN

The two-part first-species model can help us better understand other passages of free composition. Example 6 shows measures 43-53 of the second movement of Aaron Copland's Piano Sonata. While we can conceive of the parts as both being in B Ionian, the ending of the lower part on G#, the repeated bass motions of D#3 to G#3, the third created by G#3 with the upper B (measures 2 and 6), together with the modal

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consonance G#-F# ending the passage, suggest that the lower part be read as G# Aeolian. Hence, this passage can be heard as an instance of bimodality.

Example 7 is a reduction of the passage to two-part first-species counterpoint. The reduction is suggested by the note-against-note correspon- dence of the parts in the original: octave doublings and phrase repetition are eliminated, and registers are compressed. For example, the note-to- note dyads of Copland's measures 43-44 are reproduced in order as measures 1-5 of Example 7; likewise, measures 51-52 of Example 6 are reproduced as measures 5-10 of Example 7.

B Ionian

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T M T M T T M T M M

EXAMPLE 7: FIRST SPECIES REDUCTION OF EXAMPLE 6

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EXAMPLE 8: F LYDIAN AND G MIXOLYDIAN

As suggested by Example 7, it is useful to practice first-species exercises in which the projected modal centers of each voice differ. Such exercises assist the student with practice in compositional models that, in free composition, approximate the effect of the Copland excerpt in Example 6. Example 8 models simple bimodality in a first-species setting: a C. F. in F Lydian-the same one that appeared in the bass of Example 5-is in the soprano, while the lower counterpoint of the alto is in G Mixolydian. The parallel sevenths that conclude the example help establish the modal

141




center of G: since tonal consonances are avoided in measures 8-11, the ear can more easily distinguish the -3 - cadence of each voice alone. (By playing the lines separately, the bimodal implications are clearer.)1l

In both the species exercises thus far presented (Examples 5 and 8), each part is written with the same pitch classes; all that differs is the coin- cidence or noncoincidence of modal centers in each part. More specifically, the C Ionian scale generates the modes in both Examples 5 and 8. The underlying scale from which different modes are generated is called the parent scale of the modes.1l For simplicity's sake, parent scales are designated as the Ionian (first mode) of the diatonic scale. The other diatonic modes are then seen as derived from the Ionian.

Since each part in Examples 5 and 8 uses the same parent scale, these examples of modal counterpoint are called "ic 0" (or "Type A"). The "ic" (again, interval class) represents the distance between the parent scales of each part-in this case a distance of 0 pcs. The diagram in Ex- ample 4 represents the potential consonance-dissonance environment of any two-part modal counterpoint in which the parent scales are identical.

While bimodality is suggested by the Copland Piano Sonata (Example 6), twentieth-century practice also features numerous instances of what is usually called bitonality, in which nonidentical pitch-class collections retain strong internal independence, usually within self-contained registers. Examples include the F# major-C major of Stravinsky's Petrushka, among much familiar literature. In the second movement of the Copland Piano Sonata, there is an intriguing passage in measures 29-33, given in Example 9. Since the left-hand and right-hand parts are presented antiphonally, this passage cannot be reduced to the first species.12 How- ever, the two parts do not use the same pc collection. In particular, there is a strident dissonance between the D ~ of the right hand versus the D of the left. Assembling the pcs of each hand into a scale, we obtain the two collections shown in Example 10, which can be conceived as an A Ionian (r. h.) versus E Ionian (1. h.). These scales are a distance of ic 5 from one another.

(Vivace)

IicMP lr r r r y^~" ?ydj m G t y. t r

EXAMPLE 9: COPLAND PIANO SONATA, II, MEASURES 29-33

142




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EXAMPLE 10: PITCHCLASS COLLECTIONS FROM EXAMPLE 9

We can model such usage in a species counterpoint setting with appropriate parent scales. The simplest alteration of the ic 0 (Type A) counterpoint in Example 4 would be to change F to F# or B to Bs in one of the scales. In either instance, one parent scale is shifted a distance of ic 5 relative to the other, i.e., either C Ionian and G Ionian, or C Ionian and F Ionian. Example 11 shows the effect of the consonance-dissonance relationships between pcs of the C Ionian and G Ionian scales at "ic 5" counterpoint, also called "Type B."'13

Pitch classes:

(Lower voice)

(Upper voice)

C D E F# G A B

C: T M T D T T D D: M T M T M T T E: T M T M T M T F: T T D Dxr M T D G: M T T D T M T A: T M T T M T M B: D T M T T M T

EXAMPLE 1 1: IC 5 DIATONIC MODAL COUNTERPOINT (TYPE B)

A new relationship, not occurring in ic-0 counterpoint, obtains between the parts: an F-F# cross-relation, marked as xr.

The ic 5 model helps explain the Copland excerpt of Example 9. Copland creates a pungent effect by emphasizing the single cross-relation (D#-Dl) that is found between the parts in ic-5 counterpoint. The cross-relation is marked with a connecting line and xr in Example 10.

