Symmetries in Counterpoint Patterns...Generalized Musical Intervals and Transformations (Lewin,...

Symmetries in Counterpoint Patterns

Jingwei Liu

Abstract

This article explored eight types of symmetries in counterpoint patterns. Inspired by the

observation of dual patterns in the second species counterpoint, the searching of symmetries is

then extended to one-to-one counterpoint, where eight symmetries are discovered in total. The

relationships of these symmetries fit in a cubic structure of which every facet represents a Klein

four-group. Later on, by mathematical abstraction, the entity of species counterpoint is stripped

and the results are reformulated under Generalized Interval Systems (GIS), where we prove the

existence and uniqueness of the eight symmetries. Finally, we generalize our system from two-line

counterpoint to multiple layers, thus taking chords and other more complicated musical systems

into consideration.

Keywords: symmetry, counterpoint, group theory, musical abstraction, mathematical music

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Content

Introduction 1

1 Previous Work Summary 1

2 Mathematical Preliminaries 2

2.1 Generalized Interval System and Modular Operation . . . . . . . . . . . . . . . . . 2

2.2 Equivalence Classes and Transpositions . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Inverse Pairs and Symmetric Elements . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Second Species Counterpoint 6

3.1 Corresponding Cantus Firmus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Dual Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Swap and Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 First Species Counterpoint 12

4.1 Klein 4-Group of Symmetric Relationships . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 D6 Group, Retrograde and Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Dual and Redirection of D6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Mathematical Generalization 21

5.1 Abstract Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Lattice, Order and Multi-Layer Counterpoint . . . . . . . . . . . . . . . . . . . . . . 24

References 28

1 PREVIOUS WORK SUMMARY Symmetries in Counterpoint Patterns

Introduction

The discovery of symmetries was an accident at first. When I was studying the second species

counterpoint, I found that it’s hard to compose a satisfying piece sometimes. It’s preferable to use

the melodic schemas such as passing motion, neighbor motion and subdivided leaps to make the

melody songful, but in some situations, it’s just impossible to adopt them. However, when I started

over with a different choice of several notes, the problem solved itself. It made me wonder why

this situation happened and how the problem was solved. With this question, I looked into the

correspondence between these melodic schemas and the cantus firmus, and I found the answer! It

turned out that if the cantus firmus is in a descending stepwise motion and we happened to compose

a 6th above the first note, then the situation became tricky. However, if we changed it to a 3rd,

it’s highly possible that the melody is going to flow smoothly. This discussion is given in section

3.1. At the same time, as a byproduct, a symmetry called dual pattern was discovered. It intrigued

my interest in studying the symmetric structure. Unlike the compositional question, symmetry is

universal and much more profound. I found other symmetries like swap and mirror immediately.

Then the relationships among the three types of symmetries became the focus of my study (section

3).

Later on, retrograde and reflection were added to my blueprint (section 4). Then the relation-

ships among symmetries became more complex and a more systematic illustration was in need to

put them in place. At the same time, my music-theory professor pointed me toward David Lewin’s

Generalized Musical Intervals and Transformations, which provided me with the base to generalize

my work (section 5.1).

Not satisfied with two-line counterpoint, I generalized the system into a larger framework with

multiple layers, thus took chords and other musical objects into consideration (section 5.2). My

work is based on a comprehensive knowledge of mathematics, especially in abstract algebra, so it

may take some efforts for readers without mathematical background to understand it. Besides, this

work is more about math than music. Music is just a concretization of mathematical concepts. The

musical system provides mathematicians with a distinct algebraic structure to work with, and this

paper makes an attempt on it.

1 Previous Work Summary

I believe there should be work about symmetries in music or related topics, but I’m not fully

aware of it. This paper is simply based on my knowledge of mathematics and music. Started

from a practical problem in composition, the research was turned to symmetries by chance. In the

process, I did refer to the book Generalized Musical Intervals and Transformations (Lewin, 2011)

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2 MATHEMATICAL PRELIMINARIES Symmetries in Counterpoint Patterns

and borrowed the concept of Generalized Interval Systems (GIS). Everything I adopted from this

book is included in section 2.1.

2 Mathematical Preliminaries

In this section, we are going to introduce some mathematical concepts related to our discussions.

Theorems in section 2.1 are stated without proof. Readers can refer to the first three chapters of

Generalized Musical Intervals and Transformations (Lewin, 2011) for details. All concepts given

here fit in with the context, so it’s possible for the readers to refer to this section from time to time

with the flow of this paper.

2.1 Generalized Interval System and Modular Operation

Definition 2.1. A Generalized Interval System (GIS) is an ordered triple (S , IVLS , int), where

S , the space of the GIS, is a mathematical set of elements, IVLS , the group of intervals for the

GIS, is a mathematical group, and int is the function mapping S ×S into IVLS , all subject to

the following two conditions:

• For all r, s and t in S , int(r, s) ◦ int(s, t) = int(r, t).

• For every s in S and every i in IVLS , there is a unique t in S which lies an interval i from

s, which satisfies the equation int(s, t) = i.

Our discussion in the first four sections is based on diatonic scales, so the space S is the

7-pitch classes {A,B,C,D,E,F,G} cycling octaves up and down. The concrete examples of species

counterpoint give us good intuitions and musical interests in discovering symmetric relationships,

but it’s also important to remember that almost all of our results can be generalized in abstract

formulation, which is discussed in section 5.1.

As described in (Lewin, 2011), int(s, t) denotes “a directed measurement or motion behaving

like an interval from s to t”. In our space S (diatonic), for example, int(C,D) = 1̄, int(A,E) = 4̄.

Unlike the traditional musical measurements which denote the interval between C and D as a “2nd”,

and the interval between A and E as a “5th”, “int(s, t)” interacts effectively with “+” (here we

use modular addition). For example, int(D, A) = int(D, C) + int(C, A) = 6̄ + 5̄ = 4̄. In our

traditional interval denotation, 7th + 6th = 5th, which counters the algorithm of addition. Using

the cycling property of the pitch class, as described vividly in the book, “If we wrap the scale

around the face of a seven-hour clock, then int(s, t) is the number of hours that we traverse on that

clock, in proceeding clockwise from s to t”, it’s easy to tell that it’s a ”mod 7” addition endowed

on IVLS (diatonic).

