Flip-Flop Circles and their Groups

Flip-Flop Circles and their Groups John Clough I. Introduction We begin with an example drawn from Richard Cohn s 1996 paper Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions. Example 1 shows four circles of six triads each. The circle at the top symbolizes the chord progression C major - C minor - Ab major -Ab minor -E major - E minor, shown also in musical notation. The moves from one triad to the next make a regular pattern: reading clockwise, each major triad is followed by the minor triad on the same root (for example, C major to C minor); and each minor triad is followed by the major triad whose third is the root of the preceding minor triad (for example C minor to Ab major). The first of these moves is Hugo Riemann s Parallel transformation, symbolized P, and the second is Riemann s Leittonwechsel, symbolized L. Hence the label neo-riemannian theory that is sometimes used to describe the recent work of Cohn and others that harks back to nineteenth-century theory. A passage from Brahms, given as Example 2, exemplifies the top circle of Example 1. [Play] Like the top circle of Example 1, the other three circles of the example have six triads each, and they can each be circumnavigated by means of P and L. Cohn refers to each individual circle as a hexatonic system and to the four circles taken together as a the hyper-hexatonic system. Example 3 shows that each of the two transformations, P and L, may be realized by 12 pairs of triads. And each is reversible it applies in both directions. So P is the transformation that takes C major to C minor, or C minor to C major; P also takes C#

Flip-Flop, p.2 major to C# minor, or C# minor to C# major, etc. Likewise, L takes C major to E minor or E minor to C major; also L takes C# major to E# minor, or E# minor to C# major, etc. Transformations such as P and L that reverse themselves when applied twice are called involutions. Their role is central in the present study. Looking at Example 4, we see another transformation defined by Riemann called the Relative, symbolized R, that takes C major to A minor or the reverse, C# major to A# minor or the reverse, etc. Like P and L, R applies to 12 pairs of triads. Note that P, L, and R, all support connection of two triads by means of two common tones. And they are the only involutions that do so. P connects two triads by retaining the pair of notes that form a perfect 5th and moving the other note by a half-step; L connects two triads by retaining the pair of notes that form a minor 3rd and moving the other note by a halfstep; and R connects two triads by retaining the pair of notes that form a major 3rd and moving the other note by a whole-step. However, we emphasize that while P, L, and R are conceived largely on the basis of voice leading, they are defined here as operations on sets of notes (pitch-classes). As such, P, L, and R do not require any particular voiceleading. We can define additional operations as strings of P, L, and R. For instance, referring now to Example 5a, we see that the motion from C major to D minor results from doing R, then L, then R once again. This same operation, call it RLR, also takes us from C# major to D# minor, from D major to E minor, etc. In each case, the operation is reversible: RLR takes us from D minor to C major, etc. like P, L, and R, RLR is an involution. However, not all strings of P, L, and R are involutions. Example 5b shows that R followed by L, call it RL, takes us from C major to F major, but RL applied to F major does not return us to C major; it leads to Bb major. Involutions are special, though, because circles based on them can be traversed equally well in either direction. There are many historical precedents for circles like Cohn s hexatonic systems considered above. Consider Example 6, a diagram from Heinichen s 1728 treatise Der Generalbass in der Komposition. The example portrays a circle of major keys alternating with minor keys: reading clockwise from the top, C ma-

Flip-Flop, p.3 jor - Aminor - Gmajor - E minor - Dmajor - Bminor, etc. For our purposes today, we shall view it as a circle of major and minor triads. As we move clockwise around the circle, R and RLR apply alternately. As with Cohn s circles, the pattern of Example 6 is uniform and the circle is closed. In contrast to Cohn s circles, however, Heinichen s touches all 24 consonant triads. It goes without saying that Cohn s and Heinichen s circles are responsive to tonal music the latter applying quite generally and the former more particularly to nineteenth-century music. But looking beyond the tonal frame of reference and thinking of the matter quite generally, it seems natural to ask: What is the space of all such circles? It is this question that motivates the present investigation. II. The one hundred sixty-eight UFFCs Before proceeding we establish some terminology and notation. In Cohn s hexatonic systems and in Heinichen s circle, each circle may be formed by the overlay of two smaller circles one of major and the other of minor triads, both based on the same interval of transposition in the same direction. Ex. 7a shows how one of Cohn s hexatonic systems can be formed from two smaller constituent circles. Each constituent circle is based on transposition by 4 semitones, symbolized T4; hence, reading counterclockwise, T4 from C major to E major, T4 from E major to Ab major; and T4 from Ab major to C major, closing the circle, and the same for the constituent circle of three minor triads. Example 7b shows a comparable scheme underlying Heinichen s circle, but here the large circle is composed of two interleaved circles of 5ths (or intervals of 7 semitones); so the smaller constituent circles are generated by repeated application of T7, reading clockwise. I shall refer to circles of this kind as uniform flip-flop circles (or UFFCs). We can represent UFFCs and their constituent circles as sequences of symbols enclosed in parentheses, understanding that they wrap around from last to first element. In this nota-