Example 12 shows a two-part first species exercise realized as ic-5 counterpoint. The C. F. is in the soprano, written in E Phrygian (parent scale C Ionian). The 2 - i half-step characteristic of the Phrygian is

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emphasized by the C. F. The counterpoint in the bass is written in A Dorian, whose parent scale is G Ionian. The C Ionian-G Ionian parent scales establish ic 5 counterpoint between the parts. The counterpoint is written so as to emphasize the potential for cross-relations-these are marked xr with lines connecting the relevant pcs.14

1 2 3 4 5 6 7 8 9 10 11

C.F. al Ks 0 a'

5 7X1/xr xr 5 7 10 10 6 6 9 6 10 7 5

O? 0 0 o #0 O ?0

T M T T T T M T T M T

EXAMPLE 12: E PHRYGIAN AND A DORIAN

It should be emphasized that the ic-O and ic-5 relationships presented so far have various potentials for consonance and dissonance. The realization in Example 12 is particularly dissonant because cross-relations are conspicuous. An ic 5 counterpoint could also be written to the same C. F. to minimize potential dissonance. Another measure of potential dissonance in an exercise is the interval between the modal tonics. In Example 8, the dissonance is higher because of the modal consonance (the G-F seventh) between the modal centers. A factor mitigating the dissonance in Example 12 is that the modal centers are a perfect fifth apart, i.e., a tonal consonance.

A range of potential dissonance between the two parts can be developed by cycling one of the parent scales sequentially through the circle of fifths while fixing the other. For example, ic-2 counterpoint (Type C) features the two parent scales C Ionian and D Ionian, for which there are two potential cross-relations (F-F#, C-C#).15 In similar fashion, a series of parent-scale relationships can be developed that proceeds: ic-3 (Type D), ic-4 (Type E), ic-l (Type F) to ic-6 (Type G) counterpoint.

Example 13 summarizes the seven types of modal two-part counterpoint, a series of "seven steps" that varies from the extremely consonant ic 0 to the potentially very dissonant ic 6. That is, as the parent scales grow more dissimilar in pc content, so does the potential for dissonant interaction between the two parts.

Example 13 also summarizes other features that show the widening gulf between the parents scales. The tonal intervals diminish while the

144




IC 0 (Type A) IC 5 (Type B)

27 T: 16 M: 6 D: 0 XR: 12 ic 5s:

IC 4 (Type E)

T: 20 M: 14 D: 15 XR: 4 ic 5s: 6

IC 2 (Type C)

27 T: 15 M: 7 D: 1 XR: 12 ic 5s:

IC 3 (Type D)

25 T: 15 M: 9 D: 2 XR: 10 ic 5s:

IC 1 (Type F) IC 6 (Type G)

T: 16 T: 11 M: 15 M: 18 D: 18 D: 19 XR: 5 XR: 6 ic 5s: 4 ic 5s: 2

EXAMPLE 13: THE SEVEN TYPES OF DIATONIC MODAL COUNTERPOINT

dissonant intervals grow. The modal intervals fluctuate within a narrow range, from 18 to 22.16 The number of cross-relations grows with each ic type, while the number of ic 5s (perfect fourths or perfect fifths: a measure of potential consonance) diminishes.

It must be emphasized that the counterpoint types always measure potential dissonance; two-part pieces exist in which the voices are in a dissonant ic relationship, but passages may be fairly consonant. For example, in Bartok's Mikrokosmos 103, "Minor and Major," the left hand (the "minor") is written in A Dorian (or Aeolian), while the right hand (the "major") is in B Ionian. The parent scales can be construed respectively as G Ionian and B Ionian-an ic 4 relationship with potential for considerable consonance or dissonance. The sketch in Example 14 shows how Bartok exploits both possibilities: tonal intervals predominate in the more consonant first and final sections (Examples 14a and 14c), while

b. Middle section: emphasized dissonances

c. Final section: emphasized consonances

0 #,- #- ? # o- # #_ # ? . 0# #o 0 ? 0

9: o . o .* ? L r ? o ?? o ? r T T T T T T

T: M: D: XR: ic 5s:

23 14 12 3 8

a. Opening allegro: emphasized intervals

T T T T T D Xr Xr D

EXAMPLE 14: BART6K MIKROKOSMOS, NO. 103

145




the contrasting middle Lento emphasizes the more dissonant possibilities of ic-4 counterpoint (Example 14b). The Lento is prepared by a dissonant D3-C#4 major seventh, which ends the first section. Lending a characteristic sound to the piece throughout are the cross-relations available in ic-4 counterpoint; nonetheless, the overall effect of the piece is one of intervallic consonance.

On the other hand, Bartok's Mikrokosmos 148, "Six Dances in Bulgar- ian Rhythm-No. 1," features an interesting bimodal strategy that stresses potential dissonance (Example 15). After the introductory three bars of

1 JS0 (,J J-s, EIonian

..z- . -* t-

14S

2 . _ .- t

n!f

l`(E Phrygian) L

(E Phrygian)

6 _ 7 . r-' 8

jl#v Jr ̂ j S

. --- Ir'

EXAMPLE 15: BARTOK, MIKROKOSMOS, NO. 148,"SIX DANCES IN BULGARIAN RHYTHM," NO. 1

( .i^ t fl ziL I- V I Lj t'

146

bL\




12 13 14 CAeolian

{4 C^":-EJ r r I

Lj$gg : rA L yda

C Phrygian 180 L ^^- 19

f Ly'iT t

J- L^A Lydian

EXAMPLE 15 (CONT.)

the piece, scales in the left hand are juxtaposed against a right-hand melody. From measures 4-13, the left hand features an E Ionian scale. But at measure 14, the left-hand mode changes to C Lydian at measure 14, then to A Lydian at measure 18. (The Db3 and Bb2 are chromatic upper neighbors.17) The modal change at first seems arbitrary, but exam- ining the interaction of the counterpoint clarifies matters.

