Symmetry in Music

1. Symmetry in Music Mathematics Institute 2005 SACNAS National Conference Denver, CO September 29, 2005 John B. Little College of the Holy Cross Worcester, MA 01610 little@mathcs.holycross.edu

2. Plan for this Session • Organizing principles in music • Visual analogies • The language of symmetry • How does this apply to music? • Two extended examples • J. S. Bach's The Musical Offering • Nzakara harp music • Discussion

3. Organization of music • Pieces of music are composed and experienced as time-sequences of “events” or “gestures” • melodies or motifs • harmonic progressions • rhythmic patterns, etc. • organized in various ways, using • repetition • grouping • variation

4. Describing musical forms • Ternary song form: ABA • Verse, refrain song form: VRVRVR ... (any number of verses) • Sonata allegro form: Exposition, Development, Recapitulation • Rondo form: ABACABA (one type, even more episodes sometimes occur) • Variation forms (repetitions of a basic pattern, but altered on each repetition)

5. Balance and symmetry • Note that some of these are very “symmetric” (ABA, ABABCBA) • Using ABA form gives a feeling of balance or closure at end -- we return to the original music after a varied middle section. • Many of these forms have close analogs in visual arts too.

6. A visual analogy

7. Frieze and wallpaper patterns • In architecture, textiles, jewelry, other decorative work, repeating patterns of elements arranged along a line or in a strip are often used. • called frieze patterns, from term used in architecture • if the repeating elements are arranged not just along a single line, but cover (a portion) of a plane, then we have a “wallpaper” pattern • some extremely interesting patterns occur in drawings and prints of M. C. Escher

8. An Escher “wallpaper” pattern

9. Language of symmetry • In his famous book, Symmetry, the mathematician Hermann Weyl says that the word symmetry is used in everyday language in two meanings: “In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole ... the second sense in which the word is used in modern times [is] bilateral symmetry, the symmetry of left and right, ... conspicuous in the structure of higher animals, especially the human body.”

10. Leonardo DaVinci's “Vitruvian man” is probably the best known image representing this meaning of symmetry today.

11. Mathematical symmetry • Weyl then goes on and articulates the more precise mathematical understanding of symmetry: invariance under a transformation. • For instance, bilateral symmetry is invariance under reflection across a plane in three dimensional space (or across a line in the plane). • Other types of transformations can be considered and other symmetries studied. • Geometrical examples: translations, rotations • Each gives a corresponding form of symmetry.

12. Symmetries in Escher drawing • Imagine the Escher drawing extended to cover the whole plane (“idealized” version). Then, the whole pattern is invariant under: • translations by vectors in a 2-dimensional lattice L = {m v + n w : m, n in Z} (subdivides plane into squares in this case) • rotationsby 90 degrees around the points in L and at centers of squares. • rotationsby 180 degrees around the points such as (½) v, (½) w, and so forth.

13. A link to abstract algebra • Let G represent the collection of all distance-preserving mappings of the Euclidean plane to itself. (G is a group under composition of mappings, generated by translations by all nonzero vectors, rotations about the origin, and reflection across the x-axis.) • If X is a figure in the plane, let H(X) represent the collection of all elements T of G mapping X to itself (as a set). • Then H(X) is a subgroup of G, called the group of symmetries of X.

14. Group of symmetries • Reason this is true: • The identity element of G is the identity mapping I on the plane, and this is in H(X) since I(X) = X. • If S, T are elementsof H(X), they satisfy S(X) = X and T(X) = X. The operation in G is composition of mappings so (ST)(X) = S(T(X)) = S(X) = X. This shows ST is in H(X), so H(X) is closed under the operation in G. • If T is in H(X), and S is the inverse mapping of T (that is, ST = I), then S(X) = X also, so H(X) is also closed under inverses.

15. The Escher drawing again • In fact the symmetry group of the Escher drawing is generated by the translations by two linearly independent generators of the lattice L, and the 90 degree rotation around (0,0) in the plane. • To see this: • In coordinates, say v = (1,0), w = (0,1). • Translation by v: T(x,y) = (x + 1, y) • Rotation by 90 degrees about (0,0), counterclockwise: S(x,y) = (-y,x).

16. Escher symmetries explained • Then, for instance, TS(x,y) = (-y+1,x) gives rotation by 90 degrees about the point (½ , ½) • Similarly, STS(x,y) = (-x, -y+1) is the 180 degree rotation about (0, ½). • “Moral:” All of the symmetries we see in the drawing are consequences of the translation invariance together with the rotation invariance for the rotation about one of the four-fold centers such as the corner (0,0) of the fundamental parallelogram.

