**copyright ©**
**Larry J. Solomon, 1973, revised 2000**

How may symmetry be discovered in pre-existing music? In order to answer this question the common properties and manifestations of symmetry must be determined and a clear procedure followed to obtain an exhaustive exposure of symmetric properties.

The most fundamental property of musical symmetry is repetition. All types of symmetry are forms of repetition, and all repetitions contain at least an element of symmetry. Translations are open ended repetitions, rotations are cyclic repetitions, and reflections are retrograde or inverted repetitions. Automorphisms are normally temporal translations in which the pattern repeated is linearly expanded or contracted in at least one of its dimensions.

The recognition of symmetry is preceded by the recognition of repetitions. All music, except perhaps for chance music, contains some repetition, and most music contains a large amount. Order and symmetry are literally built into our tuning systems, our instruments, and the conventions of our music. The mere fact of a limit of twelve tones in the equal tempered scale necessitates their repetition in music.

"Elemental" repetition refers to the repetition of single elements, such as that of a pitch. This type of repetition is the most common and most primitive. While the repetition of a single pitch class is likely to be trivial observation in the music based on the twelve tone scale, the single element repetition of a key in a large scale structure is not.

A ternary form, A B A, may seem to be elemental; however, the recurrence of A usually represents more than a recurrence of a single element but that of a complex of elements. Thus, the significance of elemental repetition is dependent upon the duration and complexity that the element contains.

All repetitions need not be accounted in an analysis, depending upon the result desired.. Of primary importance, however, is group repetition, which is more difficult to recognize than the elemental type. A group consists of two or more adjacent, ordered elements and may be observed in music as sections, pitches, dynamics, tempo, etc.

The following form is an example:

A A B C D A A B C

The repeating groups may be bracketed as follows:

[A A B C] D [A A B C]

Thus, partitioning the group is the crucial step in outlining repetitions.

**Exercises:** The reader may want to clarify the preceding exposition
by identifying the repeating groups or their repeating aspects in the following
examples.

1.

2.

3.

4.

5.

6.

7. A B C D A E B C D A

**Answers:**

1.

2. quartal chords

3. 3/4 meter

4. . . . _

5. * p f*

6. scalar: up 2, down 1

7. A [B C D A] E [B C D A]

The plan of group repetitions may be reflected by divisions of the form on a higher level, and, for this reason, it is suggested that the analysis proceed from the large scale to the small scale. For example, number 7 of the previous excercise is grouped without considering any large scale divisions:

1. A [B C D A] E [ B C D A]

However, if the composition is divided differently on a higher level, the above grouping may be less valid. For example:

2. A B C D | A E | B C D A

Because the divisions are established by other means than repeating sections of #1, the groups are more logically outlined:

A [ B C D] A E [ B C D] A

Once repetitions are identified, look for changes in at least one dimension by one of the defined operations. One way to proceed is to locate repeating unordered sets of elements, such as

A B C C B A = [A B C] [C B A].

Here the set elements are in reverse order, but repeating sets may yield less symmetry, as in the following:

A B C B A C = [A B C] [B A C].

Once the unordered sets are found, the operations of reflection, translation, and rotation may be tested. In addition, it is suggested that operations be preperformed on repeating groups of step one and their results checked for recurrences.

Rotations are often synonymous with simple repetitions, such as in meter,
isorhythm, isomelos, trills,rounds and any such cyclic structures. They
are among the easiest operations to recognize. A checklist of common musical
structures is given in Table I at the end of this chapter. These should
help to identify some of the common uses of symmetry and to classify the
types of variant operations.

**Exercises:** The reader may wish to clarify, once more, the preceding
exposition by analyzing the following variants. Note that reflections may
be regarded as backward or inverted repetitions.

1. A B C D B C B A C D

2.

3.

4.

5.

6.

7.

**Answers:**

1.

2. melodic intervals reflected

3. pitches in measures:

4.

5. intervals:

6.

7. scalar motion:

When translations of order or duration are solely temporal, the result is an automorphism. Augmentation and temporal contraction are examples. A tempo or dynamic pattern may also be translated temporally to yield a "transposed" pattern, such as:

*allegro andante moderato adagio
andante*

|-----------------------------------|

|-----------------------------------|

Temporal translations are recognized in group proportions. Development
by fragmentation is common in the works of Beethoven. Frequently, the automorphism
results from repeated phrase bisection. In the Seventh Symphony, bars 88-110
of the first movement the phrase length is contracted from eight (4+4)
to two, and,finally, to one measure by fragmentation. In the piano sonata,
Op. 90, bars 130-144 of the first movement, a five note figure in sixteenths
is expanded to eighths, then to quarters, and, finally, to half notes,
at which time the five notes are reduced to three. Next a temporal contraction
occurs, returning to the eighths. At the same time, the dynamics change
from ff to p to pp and crescendo to f.