Measures 1-8 are in ic-4 counterpoint, E Phrygian (parent scale C Ionian), against E Ionian (parent scale E Ionian). At measure 14, when the left hand switches modes to C Lydian, ic 4 counterpoint is maintained: in the right hand, we have C Aeolian or Al Lydian (parent scale Eb Ionian) and in the left hand C Lydian (parent scale G Ionian). Though the scale types have changed, continuity is maintained through ic-4 counterpoint between the Eb and G parent scales.

From the continuity of ic-4 relationship, we can begin to understand the climax of the passage at measure 18: a C Phrygian scale in the right hand (parent scale Ab Ionian) is juxtaposed with an A Lydian scale in the left hand (parent scale E Ionian): yet again an ic-4 relationship. The net- work of cross-relations in ic-4 counterpoint is further demonstrated in

147




the introductory three bars: there are four potential cross-relations between E Ionian and E Phrygian, F#-F, C#-C, G#-G, and D#-D. Three of these four potential cross-relations dramatically enter on the last three notes of each bar in the left hand. The cross-relation with C is saved for prominent melodic use beginning in measure 4.

As the types of two-part counterpoint increase in potential dissonance, the music becomes less tonally focused. This follows from the increas- ingly smaller number of pc intersections between the modes. Note as well that ic-1 and ic-6 counterpoint, the most dissonant, have a dissonant relationship between their modal centers. With the modal distances of ics 4, 1, and 6 between the counterpoints, the music can verge on atonality. It is very close to a counterpoint in which potential modal centers have no identity and the chromaticism is pervasive-a practice very similar to Charles Seeger's "dissonant counterpoint."'8

This theory of two-part counterpoint thus provides graduated exercises in composition as well as a broad-based method for the analysis of twentieth-century music in two parts. Extension of the model to the second, third, fourth, and fifth species will be presented in a follow-up paper. The key idea of the two-part method is the modal consonance; this concept also sets the stage for three-part counterpoint.

THREE-PART FIRST SPECIES COUNTERPOINT

As argued above, many works of the twentieth-century literature can be placed on a "sliding scale" in which pieces range from tonal to vaguely tonal to nontonal-an experience modeled in the "seven steps" theory. Levels of bimodality and bitonality are strongly established in much of this literature, and the two-part theory attempts to capture its overall effect. Yet composers have shied away from three or more tonal centers operative simultaneously. It is too difficult to project the modal or tonal individuality of each part within the pitch-frequency spectrum. Hence, not surprisingly, the "seven steps" model does not translate effectively into three parts.

For three-part writing, it is simpler and more direct to deal with the vertical simultaneities (trichords) at the outset. This is Schenker's strategy as he begins the exposition of three-part writing in Volume 2 of Counter- point with a description of the permissible sonorities.19 The verticals are of utmost importance in three-part writing; in twentieth-century counterpoint, they take precedence over the linear projection of a mode in each part, although these modal linear projections can be important ele- ments of musical structure.

148




For an approximation of twentieth-century practice, we begin with observing vertical sonorities in fairly "conservative," quasi-tonal writing. As an example, consider Hindemith's "Fuga Quarta in A" from the Ludus Tonalis (Example 16). Measures 7-8 contain a three-part texture with an added bass voice completing the fugal exposition. In measures 7- 8, the prime trichords are labeled for each created vertical.20 Conspicu- ous is the recurrence of trichords 037, 025, and 027. The prime trichord 016 only occurs once and is scarcely emphasized, as the D1 can be conceived as an escape tone. A 015 trichord occurs at the beginning of measure 7, but it can also be conceived as a suspension to the 037 that follows. A 015 trichord also occurs in measure 8; its function will be explained more fully at the end of this paper when the Hindemith fugue is reexamined.

The predominance of such prime trichords as 037, 025, and 027 in literature of this type suggests that composers may distinguish their use of trichords according to levels of consonance as well as their foreground stability, i.e., use as sonorities not requiring immediate resolution. In this regard, such trichords as these can be considered as three-part analogues to the tonal consonances described in the two-part theory. That Hin- demith features such trichords often is not surprising given his insistence on a basic tonal tradition updated to the twentieth century.21 Composers such as Webern, on the other hand, are partial to 016 trichords, given their "more dissonant" inclusion of ics 1 and 6. Thus, a theory of trichordal consonance and dissonance that parallels the two-part theory ought to show the 016 trichord as far more dissonant than trichords 037, 025, and 027.

A promising strategy for developing a three-part first-species counterpoint would be to examine the twelve prime trichords for consonance- dissonance criteria.22 In particular, the intervals of the trichords could be examined to determine frequency of interval type (T, M, or D) and to ascertain the relationship of the two upper notes of the trichord to the bass. This method of consonance-dissonance determination bears relationship to tonal practice. Paralleling the two-part theory developed earlier in this paper, a three-part first-species counterpoint could present levels of consonance, much like the T and M intervals in the two-part theory. Once the most consonant trichords are determined, then a first- species counterpoint could follow, since, in the first species, continuous consonance should prevail.