17. Classification by symmetry group • Mathematicians and crystallographers have developed a complete classification of frieze and wallpaper patterns based on the groups of symmetries. • There are exactly 7 different classes of frieze patterns • There are exactly 17 different classes of wallpaper patterns • Technical note: the equivalence relation here is a bit finer than just isomorphism of groups (also uses the geometric nature of the symmetry transformations).

18. The 17 wallpaper groups • Can you see where our Escher drawing fits?

19. How does this apply to music? • The idea is that various “dimensions” of a musical composition • the points in time when different events occur • duration and spacing of events • the pitches that sound • others, too(!) • can also be subjected to various types of transformations, so a musical composition can also exhibit the kind of invariance (or “idealized” invariance) under a transformation that we saw in geometric patterns.

20. The time dimension and rhythm • Western music is represented by a notational system that has some common features with mathematical graphs. • Reading notation left to right gives the time-sequencing of events. For instance, here is a representation of the 3-2 son clave rhythmic pattern that forms the underpinning for salsa dance music, played by the percussion instrument called the claves:

21. One form of symmetry in time • For long stretches of time in salsa, this pattern repeats, unchanged (underneath rest of the music) • So, if we transformed the music by translating forward or backward in time by the length of time in one instance of the basic pattern (hence any integer multiple of them), then that rhythmic structure would be unchanged – it is invariant under time translations (like periodicity of a frieze pattern or functions like sin). • Of course a real piece of music has a beginning and an end in time, so this is a mathematical idealization of the actual properties of the music.

22. Another form of time symmetry • Another type of transformation in time is analogous to reflection. Namely, we could imagine reflecting a piece of music about its midpoint in time. This would have the effect “running time in reverse” and transforming the sequence of events in a piece of music accordingly. • Symmetry under this sort of transformation would be similar to the property of palindromes in language: “able was I ere I saw Elba” (ignoring the capital letter anyway!)

23. Adding a second dimension • The claves and most other percussion instruments produce sounds of one basic pitch (frequency of sound waves). • Other instruments such as guitars, pianos, wind and stringed instruments can produce sounds of different pitches too. • To indicate the correct pitch to the performers, a second, vertical dimension is added in Western musical notation.

24. The pitch dimension • Here is the notation for a theme from J. S. Bach's A Musical Offering that we will hear in a bit. • Symbols at left are the clef and key signature, which set up the correspondence between lines and spaces of the staff and pitches. • The C with a slash is called the time signature. • Durations of notes indicated as before.

25. Transformations in pitch • Music can also undergo transformations in the pitch direction. • Translating a piece up or down in pitch is a very common musical operation known as transposition. (Often done to facilitate performance for instruments or singers with a different pitch range than originally intended.) • Music can also be reflected across a pitch level (called inversion).

26. An inversion example • The following short phrase from music by the 20th century Hungarian composer Bela Bartok shows two voices, each the inversion of the other across the pitch level represented by the middle line of the staff (B-flat):

27. What about rotations? • We haven't said anything about rotations in the time or pitch dimensions. Are there any such transformations in music? • There is a piece attributed to W. A. Mozart to be played by two violins reading from the same sheet of music, but on opposite sides of a table. What one player plays is a physical 180-degree rotation of what the other one plays(!) • In the pitch dimension, the names of notes repeat after the interval of an octave: A,B,C,D,E,F,G,A. • So “pitch classes” could be transformed cyclically, or “rotationally”. Not that common, though.

28. Two extended examples • We have been discussing possible transformations of music and symmetry properties from a more or less “theoretical” point of view so far. • The next goal is to explore two extended examples to see some of these ideas “in action”. • The first piece we will look at is called A Musical Offering, by Johann Sebastian Bach (1685-1750). • There's an interesting story attached to the creation of this piece.

29. A visit to Potsdam • In 1747 (just three years before he died), Bach visited the palace of Frederick the Great, the King of Prussia at the time. Frederick was a great lover of music as entertainment, was himself an enthusiastic flutist, and composed a substantial amount of music for himself and for others to play. • On Bach's arrival, Frederick apparently immediately called on him to try out several newly delivered pianos in the palace (the piano was a very recent invention at that time; the harpsichord was still the most common keyboard instrument).

30. A royal challenge and a “royal theme” • Knowing of Bach's reputation for supreme skill as a composer, the King asked Bach to improvise (that is, make up and perform on the spot) a fugue (a type of polyphonic composition that is based on combining a theme and various transformations of that theme together, overlapping in time). • The King in fact specified the theme to be used in the fugue, one he may have composed himself. This is the same music we saw earlier.