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Theorem 2.2. In any GIS, for every s and t in S , int(s, s) = e, where e denotes the identity of

IVLS group, and int(t, s) = int(s, t)−1, where int(s, t)−1 denotes the inverse element of int(s, t) in

IVLS.

IVLS (diatonic) is isomorphic to Z7 = {0̄, 1̄, 2̄, 3̄, 4̄, 5̄, 6̄}. For example, int(D, D) = 0̄, where 0̄

is the identity element. int(C, B) = 6̄, int(B, C) = 1̄, where 6̄ and 1̄ form an inverse pair in Z7 as

6̄ + 1̄ = 0̄.

By fixing one element in S as a referential member, we get the Referential GIS.

Definition 2.3. Given a GIS (S , IVLS , int) and a fixed referential member ref of S , the function

label, mapping S into IVLS , is defined by

label(s) = int(ref, s)

Theorem 2.4. Regardless of the element ref, the function label maps S 1-to-1 and onto IVLS,

and it satisfies

int(s, t) = label(s)−1 ◦ label(t)

2.2 Equivalence Classes and Transpositions

Let’s interpret Definition 2.1 under diatonic settings into mathematical languages. We defined

a map int from S ×S into IVLS , namely,

int : S ×S −→ IVLS

(s, t) 7−→ t− s

int is a dimension reduction map reducing a 2-dimensional space into a group. Besides, int is

surjective. For any c ∈ IVLS , int(s, t) = int(s + c, t + c), which corresponds to transpositions of

note patterns. S ×S is divided into equivalence classes by the image of int. Therefore, we say

two patterns are ”equivalent” up to transpositions.

Similarly, we can generalize the theorem into n-dimensional space S n, the map would be

intn−1 : S n −→ IVLSn−1

(s1, s2, . . . , sn) 7−→ (s2 − s1, s3 − s2, . . . , sn − sn−1)

The equivalence classes for S n are {(s1 + c, s2 + c, . . . , sn + c)|c ∈ Z7}.Another notice is that, due to the isomorphism of IVLS and Z7, we don’t distinguish notes

up to octaves. To be specific, we consider C3, C4, C5... all to be the same C. We distinguish

relative positions of notes between staff lines, but we don’t distinguish that of sequential notes in

the same line. As shown in Figure 1 Ex.1, fixing C, E is above or below C as a counterpoint. Since

the interval changes from 3rd to 6th, these two positions are different, which means we evaluate

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intervals from the notes below to above in a counterpoint. In Ex.2, although E is an octave up, the

relative position of C and E doesn’t change, so they are equivalent. In Ex.3, the relative position

seems to be different, E is higher then lower than C. However, since they are in the same line, we

always evaluate the interval from left to right. int(C,E) = 2, so the two motions are equivalent.

Figure 1: Relative Positions of Two Notes

The results in Ex.3 directly affect our definition of transposition in this context. In Figure 2,

we give an example of transposition. These two patterns are not parallel on staff but due to the

modular operation, high B and low B are indeed indistinguishable. Due to this property, we can

always replace the low B in Figure 2 with a high B, which gives a parallel transposition.

Figure 2: An Example of Transposition

Claim 2.1. Two n-note patterns are equivalent if and only if they are transpositions to each other.

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2.3 Inverse Pairs and Symmetric Elements

Definition 2.5. Given a group G, for any a, b ∈ G, if a ◦ b = e, where e is the identity of G, a and

b form an inverse pair.

Claim 2.2. Any element in a group has a unique inverse in this group.

Definition 2.6. Let G be a group, given a property P , Pp is a set of elements in G which satisfies

P . We say Pp is the property set of P .

Definition 2.7. Let G be a group, given a property P . For g ∈ Pp, if g−1 ∈ Pp, then g is a

symmetrical element of G with property P . All symmetric elements form the symmetric set Dp.

Corollary 2.7.1. Dp ⊆Pp ⊆ G

In our diatonic space S , the group G is induced by int acting on S ×S , namely, the IVLS

group. In GIS, if a, b ∈ IVLS form an inverse pair, let a = int(s, t), then b = int(t, s). It means

the definition of inverse pair in IVLS corresponds to the exchange of elements in S ×S (up to

transpositions), which is a symmetric operation.

The property P is usually given by the classification of consonances. If P is all consonances,

which is denoted by Pc, then the property set of Pc is

Pc = {0̄, 2̄, 4̄, 5̄}

2̄ and 5̄ form an “imperfect consonant” inverse pair. 0̄ is its own inverse since it’s the identity.

However, the inverse for 4̄, 3̄, is excluded from Pp. It indicates that perfect 5th cannot be inversed,

which is obvious since its inverse, perfect 4th, is not a consonance. According to definition 2.7, 4̄ is

not a symmetric element. Thus,

Dc = {0̄, 2̄, 5̄}

Since IVLS ∼= Z7, it’s immediate that Dc ⊂Pc ⊂ IVLS .

It’s worth noticing that the property P is defined pointwise. In a system, different elements

may have different functions, thus require different properties. Based on a united group G, with

different properties P , Pp and Dp change accordingly. One example in the diatonic GIS is that,

instead of choosing consonances, P can be “consonances and dissonances”, corresponding to the

situation where we can use either of them. In this case, P is denoted by Pa and

Da = Pa = IVLS

The property set and the symmetric set are both groups.

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3 SECOND SPECIES COUNTERPOINT Symmetries in Counterpoint Patterns

Another situation is that, instead of choosing all consonances, P only includes imperfect con-

sonances. The property set of imperfect consonances is denoted by Pi,

Pi = {2̄, 5̄}

Since 2̄ and 5̄ form an inverse pair,

Di = Pi ⊂ IVLS

The property set is equal to the symmetric set, which means every element satisfying the property

Pi is symmetric. This condition guarantees the existence of all kinds of symmetries (Theorem 3.1).

