Nomos Alpha
Encyclopedia
Nomos Alpha is a piece for solo cello
composed by Iannis Xenakis
in 1965
, commissioned by Radio Bremen
for cellist Siegfried Palm
, and dedicated to mathematicians Aristoxenus of Tarentum
, Évariste Galois
, and Felix Klein
. This piece is an example of a style of music called, by Xenakis, symbolic music -- a style of music which makes use of set theory
, abstract algebra
, and mathematical logic
in order to create and analyze musical composition
s. Along with symbolic music, Xenakis is known for his development of stochastic music and strategie music.
During his lifetime, Xenakis was a vocal critic of , since the development of polyphony
for its diminished set of outside-time structures, especially when compared to folk
and the Byzantine
musical traditions. This perceived incompleteness of western music was the main impetus for the development of symbolic music and for composing Nomos Alpha, his most well known example.
Nomos Alpha consists of 24 sections divided into two layers. The first layer consists of every section not divisible by four, while the second layer consists of every fourth section.
to the rotations of a cube
. The elements of the set are permutations and the binary operator is composition
.
, but is instead, a "continuous motion of registral evolution".
Cello
The cello is a bowed string instrument with four strings tuned in perfect fifths. It is a member of the violin family of musical instruments, which also includes the violin, viola, and double bass. Old forms of the instrument in the Baroque era are baryton and viol .A person who plays a cello is...
composed by Iannis Xenakis
Iannis Xenakis
Iannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...
in 1965
1965 in music
-Events:*January 4 – Fender Musical Instruments Corporation is sold to CBS for $13 million.*January 12 – Hullabaloo premieres on NBC. The first show included performances by The New Christy Minstrels, comedian Woody Allen, actress Joey Heatherton and a segment from London in which Brian Epstein...
, commissioned by Radio Bremen
Bremen
The City Municipality of Bremen is a Hanseatic city in northwestern Germany. A commercial and industrial city with a major port on the river Weser, Bremen is part of the Bremen-Oldenburg metropolitan area . Bremen is the second most populous city in North Germany and tenth in Germany.Bremen is...
for cellist Siegfried Palm
Siegfried Palm
Siegfried Palm was a German cellist who is known worldwide for his interpretations of contemporary music. Many 20th-century composers like Kagel, Ligeti, Xenakis, Penderecki and Zimmermann wrote music for him....
, and dedicated to mathematicians Aristoxenus of Tarentum
Aristoxenus
Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...
, Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
, and Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
. This piece is an example of a style of music called, by Xenakis, symbolic music -- a style of music which makes use of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, and mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
in order to create and analyze musical composition
Musical composition
Musical composition can refer to an original piece of music, the structure of a musical piece, or the process of creating a new piece of music. People who practice composition are called composers.- Musical compositions :...
s. Along with symbolic music, Xenakis is known for his development of stochastic music and strategie music.
During his lifetime, Xenakis was a vocal critic of , since the development of polyphony
Polyphony
In music, polyphony is a texture consisting of two or more independent melodic voices, as opposed to music with just one voice or music with one dominant melodic voice accompanied by chords ....
for its diminished set of outside-time structures, especially when compared to folk
Folk music
Folk music is an English term encompassing both traditional folk music and contemporary folk music. The term originated in the 19th century. Traditional folk music has been defined in several ways: as music transmitted by mouth, as music of the lower classes, and as music with unknown composers....
and the Byzantine
Byzantine music
Byzantine music is the music of the Byzantine Empire composed to Greek texts as ceremonial, festival, or church music. Greek and foreign historians agree that the ecclesiastical tones and in general the whole system of Byzantine music is closely related to the ancient Greek system...
musical traditions. This perceived incompleteness of western music was the main impetus for the development of symbolic music and for composing Nomos Alpha, his most well known example.
Nomos Alpha consists of 24 sections divided into two layers. The first layer consists of every section not divisible by four, while the second layer consists of every fourth section.
Layer 1
Layer 1 of Nomos Alpha is determined by the 24-element octahedral group isomorphicIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the rotations of a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
. The elements of the set are permutations and the binary operator is composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
.
Layer 2
Layer 2 of Nomos Alpha—in contrast to Layer 1—is not determined by group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, but is instead, a "continuous motion of registral evolution".
Sources
- DeLio, Thomas, ed. (1985). Contiguous Lines: Issues and Ideas in the Music of the '60's and '70's. ISBN 0-8191-4330-8.
- DeLio, Thomas. "The Dialectics of Structure and Materials: Iannis Xenakis' Nomos Alpha". Cited in DeLio (1985).
- Xenakis, Iannis (1971). Formalized Music, p. 219-236. Bloomington, Indiana: Indiana University Press.
External links
- Nomos Alpha performed by Moritz Müllenbach