Determination of trichordal consonance and dissonance proceeds as follows. A trichord has three intervals, each of which is T, M, or D. Depending on the types and voicings of the three trichordal intervals, a number, called the consonance rating, (cr) may be computed that is a

149



6

With energy

1 2 3 4

( I - - J 1j n t I J I I L P iano

Piano J f _Ix a J~ --.& -

f r rFF I $ F rI f rFr zpr #- r-r rLL

@--X r 1 ; bJ ri I tI I.-,r I r i r

()r r 1-Yrr r i I _

7 8

015 037 037 037 016 025 027 025 027 015

EXAMPLE 16: HINDEMITH "FUGA QUARTA IN A"

Ln O

-o CD

cr

t,

0 CD

z

.00v 40

I

4- - -0 g




rough gauge of a trichord's consonance or dissonance. The cr will depend on the voicing of the trichord, as is often the case in tonal practice.23

In further analogy to voicings in tonal practice, it was determined that a finer discrimination should be preserved among types of ic-l intervals: that within a trichord, the perceived consonance-dissonance status of an 11 (a major seventh) can be quite different from that of a 1 (minor second) or 13 (a minor ninth). After trial and error, the following procedure emerged to compute the cr of a given trichord:

step 1. analyze the three intervals: for each T, add 2; for each M, add 1, for each D, add 0;

step 2. analyze the two intervals with the bass: for each Twith the bass, add 2; for each Mwith the bass, add 1; for each D with the bass as a 6 or 11 interval, subtract 1; for each D with the bass as a 1 or 13 interval, subtract 2;

step 3. analyze the lowest interval: if the lowest interval in the trichord is T, add another 1;

step 4. if the highest interval in the voicing is an ic 1, subtract 3; step 5. add the results from (steps 1-4) to determine the cr.

As an example, let us determine the cr for an 037 trichord. Any trichord can be voiced in six different ways. Assume that moving left to right through the pc numbers of the prime form in all of its six permutations describes an ascending voicing; then 037, 073, 307, 370, 703, and 730 are the six voicings of 037. (These are shown in Example 17a.) If we take the first voicing, 037, we determine the cr as follows:

step 1. there are three Tintervals (+2 +2 +2=6); step 2. there are two Tintervals with the bass; no Ms or Ds (+2 +2=4); step 3. the lowest interval in the trichord is a T(+1); step 4. the highest interval in the trichord is not an ic 1 (+0); step 5. hence, the cr is the sum +6 +4 +1 +0 = 11.

In addition, since 037 is not symmetrical, its inversion, 047, is distinct, and it too has six different voicings (Example 17b). Hence, there are twelve voicings of the trichord to be considered. The following diagram (Example 18) shows them with the steps labeled that sum to the cr.

The average consonance rating for a prime trichord is computed by adding the crs of each voicing (including the voicings of the inversions) and dividing by 12.24 An interesting confirmation of the cr algorithm within the family of 037 trichord voicings is that the second inversion of the trichord (with 7, or the fifth, as bass) has lower crs than the root- position and first-inversion trichords. This confirms experience:25 since

151




I I -

f ? I ?I lbo 1 8 I o? 1 I O

037 073 307 370 703 730

b.

I I C'

I ' I a" I H? I a"

I

074 407 470 704 740

EXAMPLE 17: VOICINGS OF 037 AND 047

037 Prime Trichord

Step 2

+2 +2 +2 +2 +2 +2 +2 +2 +1 +2 +2 +1

+2 +2 +2 +2 +2 +2 +2 +2 +1 +2 +2 +1

Step 3

+1 +1 +1 +1 +0 +1

Step 4

0 0 0 0 0 0

CR

11 11 11 10 8 9

+1 0 11 +1 0 11 +1 0 11 +1 0 10 +0 0 8 +1 0 9

Average cr: 10.00

EXAMPLE 18

these exercises can often be "quite tonal," we would like the second inversion's dissonant status in traditional counterpoint to be reflected somehow in the more modern theory.

The 037 trichord, in its twelve voicings, has an average cr of 10.00. For each of the twelve prime trichords, all possible voicings and their crs are shown in the Appendix. The cr measure creates a ranking of the twelve trichords on the basis of average consonance and dissonance, as

a. A

C) 04

047

All Possible Voicings

037 073 307 370 703 730

Step 1

+2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +1 +1 +2 +2 +2 +1 +2

047 Inversion

All Possible Voicings

047 074 407 470 704 740

+2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +1 +1 +2 +2 +2 +1 +2

152

I an




summarized in Example 19. In general, the range of crs is fairly even, although there are wider gaps at the top and bottom. The enumeration can be broken down into three subsets of trichords: the "fully consonant," the "partially consonant," and the "fully dissonant." These groups are analogous to the intervallic classifications of tonal, modal, and dissonant.26

Trichord Average cr

a. 037 [inversion 047] 10.00 b. 025 [inversion 035] 8.00 c. 027 7.00 d. 024 7.00

... Fully Consonant

e. 036 6.67 f. 014 [inversion 034] 5.33 g. 048 5.00 h. 026 [inversion 046] 4.67 i. 015 [inversion 045] 4.33

... Partially Consonant*

j. 013 [inversion 023] 3.33 k. 016 [inversion 056] 0.00 1. 012 -2.33

... Fully Dissonant

*(The 6 inversions of 036 are also Fully Consonant)

EXAMPLE 19

The trichords 037, 025, 027, and 024, as well as the 360 and 306 voicings of the 036 trichord, are fully consonant. The 037, 025, 027, and 024 trichords have no dissonant intervals; the two trichordal voicings of 036 are the first-inversion diminished triads, familiar as consonances from traditional counterpoint. The first-inversion diminished triads also have a high cr of 9, both reflecting their consonant status and lending corrobo- ration to the algorithm for determining trichordal consonance and dissonance. Exercises with the fully consonant trichords are designated "Level One" counterpoint. These exercises approximate the harmonic texture of measures 7-8 of Hindemith's "Fuga Quarta in A," as given in Example 16.