Example 75. Automorphism in Beethoven's piano sonata, Op. 90, first movement.

The following exercises are offered for those who may wish to clarify the preceding exposition. Analyze and identify the variant operations in the following:

1. A B B C D B C D B C A B

2. *pp mp p ff
mp mf*

3.

4.

5.

**Answers:**

1. Reflections:

2.

3. Interval translation:

4. Isomelos, after 4 pitches (translation)

5. translation + tonal reflection: 3-voice imitation at 2 semiquavers at the fifth above and below A (pc).

final step of analysis is to determine the fitness of symmetric operations to the patterns of repetitions and variants on various levels of the compositional structure. A scanning technique can be employed in the search for these relationships. As an example, determine the symmetric structure of the following:

A B A C D C A B A

Answer:

Step 1. **[A B A] [C D C] [A B A]**

Step 2.

Step 3.

**X Y X**

Step 4.

a. X Y X is rotational and reflective.

b. A B A translates to X Y X.

c. C D C translates to X Y X.

Thus, self-similarity emerges as the pattern X Y X.

**Determine the symmetric structures in the following in a similar
manner: for clarification. **

1. A B B C D C D B C A B

2. Row from Anton Webern's Op. 30.

**Answers: **

Step 1: [A B] [B C] [D C D] [B C] [A B]

Step 2: [A B] B C D C D B C [A B]

|_____________________|

[A B] [B C] D C D [B C] [A B]

x
y | | y x

|______|_____
|______|

Steps 3-4: [A B] [B C] [D C D] [B C] [A B]

|_____________|____________|

2. Step 1:

Interval order: translation

1 3 11 11 3 1
3 11 11 3 1

|______________|

|_______________|

Steps 2-3: (variant groups)

Step 4:

P= RI a and c = p

R= I b and d = ri

a+b= (c+d)5 (translation by five semitones)

Analyze and identify the types of symmetry in the following examples:

1.* p f pp p mf
ff p mf f pp |
p *

2. A B | A C B D A C B A

3. A B C D C A C D B C A

4. A B B C | D B C D | B C A B

5.

6.

7. *ff mf f mp mf p mp
pp p*

8.

9.

**Answers: **

1. Reflection:

2. Reflection:

3.

4.

5.

6.

7. Automorphism and Translation

ff mf
f mp mf p mp pp
p

|_______| |_______|

|________| |________|

8. Automorphism and translation of intervals (+2 semitones)

9. Rotational intervals and pitch classes

(up3, down1, up1, for intervals and and 12 eighths for pcs)

In summary, the method for symmetric analysis may be outlined as follows:

1. Locate group and elemental repetitions within large and small scale compositional divisions.

2. Locate set variants. a. locate unordered sets and reflections. b. locate group proportions.

3. Identify automorphisms.

4. Determine the similarities of symmetric operations and patterns at various levels of the compositional scale.

1. timbre identity | T | p |

2. transposition | T | p |

3. parallel organum | T | tp |

4. fauxbourdon | T | tp |

5. sequence | TO | tp |

6. imitation | TO | tp(s) |

7. antiphony and concertato | T(OR) | ts |

8. rounds, canons | TP | tp |

9. cantus firmus composition | O | tp |

10. ostinati and chaconne | O | tp |

11. Alberti bass, arpeggios | O | tp |

12. trills | O | tp |

13. vibrato | O | tp |

14. pitch identity | O | t |

15. meter and pulse | O | t |

16. isorhythm | O | t |

17. circle of fifths | O | p |

18. scale formation | OT | p |

19. serial operations | ROT | tp |

20. invertible counterpoint | R | p |

21. melodic inversion | R | p |

22. mirror chords | R | p |

23. cancrizans | R | t |

24. arch forms | R | t |

25. sonata-rondo | R | t |

26. compound rondo and compound ternary | R | t |

Symmetry type: Translation = T, Reflection = R, Rotation = O

Dimension: Pitch = p, Time = t, Space = s

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