Symmetry as a Compositional Determinant

copyright © Larry J. Solomon, 1973, revised 2002

Chapter II. Definitions and Principles

A dictionary defines symmetry as "beauty of form arising from balanced proportions. 2. corresponding in size, shape and relative position of parts on opposite sides of a dividing line or median plane or about a center or axis. {Merriam Webster's Collegiate Dictionary, 1194} Often the term is meant to be synonymous with balance, proportion, or a pleasing arrangement of the parts to the whole, especially in the arts and music. In order to arrive at a systematic treatment of symmetry we need a more rigorous definition and one which, at the same time, is most generally applicable.

Symmetry is here to be defined as a congruence which results from the operations of reflection, rotation, or translation.

One of the important types of symmetry, and one familiar in everyday life, is called bilateral or reflective. A figure in two dimensions said to possess such symmetry can be reflected about a linear axis and made to correspond point for point with its original image. The following geometric figure is an example.

Figure 5. Illustration of the Reflection Operation.

If figure 5a is reflected, or flipped, about the vertical axis, y, 5b results. Point x1, y1 of figure 5a is projected into -x1,y1 in 5b, and x2,y2 is projected into -x2,y2. Figure 5b is congruent to 5a. The operation is mathematically defined as:

T(x,y)= (-x,y)

meaning a transformation upon (x,y) such that it becomes (-x,y). The figure is symmetric only with respect to one axis, the one bisecting it vertically. The same figure is not symmetric about an axis such as:

Figure 6. An asymmetric axis.

The following is asymmetric with respect to any axis.

Figure 7. An asymmetric figure.

A test for symmetry involves an operation upon a subject, such as reflection about an axis, which makes it congruent with the original form. A musical figure such as the one in the following example is symmetric with respect to an axis of time in a pitch-time context. The axis is vertical with respect to the notation.

Example 3. Temporal reflection in a musical figure

The axis falls on the fourth sixteenth note. In musical notation, time is represented horizontally. If the entire figure is reflected about this axis, the result is congruent with the original, and therefore satisfies the condition for reflective symmetry. In three dimensions (or three variables such as pitch, time, dynamics) the axis of reflection becomes a plane. Most musical manifestations, however, will be confined to two variables. We may refer to their graphic representation in two-dimensional space. Musical figures may also be symmetric with respect to a tonal axis, or, an axis of pitch. Pitch is represented vertically in musical notation.

Example 4. Tonal reflection in musical figures.

Example 4a is a reflective chord with an axis of pitch, A4. All the tones sounding above the A4 have corresponding tones sounding the same distance' below A4 (distance = interval). If the chord is reflected about the axis tone A, the same chord will result. Example 4b is a melodic figure sounded with its exact inversion. If the entire example is flipped about the axis tone A4, an identical figure will result. The following is an analogous visual example with symmetry about a horizontal axis.

Figure 7. A visual form reflective about the x axis

Reflection again results in an identical figure. The general transformation is:

T(x,y)= (x,-y)

The same principles may be applied to most of the parameters of music which can be ordered and controlled, e.g., dynamics, texture, tension, tempo, form, etc.

Example 5.  Temporally reflective musical figures.

  a. p < f > p         b. a tempo ---- accel.---- decel.---- a tempo  

  c. A B A

All of the above figures are common symmetrical forms encountered in music, all of which are reflective about a temporal axis.

Another possible symmetry operation is a rotation around a point, or a line in three variables, exhibited by the letter Z. This letter may be rotated 180 degrees around a point to create an image identical to the original.

Figure 8. Symmetry by rotation

This letter is said to have an axis of two-fold symmetry since it duplicates itself twice in a complete revolution. An analogous musical figure is:

Example 6. A rotational figure equivalent to the previous graphic representation

The operation which transforms both of these figures into themselves may be defined as the 180 degree rotation.

T (x, y) = (-x, -y )

Another example of this type is:

Example 7. Another rotational figure

These figures have an axis of two-fold symmetry. The last is similar to the fugue subject of The Well Tempered Klavier, Volume I in g minor (the exception being the third eighth).

A notable rotational form is the retrograde inversion. The 180 degree transformation is identical to the combined reflection operations; this points out their fundamental relationship.

There may be occurences of 90 degree rotations in music which result in an axis of four-fold symmetry. Consider the following visual form:

Figure 9. A visual form having 90 degree, four fold symmetry

If we revolve this about the point 0, the figure goes into itself four times, once every 90 degrees of rotation. The transformations may be defined as follows for 90, 180 and 270 degree positions respectively.

T (x,y) = (y -x)

T(x,y) = (-x,-y)

T (x,y) = (-y,x)

The use of this operation upon the variables of pitch and time results in the mutual exchange of their functions. This creates new transformations which are reserved for discussion in a later chapter.