Flip-Flop, p.4 tion, Ex. 8a shows how Heinichen s circle may be represented as a conjunction of its component circles, and likewise Ex. 8b for each of Cohn s four hexatonic systems. To recapitulate, a UFFC has two component circles, a circle of major triads and a circle of minor triads, both with the same interval of transposition in the same direction, and each containing at least two triads. When the two component circles are interleaved, the result is a UFFC. There are 168 UFFCs in the usual 12-pc universe, and the process of enumerating them is instructive. Let us take as Ex. 8b as a point of departure. For each UFFC in the example, there is one circle of major triads and one of minor, and these triads are all different; so we have four different component circles of three major triads each and four of three minor triads each. Based on T4, these are all the possible circles of major and minor triads; they are arrayed in Ex. 9 on the left. However, each of the four UFFCs of Ex. 8b represents just one of the ways that a particular major and a particular minor circle can be paired to form a UFFC. Since we have four each of major and minor circles, there are in fact 4 X 4 = 16 different pairs of component circles that can support a UFFC. Ex. 9 illustrates one of these 16 a pairing of (C E Ab) with (db f a) to form the UFFC (C db E f Ab a). But there are more than 16 UFFCs based on T4! To see how this is so, consider the UFFC of Ex. 9, reproduced in Ex. 10. By rotation of one component circle against the other, we can generate two additional UFFCs. It should be clear that, based on T4, each and every one of the 16 possible major-minor pairs of circles yields three UFFCs, for a total of 48. Ex. 11 gives a format for a roster of these 48 UFFCs, which the student may wish to complete as an exercise. To proceed with the enumeration of UFFCs in the 12-pc universe, we need to consider the operators T t, where t ranges from 1 through 6. The value t = 0 does not generate UFFCs (we will discuss this case later), and values of t greater than 6 merely replicate UFFCs with smaller values for t. For example, T8 reproduces the circles that we have already accounted for above with T4, since 4 and 8 sum to 12, and similarly for

Flip-Flop, p.5 other pairs of pc intervals that sum to 12 (that is, pairs of intervals in the same interval class). Table 1 shows the distribution of the 168 UFFCs with respect to the values of t ranging from 1 through 6. The shaded row reflects the count we have just completed of the 48 UFFCs for t = 4. Scanning that row, we see in the first column the value t = 4. The second column, headed #C = GCD (t, 12) gives the number of distinct circles of major triads, also the number of distinct circles of minor triads, four of each, as we have seen. The expression GCD (t, 12) means greatest common divisor of t and 12. Note that, for values of t that are divisors of 12, namely, 1, 2, 3, 4, and 6, this expression is equal to t. Thus, for t = 1, there is just one circle of major triads and one of minor, based on the circle of the chromatic scale. For t = 2, there are two of each, whose chord roots comprise the two distinct whole-tone scales; for t = 3, there are three of each, based on the three diminished 7th chords; for t = 4, four of each based on the four augmented triads; and finally for t = 6, there are 6 of each based on the six distinct tritones. The third column, headed by C enclosed in vertical lines read cardinality of C or simply card. C gives the number of triads in each of the two component circles. As we have seen, for t = 4, this number is 3: three major triads and three minor triads. Since, taken together, the chord roots of all the major circles exhaust the 12 pcs without repetition (and the same for minor circles), it is plain that the expression 12 / #C yields the desired value. The fourth column, headed #Cprs counts the number of distinct pairs of component circles that can form a UFFC, that is, pairs of one major and one minor circle. For t = 4, that number is simply 4 squared, or 16, as we have seen. The next column, headed #UFFC gives the total number of UFFCs. As computed above for t = 4, there are 48 UFFCs. At the head of the column, following the first equals sign, in the expression C #Cprs, the term C accounts for distinct rotations of the two component circles. However, in the case of t = 6 this number must be halved (a kind of tritone exception familiar to students of atonal set theory). Note that the total for all entries in this column is the promised 168. For the moment, we defer consid-

