Volume 1, Number 1 (Spring 2008)

The Death and Resurrection of Function1

by Gabriel Miller

[1] Function is a suggestive term which is still inspiring creative work in theory after over a century of use. But careful definition and elaboration is crucial. ….The term carries so many associations for us that it is difficult not to read some of them into the historical subject, thereby occluding perception of subtle yet important differences from our own views. Even if we know exactly what we mean, there is no guarantee that our reader will accurately grasp our meaning when we use the term without scrupulous qualification, since there are so many acceptable interpretations of the concept. Furthermore, the widespread and casual use of the term nowadays has diminished its descriptive power. It may prove helpful to investigate our assumptions and more clearly articulate and differentiate the myriad concepts which function has come to represent for us.2

[2] As proof of the "subtle yet important differences" David Kopp notices in the term function, consider the secondary dominant chord in Example 1. Imagine that this progression adorns the chalkboard of a freshman theory classroom. The teacher has just explained the concept of secondary dominant, and now wishes to give it functional significance…

Example 1. The functional ambiguity of V7/V.

Example 1
[3] Teacher: Notice how the secondary dominant, V7/V, is the same as a ii7 chord, except the F is sharp instead of natural. In other words, according to our TSDT model, the chord still functions as S.
[4] Student: I kind of get it, but can you explain why it's called secondary dominant again?
[5] Teacher: Sure, the secondary dominant functions as a dominant because, as opposed to ii7, V7/V has a leading tone. It is the dominant of the dominant.
[6] Student: Wait, I thought you said it functions as S. Does it function as S or D?

[7] Well, here we have a problem. You may imagine some uncomfortable response by the teacher to this dilemma, such as "… function in this sense of the word, but … function in this other sense of the word." The teacher may even deny making an error. But an error has occurred: the teacher is ensnared by language. Four aspects of function, some of which are contradictory, have converged on the same harmony, yet the teacher only has one word to describe all four aspects. First, the chord functions as subdominant since it shares three scale degrees with IV. Second, the chord functions as subdominant (or pre-dominant) since it precedes the dominant in the prototypical harmonic progression, TSDT. Third, the chord functions as dominant since, as opposed to ii7, it has Mm quality. Fourth, the chord functions as dominant in that its root moves up by fourth (to a harmony of higher status) and its seventh points down by step. All four of these characteristics of V7/V are important, but if all four are called by only one name, function, contradictions and confusion result.

[8] I agree with Kopp that it is incumbent upon us to clear up these contradictions and confusions. In this article I accept his challenge to "more clearly articulate and differentiate the myriad concepts which function has come to represent." I will first provide a brief historical perspective for each of the four aspects of function described above. I will then suggest a plan to rid ourselves of the problem of ambiguity in the term function, once and for all.

A Brief Historical Perspective on Four Aspects of Function

[9] Ambiguity in function is not a new phenomenon. Three of the four aspects of function described above may be traced all the way back to Riemann himself; the other may be traced back even before that, to Rameau.

Three Riemannian Aspects of Function

[10] Identifying one clear and consistent meaning of harmonic function in the writings of Hugo Riemann is impossible. Instead, it is convenient to view Riemannian function as comprising three aspects.3

[11] First Aspect of Function (F1): Function involves the grouping together of harmonies that share scale degrees. Harmonies that are not primary triads may be derived from or associated with one (or more) of the primary triads.
[12] Second Aspect of Function (F2): (a) Function implies that the three pillars of harmony maintain a sphere of authority or field of activity over portions of harmonic progressions, and that (b) These spheres of authority or fields of activity tend to be organized according to the prototypical progression, TSDT.
[13] Third Aspect of Function (F3): Function may be transferred to different scale degrees.

[14] F1. The idea that a chord may function as T or S or D if it shares at least two scale degrees with its corresponding primary triad is implied by the Riemannian parallel and leading-tone changes. For example, IV is transformed into ii by the parallel change; I is transformed into iii by the leading-tone change.4 Several of Riemann's successors have clarified this aspect of function in various ways, two of which are shown in Example 2. In both the staff representation of Joel Lester and the circular diagram of Eytan Agmon, the primary triads are positioned adjacent to those secondary triads that have two common scale degrees, and are therefore, most alike.

