Two C Major Preludes

In an earlier post, I briefly described a computational method that can be used to classify and visualize perceived tonality in a piece of music.  I'm interested in using techniques like this to extract mathematical information from musical compositions.

Here I will look at two piano works:  The C major prelude from Book 1 of  J. S. Bach's Well Tempered Clavier, and the C major prelude from Shostakovich's  24 Preludes and Fugues, opus 87.  The story is often told that Shostakovich composed this work after being inspired by Bach's W.T.C. during a  1950 commemoration of  the 200th anniversary of Bach's death .

Separated by two hundred years, it is easy to discern that these two pieces are harmonically very different from one another.  I am curious to quantify that difference in some way.

Here is a video showing the tonality map along with a performance of the Bach prelude:

And a corresponding video for the Shostakovich prelude:




In these visualizations, what characteristics distinguish the pieces from one another?  The Shostakovich features a wider variety of tonal centers, tonal centers which are sometimes less distinct --  less pronounced dark red  regions on the map, and tonal centers that move by greater distances from measure to measure.  This last point suggests a way to characterize the harmony of the pieces:  the distribution harmonic distance.  Here is a plot that shows the harmonic distance, measured on the tonality map, for the  Bach and Shostakovich pieces.

It is striking that distances 5, 7, and 9 are especially pronounced in the Bach prelude.  These distances on the tonality map correspond, for the most part, to  occurrences of the  ii7 -- V7 -- I  sequence of chords.  This harmonic cadence establishes a tonality in the I key very strongly.   The Shostakovich harmonies avoid this progression.  This comes as no surprise to the listener, who clearly  gets a weaker sense of tonality and systematic modulation between tonal centers.

Percussion Instruments

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Percussion instruments are characterized by vibrations initiated by striking a tube, rod, membrane, bell, or similar object.  Percussion instruments are almost certainly the oldest form of musical instrument in human history.  The archeological record of percussion instruments, in particular the bianzhong bells of ancient China, give clues to the history of music theory.   From a mathematical point of view, percussion instruments are of special interest because unlike other types of instruments such as strings and winds, the resonant overtones typically do not follow the harmonic series. (See Harmonics entry in this volume.)   In the last fifty years, a question of great interest in applied mathematics has been the famous inverse problem:  Can one hear the shape of a drum? 

Rods and Bars

Some percussion instruments produce a distinct pitch by the vibration of a rod or bar.  Examples included the tuning fork (a U-shaped metal rod suspended at its center), a music box (a metal bar suspended at one end) and the melodic percussion instruments such as the xylophone and marimba (suspended at two non-vibrating points or nodes along the length of metal or wooden bars).   Like vibrating strings, the frequency of the bar’s vibration, and hence the pitch of the musical sound it produces, are determined by its physical dimensions.  In contrast to the string, in which the frequency varies inversely with the length, the vibrating bar has a frequency that varies with the square of the length.  The resonant overtone frequencies f_n of the vibrating bar are related to the fundamental frequency f₁ by the formula

f_n = a(n+1/2)² f₁,

where the constant a is determined by the shape and material of the bar.  In contrast with the harmonic overtone series of vibrating strings

f_n = n f₁,

these inharmonic overtones give percussion instruments their distinct metallic timbre.  The overtones of vibrating bars decay at different rates, with rapid dissipation of the higher overtones responsible for the sharp, metallic attack, while the lower overtones persist longer. The bars of the marimba are often thinned at the center, effectively lowering the pitch of the certain overtones, in accord with the harmonic series.

Bells

Like the vibrating bar instruments, the classic church bell possesses highly non-harmonic overtones.  These are typically tuned by thinning the walls of the bell along the circumference at certain heights.  A distinctive feature in the sound comes from the fact that apart from the fundamental pitch, the predominant overtone of the church bell sounds as the minor third above the prevailing tone.  This accounts for the somber nature of the sound.

The bianzhong bells of ancient China were constructed in a manner that produced two pitches for each bell, depending on the location at which it was struck.   In the 1970s, a set of 65 such bells were discovered during the excavation of the tomb of Marquis Yi  in Hubei Provence.  The inscriptions on the bells make it clear that octave equivalence and scale theory were known in China as early as 460 BCE.

Membranes

Drums are perhaps the most common percussion instrument.  Consisting of vibrating membranes (the drum heads) stretched over one or both ends of a circular cylinder, drums exhibit a unique mode of vibration which accounts for their characteristic sound.  Mathematical models of vibrating drumheads provide a fascinating application of partial differential equations.  The inharmonic overtone frequencies are distributed more densely than for vibrating strings or rods.  Further, each overtone is associated with a particular vibration pattern of the drum head.  These regions can be characterized by the  non-vibrating curves (nodes) that arrange themselves in concentric circles and diameters of the drum head. 