Flip-Flop, p.6 eration of the expression following the second equals sign at the top of the column, also of data in the remaining two columns. For lack of time, we omit consideration of many aspects of Table 1, but those interested may find additional information regarding the table in the notes on the same page. As noted above, the value t = 0 requires special comment. The data corresponding to t = 0 are given in a separate row at the foot of Table 1. Here card. C equals 1, which tells us that a constituent circle contains just one triad; hence, the two constituents together form a degenerate circle with just one major and one minor triad. We do not count these cases as UFFCs. Yet, regardless of terminology, we need to consider the case t = 0 for the sake of completeness, for reasons that will become clear later. The large number of cases, 144, simply represents all the possible pairings of one major and one minor triad, that is #Cprs = 12 squared. (Obviously, in this case, rotation does not produce anything different.) III. UFFCs and the Schritte-Wechsel Group Now that we have enumerated the UFFCs in the usual 12 pitch-class universe, we seek a deeper understanding of their structure. Underlying much of atonal set theory and also neo-riemannian theory is the concept of a mathematical group. How can the theory of groups advance our understanding of UFFCs? This question will be our major concern throughout most of the remainder of this lecture. To begin answering it, we review the definition of a group by means of a construct that is familiar in music theory. With this preparation, we then move on to what is called the Schritt-Wechsel group, after Riemann, and explore its close relationship to the notion of UFFC. The group of transpositions and inversions, acting on the 12 pcs, and by extension on sets of pcs, is well known in atonal set theory. I will refer to this group as the T/I group. It consists of 12 transpositions and 12 inversions, often symbolized T0, T1,..., T11; and T1I, T2I,..., T11I, as shown in Ex. 12a. These are the elements of the group.

Flip-Flop, p.7 Examples 12b and c give the rules by which these transformations operate, singly and in combination. A word on notation: in Ex. 12c, the transformations within parentheses are performed first. The set of 24 transformations forms a group since it satisfies the following axioms: 1. Closure: any series of transformations is equivalent to one of the 24 transformations. Ex. 12d shows that if we first perform T3I on pc x, and then perform T1 on the result, then we have effectively performed T4I on x. 2. Identity: There is a transformation that leaves its argument unchanged. This transformation is, of course, T0. See Ex. 12e. 3. Inverses: For every transformation a, there is a transformation b, not necessarily distinct from a, such that b undoes the effect of a. See Ex. 12f, where T4I is shown to be its own inverse. There is another axiom, namely associativity, that is necessary to the definition of a group. However, for sets of transformations such as we are considering today, that axiom is automatically satisfied, so we ignore it here. Mathematical groups may be as complex as we might wish, but today we shall be concerned with only two elementary types of groups: cyclic groups and dihedral groups. The structure of cyclic groups is quite simple. The are generated entirely by repetition of a single element. Consider the set of 12 transpositions by themselves, without the 12 inversions. These transformations, T0, T1, T2,..., T11, satisfy the required axioms, and therefore they form a group. Since we can generate all elements of the group from a single element, the group is cyclic. Ex. 13 shows how this is done with T1 as the generating element. In the equations of Ex. 13 and elsewhere, repetitions of an element are indicated as powers; for example, T1 squared equals T2, T1 cubed equals T3, etc. By the way, T1 is not the only element that generates the group of transpositions. T5, T7, and T11 will also serve. An important concept in group theory is that of a subgroup. For the present purpose, a subgroup is a subset of a group that satisfies all three of the above axioms. The subgroups of the group of 12 transpositions (without the inversions) are listed in Ex. 14a.