3 Second Species Counterpoint

In this section we will introduce three types of symmetries discovered from the second species

counterpoint. Our motivation was to study the correspondence between cantus firmus and the

melodic schemas in counterpoint, such as passing motion, neighbor motion and subdivided leaps.

In the process, we will find symmetries as a by-product generated by these motions. By observing

the symmetric relationships, we are led to a more profound and universal topic – symmetries in

counterpoint system.

3.1 Corresponding Cantus Firmus

Let’s take a look at the motions given in Figure 3. Ex.1 and Ex.2 are passing motions, Ex.3

and Ex.4 are neighbor motions, Ex.5 and Ex.6 are incomplete neighbor motions, Ex.7, Ex.8, Ex.9

and Ex.10 are subdivided leaps. The pattern selection is independent of discovering symmetries,

which means no matter what pattern we choose, symmetries can always be constructed under cer-

tain conditions. However, to make our results more representative – applicable in second species

counterpoint composition – we choose these 10 patterns as our melodic schemas. First, they fre-

quently occur in the second species counterpoint and are identified as “remarkable” motions (we

use all-in-one slurs to denote them). Second, our selective choices in incomplete neighbor motions

(only go a third up or down then change direction to fill in the note in between) and subdivided

leaps (only the cases 3rd comes first) avoid “awkward” leaps which may not be compensated by the

following moves. The possibility of composing a “songful” melody is increased by these choices.

According to the compositional rules of the second species counterpoint, only consonances can

appear on the strong beat. These consonances include imperfect consonances 3rd, 6th and perfect

consonances perfect 8th, perfect 5th and unison. Because direct and parallel perfect consonances

are urged to be avoided, and imperfect consonances are more euphonious, we would only include

intervals 3rd and 6th in our discussion. We shall also notice that the choice of consonances doesn’t

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Figure 3: Slected Motions for Second Species Counterpoint

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affect our settings of symmetries in general. However, it does facilitate our discovery of symmetries

because of the invertibility of imperfect consonances.

To account for the numbers on each exercise in Figure 3, we will use referential GIS introduced

in section 2.1. According to our choice, {2̄, 5̄}, which appears above the initial note of each pattern,

stands for the possible intervals between counterpoint and cantus firmus. Since we are in search of

corresponding cantus firmus for these patterns, taking the first note (A in this case) as the referential

member, 2̄, 5̄ are the results of label function, and the labeling notes (s of label(s)) are the possible

notes in cantus firmus. Since {2̄, 5̄} forms an inverse pair, it doesn’t matter we take ”+” or ”−” in

calculating labeling notes. To keep consistency with the convention of group operation, regardless

of the cantus firmus being below the counterpoint, we choose ”+” as our operation. Take Ex.1 for

example, since it’s a passing motion, B on the weak beat can either be consonance or dissonance,

so no restriction is imposed here. For the note C, it’s on the second strong beat, so it’s supposed

to form a imperfect consonance with the cantus firmus, an initerval 2̄ or 5̄. However, since the

referential member is not C but the initial note A, the labeling set for the downbeat of the second

measure is evaluated as

label(s) = int(A,C) + int(C, s) = 2 + {2̄, 5̄} = {4̄, 0̄}

We can compute the labeling sets in Ex.2, Ex.3 and Ex.4 in the same way. For the arrow

between the staff lines, the numbers below it are computed by taking difference between two labeling

sets. Denote the labeling sets as A and B, the difference set D is

D = B − A = {b− a|a ∈ A, b ∈ B}

In Ex.1,

D1 = {4̄, 0̄} − {2̄, 5̄} = {4̄− 2̄, 4̄− 5̄, 0̄− 2̄, 0̄− 5̄} = {2̄, 5̄, 6̄}

Note that in cantus firmus, two sequential notes won’t either be in oblique motion (repeated

notes) or form an octave leap, so we remove 0̄ if it appears in a difference set D.

Now let’s move on to Ex.5 – Ex.10. In each case, the second half note on the weak beat form

a disjunct motion, so it has to be consonant with the cantus firmus. However, since this note is

“in between” in the 3-note pattern, where the first and third notes form imperfect consonances,

it’s impossible to compose direct or parallel perfect consonances here. Therefore, We relax the

restrictions of consonances and the interval set becomes {0̄, 2̄, 3̄, 5̄}. 2̄ and 5̄ stand for imperfect

consonances as usual; 0̄ represents perfect octaves or unison; 3̄ is perfect 5th. In this case, since we

compose above the cantus firmus and the group operation is +, 3̄ means a perfect 4th above the

given note, which is exactly a perfect 5th below it.

Same as before, to calculate the labeling set for the middle note, take Ex.5 for example,

label(s) = int(A,C) + int(C, s) = 2 + {0̄, 2̄, 3̄, 5̄} = {0̄, 2̄, 4̄, 5̄}

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Since two half notes in one measure correspond to a whole note in the cantus firmus, take the

intersection of the two labeling sets,

{2̄, 5̄} ∩ {0̄, 2̄, 4̄, 5̄} = {2̄, 5̄}

which is the set between the staff lines. Then we take the difference between the two labeling sets

in two measures,

D5 = {3̄, 6̄} − {2̄, 5̄} = {1̄, 4̄, 5̄}

Elements of the difference set D5 appear below the arrow. Therefore, the illustration of Figure 3 is

completed. The labeling sets give all possibilities of cantus firmus by taking the first note in each

pattern as the referential member. The difference sets give the intervals between two notes in the

cantus firmus. For example, in Ex.1, the differences are 2̄, 5̄, 6̄. If we denote the two notes in cantus

firmus as s and t, then

int(s, t) = 2̄, 5̄ or 6̄

Now we already know how to calculate the cantus firmus for patterns in Figure 3. To take a

close look at these counterpoints, we enumerate all of these possibilities in Figure 4. As we can see,

Ex.11 – Ex.16 correspond to 7 total modes of cantus firms. For every mode, we divide the patterns

into two groups – starting with a 6th or a 3rd. To illustrate how we assign the letters a through f

to these motions, we will introduce our first type of symmetry – dual patterns.

