''Pithoprakta'' (1955–56) is a piece by Iannis Xenakis for

string orchestra A string orchestra is an orchestra consisting solely of a string section made up of the bowed strings used in Western Classical music. The instruments of such an orchestra are most often the following: the violin, which is divided into first a ...

(with 46 separate solo parts), two

trombone The trombone (, Italian, French: ''trombone'') is a musical instrument in the Brass instrument, brass family. As with all brass instruments, sound is produced when the player's lips vibrate inside a mouthpiece, causing the Standing wave, air c ...

s, xylophone, and wood block, premièred by conductor Hermann Scherchen in Munich in March 1957. A typical performance of the piece lasts about 10 minutes. The word ''Pithoprakta'' translates to "actions through probability". This relates to

Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibniz ...

law of large numbers In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...

which states that as the number of occurrences of a chance event increases, the average outcome approaches a determinate end. The piece is based on the

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

of gases, Gauss's law,Xenakis, Iannis (1992). ''Formalized Music: Thought and Mathematics in Composition'', p.15. . or

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

.Emmerson, Simon (2007). ''Living Electronic Music'', p.48. . Each instrument is conceived as a molecule obeying the Maxwell–Boltzmann distribution law,Randel, Don Michael (1996). ''The Harvard Biographical Dictionary of Music'', p.999. . with

Gaussian distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...

of temperature fluctuation. This theory states that "the temperature of a gas derives from the independent movement of its molecules." Xenakis drew an analogy between the movement of a gas molecule through space and that of a string instrument through its pitch range. To construct the seething movement of the piece, he governed the 'molecules' according to a coherent sequence of imaginary temperatures and pressures. Brownian motion is a four-dimensional phenomenon (three spatial dimensions and time), and Xenakis created the score by first creating a two-dimensional graph, necessitating some simplifications. The abscissa () represents time at 5 cm = 26 MM, while the

ordinate In mathematics, the abscissa (; plural ''abscissae'' or ''abscissas'') and the ordinate are respectively the first and second coordinate of a point in a Cartesian coordinate system: : abscissa \equiv x-axis (horizontal) coordinate : ordinate \e ...

s () represent pitch at 1

semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...

= .25 cm, when transcribed into musical notation 5 cm = 1 measure. This length is subdivided into three, four, and five equal parts, allowing for fine differences in duration but also creating a steady pulse in each instrumental part, reducing the rhythm of the random walk to steady footsteps. Composer Simon Emmerson notes that this regularity is "impossible in reality and fails to use his ‘

stochastic Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; i ...

’ approach in this instance." Each part gets nowhere, but as a whole the mass's pitch is freely modulated and its speed-determined "temperature" varies. This piece was made into a ballet by

George Balanchine George Balanchine (; Various sources: * * * * born Georgiy Melitonovich Balanchivadze;, Romanization of Georgian, : April 30, 1983) was a Georgian-American ballet choreographer, recognized as one of the most influential choreographers ...

, '' Metastaseis and Pithoprakta'', along with the earlier '' Metastaseis''.

References

Sources

Compositions by Iannis Xenakis 1956 compositions Modernist compositions {{composition-stub - Treble