Flip-Flop, p.8 All of these subgroups are cyclic. Among them there are two that qualify as subgroups automatically: the subgroup consisting of only the identity element, in this case T0, and the full group itself, in this case all the transpositions. Any group has two such trivial subgroups. If we wish to exclude them, we may refer to the proper subgroups of the group we are concerned with. The full set of 24 transpositions and inversions is an example of a dihedral group. Mathematicians usually describe such groups in terms of the congruence motions of a regular polygon. We will describe it here as a group as arising from a cyclic subgroup in combination with a flip or inversion which doubles the size of the cyclic subgroup. Thus for the T/I group, we start with the cyclic subgroup consisting of the 12 transpositions, and then take a specific inversion called I, arbitrarily defined to be the inversion around the pitch class C. When combined with the 12 transpositions, I yields 12 additional transformations, namely the 12 distinct inversions around the 12 possible pitch-class centers. The 28 dihedral subgroups of the T/I group are listed in abbreviated form in Ex. 14b. I leave it as an exercise for the interested student to verify that the list is complete and correct. In carrying out this exercise, it may help to bear in mind the equations given in Ex. 15. These show the order of each element in the T/I group, that is the number of times it must be repeated to equal the identity. Meanwhile, what has become of our old friends L, P, R, and sequences such as RL and RLR formed from them? Let us see how these transformations induce their own groups. Ex. 16 shows a circle of the 24 major and minor triads. Here, as with the circle of Heinichen examined earlier, we can traverse the circle in either direction by alternation of two transformations, in this case L and R. By means of an appropriate sequence of L s and R s, we can navigate the circle at will: given any two triads we can go from one to the other. Say we wish to go from C major, at the top of the circle, to C# minor, clockwise between 3 and 4 o clock. The sequence LRLRLRL will take us there. As shown on the example, we can rewrite this sequence as (LR) 3 L. In this notation, the pair LR may be conceived as a single transformation which takes us from C major to G ma-

Flip-Flop, p.9 jor, or from G major to D major, etc. In fact we can conceive of any arbitrary sequence of L s and R s as representing a unique single transformation. Moreover, as shown in Ex. 17a, we can rewrite any such sequence in the form (LR) n or (LR) n L, where n ranges over the integers 0 through 11. This is, in essence, the Schritt-Wechsel (S/W) group, presaged in the work of Hugo Riemann (1882) and first presented in explicitly grouptheoretic terms by Henry Klumpenhouwer (1994). The powers of (LR) are the 12 Schritte, which move from one triad to another of the same mode, while the powers of (LR) followed by an additional L are the 12 Wechsels, which move from one triad to another of the opposite mode. It is easy to see that this set of transformations is closed, and the other group axioms are not difficult to verify. As shown in Ex. 17b, the identity element may be written as (LR) 0 or L 2 or R 2 or in many other ways. Inverses are shown in Ex. 17c. Ex. 17d gives an equation showing the order of each element. Note that, while Wechsels, the elements with the extra L are their own inverses, this is not generally true of Schritte, the powers of LR. But what of the transformations R and RLR that formed the basis for Heinichen s circle of Ex. 6? These too can be expressed in terms of LR and L, as shown in Ex. 17e. In fact there are infinitely many ways to express any of the 24 transformations of the group at hand. Ex. 17f shows just four of the ways to express LR. Which one should we choose? It depends on our objectives. Much of the neo-riemannian work to date has focused on the analysis of 19th-century music, where it is natural to choose representations that reflect the actual path from one triad to another. As realized in music, these paths have tended to feature smooth voice-leading; hence the emphasis, in this analytical work, on P, L, and R. Certainly, if we find a C major triad followed by an A minor triad, it is natural to write R instead of, say, (LR) 11 L, as in Ex. 17e. But my program here is more abstract, and, as I will show, there are certain advantages to the apparently more mechanical notation used in Example 17 and another much simpler notation which I will now introduce.