3.2 Dual Patterns

In this section, our introduction to symmetries would be observation-based. This gives us a

good intuition about how these symmetries come into play in compositions and a way to define

these symmetries.

Let’s take a look at Ex.11-1a and Ex.16-2a in Figure 4, namely, the “a” pair. First, we notice

that the types of motions in both exercises are the same: passing motion (P), incomplete neighbor

motion (IN) and subdivided leaps (SL). In these matching motions, treble line and bass line “flip

over” respectively, and harmonic intervals are inverted. It happens in every matching pair from a

to f. We call these pairs dual patterns.

Dual patterns don’t always exist. In Ex.11-2b and Ex.16-1b, due to the presence of 5th, the

patterns in Ex.11-2b don’t have dual patterns. It happens because the 5th in this case is not

invertible. Its inverse, the 4th, is not a consonance. This situation leaves Ex.11-2b blank. In other

words, when the cantus firmus steps down, it’s impossible to compose any pattern listed in Figure 3

by starting with a 6th above. However, if we start from a 3rd, as shown in Ex.16-2a, there are three

motions available and they serve various functions as upward move (P), downward move (SL) or

direction change (IN). It indicates that in our composing practices, when encountering a stepwise

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Figure 4: Dual Patterns

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downward cantus firmus (which is frequent), if we could manage to start from a 3rd instead of a

6th above it, we are more likely to compose a satisfying piece.

However, the presence of 5th is not an absolute taboo in forming dual patterns. For example,

in the “e” pair Ex.13-1e and Ex.14-2e, the 5th appears in the lower-neighbor motion in Ex.13-1e.

Its dual pattern is given by the upper-neighbor motion in Ex.14-2e with 5th inverted to a 4th. To

account for this phenomenon, musically speaking, the transition note in a conjunct motion can be

consonant or dissonant. Hence, as marked in Figure 3, 5th is an asymmetric element if and only if

it appears in a disjunct motion, which eliminates the existence of dual patterns.

Now we want to generalize this phenomenon and put it into mathematical languages. For the

3-note patterns in Figure 4, according to the notations in section 2.3, the property sets for three

counterpoints in a conjunct motion are Pi −Pa −Pi. We argued that Di = Pi, Da = Pa, so

Di−Da−Di also holds. In disjunct motions, the property sets are Pi−Pc−Pi. Since Dc 6= Pc,

where 4̄ plays a role as asymmetric element, Di − Dc − Di doesn’t hold, where dual patterns may

fail to exist.

Theorem 3.1. Given a pattern X = (~x1, ~x2, · · · , ~xn), where ~xj is a counterpoint pair, 1 ≤ j ≤ n,

the dual pattern X∗ exists if for every ~xj, its property set Pj equals its symmetric set Dj.

Corollary 3.1.1. If the dual pattern exists for a pattern X, there is no asymmetric entry in X.

Theorem 3.1 will be proved in section 4.

3.3 Swap and Mirror

Our discussion so far only covered the situations composing above the cantus firmus. We can

also compose below it. Taking the ”c” pair in Figure 4 for example, by exchanging the relative

position of cantus firmus and counterpoint, the results are given in Figure 5.

In Figure 5, we take Ex.12-1c as prototype, which is the identity operation on pattern X.

Ex.15-2c includes its dual pattern. By exchanging the staff lines in Ex.12-1c, we get Ex.15-2c*. We

call this operation a swap. Applying swap to Ex.15-2c, we get another pattern in Ex.12-1c*, called

the mirror of X.

To understand the relationships between these operations would take some efforts, we will cover

this part in section 4.1. Now let’s observe the 4 exercises in Figure 5 and list some fun facts.

First, we should understand the meaning of each operation. dual takes the inverse of all

melodic and harmonic intervals while keeping the relative position of staff lines; swap exchanges, or

swaps the two staff lines, translating the counterpoint below (or above) the cantus firmus; mirror

is a horizontal reflection of staff lines, which flips over the two lines while keeping the harmonic

intervals. They are all symmetric operations.

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4 FIRST SPECIES COUNTERPOINT Symmetries in Counterpoint Patterns

Figure 5: Symmetries for Second Species Counterpoint

Second, we take dual to Ex.12-1c to get Ex.15-2c, then swap them to get Ex.15-2c* and

Ex.12-1c*. However, by observing these patterns, it’s not difficult to tell that these operations are

“commutable”: Ex.12-1c* is also a dual to Ex.15-2c*; Ex.15-2c* mirrors Ex.15-2c.

Third, although dual and swap may not exist for a given pattern, mirror is always there. It

means mirror can be induced by a composition of two symmetric operations and itself is symmetric,

but it “gets away with” the restrictions of symmetric elements. Even if Dp 6= Pp, mirror still applies.

Moreover, mirror doesn’t rely on the “transitional” state. Ex.15-2c mirrors Ex.15-2c* 3-by-3 even

if some patterns lose their dual patterns or swap patterns in transition.

4 First Species Counterpoint

In this section we are going to study 8 symmetries and their relationships based on a 3-note first

species pattern. Notice that any two-line motion can be transferred to a one-to-one correspondence

as long as the note durations are divisible. In a general counterpoint, we can divide the note

with longer duration to an oblique motion, forming a one-to-one match with its corresponding

notes. Therefore, we introduce symmetries through an one-to-one counterpoint — first species

counterpoint without loss. To follow the rules of first species counterpoint and ensure the existence

of all symmetries, our example is given when Di = Pi = {2̄, 5̄}.