The middle group features partially consonant trichords whose voicings sometimes vary significantly in cr.27 For example, with trichord 015,

153




the crs of the voicings 015 and 051 are 2, while the crs of 150, 501, and 510 are 6 (see the Appendix). These latter three voicings are permissible in "Level Two" exercises, but the dissonant voicings 015 and 051 are restricted to "Level Three" (see below), since their cr range conforms to those of the fully dissonant trichords. The 014 trichord, favored by Bar- t6k and many others, appropriately ranks in the middle of the list: although it is not a stand-alone consonance in tonality, it can be linked to tonality through the M7 and M5 operations.28 Yet, its ic-l interval may account for its popularity with composers wishing to avoid tonal implication.

The fully dissonant trichords have crs significantly below the others, especially the 016 and 012. The 016 has only two voicings that include a tonal interval. The large drop-off between 016 and 012 arises from the fact that 012 is the only trichord whose voicings exhibit no tonal intervals. Interestingly, the much lower crs in this group delimit those trichords known to be favored by the Second Viennese School. Exercises that include the full range of trichords are designated "Level Three" counterpoint.

It is curious that the four fully consonant trichords all have an even (nonfractional) average cr; it conspicuously separates them from the 036 trichord that begins the second grouping. Moreover, it seems proper that there be a significant drop-off in average cr between the 037-the Urklang of tonality (at least in its inverted major-triad form)-and the other fully consonant trichords.

Example 20 shows a Level One first-species exercise with the fully consonant trichords. The overall mode is G Ionian, with a C. F. in the soprano that defines the mode unambiguously. The prime trichord type is written below the staff for a visual check that a variety of the permissible trichords is used. To create the Level One contrapuntal lines with the C. F.:

1. only the four most consonant trichords may be used;

2. the standard voice leading rules from traditional tonal species counterpoint generally continue to apply;

3. the counterpoints must conform to the mode of the C. F. with occasional chromaticisms permissible;

4. the occasional chromaticisms must immediately be resolved stepwise to a diatonic note of the mode.

As examples of chromaticisms in Example 20, C#4 resolves to D4 (the alto, measures 5-6); D#4 resolves to E4 (the alto, measures 8-9); finally,

154




1 2 3 4 5 6 7 8 9 10

C. F. # a ? o ?, aKID 0? .

-J

Dg# t o K1 o KI 40 KI all o4 as

A? . o o a- A\ . o . Ii.t I -- _ b- () -

, 11 I ..

027 027 025 027 037 025 024 025 025 027

EXAMPLE 20: G IONIAN (LEVEL ONE)

F3 proceeds chromatically to F#3 (the bass, measures 8-9). A cross- relation occurs between measures 7-8, which is permissible. With the consonance of the trichords clearly laid out in the exercise, the cross- relation is not disruptive.29

As pointed out, three-part first-species counterpoint becomes increas-

ingly dissonant as the other types of trichords become permissible. Since for Level Three exercises all trichord types are permissible, the "continual use of consonance" as a first-species criterion no longer applies. In Ex-

ample 21, a Level Three realization, a strategy is shown whereby the exercise progresses quickly from consonant to dissonant as the climactic 012 trichord is approached at measure 7. In heading toward resolution at measure 17, the exercise generally returns to more consonant trichord types (although note the 016 in measure 15). This species exercise models a Hindemith-like approach, in which mid-piece textures may be dissonant and dramatic, while the beginning and end are more modally conceived, i.e., more consonant.30

The opening exposition of Hindemith's "Fuga Quarta in A" from the Ludus Tonalis (given above in Example 16) provides a fitting conclusion to the two- and three-part models dealt with in this paper. Example 22 presents a reduction of the exposition to a generally first-species setting. The shifting modal centers and pervasive chromaticism are typical of free composition in this style. Measure numbers in the comments to follow refer to the reduction in Example 22, not the original of Example 16.

The opening statement of the subject is in A Phrygian. While this is not clear from the first six bars, in which an F modal center is equally viable, the convergences to the A-E open fifth in measure 7 and in measure 29 confirm the mode.

155




1 2 3 4 5 6 7 8 9

C.F.

024 027 024 025 027 013 012 027 014

10 11 12 13 14 15 16 17 ^ IL aI_

. S o ,

i41 #0#"0 " ? 0? h #0

o

i0.&. tt "k bo3 ns "h # ff ffI

024 025 026 025 015 016 024 025

EXAMPLE 21: A IONIAN (LEVEL THREE)

With the entrance of the soprano in measure 7, T and M intervals occur almost exclusively during the two-voice texture that extends through measure 22. Important exceptions include the F3-E4 major seventh (ic 1) in measure 8 and the C4-F#4 tritone (ic 6) in measure 9. A function of the F3 in measure 8 is to emphasize the cross-relation with the F#4 in measure 9 in shifting the overall modality (in terms of parent scales) from F Ionian (measures 1-8) to G Ionian (measures 9-13). A further shift to D Ionian occurs at the C#4 in measure 14. The "modula- tion" bars from measures 14-20 gradually undo the chromaticism: first C# is canceled at measure 17, then F# at measure 20, then B~ at measure 21. This would return us to an F Ionian parent scale, but for the El at measure 21, which prepares the entrance of EL Ionian for the presentation of the bass subject in measure 23. The El Ionian is "undone" by the A4 at measure 28, then the E4 at measure 29, so as to collapse back into A Phrygian.

156

, 11 ffliT -I ff- - T J I t"




1 2 3 4 5 6 7 8 9 10

. --. -- o o J J , , 5 6 7 3 4+ 7 8

I8 o r r 0 J o 0 J ? J J T T D TD MT

11 12 13 14 15 16 7 18 19 20

CJ. ? #" #e Jc d ?

