Symmetry as a Compositional Determinant

copyright © Larry J. Solomon, 1973, revised 2002

Chapter IV. Translation and Rotation

The operation of translation has been defined in Chapter II as follows:

T (x,y)  = (x+a,y)
T (x,y ) = (x,y+a)

This transformation is closely allied with rotation, and, in fact, translation may be regarded as open ended rotation, i.e., continuous. This will be clarified in the following section on rotation.

Translation

There are two types of translation; these will be called simple and compound. A simple translation contains a single operation in one variable, while a compound translation contains two or more variables and operations. Examples of the simple type are theme and variations and refrain forms:

A1 A2 A3 A4 A5 A6
A1 B A2 B A3 B ...

The third Bagatelle of Bartok demonstrates another manifestation.

Example 46. Temporal translation in the third Bagatelle by Bartok.

Translation is necessary here in only the temporal dimension to yield symmetry, and the interval of translation is one quarter note.

Scale formation is also a manifestation of simple translational symmetry. A C major scale, for instance, may be translated at the octave to yield a congruence of pitch class.

Example 47. Translation in the C major scale.

Using the scale step as a unit of measure, this translation may be expressed mathematically:

S (x,y) = (+-x+al, +-y+a2)
S (x,y) = (x+a, y+b)
S (x,y)= (x,y+7)

Octave identity itself is a translational operation in pitch which depends upon our perception of pitch class identity and non-identity.

Example 47b. Translation of octave identities.

Chord structure identification and transposition are dependent upon the translation concept. A tertian triad built on a single pitch may be translated to any other pitch and maintain its identity, and so may less orthodox structures.

Example 48. Temporal translation of secundal harmonies in Bartok's "The Night's Music" from the Out of Doors Suite. Copyright 1954 by Boosey and Hawkes.

Most symmetry translations in music are compound rather than simple. This is because of the constant temporal motion of music which obliges translation in at least that dimension if it occurs at all. Melodic sequences consist of translations in two dimensions, pitch and time.

Example 49. Translation in melodic sequences.

The time interval for translation of example 49a is one quarter note while for 49b it is two quarters. The interval of pitch translation for both 49a and 49b is a perfect fourth, or five semitones. Such operations can involve many other parameters, such as dynamics, rhythm, tempo, etc.

Example 50. Temporal translation of dynamics (a), rhythm (b), and tempo (c).


  andante    allegro    andante    allegro    andante ...

In some folksongs simple musical repetition is used with a change of text on each repeat, commonly known as strophic form. In the performance of these songs, emotional intensity is sometimes heightened by speeding up each section and in slightly raising the pitch level, e.g., in "Pretty Polly," on Folkways Records Music of the World's Peoples, compiled and edited by Henry Cowell. These translations are examples of tempo and pitch symmetry. Translation occurs in three dimensions: time, tempo, and pitch. Here, the alliance of translation and automorphism becomes apparent.

Consider the following line:

Example 51. A musical line and its graphic notation

A further translation can occur in a third dimension, z = tempo.

Example 52. Tempo translation, the equivalent of an automorphism

Melodic imitation is a result of a translation operation. In this case a shift is made in pitch and time, but may also occur in space and timbre, between vocal or instrumental groups.

Example 53. Graph of imitative entries in a typical contrapuntal choral work, where S= soprano, A= alto, T= tenor, B= bass, showing translations

This type of operation is actually required to be done by the performer of J.S. Bach's puzzle canons.

Example 54. Puzzle canon by J.S. Bach requiring the translation operation

realization:

Symmetry of textural order has often been the result of translation in imitative counterpoint following the technique of Josquin des Prez, including the instrumental ricercare.

Example 55. Graph of Josquin's imitative formula, showing translation and textural expansion

Johannes Ockeghem's Missa Prolationum is a large scale result of automorphism plus translation. In this canonic work, voices imitate each other in different mensurations.

Example 56. Automorphism and translation in Ockeghem's Missa Prolationum

A common form of translational automorphism occurs in scale formation and octave identity. Scales are formed by what are considered constant units, the intervals. However, as pitch ascends, the frequency difference expands between identical intervals. We know this because the ear's response to frequency is multiplicative rather than arithmetic.

Example 57. A graph showing the relationship between frequency and octave perception

The ear's response to loudness is also nonlinear and exponential with respect to measured sound levels. Thus, even though an increase from p to mf may seem equivalent to the change from mf to f, this expansion of loudness is automorphic with respect to measured sound levels.

Example 58. A graph showing the relationship between measurable intensity in decibels and loudness units

The overtone spectrum, timbre and amplitude envelopes are all results of translation operations. An oboe sounding a C generates an overtone spectrum which is essentially congruent with its spectrum on C#, D, etc. This is a special case of transposition or pitch translation.

A very thin line divides harmony from timbre. Even a loud tone from a recorder, an instrument with few overtones, stimulates the ear's mechanism to generate overtones and difference tones, once called "subjective tones." In effect, every tone is heard as a complex "chord," and, as the pitch changes, these chords move in parallel motion. Since the first six of the overtones are predominant in the spectra of most instruments, these "chords" are essentially parallel five-three, including parallel octaves and fifths. Translational symmetry is ubiquitous. When different instruments are combined, each having an independent line, the harmonic results are complex.

Although timbre is rarely discussed in this way, we can see its relationship to parallel harmonic motion in composition. Parallel organum and fauxbourdon both possess translational symmetry.

Example 60. Parallel organum, from: Apel and Davidson, Historical Anthology of Music, Volume I, Cambridge, Mass.: Harvard University Press, 1959), No. 25b

Such parallelisms are not restricted to the Middle Ages.