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4.1 Klein 4-Group of Symmetric Relationships

As shown in Figure 6, we choose a 3-note first species pattern as our prototype. In this pattern,

the cantus firmus is in an incomplete upper-neighbor motion and the counterpoint is in a downward

passing motion. The two lines form a 6-3-3 harmonic interval pattern.

Figure 6: Dual, Swap and Mirror

Claim 4.1. Given a note-sorting method, any pattern with same counterpoint structure consisting

of finite notes can be uniquely determined by an interval chain up to transpositions.

By sorting the notes in a pattern to a sequence, we connect every two adjacent notes with a

unidirected interval, then we get a unidirectional chain called interval chain, denoted by Ic, which

fully represents this pattern.

The property of interval chain comes from the transitivity of int function in a general GIS. In

Definition 2.1, the first condition for int is “for all r, s and t in S , int(r, s) ◦ int(s, t) = int(r, t)”.

It gives us a gateway to reduce a two-dimensional pattern to a one-dimensional chain and study it

in this simplified manner.

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In the first species counterpoint, we give a sorting method as shown in Figure 6 and the interval

chain is given below the staff. We sort from left to right in the cantus firmus until the last note,

then move one harmonic interval up to the counterpoint, continuing from this note to the left-most.

In this way, we traversed all the notes through a path. Besides, in Figure 6, we use a duple (s, t) to

denote int(s, t) for short.

To get the dual, we have to keep the types of motions in cantus firmus and counterpoint (IN

and P) while keeping the harmonic intervals consonant. We shall prove Theorem 3.1 and Corollary

3.1.1 now.

Proof. The pattern is given by X = (~x1, ~x2, · · · , ~xn), where ~xj is a counterpoint pair (xj, x′j) with

xj in the cantus firmus and x′j in the counterpoint, 1 ≤ j ≤ n.

First, we are not allowed to change the relative position of two staff lines. In order to get a

dual pattern, we need to invert at least one motion in X. Without loss, let’s say the counterpoint is

inverted while keeping the cantus firmus invariant. We use * to denote dual. Take the last harmonic

interval int(x∗n, x′∗n ) to be an unknown c ∈ Pc, then the rest of intervals can be evaluated by the

transitivity of int. Let’s compute int(x∗j , x′∗j ), ∀j, 1 ≤ j < n.

int(x∗j , x′∗j ) = int(xj, xj+1) + · · ·+ int(xn−1, xn) + c+ int(x′n, x

′n−1)

−1 + · · ·+ int(x′j+1, x′j)−1

= int(xj, xj+1) + · · ·+ int(xn−1, xn) + c+ int(x′n−1, x′n) + · · ·+ int(x′j, x

′j+1)

= int(xj, xn) + c+ int(x′j, x′n)

We need int(xj, xn) + c + int(x′j, x′n) ∈ Pc. The only way to connect these three intervals is

setting c = int(xn, x′j) or c = int(x′n, xj), then int(x∗j , x

′∗j ) = int(xj, x

′n) or int(x∗j , x

′∗j ) = int(x′j, xn).

These intervals are out of our consideration for property P since P is only defined on counterpoint

pairs.

Then we start our second try by inverting both cantus firmus and counterpoint. By the same

argument,

int(x∗j , x′∗j ) = int(xj, xj+1)

−1 + · · ·+ int(xn−1, xn)−1 + c+ int(x′n, x′n−1)

−1 + · · ·+ int(x′j+1, x′j)−1

= int(xj+1, xj) + · · ·+ int(xn, xn−1) + c+ int(x′n−1, x′n) + · · ·+ int(x′j, x

′j+1)

= int(xn, xj) + c+ int(x′j, x′n)

In this situation, c can be int(xj, x′j) or int(x′n, xn), then int(x∗j , x

′∗j ) = int(xn, x

′n) or int(x∗j , x

′∗j ) =

int(x′j, xj). Since j is arbitrary, the first possibility is eliminated because c needs to be fixed to form a

dual pattern X∗. Hence, c = int(x′n, xn) = int(xn, x′n)−1, int(x∗j , x

′∗j ) = int(xj, x

′j)−1,∀j, 1 ≤ j < n.

Every harmonic interval is inverted.

Therefore, the dual pattern X∗ exists if and only if every harmonic interval in X can be

inverted.

14


Since every melodic and harmonic interval is inverted in the dual pattern X∗, it’s not difficult

to tell that the interval chain Ic∗ of X∗ inverts the direction of Ic under our sorting method. This

is exemplified in Figure 6.

Swap is much more straightforward than dual. By exchanging the position of cantus firmus

and counterpoint, every harmonic interval is inverted. Hence, the existence condition for swap X ′

is the same as that for X∗. In other words, dual and swap are geminate. As for the interval chains,

Ic′ inverts the elements of Ic. Mirror is a composition of dual and swap, and its interval chain Icm

combines the changes of Ic∗ and Ic′. Namely, Icm is inverted by elements and direction of Ic.

As for the relationships among these symmetries, it’s easier to deduce them from the interval

chains. Since the interval chain is a full representation of its pattern, we can shift our attention

from the counterpoint to its interval chain without loss.

First, the three operations, dual, swap and mirror are all symmetric. Symmetric relation is

denoted as ↔. If a ↔ b then b ↔ a, we say a and b are in a symmetric relation. It’s easy to

tell from the interval chains that X∗, X ′ and Xm are all in symmetric relations with X. Besides,

symmetric relation is its own inverse.

Claim 4.2. X = (X∗)∗ = (X ′)′ = (Xm)m

Second, these symmetric operations are commutative. Commutativity is given in two senses.

The first one is that any two different operations are commutative and their composition gives the

third operation. For example,

dual ◦ swap = swap ◦ dual = mirror

Another interpretation is that these operations preserve each other in maps. Take dual for example.

