ThatsMaths

Motif: A short musical unit, usually just few notes, used again and again.  

A recurrent short phrase that is developed in the course of a composition.

A motif in music is a small group of notes encapsulating an idea or theme. It often contains the essence of the composition. For example, the opening four notes of Beethoven’s Fifth Symphony express a musical idea that is repeated throughout the symphony. 

The motif, comprising just four notes, is instantly recognizable, and evokes all the developments and dynamics that occur in the piece.

Beethoven’s motif is simplicity itself: just three short notes followed by one long one; but he uses this motif in a masterly manner. It recurs throughout the work in numerous variations and is an icon of the symphony.

Transformations and Manipulations

A motif is the minimum unit that conveys the character of a musical piece, the smallest independent particle with a definite identity. Thus, we may call a motif a molecule of music. It is most often thought of in melodic terms, but may be a harmonic chord sequence without an explicit melody, or even simply a rhythmic pattern.

The composer can develop a piece of music by using variations based on the motif. Essentially, the motif is subjected to a series of mathematical transformations. These transformations are organized into (algebraic|) groups. The motif may be transposed (raised or lowered in pitch), inverted, reversed, etc. These transformations are especially important in twelve-tone music.

The Klein 4-Group

Take a book, place it on the table and draw a rectangle around it. How many ways can the book fit into the rectangle? There are four ways that this can be done. They are shown — for a randomly chosen book — in the figure below. The four orientations can be described in terms of simple rotations, starting from the upright configuration on the left:

1. 1. Place the book upright with front cover up (that is, do not move it).

   2. Rotate through 180º about X-axis (horizontal line through centre).

   3. Rotate through 180º about Y-axis (vertical line through centre).

   4. Rotate through 180º about Z-axis (line through centre perpendicular to book).

We use the four symbols {I, X, Y, Z} for these geometric operations. They combine according to the multiplication table shown here. They form a group called the Klein 4-Group, and the figure below shows how the transformations are applied to a tone row.

Rhythmic Motifs

A motif usually carries the seed of a melody, or a crucial chord sequence. However, it may simply be a rhythmic structure, like the motif of the One Note Samba, shown here.

The B-A-C-H Motif.

The B-A-C-H motif is the four-note motif, B-flat, A, C, B natural. In German musical notation, in which the note B natural is named H and B-flat is named B, it spells out Bach’s name. It is an example of a musical cryptogram. The B-A-C-H motif has been borrowed and re-used by many composers. Bach himself used the motif in a number of works; indeed it appears so frequently in the bass lines of his fugues that its use had to be intentional.

A motif reminiscent of Bach’s motif is used in the song It’s Impossible, popularised by Perry Como.

Rachmaninov Modifies a Molecule to Make Magical Music.

Niccolò Paganini’s Violin Caprice 24, composed in 1817, was the basis for Rachmaninov’s Rhapsody on a Theme of Paganini. The motif in the opening bars is shown below (left). Rachmaninov performed his magic, converting this molecule into a new and wonderful form by a few simple transformations. For his 18th variation, he inverted it (a flip about the horizontal) and changed from minor to major. He also modulated to another key, D-flat major (not shown). This composition is described by Stephen Hough in a video (link below).

Sources

Proms, 2013: Stephen Hough on Rachmaninov’s Rhapsody on a Theme of Paganini. Video