Flip-Flop, p.10 Returning to Ex. 16, note that, if we begin on C major, at 12 o clock on the circle, and perform the transformation (LR) seven times, we arrive on C# major, at 7 o clock on the circle, seven hours later. On the other hand if we begin on C minor, at 8:30 o'clock on the circle, and perform (LR) seven times, we arrive on B minor, at 1:30 o clock, seven hours earlier, or five later if you prefer. Let us give this transformation its own special symbol and write τ1 (pronounced tau one ) for it, as shown in Ex. 18a. Note that τ1 has the effect of transposing a major triad up by one semitone and a minor triad down by one semitone. In the same vein, we write τ2, τ3, etc., for transformations that transpose the major triads up, or the minor triads down, respectively, 2, 3, etc. semitones. These are the Schritte of the Schritt-Wechsels group. Using τ1, τ2, etc., in conjunction with P, it is possible to express the 24 transformations we are dealing with in these terms, as indicated in Ex. 18b. Ex. 19 shows how the 24 transformations operate on the major and minor triads. Again, it is not difficult to verify that the 24 transformations form a group. Indeed they are the same transformations as those represented in Example 17, so they are simply another way to represent the Schritt-Wechsel group; only the notation is changed. I leave this verification as an exercise for those who are interested. Examples 20 and 21 are provided as an aid. It is easy to see that each and every UFFC is based on the alternation of two distinct Wechsels. For example, the circle of Ex. 6, reproduced in compact notation as Ex. 22, is generated by alternation of τ9p and τ2p. I will call such a pair of distinct Wechsels a pattern. How many different patterns are there? Since we have 12 distinct Wechsels, there are (12 11) / 2 = 66 different patterns. These are distributed as shown in the next-to-last column of Table 1. For each value of t, t = 1 through 5, there are 12 patterns. To see why this is so, note that for a pattern to produce a UFFC with t equal to one of these values, we must alternate τmp and τnp where n - m is congruent to t, mod 12, and m - n is congruent to -t, mod 12, For each of these values of t, there are 12 distinct pairs of values, m and n, that suit this condition. However, for t = 6, there are just 6 such pairs, since if n - m s congruent to 6, then m - n is also congruent to 6, mod 12. (leave for student instead?)

Flip-Flop, p.11 How do patterns correspond to UFFCs? The relevant formula is given at the top of the column headed #UFFC, following the second equals sign. For any given value of t, the number of UFFCs is equal to the number of circles for each mode times the number of patterns. Example 23 lists, in abbreviated form, the 12 patterns that correspond to t = 4, and gives a partial display of how these generate the 48 UFFCs for that value of t. Note that, for t = 6, no tritone exception is necessary in the application of this formula, since it has already been factored into the value for #P. The subgroups of S/W are as shown in Ex. 24. They are listed in two categories: cyclic groups and dihedral groups. There are six cyclic subgroups and 28 dihedral subgroups, for a total of 34 subgroups. (Do these numbers ring a bell?). How do these subgroups correspond with UFFCs? To begin with, let us stipulate that, for our purposes here, we shall recognize a subgroup as corresponding to a UFFC if and only if, for any ordered pair of triads selected from the UFFC (but not necessarily adjacent in the UFFC), there is a unique transformation in the subgroup that transforms the first triad into the second. (In mathematical terms, such a group is said to be simply transitive.) With this condition, it is clear that, for any given value of t, the number of elements in a UFFC is the same as the order of the corresponding subgroup, which equals 2 times C. This situation is illustrated in Ex. 25, a kind of multiplication table for the set of the now familiar hexatonic cycle (C major E minor E major Ab minor Ab major C minor). If we first choose a triad in column to the left of the table, and then choose a triad in the row at the top, the entry at the intersection of the appropriate row and column is the transformation that takes the first triad to the second. For example, the transformation that takes C major to Ab minor is τ8p. Beyond that, without getting into too many complications, we simply observe that, for any given value of t, the number of subgroups is equal to #C, that is, to the number of constituent circles of each mode (we leave it to the student to discover why this is so) and these subgroups are distributed equally among the UFFCs for that value of t.