*

J0 J

2 5 7 9 2 3 6 5 10 12 11 10 6 5 10

M T M M M T T T T TM T T T T

21 22 23 24 25 26 27 28 29

& so _o -J dlJ~ ,l 60 - J" o0

12 11 10

I j J J J 0 0 ? W .. o T T

I I .. 1 1 I 037 037 037 016 025 027 025 027 015

(as 510 voicing)

EXAMPLE 22: FUGA QUARTA IN A (REDUCTION)

The sequence of harmonic implications in the opening exposition is complex and is underscored by the subject itself, which can be inter- preted in several modes: compare its A Phrygian opening (measures 1-7) with the same pcs in measures 23-27, in which the harmonic implication is El Ionian. In measure 28, the Ai in the soprano anticipates the return to the A-E fifth at measure 29.

Once all three parts have entered at measure 23, Hindemith focuses on fully consonant trichords. The dissonant vertical created at the upbeat of measure 24 is an 016 that seems to contradict the prevailing texture. However, the D1 (as noted earlier) can be thought of as an escape tone. The dissonance of this particular note of the subject is in itself thematic: compare its occurrence in measure 9, where the soprano F 4 creates the dissonant tritone with C4 (and cross-relation with the F3). Hence in a mostly consonant environment, Hindemith chooses to make this particular upbeat dissonant.

157




Also of interest is the 015 trichord in measure 28. As discussed above, the 015 trichord has a wide variability of cr in its voicings. The particular voicing in measure 28 is 510, which has a cr of 6, virtually qualifying it for consonant status. Moreover, its slight edge, relative to the surround- ing trichords, creates an appropriate tension for resolution to the open fifth A-E in measure 29. Further, the dissonance of the 015 is also miti- gated by the retention of the B b3 in measure 28-a kind of suspension- so that the parallel fifths of A4-E4 in the soprano with D1-AO in the bass predominate in the cadence. All in all, Hindemith provides a satisfyingly rich exposition that can be read in terms of the theory developed in this paper: the subject cycles through modal contexts of A Phrygian, G Ion- ian, D Ionian, (modulating passage), then Eb Ionian before returning to A-E.

* * *

As with the two-part theory, the three-part counterpoint can be extended into the remaining species. This, again, will be the subject of a follow-up paper. In extending the theory to other species, it is important to note that one of the four fully consonant trichords is required on downbeats, since it is primarily through them that the existence of nonchord tones can be related and understood. If a broader use of trichords is permitted on downbeats, the concept of nonchord tones, as either dissonant passing tones or suspensions, is much less persuasive.

For textures of more than three parts, the theory would have to be extended. Investigating the twenty-four (forty-eight with inversions!) voicings of the twenty-nine tetrachordal set classes is an intimidating prospect. Still, the procedure for determining a consonance rating could be extended to chords with more than three pcs. It remains to be seen whether such an exercise would be fruitful or what modifications would need to be made to the algorithm for determining a persuasive cr. Should a cr for more complex chords prove useful, a computer program could be written to generate the permutations of the voicings with accompanying crs.

A simpler, more direct approach to handling more complex harmony would be to conceive of textures of more than three parts as trichords plus added pcs. This method parallels the tonal model, where complex harmonies are usually conceived as triads extended by sevenths, ninths, elevenths, and thirteenths. Further, a trichordal model based on the diatonic modes at least helps one get some tonal-modal bearings on a work. For more in-depth analysis of harmonically complex textures, a more par- ticularized model may be necessary.

158



Seven Steps to Heaven 159

ACKNOWLEDGMENT

I would like to thank Richard Hermann, Steve Larson, Richard Swift, and Allen Forte for their insightful comments on an earlier draft of this paper.




APPENDIX

TRICHORDS IN ORDER OF DECREASING CONSONANCE

WITH ALL VOICINGS

a. 037 (Forte 3-11)

Prime 037 11 073 11 307 11 370 10 703 8 730 9

Inversion 047 11 074 11 407 11 470 10 704 8 740 9

Average cr: 10.00

b. 025 (Forte 3-7)

025 6 052 6 205 7 250 9 502 10 520 10

035 8 053 7 305 8 350 8 503 9 530 8

Average cr: 8.00

c. 027 (Forte 3-9)

027 7 072 9 207 6 270 5 720 8 702 7

Average cr: 7.00

d. 024 (Forte 3-6)

024 7 042 8 204 6

160




240 6 402 8 420 7

Average cr: 7.00

e. 036 (Forte 3-10)

036 6 063 5 306 9 360 9 603 5 630 6

Average cr: 6.67

f. 014 (Forte 3-3)

014 4 041 5 104 5 140 6 401 6 410 6

034 6 043 6 304 5 340 4 403 6 430 5

Average cr: 5.33

g. 048 (Forte 3-12)

048 5 084 5 480 5 408 5 840 5 804 5

Average cr: 5.00

h. 026 (Forte 3-8)

026 3 062 3 206 6 260 7 602 4 620 5

046 5 064 4 406 7 460 6 604 3 640 3

Average cr: 4.67

161




i. 015 (Forte 3-4)

015 2 051 2 105 4 150 6 501 6 510 6

045 054 405 450 504 540

4 3 4 4 6 5

Average cr: 4.33

j. 013 (Forte 3-2)

013 3 031 4 103 3 130 3 301 4 310 3

023 032 203 230 302 320

3 4 2 2 5 4

Average cr: 3.33

k. 016 (Forte 3-5)

016 -2 061 -1 106 1 160 1 601 0 610 1

056 065 506 560 605 650

-2 -2 3 2

-1 0

Average cr: 0.00

1. 012 (Forte 3-1)

012 -3 021 -3 102 -2 120 -2 201 -2 210 -2

Average cr: -2.33

162




NOTES

1. Johann Joseph Fux. Gradus ad Parnassum [Steps to Parnassus], 1725. The portion on species counterpoint is available as The Study of Counterpoint, trans. and ed. by Alfred Mann (New York: W. W. Norton, 1965).

