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 Pro-

gressions.” Example 1 shows four circles of six triads each. The circle at the top sym-

bolizes 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, sym-

bolized “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 Ex-

ample 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 half-

step; 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 voice-

leading.

We can define additional operations as strings of P, L, and R. For instance, refer-

ring 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 re-

versible: 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 to-

nal music— the latter applying quite generally and the former more particularly to nine-

teenth-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 hexa-

tonic 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 in-

terval 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 direc-

tion, and each containing at least two triads. When the two component circles are inter-

leaved, the result is a UFFC.

There are 168 UFFCs in the usual 12-pc universe, and the process of enumerat-

ing 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 cir-

cles, 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 Tt, 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 tri-

ads; 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 cir-

cles. As we have seen, for t = 4, this number is 3: three major triads and three minor tri-

ads. 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 com-

ponent 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 com-

puted 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 to-

tal 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 correspond-

ing 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 constitu-

ents 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 re-

view 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, af-

ter Riemann, and explore its close relationship to the notion of UFFC.

The group of transpositions and inversions, acting on the 12 pcs, and by exten-

sion 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 follow-

ing axioms:

1. Closure: any series of transformations is equivalent to one of the 24 transfor-

mations. 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 neces-

sarily 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, with-

out 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 pur-

pose, 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 sub-

group in combination with a flip or inversion which doubles the size of the cyclic sub-

group. 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 com-

plete 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 num-

ber 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 alterna-

tion 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)3L. 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)nL, 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 group-

theoretic 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 axi-

oms are not difficult to verify. As shown in Ex. 17b, the identity element may be written

as (LR)0 or L2 or R2 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 transforma-

tions 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 mu-

sic, 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)11L, as in Ex. 17e. But my pro-

gram here is more abstract, and, as I will show, there are certain advantages to the ap-

parently more mechanical notation used in Example 17 and another much simpler nota-

tion 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 cir-

cle, 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. semi-

tones. These are the Schritte of the Schritt-Wechsels group. Using τ1, τ2, etc., in con-

junction 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 transforma-

tions form a group. Indeed they are the same transformations as those represented in Ex-

ample 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 inter-

ested. 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 dis-

tinct 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 Wech-

sels a pattern. How many different patterns are there? Since we have 12 distinct Wech-

sels, 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 pat-

terns. 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 dis-

tinct 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 num-

ber 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 sub-

groups, 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 trans-

formations among all the triads of any of the three UFFCs containing the major and mi-

nor 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 sub-

groups 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 transforma-

tions 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 ba-

sically from the alternation of two inversions, then only the S/W groups and its sub-

groups will serve. Example 27 shows a UFFC that we looked at above (C major - E mi-

nor - 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 “re-

versed” 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 anti-

isomorphism, 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 pro-

ceed 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 re-

produces 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) semi-

tones, 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 in-

cludes 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 percep-

tions in a particular musical context.

VI. UFFCs and Voice Leading

In our readings of Examples 3 and 4, we took note of voice-leading considera-

tions, 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 connec-

tions 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 num-

ber, 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 never-

theless 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 ex-

plored 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 in-

cluding 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 investi-

gating 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 the-

ory and practice of microtonal music, extension of neo-Riemannian theory to equal-

tempered 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 mir-

ror 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 treat-

ment 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 Ton-netz 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 Transforma-tions.” 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: Breit-kopf 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.

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