Example 2. Contemporary models for function.

  1. Joel Lester's Model5

    Example 2a
  2. Eytan Agmon's Model6

    Example 2b

[15] F2. Riemann posits a prototypical progression, TSDT, in which function implies that the three pillars of harmony maintain a relationship of order through time.7 Furthermore, F2 also implies that these primary harmonies maintain a sphere of authority or field of activity over portions of harmonic progression. That is, T, S and D each maintain the highest status over their particular section of harmonic progression; within that section, all harmonies are subject to the authority of that particular pillar of harmony. This is perhaps most clearly exemplified in Riemann's notation for the cadential six-four chord, D6/4-5/3, which indicates that the tonic six-four chord is subject to the sphere of authority of the dominant.8 Other possibilities for the expansion of these T, S and D windows of harmony include but are not limited to: non-harmonic tones, including suspensions; other six-four paradigms; and harmonic inversion.

[16] F3. The notion that function may be transferred to different scale degrees is found in Riemann's notations, D/D and S/S.9 That Riemann chooses to notate V/V with D/D rather than S shows that he understands function to be transferable to local levels.

The Ramellian Aspect of Function

[17] Long before Riemann proposed harmonic function as an explanation of tonal music, Rameau had already done much work in function theory, though he never used the term. Among other important contributions, Rameau redefined the subdominant, and was the first to juxtapose it against the dominant as rivals for second place status compared with tonic.10 But the aspects of the three pillars of harmony that Rameau emphasized were different from those aspects Riemann developed. Whereas Riemann was more concerned with the essence of tonic, dominant and subdominant (what they are); Rameau seemed to be more concerned with their behavior (what they do). In particular, Rameau was interested in the relationships of D to T and S to T, relationships he called perfect cadence and irregular cadence, respectively.11

[18] For Rameau, dominant was not necessarily associated with scale degree 5, but rather was applied to any harmony, the root of which moves up by fourth. In other words, dominant was always a local concept. If the dominant in question were the real dominant (i.e., the one built on scale degree 5), it was called dominant-tonic.12 So a perfect cadence is not translated into modern parlance as authentic cadence, but rather as any adjacent pair of harmonies, the roots of which are separated by an ascending fourth. Likewise, an irregular cadence is not limited to plagal cadences, but to any adjacent pair of harmonies, the roots of which are separated by a descending fourth. Thus, in addition to the three aspects of the Riemannian concept of function, a fourth aspect of function can be found in the work of Rameau.

[19] Fourth Aspect of Function (F4): Function connotes a relationship between two adjacent harmonies whose roots lie a fourth apart. If the roots are separated by an ascending fourth, D-T is implied; if the roots are separated by a descending fourth, S-T is implied.

[20] Gregory Proctor follows Rameau in conceiving of function as related to a chord's behavior rather than its essence. He situates local function in the context of voice-leading and root-movement possibilities. Example 3 shows the primary triadic voice-leading moves. These models show that roots may move up or down by third, up or down by fourth, or up or down by step. The corresponding voice-leading moves are: one note displaced by step, two notes displaced by step, and three notes displaced by step, respectively. Of the six models, the most important are those in which the root moves by fourth, and two notes are displaced by step; these yield Rameau's perfect and irregular cadences.

Example 3. Proctor's voice-leading paradigms.13

Example 3

[21] Example 4 shows Proctor's functional voice-leading paradigms. In this example, the t-d-t model features lower-neighbor motion whereas the t-s-t model features upper-neighbor motion. We could say, then, that d is a lower-neighbor/lower-status element with respect to t, and that s is an upper-neighbor/lower-status element with respect to t.

Example 4. Proctor's functional voice-leading paradigms.