An important question in the study of partial differential equations asks “Can one hear the shape of a drum?” That is, can mathematical techniques be used to work backwards from the overtone frequencies to determine the shape of the drum head that caused the vibration?  The answer, as it turns out, is “not always”.

FURTHER READINGS:

Cipra, Barry.  ”You Can’t Always Hear the Shape of a Drum” in What’s Happening in the Mathematical Sciences, volume 1. New Haven: American Mathematical Society, 1993. 

Jing, M. “A Theoretical Study of the Vibration and Acoustics of Ancient Chinese Bells.”

Journal of the Acoustical Society of America.  (v.114/3, 2003).

Rossing, Thomas, D. The Science of Percussion Instruments. Hackensack, NJ: World Scientific, 2000

Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991

Wind Instruments

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Wind instruments convert the energy of moving air into sound energy --- vibrations that are perceptible to the human ear.  Under this definition, wind instruments include the human voice, pipe organs, woodwind instruments such as the clarinet, oboe, and flute, and brass instruments like the trumpet.  The nature of this vibration and of the associated resonator tube are responsible for the unique timbre of each type of wind instrument.

The sources of vibrations

In the human voice, the flow of air from the lungs causes the vocal cords (also called vocal folds) in the larynx to open and close in rapid vibration.  This periodic stopping of the air stream creates oscillatory pulses of air pressure, or sound.  The frequency of this vibration and thus the pitch of the resulting sound are determined by the length and tension of the cords.  The singer controls these factors using the musculature of the larynx. 

The rapid open-close vibration of the vocal cords is present in many wind instruments.  In brass instruments such as the trumpet, trombone, French horn and tuba, the lips of the performer form a small aperture that opens and closes in response to air pressure.  These are sometimes called lip-reed instruments.  In single-reed instruments like the clarinet and saxophone, a thin cane reed vibrates in oscillatory contact with a specially shaped structure (the mouthpiece) to bring about the open-close effect.  .  The oboe and bassoon utilize two cane reeds held closely together with a small space between them that opens and closes in response to flowing air, controlled by the muscles of the lips. 

A third important mechanism for converting the energy of moving air into vibration utilized in the flute and the so-called flue pipes of the pipe organ.  In these instruments vibration occurs when flowing air passes over an object with a distinct edge that splits the airstream.  The resulting turbulence gives rise to oscillatory vibration. In the modern flute, the performer’s lip muscles actively control the interaction between the airstream and the edge.  In the recorder and other whistle-type instruments, as well as flue pipes of the organ, this is controlled by the mechanical design of the instrument alone.

Tube resonators and overtones

With the exception of the human voice, all of the wind instruments described above are constructed with a tube resonator enclosing a column of air that functions in much the same way as the vibrating string.  Oscillations in air pressure inside the tube reflect from the ends, resulting in significant feedback with the primary vibrating medium. In the Harmonics entry in this volume, the relationship between the vibration frequency and length of a string fixed at both ends is explained.  In that idealized setting, changing the string length by small integer factors (1/2, 1/3, 1/4,  for example) results in frequency changes that are recognizable as musical intervals (an octave, an octave plus a fifth, two octaves, respectively).  The resonating air column in wind instruments behaves similarly to a vibrating string.

An important performance practice on most wind instruments is overblowing. Not to be confused with simply playing overly loudly, overblowing refers to the fact that changes in the airflow can cause  the resonating air column to vibrate at an overtone above its fundamental frequency.  This allows performers on modern instruments to achieve a large range of pitches (often two octaves or more) from a relatively compact resonating tube.  Instruments with cylindrical tubes open at both ends (e.g. the flute) overblow at the octave, as do conical instruments that are closed at one end such as the oboe and saxophone.  On the other hand, cylindrical tubes closed at one end (such as the clarinet) overblow at the twelfth:  an octave plus a fifth.   The relative weakness of the overtone at the octave and other even-numbered overtones account for the particular timbre of the clarinet.

Altering the tube length in performance

Just as the length of a vibrating string determines the frequency/pitch of the vibration, the length of the resonating air column accounts for the pitch of notes played by a wind instrument.  In reed instruments, the resonating tube is perforated along its length with holes.  By systematically covering some of these holes but not others, the performer effectively changes the length of the resonating column.  This in turn causes the vibrating reed assembly to assume the frequency of the air column.   Most brass instruments have secondary lengths of tubing which are brought into play by mechanical valves by which the performer alters the length, and hence the fundamental frequency of the vibrating air column.  The exception to this is the slide trombone, which features a concentric tube arrangement by which the outer tube can move to lengthen the air column resonator.

FURTHER READINGS:

Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991

Wood, Alexander. The Physics of Music, seventh edition. London: Chapman and Hall,  1975