Dual maps the original space to its dual space, namely

dual : X −→X ∗

Swap is an operation defined in the two spaces X and X ∗. That dual preserves swap means if we

take swap for some x ∈X to get x′ ∈X , mapping x and x′ to X ∗, their dual elements x∗ and x′∗

is also a swap to each other, namely

dual : X −→ X ∗

swap(x) 7−→ swap(x)∗ = swap(x∗)

Third, as mentioned in section 3.3, although the three operations are all symmetric and com-

mutative, mirror is quite different from dual and swap. Mirror is more aligned with the identity

operation, namely, prototype. From Figure 5, we know that dual and swap may fail to exist but

mirror always accompanies its pattern. In Figure 6, from the patterns we can tell mirror keeps

the harmonic intervals without inverting them. Property P is always satisfied, thus mirror always

exists. Following the interval chains, we see that Icm and Ic are essentially the same chain.

15


Claim 4.3. A symmetric pattern exists for any given pattern under any property if and only if the

harmonic intervals are maintained.

A diagram illustrating the relationships among dual, swap and mirror is given in Figure 7.

These operations form a Klein four-group. In this diagram, ”←→” means a bidirectional relation.

The blue line means two modes it connects accompany each other.

Figure 7: Relationship of Dual, Swap and Mirror

4.2 D6 Group, Retrograde and Reflection

Here we are going to introduce another symmetric operation called retrograde. As shown in

Figure 8 label 5, retrograde reverses each staff line while exchanging their positions. This operation

also inverses the harmonic intervals. If we transform the interval chains of prototype and retrograde

into circuits by connecting their first and last notes, it’s easy to tell that retrograde rotates prototype

3 times along the chain.

Inspired by this transformation, rolling all interval chains in Figure 8, we get Figure 9. Since

they are topologically equal to regular polygons, it provides us with an opportunity to adopt dihedral

groups in our discussions.

Let ρ and ε denote rotation and reflection, a D6 group is given by

D6 = {1, ρ, ρ2, ρ3, ρ4, ρ5, ε, ερ, ερ2, ερ3, ερ4, ερ5}

In Figure 9, regardless of the length of intervals, we are given a subgroup K of D6,

K = {1, ρ3, ε, ερ3}

where ρ is counterclockwise and ε is swap — reflection by the dashed in Figure 9.

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Figure 8: Retrograde and Reflection

17


Figure 9: Symmetries of D6 Subgroup K

Figure 10: Relationship of Retrograde, Swap and Reflection

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It’s easy to check that K is a group and these three operations are symmetric and commutative

as defined in section 4.1. ερ3 is called reflection because it reflects the prototype by vertical lines.

Moreover, reflection functions like mirror. It also preserves harmonic intervals and accompanies

its original pattern all the time. However, if we look at the interval chains in Figure 8, we will

notice that unlike mirror and prototype, reflection and prototype have totally different interval

chains. However, the preservation of harmonic intervals can be read from the arrows between each

counterpoint pair. Due to the transitivity of int, an interval only depends on its starting and ending

points. Hence, regardless of the form of chains, as long as the direction between a counterpoint pair

is preserved, we know it’s an accompanying pattern.

Therefore, we can draw a similar “relationship diagram” for retrograde, swap and reflection, as

shown in Figure 10.

4.3 Dual and Redirection of D6

Figure 11: Redirection of D6 Subgroup K

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Notice that the arrows in Figure 9 are all in the same direction. By redirecting these graphs,

we get Figure 11.

As illustrated in Figure 6, the redirection of prototype and swap, id∗ and ε∗, are exactly dual

and mirror. Dual is the operation that redirects interval chains. Consequently, the redirection of

ρ3 and ερ3 are the duals of retrograde and reflection, as shown in Figure 12.

Figure 12: Duals of Retrograde and Reflection

Using Xr to denote retrograde dual, we get a third “relationship diagram” for symmetric

operations (Figure 13).

So far, we have introduced 8 symmetries (including prototype) and studied their relationships.

We have three basic operations, which are dual, swap and retrograde. They form a basis for our

symmetric-operation group. following the previous notations in “relationship diagrams”, the base

set is B

B = {X∗, X ′, X−}

It’s easy to check that B is a basis. These three operations are mutually exclusive and generate all

8 symmetries by composition. Besides, they always accompany each other since they all invert the

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5 MATHEMATICAL GENERALIZATION Symmetries in Counterpoint Patterns

Figure 13: Relationship of Dual and Retrograde

harmonic intervals. By composing any two operations in B, we get

B ×B = {X,Xm, Xf , Xr}

In this set, as we discussed before, all symmetries preserve the harmonic intervals in X, thus we

call B ×B an accompanying set. By composing the base set three times, we get

B ×B ×B = {X∗, X ′, X−, XA}

It includes an extra element XA comparing to B. XA is reflection dual, a composition of three basic

operations, X∗ ◦X ′ ◦X−. It inverts the harmonic intervals, so B ×B ×B is called a conditional

set, meaning that there are conditions exposed for the existence of these patterns.

If we take a fourth direct product, the set goes back to B×B, since the symmetric operations

are of order 2. A combination of three ”relationship diagrams” is given as a cube in Figure 14. The

blue lines connect the elements in the accompanying set.

5 Mathematical Generalization

In this section we are going to restate our results in an abstract level. It means we will get rid

of the restrictions of diatonic space and work in a general GIS. Our objects are no longer musical

notes and species counterpoint. Instead, notes will be generalized to elements in GIS and species

counterpoint corresponds to a collection of mappings, or pairs of elements. After that, we will

generalize the two-line counterpoint to a multi-layer one. By stacking more layers, we are able to

take chords and other more complicated musical systems into the framework of our model.

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Figure 14: Ralationship of 8 symmetries

5.1 Abstract Formulation

Let’s go back to Definition 2.1. We have a space S , which is a set of elements. For any two

elements in S , their relationship is depicted by a function int. int maps S ×S into a group IVLS

and is surjective. Since the image of mapping int is a group, there must be an operation defined

for int in IVLS , and this operation is closed. Besides, IVLS contains an identity and every element

in IVLS has an inverse. Since the elements in IVLS is given by int acting on a duple (s, t), where

s, t ∈ S , it’s necessary to connect the group operation to this map int, so we have the following

two conditions:

• For all r, s and t in S , int(r, s) ◦ int(s, t) = int(r, t).