Example 4

[22] Alternative voice-leading moves may be achieved through passing motion. Example 5 shows how the lower-status element of each voice-leading paradigm may result from passing motion instead of neighbor motion. A d may result from passing motion from scale degree 1 to scale degree 3; s may result from passing motion from scale degree 3 to scale degree 5. We can, therefore, revise the earlier statement regarding the status of d and s with respect to t to read: d is a lower-neighbor (or 1-3 passing), lower-status element with respect to t; s is an upper-neighbor (or 3-5 passing), lower-status element with respect to t.

Example 5. Functional voice-leading paradigms with passing motion.

Example 5

[23] Thus far the voice-leading paradigms have comprised only triads. When characteristic dissonances are added to these triads, new voice-leading possibilities emerge. Example 6 shows the primary voice-leading moves for the dominant seventh and subdominant added-sixth chords.14

Example 6. Functional paradigms for seventh chords and added-sixth chords.

Example 6

[24] If seventh chords and added-sixth chords are treated as the complete version of d and s respectively, then the triadic paradigms from Example 4 may be seen as subsets of the complete version. In addition, another triadic subset emerges that takes on either d or s characteristics. This subset is the one found in the last two paradigms of Example 3, in which the root moves down or up by step and all three voices are displaced by step. In Example 7 these two moves are added to the list of functional root moves.

Example 7. Functional models expanded to include root movement by step.

Example 7

[25] This account of the history of function theory has shown that the term function is and always has been ambiguous. Riemann's concept of function included at least three distinct aspects: (F1) function as similarity in essence among harmonies as exemplified by two or more shared scale degrees, (F2) function as place holder and sphere of authority within a prototypical harmonic progression, and (F3) function as transferable to local harmonic levels. A case has been made for the inclusion of a fourth aspect of function theory, courtesy of Rameau, for which function involves the motion of harmonies whose roots are separated by fourth or by step (F4). This fourth aspect of function loads the term with even more ambiguity. All four aspects reveal important information about harmonic characteristics, and all four are therefore worthy of usage as explanatory tools for tonal harmony. However, if all four aspects go by only one name, function, ambiguity remains, and function theory can never reach its maximum potential for explanation. This problem must be solved.

A Solution to the Problem of Ambiguity

[26] I propose a solution: the elimination of the term function. In place of function I suggest invoking four new terms—kinship, province, quality and behavior—each of which uniquely corresponds to the four distinct aspects of function described above.

Kinship

[27] Kinship (which has thus far been called F1) yields an array of harmonies categorized together based on their similarity in essence. Harmonies that share two or more scale degrees with one of the pillars of harmony have kinship with that harmony. A healthy, but not comprehensive list of harmonies grouped by kinship is given in Table 1 and Table 2.

Table 1. Diatonic triads grouped by kinship.

T S D
I, i IV, iv V, v
vi, VI ii, ii° vii°, VII
iii, III vi, VI iii, III

Table 2. Chromatic harmonies grouped by kinship.

T S D
V/IV, vii°/IV V/V, vii°/V vii°7
Gr6 It6, Fr6, Gr6, Sw6 Gr6/I
N6

Province

[28] Province is defined as a "sphere or field of activity or authority."15 I have used these phrases, sphere of authority and field of activity, throughout this article when referring to the second of the Riemannian aspects of function (F2). Province implies two characteristics: first, that tonic, dominant, and subdominant maintain spheres of authority over portions of harmonic progressions, and second, that these spheres of authority are arranged according to the prototypical progression, TSDT.

Quality

[29] When diatonic seventh chords are built on all scale degrees in the major mode, the Mm7 appears only once, and is unique to scale degree 5. It is called the dominant seventh chord. Since Mm quality is unique to the dominant in the diatonic system, we can say that Mm7 chords have dominant quality, even if they are not built on the dominant (i.e., if they are secondary dominants created through tonicization).

[30] In the same way, when diatonic added-sixth chords are built on all scale degrees in the minor mode, the minor (triad with major) added-sixth appears only once, and is unique to scale degree 4. Since madd6 quality is unique to the subdominant in the diatonic system, we can say that madd6 chords have subdominant quality, even if they are not built on the subdominant.