Many other types of rotational symmetry occur in nature but seem to have little application in music. Notable among these are the hexagonal six fold symmetry of snowflakes, flowers, and the cells of beehives. In three dimensions this principle is strikingly represented in the science of crystallography where complex systems of nomenclature and classification have been developed for various levels and manifestations of symmetry in natural crystals.

Another symmetry operation is exemplified by what some artists call "infinite rapport" and what will here be called translation. The following ornamental figure will serve as an example:

Figure 10. Translational symmetry in an ornament


There is a pattern in this which is repeated at a regular spatial rhythm. Similar manifestations which are common in nature are called metamerisms by zoologists. The legs of a centipede or the leaflets of a fern are examples. Of course the pattern does not continue infinitely in its physical manifestation in any of these examples, but one may call them potentially infinite. An exact image may be moved linearly upon the original and result in congruences at regular intervals. This operation may be precisely defined.

T(x,y) = (x+na,y) for horizontal translation

T(x,y)= (x,y+na) for vertical translation

Since the operations are iterative, i.e., an operation performed upon itself yields a similar expression, they may be expressed:

T (x, y) = (x+a, y)

T (x, y ) = (x, y + a)

Translational symmetry is often combined with reflective symmetry in music.

Example 8. Figures with translational and reflective symmetry

a.                                                                                                                   b.

Both examples 8a and 8b exhibit translational symmetry at an interval of four eighths, i.e., T(x,y)= (x+4,y). While example 8b has reflective symmetry as shown by the axes, example 8a does not.

Of course, it is possible to employ the translation operation in the tonal dimension as well as the temporal dimension. Such translations are normally combined, i.e., both tonal and temporal translation simultaneously, as in the following.

Example 9. Translational symmetry in Chopin's Etude, Op. 10, No. 12

The combined iterative transformation may be expressed:

T(x,y)= (x+a,y+b)

Spatial dilation or contraction is a special type of symmetry transformation which will here be called an automorphism. Two photographs of the same image in different sized prints have a point for point correspondence. In geometry they would be called similar figures. A musical idea maintains a similarity if it undergoes a temporal or tonal automorphism. Augmentation and diminution are traditional examples of this operation.

Relativity of size is a well known phenomenon in the art of perspective and is a phenomenon of our perception. {Gregory, 147-163} An object may be made to look distant by making it smaller on a flat surface. In some kinds of space it would be impossible to tell if an object is smaller or more distant than another.

Automorphisms are linear and multiplicative in function. Consider the following:

Figure 11. A diagram of a horizontally grown branch of maple showing automorphic sizes of leaves.

This figure shows the vertical-horizontal orientation of growth in maple leaves. Notice the difference in the size of the leaves 11a and 11d, but they are similar in shape. This illustrates automorphism across a horizontal axis. If leaf 11a is made larger by the proper amount, or leaf 11d smaller, their symmetry can be shown by reflection. Leaves 11b and 11c grow larger on the bottom than on the top half, and their symmetry can be shown after a similar automorphism across a horizontal axis. In fact, symmetry can be demonstrated for the entire figure if an automorphism is performed on one side of the horizontal axis. If any point above the axis, x1,y1, is expanded by the proper multiplier, x1,y2 will result. The transformation may be generalized to:

T(x,y)=(x,c1y) where y < 0, (x,c2y) where y >= 0; c1,c2>0

In our example, c2= y1/y2. The transformation can also apply to x, a horizontal dilation.

T (x, y) = (c1x,y) where x<0, (c2x,y) where x>=0; c1,c2>0

A musical example of the last transformation would be:

Example 10. Reflective symmetry by horizontal dilation.

Time is halved, in this example, after the vertical axis and may be expanded by the multiplier 2.

T(x,y)= 2x,y where x=time and y=pitch

Symmetry is then demonstrated by reflection about the axis. If y=pitch, the following is also illustrative:

Example 11. Reflective symmetry after a vertical dilation.

Expanding the intervals after the axis by a factor of 2 will yield symmetry by reflection again.

Automorphisms are not enough by themselves to confirm symmetry. They must be combined with some previously described operation, such as reflection or translation. Automorphisms are, therefore, called auxiliary transformations. The operation is normally carried out before an accompanying primary transformation, i.e., those previously described.

A striking non-linear type of automorphism occurs in the form of shells of the chambered nautilus and other animals and plants which can leave a record of their growth patterns.

Figure 12. Diagram of the structure of the nautilus. shell.

The symmetry here is rotational combined with an automorphism. Each chamber is an exact copy of another except for size. Similarly, in music, such dilations exist in the equal tempered scale since the semitone becomes increasingly expanded with respect to frequency difference the higher we go in pitch.

Figure 13. Automorphic rotation in the equal tempered scale.