Flip-Flop, p.12 IV. UFFCs and the T/I Group Consider Ex. 26. Like Ex. 25, it shows how a subgroup may account for transformations among all the triads of any of the three UFFCs containing the major and minor triads on roots C, E, and Ab. Each of these subgroups has the same structure: it is a dihedral group of order 6. However Ex. 26 shows a subgroup of the T/I group while Ex. 25 shows a subgroup of the S/W group. It turns out that for any UFFC, there are subgroups of identical structure mathematically they are said to be isomorphic one a subgroup of the S/W group and the other of the T/I group, that account for transformations that take us from any triad of the UFFC to any other triad. So why do we need the S/W group with its τ1, τ2, etc., when we could employ the more familiar T/I group to support the UFFCs? The answer lies with the notion of pattern. If we conceive UFFCs as arising basically from the alternation of two inversions, then only the S/W groups and its subgroups will serve. Example 27 shows a UFFC that we looked at above (C major - E minor - E major - Ab minor etc.). Marked outside the circle is the pattern of two Wechsels, τ0p and τ4p, that alternate to support the UFFC. Now what happens if we employ the subgroup of Ex. 26 to move around this same UFFC? As shown inside the circle, three transformations, T11I, T3I, and T7I are required to account for the adjacencies. And similarly for any UFFC, save those where t = 6: if we are limited to the T/I group, then more than two inversional transformations are required to generate the adjacencies of the circle. What happens if we export to the T/I group our notion of pattern developed for the S/W group? Suppose we select two inversions, say T0I and T3I, and an arbitrary starting triad, say C major. The result of alternating these two inversions is shown in Ex. 28, where T0I and T3I are marked outside the circle. Note that each of the constituent circles, one of major triads and the other of minor, is regular, but the circles are reversed so to speak: as we move clockwise from one major triad to the next, the triads

Flip-Flop, p.13 are transposed up by three semitones (i.e., t = 3), but when we do the same for minor triads, they are transposed down by three semitones (i.e., t = 9). It is as though the two component circles were formed so as to make a UFFC, but then one was flipped over, reversing the flow. In this case, we have a pattern of two inversions drawn from the T/I group. Now suppose we approach this circle with the S/W group instead of the T/I group. As shown inside the circle, four different Wechsels are required to generate the adjacencies. These observations, pertaining to what David Lewin calls an antiisomorphism, are pursued in Lewin s 1987 book Generalized Musical Intervals and Transformations and in my paper in Journal of Music Theory, vol. 42 (2), 1999. V. UFFCs and Julian Hook s UTTs It is plain that any UFFC is indeed uniform in the following sense. As we proceed around the circle in one direction or the other, the roots of major triads will all be transposed by the same interval to produce the roots of the next minor triads; and the same is true for transposition of the roots of minor triads. Please see Ex. 29, which reproduces Ex. 6 and adds some notation. If we proceed clockwise about the circle, the roots of major triads are transposed up 9 semitones (or down 3), giving the roots of the next minor triads; and the roots of minor triads are transposed up 10 (or down 2) semitones, giving the roots of the next major triads. In a notation based on that of Julian Hook s 1999 paper A Unified Theory of Triadic Transformations, we write <9, 10>, where the numbers 9 and 10 represent the intervals of transposition applied, respectively, to major triad roots and minor triad roots when moving to a triad of the opposite mode. We can also write <2, 3>, to represent motion counterclockwise about the same circle. Hook s system of uniform triadic transformations (or UTTs), as he calls them, also includes mode-preserving transformations, but we are not concerned with those here. Let us compare the two transformational schemes, based on S/W and Hook s UTTs. As we know, Wechsels are involutions, so τ9p and τ2p, as they appear on Ex. 29, work in both directions. On the other hand, Hook s UTTs are not, in general involu-

Flip-Flop, p.14 tions; The transformations <9, 10> does not reverse itself when applied twice, and the same for its inverse, <2, 3>. As a consequence, the groups that they induce are different. The pair of transformations, τ9p and τ2p, induces the dihedral group of order 24. By contrast, <9, 10> induces the cyclic group of order 24; any move from, say, the C major triad to any triad in the UFFC of Ex. 29 may be defined in terms of <9, 10> repeated an appropriate number of times. The same is true of <2, 3>. Which group is preferable? I refrain from addressing that question in detail here, but I will say that it is surely a question involving one s objectives, and one s perceptions in a particular musical context. VI. UFFCs and Voice Leading In our readings of Examples 3 and 4, we took note of voice-leading considerations, particularly of connections between triads with two common tones, which are given the Riemannian labels L, P, or R. These connections are special; they may be characterized as ultra-smooth in that each of them may be realized by motion of one or two semitones in a single voice. Suppose we match the pitch classes of a C-major triad to those of its L-companion, an E minor triad, as shown in Ex. 30, and sum the resulting pitch-class intervals. The minimum total is 1, and the same for a P connection. For the R, connection, also shown in Ex. 30, the minimum is 2. For the moment, I will refer to this as the voice-leading minimum. While L, P, and R all three play a role in Western tonal music, so do connections that are at the opposite end of the spectrum in terms of voice leading economy. Consider the progression I - IV - V- I. In either major or minor, there are no common tones between IV and V, and the minimum sum of pc intervals spanned is 6. This number, 6, is also the greatest minimum between any two triads. How is this relevant to the present topic? It is clear that transposition of a pair of triads does not affect the voice-leading minimum. Any two triads in the L relationship will have a voice leading minimum of one, and any two in the IV-V relationship will