[31] One could suggest that IVadd6 should also have subdominant quality, but this fails on two points. First, there would then be two qualities known by the same name. Second, if IVadd6 is a subdominant quality, then v7 would have to be included as a dominant quality. Most readers will intuit the problems with identifying v7 as dominant quality; it destroys the whole reason for having the term quality in the first place. In the same way, IVadd6 may not be considered to have subdominant quality. Instead, v7 and V7 are both said to have dominant identity meaning that each is a chord whose root is scale degree 5; IVadd6 and ivadd6 have subdominant identity, meaning that each is a chord whose root is scale degree 4.

[32] It might also be suggested that the half-diminished seventh chord (iiø7) be a subdominant quality. After all, it is identical to ivadd6 in essence (i.e., they share all four notes). However, this suggestion shows a bias toward seventh chords over added-sixth chords and ignores Rameau's theory of double employment. According to this theory, ivadd6 moves to (or has moved from) tonic, whereas ivadd6 must undergo a transformation into iiø7 if it moves to V. In this way the dissonances associated with each harmony resolve correctly: either the added-sixth resolves upward or the seventh resolves downward.16

Behavior

[33] Rameau's concept of function (F4) is roughly equivalent to behavior. Behavior involves the application of functional properties to local levels. Two chords behave as tonic and dominant (or tonic and subdominant) regardless of their identity (i.e., what scale degree they are built on) if there occurs between the two chords a root move of a fourth or a step. Whichever of the chords has higher status is considered the local tonic (i.e., exhibits tonic behavior). Depending on the type of root move, the other chord exhibits dominant or subdominant behavior.

[34] The cadential six-four paradigm again serves as a clear and simple example. Between the two harmonies, I6/4 and V, V has the higher status, and is therefore the local tonic. The move from I6/4 to V is a root move down a fourth to a higher-status element. Therefore I6/4 behaves as the subdominant of V.

Definitions

Dominant
  1. [35] A harmony has Dominant Identity if and only if it is built on the dominant. That is, dominant triads and dominant seventh chords (as well as dominant ninth, eleventh, thirteenth chords and even dominant added-sixth chords) have dominant identity. No other harmonies have dominant identity.
  2. [36] A harmony has Dominant Quality if and only if it is Mm (the seventh might not appear on the surface).
  3. [37] A harmony has Dominant Kinship if and only if it shares at least two scale degrees with the dominant triad.
  4. [38] A harmony exhibits Dominant Behavior if it is the lower-status element of a lower-neighbor (or 1-3 passing) voice-leading paradigm. Such a paradigm will feature root motion by ascending fourth or step from lower to higher status (d-t) or root motion by descending fourth or step from higher to lower status (t-d) or both.
  5. [39] A harmony (or group of harmonies) belongs to the Dominant Province if it is governed by the sphere of authority of the dominant. The dominant province usually occupies the position immediately before the final tonic province in a prototypical harmonic progression, TSDT.
Subdominant
  1. [40] A harmony has Subdominant Identity if it is built on the subdominant. That is, subdominant triads and subdominant added-sixth chords (as well as subdominant seventh chords) have subdominant identity. Harmonies not built on the subdominant do not have subdominant identity.
  2. [41] A harmony has Subdominant Quality if and only if it is a minor added-sixth chord (the sixth might not appear on the surface).
  3. [42] A harmony has Subdominant Kinship if it shares at least two scale degrees with the subdominant triad.
  4. [43] A harmony exhibits Subdominant Behavior if it is the lower-status element of an upper-neighbor (or 3-5 passing) voice-leading paradigm. Such a paradigm will feature root motion by descending fourth from lower to higher status (s-t) or root motion by ascending fourth from higher to lower status (t-s) or both.
  5. [44] A harmony (or group of harmonies) belongs to the Subdominant Province if it is governed by the sphere of authority of the subdominant. The subdominant province usually occupies the position immediately after the initial tonic province in a prototypical harmonic progression, TSDT.
Tonic
  1. [45] A harmony has Tonic Identity if and only if it is built on the tonic.
  2. [46] A harmony has Tonic Kinship if and only if it shares at least two scale degrees with the tonic triad. A harmony may have both tonic kinship and dominant kinship, or it may have both tonic kinship and subdominant kinship.
  3. [47] A harmony exhibits Tonic Behavior if it is the higher-status element of an upper-neighbor or lower-neighbor (or 1-3 passing or 3-5 passing) voice-leading paradigm. Such a paradigm will feature root motion by ascending fourth from higher to lower status (t-s), root motion by descending fourth from lower to higher status (s-t), root motion by descending fourth or step from higher to lower status (t-d), or root motion by ascending fourth or step from lower to higher status (d-t).
  4. [48] A harmony (or group of harmonies) belongs to the Tonic Province if it is governed by the sphere of authority of the tonic. The tonic province usually occupies the first and last positions in a prototypical harmonic progression, TSDT.