All of these symmetry operations, then, can be combined to form our general definition of symmetry. The reflection and 180 degree rotation are both of the form:

T (x, y ) = (+-x, +-y )

In order to include automorphisms on either side of the vertical axis we need:

(c1x, c2y) where x>0, y>0

T(x,y)= c1, c2, c3, c4>0

(c3x, c4y) where x>0, y<=0

This is our auxiliary transformation which may be further generalized for automorphisms in all four quadrants.

(b1x, c1y) where x, y > 0

(b2x, c2y) where x <= 0< y ; b1,b2,b3,b4>0

T (x, y ) = (b3x, c3y) where x, y <= 0 ; c1,c2,c3,c4>0

(b4x, c4y) where y <= 0 < x

The primary transformations may be generalized to include the 90 degree rotations.

S1 (x, y) = (+-x+al, +-y+a2)

S2 (x, y ) = (+-y+a1, +-x+a2)

Symmetry exists, therefore, if a congruence results after at least one of the primary operations or after a combination of auxiliary and primary operations.

By this definition, the following figure can be shown to possess svmmetry.

Figure 14. A figure having some symmetry.

First, the automorphism transformation is applied to the four quadrants independently to give:

Figure 15. An automorphism of figure 14.

S1 is then applied to reflect this about an axis, yielding a congruence. Although figure l4 is symmetric, its order of symmetry is low.

The order of symmetry is proportional to the number of primary transformations that can result in a congruence and is less than this amount by an increment for each partial transformation necessary for congruence.

The automorphism performed above is an example of such a partial transformation. In a musical ABA, the order of events in the da capo section is normally not the reverse of the first section. In this respect it is similar to the symmetry of the word DAD as distinct from that of MOM. Notice that MOM possesses point for point symmetry after the operation of reflection about an axis of vertical bisection, but DAD does not. Although DAD is less symmetric than MOM, it is more symmetric than "fly," because a partial transformation can be performed on DAD before it may be reflected, namely a 180 degree rotation of one of the D's.

The word MOM has a symmetry of order one, whereas DAD is less than one. The word "fly" has a symmetry order of zero. If no partial transformation is performed on DAD, one out of three of its letters will be congruent upon reflection. Its degree of symmetry is said to be 33 per cent, or just 33. The degree of symmetry is determined by the proportion of congruent points or parts after an operation. Therefore, after both the partial and reflective transformations of DAD previously mentioned, the degree of symmetry is 100. The tabulation of DAD's symmetry may be represented:

   OPERATIONS                            ORDER            DEGREE   

   Reflection                                          1                        33

   Partial rotation and reflection        <1                      100

SAD, however, has only a symmetry order 1, degree 33.

It should be noted that DAD is here examined for point for point symmetry, but if DAD is regarded as a simple three element figure in which the letters are construed as units (The shapes of the letters become inconsequential) its symmetry may be generalized as identical to that of MOM, i.e., order 1, degree 100. Point for point analysis is generally more thorough, however, and is preferred when possible. Consider the following:

Example 12. Symmetry types in music

a.   p < f > p           b.   pp  < ff   > p

Example 12a and d are symmetric by order 1, degree 100. Example 12b is symmetric after a partial temporal automorphism across the ff axis; the order is <1, degree 100. Example 12c must have two automorphisms, one of pitch and the other of time, to yield a lesser order of symmetry than 12b, designated <<1, 100. These may be tabulated as follows:

PLAY may be reflected about either of the letters A or Y to yield symmetry degree 25. Note that the degree here is not 50, because separate operations must be carried out for each letter. OHIO may be reflected about a vertical bisector to yield order 1, degree 50, or it may'be reflected about a horizontal bisector to yield order 1, degree 100. When two or more order designations are possible, the one with the highest degree is assigned.

To clarify the preceding exposition the reader may wish to determine the order and degree of symmetry in the following, point for point:


2. SOS

3. OHO

4. PRY


6. A B C A

7.   p < ff  > pp



    ORDER          DEGREE       OPERATIONS

1.   1                        25                Vertical reflection about T or A
2.   1                       100               Rotation, 180 degrees
3.   3                       100              Vertical and horizontal reflection, 180 degree rotation.
4.   0                           0
5.   1                         75              Translation
6.   1                         50              Vertical reflection
7. <1                      100               Partial dynamic automorphism and temporal reflection.
8. <1                      100               Partial automorphism across second bar and temporal reflection.
9.   2                      100               Tonal and temporal reflection
10. 2                      100               Translation and Temporal reflection.

Some distinction exists between real and tonal congruences, referring to real and tonal answers in fugue, sequences, and generally in diatonic compensations. A congruence is to be considered real here if the intervallic correspondences are identical and tonal if changes are made to conform to a superceding diatonic or similar framework. In either case, the congruences are strict if there is no departure outside of tonal variation. If, for example, a canonic voice is in tonal imitation rather than real imitation, it is still considered in strict congruence. If, however, the canonic voice even briefly breaks the pattern of imitation, the congruence is no longer strict. In this sense, "strictness" corresponds to symmetric degree.

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