31. A Musical Offering • Bach succeeded admirably and everyone at the court was duly impressed and satisfied. • Everyone that is, apparently, besides Bach himself! • After he left the palace at Potsdam, Bach went back to his home in Leipzig and continued working with the “royal theme”, composing a cleaned-up version of the fugue he improvised, another large fugue, ten musical canons (more on these in a moment), and a trio sonata for violin, flute, and harpsichord. He had these printed in a fancy limited edition and sent off a copy to the King.

32. Is that the whole story? • Was he trying to impress the King? • Was he trying to criticize the King for preferring ``lighter’’ music over serious compositions like Bach’s own music? • Was there a religious issue? (Frederick was an outspoken agnostic, Bach a devout composer of Lutheran church music.) We may never know for sure(!)

33. The Musical Offering canons • We'll concentrate on several of the canons that make especially strong use of musical transformations and symmetry. Here is the notation of one:

34. Structure of this canon • The piece is 18 measures long (played twice on recording). • In measures 1 – 9, the top voice plays the Royal theme, while the bottom voice plays an accompanying figure, with faster motion. • In measures 10 – 18, the top voice switches to the accompanying figure, while the lower voice plays the royal theme, but both are moving backwards in time. • Also called a canon cancrizans = “crab canon” (“crab” = “cangrejo”)

35. Structure, continued • In other words, this canon is a musical “palindrome” -- as a whole piece of music, it is symmetric under reflection in time around the barline between measures 9 and 10. It would sound the same played forwards or backward in time(!) • Another point: Who's Kirnberger? We didn't mention this before, but one curious feature of Bach's Musical Offering was that the canon sections were actually left as puzzles for other musicians to decipher. Kirnberger was the solver!

36. The canon at the unison • What symmetry is involved here?

37. The modulating canon • “As the notes rise, so may the glory of the King” -- was Bach being ironic?

38. The canon by contrary motion • What symmetry transformation is involved here?

39. Some food for thought • The earliest known examples of canons in medieval music definitely have an element of religious symbolism -- the first voice sets out The Law in musical terms and the other voices follow obediently. • What would you expect music written to flatter a royal patron and extol his “greatness” to sound like? Do these pieces sound like that?

40. Symmetry in Nzakara music • The Nzakara are an ethnic group living in a region split between Central African Republic, Sudan, and Zaire. The have a long musical tradition (which is now sadly on the verge of dying out). • Work of Marc Chemillier at IRCAM in Paris has brought to light some of the remarkable structures underlying the harp and xylophone music that was performed to accompany poetry at the courts of their kings before the arrival of Europeans.

41. The Nzakara Harp • The Nzakara music we will look at was performed on a five-stringed harp (notes sounding roughly C, D, E, G, B-flat). • Here's a picture of one of the last performers with his instrument:

42. A completely oral tradition • Interestingly enough, the Nzakara transmitted their musical compositions and performance practices completely orally – they had no system of musical notation like the one we have seen in the examples we have studied so far. • So we will use a method for describing their music developed by ethnomusicologists. Much more information on this can be found in the web site and publications of Marc Chemillier. • Also see the upcoming book Mathematics and Music by Dave Benson at U. of Georgia for more details.

43. Nzakara music • The Nzakara harpists performed accompaniments to sung poetry consisting of patterns of notes repeated in the same form many times (basic translation symmetry in time). But there was a great deal of additional structure: • the five pitches of the harp were viewed in cyclic arrangement (call these 0,1,2,3,4, so 0 follows 4). • only 5 different combinations of two pitches were ever plucked simultaneously: (1 and 4, 0 and 2, 1 and 3, 2 and 4, or 0 and 3)

44. Nzakara music, continued • Depending on the poem, one of a catalog of basic different patterns was played. • Here is a representation of one of them, called an ngbakia:

45. Representing the pattern • Let's introduce the following labels for the combinations of two strings sounding simultaneously: 1 and 4 – label 0, 0 and 2 – label 1, 3 and 1 – label 2, 4 and 2 – label 3, and 3 and 0 – label 4. (These are cyclically ordered in the same way the pitches of the harp strings are ordered.) • Then the repeating ngbakia pattern corresponds to the sequence of labels: 1403423120

46. Surprising structure • There's a rather amazing amount of structure in this and in all of the Nzakara patterns that have been studied. • Divide the label string into two halves: 14034 23120 • Do you notice something about the first and second halves? (Can you transform one into the other some way?) • What does this say about the ngbakia?

47. Questions for further thought • Not all musicians use these ideas, and even those who do don't always use them. What might be a reason for incorporating these ideas in a piece? • When they do, are they “doing mathematics”? • Knowing what you know now, let's go back to the music we played at the start of the session. Can you see more of what is happening now? • Joseph Haydn (1732 – 1809), mvmt 3 from Symphony 47.

48. The Haydn Minuet