Explanatory Ramifications of the New Terms

[49] With these new terms, the ambiguity of Example 1 is erased. The teacher could simply say that the harmony (V7/V) has subdominant kinship since it shares three scale degrees with IV. The harmony belongs to the subdominant province since it precedes the dominant in the prototypical harmonic progression, TSDT. The harmony has dominant quality since it is Mm. And finally, the harmony exhibits dominant behavior in that its root moves up by fourth (to a harmony of higher status) and its seventh points down by step.

[50] One of the most important concepts to emerge from the new terms is that behavior and provinces can be complementary. The interaction of behavior and province within a progression is the heart of functional significance. Six-four paradigms are perhaps the most straight-forward examples of behavior complementing provinces. For the sake of simplicity, I have combined all four into one phrase, shown in Example 8.

Example 8. Behavior and province in the six-four paradigms.

Example 8

[51] In the first tonic province, m. 1, the arpeggiating six-four chord simply expands the province through inversion. The I6/4 chord exhibits tonic behavior by default since it does not move to a new harmony. It also belongs to the tonic province.

[52] Within the subdominant province, m. 2, the same harmony, I6/4, is a passing chord that exhibits dominant behavior. The IV and IV6 that surround I6/4 have higher status since the subdominant maintains a sphere of authority over this section of the progression. Since IV is the local tonic (i. e., highest-status element), I6/4 behaves as a local dominant. It is a lower-adjacency (1-3 passing), lower-status harmony. Its root is approached by descending fourth, and moves forward by ascending fourth. In other words, in m. 2, I6/4 is the dominant of the subdominant. Although it has tonic identity, it exhibits dominant behavior, and belongs to the subdominant province.

[53] A third appearance of I6/4 occurs within the dominant province, m. 3. The same harmony that belonged to the tonic province in m. 1 and to the subdominant province in m. 2 now belongs to the dominant province. Since the dominant maintains a sphere of authority over m. 3, it is the element of highest status, and is therefore treated as a local tonic. The cadential six-four chord is an upper-neighbor harmony to V. As an upper-neighbor, lower-status harmony, the cadential six-four chord exhibits subdominant behavior. Its root moves down by fourth to a root of higher status. It is the subdominant of the dominant. Thus, in m. 3, I6/4 has tonic identity, but exhibits subdominant behavior and belongs to the dominant province. In the first three measures, a single harmony, I6/4, exhibits each of the three behaviors and belongs to each of the three provinces.

[54] Whereas the six-four paradigms of Example 8 reveal how behavior can complement provinces, the secondary dominant of Example 1 reveals how a behavior paradigm may cross over a province boundary. The secondary dominant is a lower-adjacency, lower-status element with respect to G7. It therefore exhibits dominant behavior. However, the secondary dominant belongs to the subdominant province; its behavior paradigm (d-t) therefore crosses over the boundary between the subdominant and dominant province.