Here ◦ is the group operation of IVLS , and ◦ is transitive for elements in S . In other words,

int is path independent. No matter how many intermediate points it passes, int only takes its

beginning and ending elements into account.

• For every s in S and every i in IVLS , there is a unique t in S which lies an interval i from

s, which satisfies the equation int(s, t) = i.

This condition indicates that S is connected and one-dimensional.

Now we have a sequence M of elements in S , namely, M = (s1, s2, · · · , sm), si ∈ S , 1 ≤ i ≤ m.

Similarly, we can give another sequence L, where L = (t1, t2, · · · , tl), tj ∈ S , 1 ≤ j ≤ l. A structure

can be formed by matching the two sequences: any element in either sequence can be matched

with one or multiple elements in the other sequence, and this matching is in order. It means if one

element is matched with multiple elements, these elements are adjacent. This is an abstraction of

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a two-clef music score. Since two sequences M and N are finite and the match is ordered, we can

always separate one element s into a repeated-element sequence (s, s, · · · , s) corresponding to the

multiple elements with which s is matched. Under this mechanism, the matching structure of the

two sequences can be transformed into a one-to-one counterpoint X, where

X =

(M

L

)=

(x1 x2 · · · xn

x′1 x′2 · · · x′n

)We can also write X = (~x1, ~x2, · · · , ~xn), where ~xi = (xi, x

′i)−1, 1 ≤ i ≤ n. Since all elements in

X belong to S , the interval between any two elements can be evaluated. However, as a counterpoint,

we are particularly interested in the intervals between two matching elements, namely, the interval

for each ~xi, int(xi, x′i) ∈ IVLS , 1 ≤ i ≤ n. They are called harmonic intervals.

Certain restriction is applied to each harmonic interval, which is called property. The harmonic

intervals have to satisfy their properties. As illustrated in section 2.3, all elements in IVLS satisfying

a property P form a property set P. P is defined pointwise. In X, every ~xi is endowed with a

property Pi, 1 ≤ i ≤ n.

Now we want to find transformations of X subject to the following two conditions:

1. Counterpoints are always kept.

2. Adjacent elements in a sequence are maintained.

In order to determine all transformations satisfying these two constraints, we deform the struc-

ture of X into Figure 15.

Figure 15: Deformation of An Interval Chain

In Figure 15, it’s easy to translate the two conditions into geometrical languages. Keeping

counterpoints corresponds to keeping these vertical parallel lines; maintaining adjacent elements in

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sequence means the position of any two counterpoints are not interchangeable. Combining the two

constraints, the only possibilities left are reflections.

We can only reflect Figure 15 by horizontal lines and vertical lines. Reflecting by horizontal

lines, due to our former definition, it’s swap; Reflecting by vertical lines, we get reflection. Since

the composition of swap and reflection is retrograde, we get two basic operations of the base set B.

As before, the set of four operations – swap, reflection and their compositions – {1, ε, ερn2 , ρ

n2 } is a

subgroup of Dn.

By changing Figure 15 into a directed graph, in which the directions are given by interval

chains, dual is defined by redirecting the graph. Dual is also subject to the two conditions of

transformations.

Claim 5.1. There are only seven transformations of X subject to condition 1 and 2, they are

generated by three basic operations: dual, swap and reflection.

For swap and reflection, as explained above, they are independent and generate an operation

group of order 4. As for dual, to account for why there are only two directions available in forming

transformations, we leave the discussion to section 5.2.

We can easily see that all our results in species counterpoint are parallel in this abstract

system. Since all transformations are subject to counterpoint pairs, their harmonic intervals are

either kept or inverted. This is the reason why we introduce symmetric set in section 2.3. Besides,

the existence conditions are the same: all transformations exist for a given counterpoint X if and

only if all harmonic intervals in X are invertible; For any X, a transformation exists if and only if

it keeps the harmonic intervals.

5.2 Lattice, Order and Multi-Layer Counterpoint

So far, we have fully discussed the two-line counterpoint. Now we want to generalize it to

multiple lines. A multi-layer counterpoint, where multiple sequences stack together, is structured

as Figure 16. There are many concrete music examples of multi-layer counterpoint, such as chords,

which can be treated as a three-line or four-line counterpoint. We can also model overlapping

instrument layers in a piece by multi-layer counterpoint.

We call the multi-layer counterpoint graph in Figure 16 a lattice. In this multi-sequence struc-

ture, as mentioned in the duple case (two-line counterpoint), the correspondence between sequences

can always be transformed into a one-to-one matching lattice. Here X = (~x1, ~x2, · · · , ~xn), where

~xi = (x1i, x2i, · · · , xmi)−1, 1 ≤ i ≤ n. The harmonic interval set of ~xi is given by

H = {int(xpi, xqi) | 1 ≤ p < q ≤ m}

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Figure 16: Lattice for Multi-layer Counterpoint Structure

The same property Pi is applied to every element in H . Related to the elements in counterpoint, let

apq = int(xpi, xqi), then H can be written as a strictly lower triangular matrix Am with apq, 1 ≤ p <

q ≤ m as its nonzero entries. If for some apq, the interval is invertible, namely aqp = int(xqi, xpi)

exists, apq is symmetric. If this condition holds for every p, q where 1 ≤ p < q ≤ m, Am is a

structurally symmetric matrix with diagonal entries 0.

Now let’s discuss the transformations for the multi-layer counterpoint. The transformations

are still subject to conditions 1 and 2 in section 5.1. Hence, similar as before, we only have reflection

and swap to form the transformations. There is only one possibility for reflection, which is by the

vertical lines. However, the stack of multiple sequences leads to multiple possibilities for swaps,

which forms a permutation group Sm. As for dual, it’s related to the orientation of the lattice Xmn,

where a concept of order needs to be introduced.