Flip-Flop, p.15 have a minimum of six. etc. etc. The point is that as a UFFC alternates two Wechsels, it alternates two voice-leading minimums as well. In the case of Heinichen s circle, these two values are, in fact, 2 and 5, as shown in Ex. 31. Clearly any particular Wechsel implies a particular voice leading minimum, but the converse is not generally true. Therefore the characterization of a UFFC in terms of pattern is stronger than that in terms of voice leading minimum, but the latter is nevertheless of interest. This topic is amply developed in Cohn s forthcoming paper Square Dances with Cubes. VII. Extensions, Connections, and Conclusion There are many more extensions and connections to pursue, some already explored in the published literature, others under examination by various scholars, and still others that, so far as I know, await investigation. Before closing. I would like to mention just a few of these. Application to the arbitrary asymmetrical chord. Everything we have looked at today will work nicely, not just with major and minor triads, but with any class of chords where there are distinct inversions. For example, we could choose the class including the Webernesque trichords {C, C#, E} and {C, Eb, E}, and produce the same kinds of circles and patterns that we studied today. Representation with lattices. Three-dimensional lattices serve well to represent the spaces where neo-riemannian transformations apply. Among the scholars investigating this is So-Yung Ahn, who has worked with lattices that can model progressions within the chord class consisting of dominant 7th chords and half-diminished 7th chords. Universes of more or less than 12 pitch-classes. For those interested in the theory and practice of microtonal music, extension of neo-riemannian theory to equaltempered systems of more or less than 12 notes per octave is fairly straightforward. Circles of objects other than chord types. Instead of chord types related by mirror inversion, other binary oppositions might be addressed. For example, Hook has sug-

Flip-Flop, p.16 gested that tonal sequences that alternate 5-3 and 6-3 chords may be amenable to treatment through neo-riemannian techniques. In conclusion I hope to have provided a modest introduction to an exciting new area of research, and a look at my own current work. It has been a great pleasure to speak to you, and I thank you for your kind attention.

Flip-Flop, p.17 Bibliography Ahn, So-Yung. 1999. Dual Structure of 4-27: Analogous to Cohn s MS-Cycle and Hexatonic System. (Unpublished paper.) Clough, John. 1999. A Rudimentary Geometric Model for Contextual Transposition and Inversion. Journal of Music Theory 42(2): 297-306. Cohn, Richard. 1996. Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions. Music Analysis 15(1): 9 40.. 1997. Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations. Journal of Music Theory 4(1): 1 66..1999. Square Dances with Cubes. Journal of Music Theory 42(2): 283-96. Heinichen, Johann David, 1728. Der Generalbass in der Komposition. Dresden. Hook, Julian. 1999. A Unified Theory of Triadic Transformations. Paper presented at Music Theory Midwest Conference, Bloomington, Indiana, USA. Hyer, Brian. 1995. Reimag(in)ing Riemann. Journal of Music Theory 39(1): 101 38. Klumpenhouwer, Henry. 1994. Some Remarks on the Use of Riemann Transformations. Music Theory Online 0(9). Lewin, David. 1987. Generalized Musical Intervals and Transpositions. New Haven: Yale University Press.. 1993. Musical Form and Transformation: 4 Analytic Essays. New Haven: Yale University Press. Riemann, Hugo. 1880. Skizze einer neuen Method der Harmonielehre. Leipzig: Breitkopf und Härtel. Special Issue of Journal of Music Theory, vol. 42, no. 2 (1999). Articles by Clifton Callendar, Adrian Childs, David Clampitt, John Clough, Jack Douthett, Edward Gollin, Carol Krumhansl, David Lewin, Steven Soderberg, and Peter Steinbeck.