Notation

[55] From the proposed new terms, an analytic system may be developed that is clearer and more comprehensive than traditional systems. This three-fold system of analysis, which is featured in Example 8, explicitly accounts for identity, behavior, and province.17 The first line of analysis is called identification. This is a low-order analysis in which harmonies are identified regardless of context (i.e., behavior, province, etc.), and labels are given based only on the key of the piece/section. The harmonies may be notated with the standard Roman-numeral system, although other systems are also acceptable.18

[56] The second line of analysis is behavior analysis. Two or more harmonies are grouped together and analyzed on the basis of status and voice leading. The highest-status element in any grouping is notated with a lower case t, which indicates local tonic. Lower-status elements are notated with lower-case d or s to indicate local dominant or subdominant. The harmonies are grouped together with a horizontal square bracket. A grouping in behavior analysis may reach into more than one province, but the behavior group is always self-contained. The lower-case t, s and d in the behavior analysis model will differentiate that notation from the upper-case T, S and D that will signify province. In the behavior level of analysis, chromatic harmonies may be distinguished from diatonic harmonies by slashes. An upward pointing slash (/) will designate a raised chromatic note; a downward pointing slash (\) will designate a lowered chromatic note.

[57] The third line of analysis is province analysis. Provinces are notated with capital T, S and D, as well as with horizontal lines that extend out to the boundary of the province. The prototypical progression for province analysis is TSDT, although incomplete versions may exist, including TSD (half-cadence), TST (plagal cadence) and TDT.

[58] Example 10 shows the beginning of a piano sonata by Beethoven analyzed with my three-fold system of analysis. Behavior analysis may be as detailed as the analyst wishes, though I find it little more than busy work to account for behavior in those places where behavior, province, and identity are identical. For example, a phrase that ends V-i, such as the one in mm. 15-16, need not be labeled d-t in the behavior analysis, since it is already labeled V-I in the identification, and D-T in the province analysis. Instead behavior analysis should at least account for any harmony whose behavior contradicts its province. By contradict I mean that the behavior label (t, d or s) is different from the province label (T, D or S) for a particular harmony. This would include six-four chords, for which the boundary of the behavior paradigm is completely within the boundary of the province. Examples of these paradigms can be found in mm. 5, 7-8, 13 and 15. It would also include behavior paradigms that cross over province boundaries, such as those in mm. 3-4 and 11-12, for which the behavior d-t contradicts the province S-D.

Example 10. Beethoven, Piano Sonata no. 9, Op. 14, no. 1, mm. 1-16.

Example 10

Conclusion

[59] A good theory has two characteristics: it is an explanation, and it is clear. Function theory is a good explanation. All four aspects of function yield important explanatory information about harmonic characteristics. Unfortunately, function theory has never been clear. As long as function is weighted down with ambiguity, its explanatory power can never reach its full potential. By replacing function with behavior, kinship, province and quality, we can "know exactly what we mean" and "guarantee that our reader will accurately grasp our meaning." Then function theory will become a clear explanation, and thus a good theory.

Selected Bibliography

Agmon, Eytan. "Functional Harmony Revisited: A Prototype-Theoretic Approach." Music Theory Spectrum 17, no. 2 (Fall 1995): 196-214.

Brasky, Jill. On Function. Paper presented at the conference of the Music Theory Society of the Mid-Atlantic, Washington D. C., 24 March 2007.

Caplin, William. "Tonal Function and Metrical Accent: A Historical Perspective." Music Theory Spectrum 5 (Spring 1983): 1-14.

Chandler, Glenn B. "Rameau's Nouveau système de musique théorique: An Annotated Translation with Commentary." Ph.D. diss., Indiana University, 1975.

Ferris, Joan. "The Evolution of Rameau's Harmonic Theories." Journal of Music Theory 3, no. 1 (1959): 231-256.

Harrison, Daniel. Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of its Precedents. Chicago: University of Chicago Press, 1994.

Kopp, David. "On the Function of Function." Music Theory Online 1, no. 3 (May 1995): 15 pages.

Lester, Joel. Harmony in Tonal Music. Vol. 1, Diatonic Practices. New York: Alfred A. Knopf, 1982.

Mickelsen, William. Hugo Riemann's Theory of Harmony and History of Music, Book III. Lincoln: University of Nebraska Press, 1977.

Proctor, Gregory. Harmonic Function and Voice Leading. Unpublished Paper.

Rameau, Jean-Philippe. Treatise on Harmony. Translated with an Introduction and Notes by Philip Gossett. New York: Dover Publications, 1971.