In section 4.1, we introduced interval chains and sorting methods for species counterpoint. A

sorting method for lattice is given in Figure 16 as red lines. Starting from x11, we follow the red lines

traversing the bottom row. When reaching the right-most element, we go one step up to the second

row and traverse it inversely. By repeating this process, we find a path through every element in

Figure 16 and the edges are parallel or perpendicular to each other. We call this sorting a normal

order. Actually, any path traversing all elements in Xmn is a sorting. We give an example of a

random order for X34 contrasting with its normal order in Figure 17.

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Figure 17: Different Sorting For Elements in X34

In Figure 17 we mark the harmonic intervals in each counterpoint ~xi, 1 ≤ i ≤ n with the same

color. As we can see, for Xmn, there are nC2m edges to keep parallel in transformations. Due to

the transitivity of intervals, this number can be reduced to n(m − 1). In other words, any m − 1

independent intervals in H generate the whole set.

In the normal order, intervals between two adjacent rows are kept parallel in the mn-gon

formed by the sorting sequence. There are m− 1 adjacent intervals between m lines, and they are

independent thus generate H . Therefore, the normal order preserves all “parallel” information we

need in the lattice.

In a random order, the parallel information is lost. Two parallel intervals may intersect in a

random-ordermn-gon as shown in Figure 17. We can still define transformations in a random sorting

sequence, while the price is that the topology is not simple anymore. To maintain adjacent elements

in every sequence, we divide the the graph into three groups by the black dashed, which represent

three rows in the lattice. The elements sorted by normal order can be linearly discriminated, as

they are divided by straight lines. However, sorted in random order, the discriminant lines are

deformed to curves. We will maintain the adjacent relations of elements as before to satisfy the

first condition of transformations.

To keep the counterpoints, it’s sufficient to keep the edges of same colors in the graph always

of the same color. In the normal order, the exchange of colors are parallel; In random order, the

map is not “conformal”, so the exchange may intersect.

Given an ordering for Xmn, we get a sorting sequence S. Now we want to add directions to

this sequence. For any two adjacent notes, the arrow can go from left to right (positive) or right to

left (negative). However, for the interval sequence, we have the following claim.

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Claim 5.2. An interval sequence S is unidirected, which converts S into an interval chain Ic.

This claim comes from the transitivity of int. We can simplify the m × n lattice into a mn

sequence because the sequence preserves all the information we need from the lattice, which is the

intervals between elements. Therefore, we say the sequence represents the original two-dimensional

structure, where we can transfer our research to a simpler form. All intervals in the lattice have to

be computable in the interval sequence S. Suppose S is not unidirected, for example

a1 → a2 → a3 → a4 ← a5 → a6

The interval chain is broken since the two parts divided by the left-pointing arrow are not transitive.

For example, int(a3, a5) is not computable. Therefore, given a sorting sequence S, there are only

two orientations, positive or negative, flowing from the beginning to the end or vice versa.

Corollary 5.0.1. Dual inverts all directions in an interval chain.

This discussion gives another proof for Theorem 3.1. The proof in section 4.1 also gives us

another argument to deduce the unidirectional property of dual. In section 4.1, we proved that for

a two-line counterpoint, its melodic intervals cannot be inverted in one line while kept invariant in

the other line. In a multi-layer counterpoint, this condition also holds. Therefore, to get dual, if we

invert a row in the multi-layer counterpoint, its adjacent rows have to be inverted too. Following

this rule, we finally invert all the rows in the lattice, which inverts the interval chain as Corollary

5.0.1 states.

Claim 5.3. For a m× n counterpoint lattice, there are 2× 2× |Sm| transformations.

The basic operations for a multi-layer counterpoint Xmn are reflection, dual and a Sm swap

group. The swap group contains m! swaps of the m lines. Noting that we are using reflection

instead of retrograde here to represent a basic operation. It’s viable since reflection is independent

of dual and swap. Besides, reflection is a more intuitive concept in this case. Following our former

notations, the base set Bm for a multi-layer counterpoint can be written as

Bm = {{1, Xf}, {1, X∗}, Sm(X ′)}

Bm is a collection of basic-operation groups, which generates the transformational group by com-

positions. Since Sm(X ′) is not abelian, swap for multiple lines is not symmetric. Operations from

different basic-operation groups are commutative. For harmonic intervals, Xf maintains all har-

monic intervals in Xmn; X∗ inverses all harmonic intervals in Xmn; Sm(X ′) satisfies the following

theorem.

27

References Symmetries in Counterpoint Patterns

Theorem 5.1. ∀g ∈ Sm(X ′), g acts on Xmn is given by

g(Xmn) = g(~y1, ~y2, · · · , ~ym)T = (~y Tg(1), ~y

Tg(2), · · · , ~y T

g(m))

where (~y Tg(1), ~y

Tg(2), · · · , ~y T

g(m)) is a row permutation of Xmn. For any i, j, 1 ≤ i < j ≤ m, if

g(i) > g(j), g(Xmn) exists if and only if int(xik, xjk)−1 ∈Pk where 1 ≤ k ≤ n.

Theorem 5.1 is a generalization of Theorem 3.1 and the existence conditions for two-line coun-

terpoint. Let T be the transformational group, then

T = {1, Xf} ◦ {1, X∗} ◦ Sm(X ′)

∀ι ∈ T, ι = α ◦ β ◦ γ where α ∈ {1, Xf}, β ∈ {1, X∗} and γ ∈ Sm(X ′). The existence condition

for ι is a combination of the existence conditions for α, β and γ. However, ι can exist without the

existence of α, β or γ separately. Same as the case in Figure 5, we say the existence of one pattern

doesn’t rely on its transitional states. Inverting an interval twice, we get the original interval which

doesn’t need to satisfy the “invertible requirement”. Thus, we have Claim 5.4.

Claim 5.4. The existence of an operation is independent of its compositions.

References

Lewin, D. (2011). Generalized musical intervals and transformations. Oxford University Press,

USA.

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