Example 61. parallelisms in Chopin's Etude in E Major, Op. 10, No. 3, consisting of a series of diminsihed seventh chords, and b. Bartok's Four Nenies, No. 3 (left hand)

b.

Parallelisms even occur in the music of Johann Sebastian Bach.

Example 62. Parallel fifths and octaves in J.S. Bach's Chromatic Fantasy and Fugue


Rotation

That rotations are simply special cases of translation may be seen by examination of their definitions:

Translation: T(x,y)= ( x+a1,y)
                     T (x, y) = (x,y+a2)

Rotation: T(x,y)= (-x,-y)    180°
                 T(x,y)= (y,-x)     90°
                 T(x,y)= (-y,x)     270°

If a1= -2x and a2= -2y the result will be our 180 degree rotation. Similar results may be obtained for the 90° and 270° rotations. The rotations are, therefore, special cases. Translation imparts movement through musical parameters such as pitch and time, and it implies indefinite reiteration. That these are also qualities of rotation may be seen by the following example. If three groups of four different instruments are positioned in space in a circle, rotational symmetry may be demonstrated by the operation at the spatial interval 4.

Example 63. Rotational symmetry in the positions of instrumental groups in a circle.

This can be done three times before returning to the origin; thus, there is a rotational axis of threefold symmetry. Considered in polar or module coordinates, the relationship to translation becomes apparent.

Translation: T(x,y) = (x+a,y)
Rotation of example: T(x,y) = (x+4 mod12, y)

Rotation, therefore, may also be called rotary translation. The basic difference lies in the closed configuration of rotation as opposed to the open one of translation. The example of an iterative scale is open-ended and translational with respect to register, but the set of pitch classes in the equal trempered scale is a closed circle.

Example 64. The circle of pitch classes in the equal tempered scale


Simple Rotations

Rotations may be classified in two types similar to translation, according to the nature of their dimensions, as, for example, in our spatial example. Others involve more than one variable, The pitch class set of a scale is symmetric by simple rotation, as in example 64. However, the pitches of a scale are not rotational, but translational, because of the change in registration. Temporal rotation is the largest single class of rotations in one dimension. The classical iterative period structure consisting of two eight measure units, each comprised of two four-measure phrases may be represented as follows:

Example 65. Rotary symmetry of phrase lengths in the classical period structure (numbers= measures).

The stimulus of pitch is symmetric by rotary translation. What makes pitch perceptible is a length of periodic vibration. No matter what the wave form, the structure of periodic motion is rotationally symmetric.

Example 66. Rotation in the sine wave.

Vibrato, the periodic fluctuation of pitch of about 6 Herz, and tremolo, the periodic fluctuation of loudness of about the same frequency, possess the same characteristic symmetry.

Many of our traditional harmonic patterns and root movements are symmetric by rotation.

Example 67. Rotational symmetry in traditional harmonic progressions and root movements.

The large scale harmonic structure of much of Western music is cyclic in the same way. The circle of fifths, a theoretical abstraction derived from the tonal structure of Western music, is indicative.

Example 68. The circle of fifths.

Any simple iterating figure is rotational. This includes the following.

Example 69. Rotational musical figures.

Other examples are isorhythms, cantus firmus repetition, color, and talea.

All row composition is symmetric by rotation or by a combination of rotation and reflection with translations. Some tone rows will exhibit a higher degree of symmetry than others. The following row, in which The second hexachord is a transposition of the first, exhibits symmetry by rotation on each hexachord.

Example 70. A twelve tone row which is rotationally symmetric on each hexachord.

All or most of the operations of serial composition may be explained as one of our primary transformations. A totally serial work may be shown to be symmetric in nearly all or all of its parameters, and hence, similar to a folded ink blot or a purely symmetric visual design.

Some types of rotational forms are also reflective, especially those which exhibit low symmetric orders. These would include ABA, I V I, and some Alberti figures. Some large scale forms, however, reveal rotary symmetry without reflection, such as in Alban Berg's Wozzeck, Morton Subotnick's Touch, Richard Wagner's Ring, and Bela Bartok's Bluebeard 's Castle, all of whose endings are linked with their beginnings.

Rounds are perfect examples of rotary symmetry. Spiral canons must be repeated a number of times to close the circle of rotation.

Example 71. J.S. Bach's perpetual canon for two voices requiring the performer to carry out rotation.


Compound Rotations

If the rotary operation involves two or more dimensions, it is termed compound. A configuration including time and pitch is an example.

Example 72, A figure which is symmetric by compound rotation.

Graphically, this may be represented:

The whole notes on C are shown in the graph as long horizontal lines indicating their sustaining while the 32nd notes ascend on the major scale from the low to high C, shown by the diagonal line. The axis is middle C, shown by the point. Symmetry may be demonstrated by rotating the figure 180° to yield a congruence. Note, however, that the figure is being rotated through a matrix of both time and pitch simultaneously, rather than just one or the other.

Compound rotations may involve variables of timbre, tempo, texture, harmony, etc. in two or more dimensions. The following shows a possible symmetric change of tempo in time.

Example 73. Graph of tempo change with respect to time which is symmetric by a compound rotation.

The compound rotation overlaps with the reflection operation in the form of the retrograde inversion, which may be regarded as either a 180° rotation or as a reflection about both a horizontal and vertical axis. Although many suitable examples exist, one of the most intricate is the relationship between the Preludium and the Postludium of Hindemith's Ludus Tonalis. If Hindemith had taken advantage of Bach's shorthand notation, it would not have been necessary to write the Postludium, since one need only perform the 180° rotation on the Preludium (read it upside down and backwards) to realize the Postludium.

Example 74. Corresponding portions of the Preludium and Postludium of Hindemith's Ludus Tonalis, illustrating their relationship by compound rotation.

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