Riemann, Hugo. Harmony Simplified or the Theory of the Tonal Functions. Translated by H. W. Bewerunge. London: Augener Ltd., [1895]; orig. German ed., London: Augener Ltd., 1893.

Smith, Charles. "The Functional Extravagance of Chromatic Chords." Music Theory Spectrum 8 (Spring 1986): 94-139.

Swinden, Kevin J. "When Functions Collide: Aspects of Plural Function in Chromatic Music." Music Theory Spectrum 27, no. 2 (2005): 249-82.

Notes

1 This article is an expanded version of a paper given at the first annual Yale Graduate Music Symposium in New Haven, CT (March 29, 2008).

2 David Kopp, "On the Function of Function," Music Theory Online 1, no. 3 (May 1995): [15].

3 William Mickelsen's summary of the "basic features of Riemann's mature harmonic system" includes four points, two of which parallel what I have termed F1 and F2. These are that "Chords other than the three primary harmonies are mixtures of notes from these chords and thus may be comprehended as representing two or even three of the primary chords" (F1) and "harmonic function (and tonality) basically involves the movement away from the tonic to chords having dominant or subdominant significance and back to the tonic chord" (F2). Mickelsen rightly does not include F3 as one of the basic features because it is tangential to the other two primary aspects. I include it here not to give it theoretical weight equal to F1 and F2, but rather to separate it from these other aspects of function in an effort to erase ambiguity. William Mickelsen, Hugo Riemann's Theory of Harmony and History of Music, Book III (Lincoln: University of Nebraska Press, 1977), 5.

4 Hugo Riemann, Harmony Simplified or the Theory of the Tonal Functions, trans. H. W. Bewerunge (London: Augener Ltd., [1895]), 71-77.

5 Joel Lester, Harmony in Tonal Music, vol. 1, Diatonic Practices (New York: Alfred A. Knopf, 1982), 251. Used with permission of the author.

6 Eytan Agmon, "Functional Harmony Revisited: A Prototype-Theoretic Approach," Music Theory Spectrum 17, no. 2 (Fall 1995): 201. Used with permission of the author.

7 Riemann, 45.

8 Ibid., 22-3.

9 Ibid., 101.

10 Jean-Philippe Rameau, Nouveau système de musique théorique (Paris: Ballard, 1726), 38; translated in Glenn B. Chandler, "Rameau's Nouveau système de musique théorique: An Annotated Translation with Commentary" (Ph.D. diss., Indiana University, 1975), 270.

11 Jean-Philippe Rameau, Treatise on Harmony, trans. with an introduction by Phillip Gossett (New York: Dover Publications, 1971), 63-70, 73-82.

12 Ibid., 237.

13 Gregory Proctor, "Harmonic Function and Voice Leading," unpublished paper, 21-28. Examples 3-7, and all discussion related to these examples, are synthesized versions of this paper and personal communication between Gregory Proctor and the author.

14 I only wish to emphasize the natural voice-leading possibilities for these chords. I have, therefore, not included unnatural resolutions of the dissonances (i.e., sevenths moving up or sixths moving down) as does Proctor. In order for dissonances to move contrary to their natural resolutions, they must be overpowered by a superior musical force. Parallel (sixth or third) motion, for example, is one musical force that overrides the tendency of the seventh to pull down.

15 Webster's New Universal Unabridged Dictionary, rev. ed. (New York: Barnes & Noble, 2003), 1556.

16 Jean-Philippe Rameau, Génération harmonique, ou Traité de musique théorique et pratique (Paris: Prault, 1737), 115; trans. in Deborah Hayes, "Rameau's Theory of Harmonic Generation: An Annotated Translation and Commentary of Génération harmonique by Jean-Philippe Rameau" (Ph.D. diss., Stanford University, 1968), 140.

17 Quality and kinship do not receive their own line of analysis in the system. This is because they remain constant regardless of their context within a progression, and therefore, do not require that analytic choices be made about them.