2. The literature on traditional counterpoint is enormous. Among the important works following Fux: Johann Georg Albrechtsberger, Grundliche Anweisung zur Composition (Leipzig, 1790); Luigi Cherubini, Cours de contrepoint et de fugue (Paris, 1835); Siegfried Wilhelm Dehn, Lehre vom Contrapunkt, dem Canon, und der Fuge (Berlin, 1859); Heinrich Bellermann, Der Kontrapunkt (Berlin, 1861); Ernst Friedrich Richter, Lehrbuch des einfachen und doppelten Contrapunkts (Leipzig, 1872); Ebenezer Prout, Counterpoint, Strict and Free (London, 1890). Among the important counterpoint stud- ies in the twentieth century: Andre Gedalge, Traite de la fugue (Paris: Enoch & Cie., 1901); Sergey Taneyev, Convertible Counter- point in the Strict Style (Leipzig and Moscow, 1909; English trans. by G. Ackley Brower, Boston: B. Humphries, 1962); Knud Jeppesen, Counterpoint-The Polyphonic Vocal Style of the Sixteenth Century (New York: Prentice-Hall, 1939); Arnold Schoenberg, Preliminary Exercises in Counterpoint (London: Faber and Faber, 1963); Felix Salzer and Carl Schachter, Counterpoint in Composition (New York: McGraw-Hill Book Co., 1969). For material on counterpoint in twentieth-century music, see, for example, Harold Owen, Modal and Tonal Counterpoint: from Josquin to Stravinsky (New York: Schirmer Books, 1992). There is usually material included on counterpoint in general guides to modern composition, for example, Leon Dallin, Techniques of Twentieth Century Composition (Dubuque, Iowa: W. C. Brown Co., 1974, especially 179-88).

3. Heinrich Schenker, Counterpoint: A Translation of Kontrapunkt by Heinrich Schenker (Volume II of New Musical Theories and Fanta- sies), 2 vols., trans. John Rothgeb and Jirgen Thym; ed. Rothgeb (New York and London: Schirmer Books, 1987).

4. When the cues are not present, the deviations are often thought of in terms of the norms; for example, when a chromatic environment is heard as an enriched diatonic environment, or when a five-bar phrase is thought of as an extended four-bar phrase.

5. For simplicity's sake, chromaticism is excluded.

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6. In many ways, these more general nonchord tones are a development of Monteverdi in his seconda prattica.

7. It can be argued that there are occasions in which consonances are seen as dependent on conceptually prior dissonances, but such occurrences are generally less common in tonal music. For an illustration, see Steve Larson, "The Problem of Prolongation in Tonal Music: Terminology, Perception, and Expressive Meaning," Journal of Music Theory 41, no. 1 (Spring 1997): 101-36.

8. It may be objected that because of the assumed harmonic background-in particular the bass line-the Parker example is not actu- ally two-part writing. Still, the timbre of the horns versus the rhythm section is, I think, so distinct that the seconds, fourths, and ninths are still perceived as standing alone. Moreover, the tempo is fast enough that it is not possible to view the "dissonances" as displaced consonances.

9. As in traditional exercise writing, students are assigned a cantus firmus to which they provide a counterpoint. Such exercises provide simple species-structured practice in the kinds of two-part writing heard in the Shostakovich and Parker excerpts. In recalling the specific features of tonal species counterpoint, the assigned exercises should be conceived vocally. In particular, this means that there should not be too many large intervals, stepwise motion should be normative, and the ambitus of each part should conform to custom- ary vocal practice. Further, after the skip of a large interval (say, a fifth or greater), stepwise motion in the opposite direction is encouraged. Finally, retaining familiar two-part voice-leading rules is encouraged, such as forbidding parallel or similar motion to perfect octaves and fifths. Such practices as these provide pedagogical continuity for students already familiar with traditional species counterpoint.

Along the same lines, students are encouraged to label the intervals created by the parts between the staves. These are labeled below as M or T, depending on the type of consonance used. An important requirement is that there not be too many Ts, since then the exercise could sound tonal, with any M intervals possibly heard as mistakes.

It is important that the music in a twentieth-century species exercise not sound "too tonal" (i.e., too many of the tonal cues should not be present). Note in the soprano's counterpoint of Example 5 that measures 2-4 feature consecutive linear fourths, which would be strongly discouraged in a traditional two-part exercise. Similarly,

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while keeping in mind the desirability of stepwise motion, a greater relaxation of this requirement is possible in order to "modernize" the writing. (Compare twentieth-century vocal music in general, which has far greater use of leaps than in common-practice melody.) The line builds to F5 in measure 6; this creates effective contrast with the C. F., which reaches its high point at measure 5. The counterpoint also outlines characteristically the Lydian ic-6 interval with the modal center: the F5 (measure 6) to B4 (measure 9) to F4 (measure 11). A - i cadence in the counterpoint establishes the F modality of the exercise.

10. Note that the alto builds to G4 in measure 7, which contrasts the climax of the C. F. at measure 5. Further, the voice leading follows classical procedures; for example, the perfect fifth at measure 6 is approached through contrary motion.

