Chiasmus Resources

Für Elise

phraseology and form of first eight bars: the music divides symmetrically into two equal halves.  The two halves, or phrases, taken together make a complete musical sentence, known as a period; the first half of this musical period is essentially ascending while the second half is descending; when two musical phrases are arranged in this way, so that the second phrase fulfills or completes the first, they are said to be related as antecedent to consequent. – Greg Pepetone, “Musical Classicism and the Newtonian World View,” IDST 2310: Fine and Applied Arts in Civilization, Georgia College & State University, online at www.faculty.de.gcsu.edu/~dvess/ids/fap/classical.htm ; he emphasized symmetry and form – Michelle Norris, “The Mathematics of Mozart’s Music,” Jan 27, 2006, interview on All Things Considered, NPR-Radio, online at www.npr.org .

Musikalisches Opfer

with 2 fugues and 10 isomorphic canons, including a “crab canon” (retrograde copy): each is a different variant on the King’s Theme, fully using the devices of inversion and augmentation – Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (N.Y.: Basic Books, 1979/1999), 8-10.

Goldberg Variations

du Sautoy Symmetry 247-248; ABA or inclusion; begins and ends the 32 movements with same aria; Bach termed the 16^th variation (of 30) as his “overture.”

fourth canon

du Sautoy Symmetry 251; second voice inverted.

fifth canon

du Sautoy Symmetry 251.

Sonate No. 16 in C, K. 545, Movt. I

opening four bars: the two halves of the musical period are separated by silence and punctuated by a temporary moment of harmonic repose known as a cadence.  While there is no discernible pattern of ascent and descent, symmetry is supplied by rhythmic repetition: Both halves begin with the same (long-short-short) motif.  So an example of direct parallelism.  – Greg Pepetone, “Musical Classicism and the Newtonian World View,” IDST 2310: Fine and Applied Arts in Civilization, Georgia College & State University, online at www.faculty.de.gcsu.edu/ dvess/ids/fap/classical.htm .  Cf. astrophysicist Mario Livio on symmetry – as used, for example, in the musical compositions of Mozart – Michelle Norris, “The Mathematics of Mozart’s Music,” Jan 27, 2006, interview on All Things Considered, NPR-Radio, online at www.npr.org .

emphasized symmetry and form – Michelle Norris, “The Mathematics of Mozart’s Music,” Jan 27, 2006, interview on All Things Considered, NPR-Radio, online at www.npr.org .

Ludus Tonalis

du Sautoy Symmetry 251; ABA or inclusion through inversion of first and last movements.

Good King Wenceslas

Hofstadter notes that, “when the original and its inversion are sung together, starting an octave apart and staggered with a time-delay of two beats, a pleasing canon results.” – Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid (N.Y.: Basic Books, 1979/1999), 9.

A simple musical analysis shows the symmetry at the fundamental level, even drawing Euclidean and Newtonian analogies: When musical pitches are arranged in a circle, they show the preservation of rotational symmetry in the diatonic scale, and “all well-formed scales show structural features.”²  Far from being accidental or happenstantial, Greg Pepetone has described several instances of deliberate chiastic and direct musical symmetry (parallelism).  Classical composers sought to utilize the full-spectrum of available characteristics: “dynamic equilibrium, linearity, symmetry, restraint, clarity, elegance, precision of detail,” etc., which just happen to be “defining characteristics of the Newtonian paradigm,”³ i.e., part of the natural order of things.  Alan Rich points to a parade example of complexity at the end of the second act of Mozart’s Figaro:

More people join in, more complex the music grows, with every line a separate, beautifully preserved personage.  And while all this is happening, Mozart is also working within the classic framework that involves our listening process with the logic of key change, key return – the design that makes it all work.⁴

Aside from the obvious complexity, Mozart also showed his complete mastery of contrapuntal devices in his Requiem and Jupiter symphonies, the last being “the real synthesis of his command over the complex musical textures that he gleaned from his contrapuntal explorations.”⁵  In light of these observations, it is more than a little relevant to quote David Noel Freedman:

Such balance or symmetry is a principal characteristic of early Hebrew poetic structure, deriving apparently from its musical framework (i.e., rhythmic dancing and singing, along with simple or complex choral antiphony).⁶

As we have seen in the Qur’an, sound-vision devices can also be used in tandem with chiasmus.

Musical canons, according to Douglas Hofstadter, can be constructed in very complex fashion, beginning with inversion of “the theme, which means to make a melody which jumps down wherever the original theme jumps up, and by exactly the same number of semitones.  This is a rather weird melodic transformation, but when one has heard many themes inverted, it begins to seem quite natural.  Bach was especially fond of inversions, and they show up often in his work–and the Musical Offering is no exception.”⁷  David & Mendel had this to say about Bach’s compositions:

His form in general was based on relations between separate sections.  These relations ranged from complete identity of passages on the one hand to the return of a single principle of elaboration or a mere thematic allusion on the other.  The resulting patterns were often symmetrical, but by no means necessarily so.  Sometimes the relations between the various sections make up a maze of interwoven threads that only detailed analysis can unravel.  Usually, however, a few dominant features afford proper orientation at first sight or hearing, and while in the course of study one may discover unending subtleties, one is never at a loss to grasp the unity that holds together every single creation by Bach.⁸

Essay on music by Robert F. Smith