18 I prefer to identify harmonies with a notation that does not mistake figured bass for inversion. Figured bass was originally used as a guide to performance, not a tool for analysis. It has since been linked to the Roman-numeral system as a way of indicating the inversion of a harmony. Unfortunately, figured bass does not indicate inversion; it only indicates intervals above the bass. Consequently, the standard Roman-numeral system, with its mistaken use of figured bass to indicate inversion, actually thwarts real analysis by eliminating the option of the added-sixth chord. For example, the figure 6/5 does not indicate a seventh chord in first inversion. It indicates a harmony arranged with a third, fifth, and sixth above a bass note. By analysis we might find that this chord is a seventh chord in first inversion or an added-sixth chord in root position. If we want to identify it as an added-sixth chord (let's say our chord is F-A-C-D in the key of C), then we cannot call it IV6/5 because the notation's bias toward seventh chords would lead us to spell that chord A-C-E-F. We cannot call it ii6/5 because we believe the root to be F. It is not so much that the figured bass does not provide us with a way to notate added-sixth chords, the notation 6/5 does that just fine. It is the fact that we have perverted figured bass notation to mean inversion, and that we have biased ourselves toward seventh chords to the exclusion of added-sixth chords, that leaves us no consistent notation for inversion in a world where seventh chords and added-sixth chords are both legitimate. (As an illustration of my point, imagine the opposite. If we were to take figured bass symbols to represent inversions of added-sixth chords instead of seventh chords, we would be identifying a root position G7 chord in the key of C as viio7: an added-sixth chord in third inversion. I am not sure that what we actually do is much less ludicrous.)

The Roman-numeral system may be refined so as to more clearly account for added-sixth chords and harmonic inversion. First, figured bass symbols may be replaced with the Arabic numbers from jazz-pop notation. These numbers—2 through 13—indicate intervals above the root, not the bass. Second, the Roman-numeral system may also borrow from jazz-pop notation its symbol for inversion, the forward slash. Inversions may be notated with slashes followed by the number of the chord member that occurs in the bass. For example, V in first inversion would be written V/3, and said "five over three." Third, invoking the slash as a notation for inversion leaves the applied chord without a notation in the modified Roman-numeral system. We may borrow from mathematics its notation for the word of: the parentheses, as in f(x). An applied chord such as "five of five in first inversion" would then be, V(V)/3. We even get the satisfaction of invoking a mathematical notation for function to become a musical notation for function!

There are two advantages to this modified Roman-numeral system. First, pedagogically speaking, learning jazz-pop notation, which usually occurs subsequent to learning the Roman-numeral system for university-trained musicians, becomes much easier. Second, and more importantly, the modified Roman-numeral system recognizes the added-sixth chord. For virtually all music from Bach to Beethoven, superscript Arabic figures would be limited to 7 and 6! Furthermore, the modified notation allows analysts to make decisions about whether a four-note harmony is a seventh chord or an added-sixth chord based on its context within a progression. Such an analysis may look like the one in Example 9. Specifically, the modified notation allows the analyst to account for Rameau's double employment of the dissonance in m. 2, where the harmony is initially an added-sixth chord as it is approached from I, and then becomes a seventh chord as it progresses to V.

Example 9. Bach, "Prelude No. 1," Das Wohltemperierte Clavier, Book I, mm. 1-6.

Example 9

I do not mean to imply that figured bass symbols are inherently bad. On the contrary, figured bass is an excellent pedagogical tool for keyboard, counterpoint, and voice-leading. Figured bass is even successful in an analytic notation system in one respect: its ability to depict melodic motion, in particular suspensions and suspension-like harmonies such as the cadential six-four chord (6/4-5/3). Fortunately, my three-fold system of analysis accounts for these motions without having to rely on figured bass. If the analyst wishes, figured bass symbols may still be marked above the notes, in much the same way as common notations for suspensions.

The modified notation described above builds on that of Carolyn Alchin, who notates inversions by placing the number of the chord member occurring in the bass under the Roman numeral, for example, I/5 instead of I6/4. Carolyn A. Alchin, Applied Harmony (Los Angeles: Carolyn Alchin, 1917), 28.