11. The term "parent scale" is borrowed from George Russell's The Lydian Chromatic Concept of Tonal Organization for Improvisation, 2d ed. (New York: Concept Publishing Company, 1959). Russell uses the term in a similar way, as a basic scale from which other modes and improvised lines may be derived.

12. The immediately following measures in the piece are in fact in a note-against-note setting, but the point made here is more appropriate for measures 29-33.

13. Example 11 shows the effect of the shift of the parent scale in the higher voice. If the lower voice were to be the one shifted, the interval counts for the modal and tonal intervals alter slightly, but not significantly. (This is because of the effect of the perfect fourth/perfect fifth under inversion.) For the enumerations given below, it is assumed that the parent scale of the upper voice is being shifted.

14. The bass counterpoint in Example 12 follows a different strategy from the counterpoints of Examples 5 and 8: it descends to a low point coinciding with the climax of the soprano at measure 7.

15. Of course, the same relationships are obtained if the shifting parent scale moves to the "flat" side, i.e., B l Ionian versus C Ionian.

16. As was mentioned previously, the conception of the modal consonance becomes fuzzier as the relationship between the parents scales grows more dissonant. That is, the classification of modal versus tonal intervals becomes less clear. For example, in ic-2 counterpoint (assuming parent scales C Ionian and D Ionian), we obtain the

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interval created by F (from C Ionian) and C# (from D Ionian). Is F- C tonal or modal? For the sake of the enumerations in Example 13, it is counted as modal, since in traditional species counterpoint it would be considered a dissonance. Even more problematically, in ic- 6 counterpoint, one encounters the interval F-E#, which, to main- tain consistency, must be considered modal! But realistically, in twentieth-century practice the context must be considered: when tonal harmonic cues are largely avoided, it is difficult to hear any distinc- tions between modal and tonal consonances as suggested by enhar- monic spellings. Hence, because there is no valid distinction between the modal and tonal consonances in the more dissonant settings, the enumerations of interval types given are rough guides. The key fac- tors are the increases in dissonances and cross-relations.

17. In a more generalized version of this theory, parent scales could be extended to include nondiatonic collections. However, since it is difficult to hear the individual tonal implications of nondiatonic sets, I have limited the theory so far to the original church modes that his- torically evolved into the tonal system.

18. Modern Music (June 1930): 25-31. Once students have experi- mented with the outer edges of modality, it is useful to jettison the "seven steps" model entirely and work with Seeger's counterpoint, in which consonance and dissonance are reversed. That is, the normative intervals are dissonant, and consonance must be "resolved" to dissonance.

19. Schenker, Kontrapunkt, 2 (1922), 1-39.

20. Prime forms will be identified by their pcs without brackets or Forte list numbers (Allen Forte, The Structure of Atonal Music, New Heaven: Yale University Press, 1973, 179-81). The corresponding numbers from Forte's list are included in the Appendix, in which each trichord is presented.

21. Hindemith's point of view is defended in The Craft of Musical Com- position (New York: Associated Music, 1937).

22. For an alternate system of analyzing the consonance or dissonance of chords, see Howard Hanson, Harmonic Materials of Modern Music (New York: Appleton-Century-Crofts, 1960), 9-16. Hanson, by rating all intervals as consonant or dissonant, develops a notational system that enumerates the underlying quality of any chord. He does not attempt a numerical ranking, however, as I will try to do. While

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Hanson accounts for all possible chords, my method, so far, only deals with trichords.

23. It is understood that in actual music, textural and registral consider- ations may render the cr questionable. However, determining the cr is a useful way to proceed and again recalls Schenker's examination of the possible voicings of the triad and three-part doubled sonorities in a simplified environment.

24. Symmetrical trichords do not have distinct inversions; the crs of the six voicings would then be added and divided by 6 to obtain the average cr. Interestingly, the sum of the crs of the prime forms always equals the sum of the crs of the inversions. The regularity shown in Example 18, however, where the cr of each prime form equals the cr of its corresponding inversion (in the permutation sequence), does not hold for all nonsymmetrical trichords.

25. Not everyone agrees with the computed crs, of course. When pre- senting this material in class, lively discussion ensues on whether cer- tain voicings are more or less consonant than others, that is, whether the computed cr seems intuitively accurate.

26. The analogy cannot be carried too far. For example, the 027 trichord is dissonant in traditional counterpoint, but is fully consonant in the twentieth-century theory.

27. For the 048 trichord, the procedure for determining the cr is modi- fied slightly: although the trichord contains three ic 4s, which are tonal intervals, any tonal implications are compromised by the trichord's division of the octave in three equal sections. Hence for determining the cr, the ic 4s are considered modal consonants. Rich- ard Hermann also points out that the 048 trichord does not occur diatonically, and hence implies a chromatic alteration to any conceptually prior diatonic collection.

28. The 014 trichord is transformed to 037 (and vice versa) through the M7 and M5 transforms, which are the multiplication operations, mod 12.

29. Chromaticism is permitted in Level One exercises for two principal reasons: (1) it is so common in twentieth-century practice generally, and (2) exercises without chromaticism tend to be tonally monoto- nous. In Example 20, for example, the chromatic 025 chord at measure 8 is perhaps the most interesting moment of the exercise.

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168 Perspectives of New Music

30. Since Example 21 is a Level Three exercise, the A Ionian C. F. (in the alto) is chromatic, with notes occurring outside the mode. As an A Ionian exercise, a proper key signature is used, although acciden- tals are redundantly marked for easier reading. Note, too, that all notes outside of A Ionian are immediately resolved stepwise to notes of the mode, a requirement that also includes the C. F.



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