''Nomos Alpha'' () is a piece for solo

cello The violoncello ( , ), commonly abbreviated as cello ( ), is a middle pitched bowed (sometimes pizzicato, plucked and occasionally col legno, hit) string instrument of the violin family. Its four strings are usually intonation (music), tuned i ...

composed by

Iannis Xenakis Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; , ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde composer, music theorist, architect, performance director and enginee ...

1965 Events January–February * January 14 – The First Minister of Northern Ireland and the Taoiseach of the Republic of Ireland meet for the first time in 43 years. * January 20 ** Lyndon B. Johnson is Second inauguration of Lynd ...

, commissioned by Radio

Bremen Bremen (Low German also: ''Breem'' or ''Bräm''), officially the City Municipality of Bremen (, ), is the capital of the States of Germany, German state of the Bremen (state), Free Hanseatic City of Bremen (), a two-city-state consisting of the c ...

for cellist Siegfried Palm, and dedicated to mathematicians Aristoxenus of Tarentum,

Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...

, and

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

. 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 is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, and

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

in order to create and analyze

musical composition Musical composition can refer to an Originality, original piece or work of music, either Human voice, vocal or Musical instrument, instrumental, the musical form, structure of a musical piece or to the process of creating or writing a new pie ...

s. Along with symbolic music, Xenakis is known for his development of stochastic music. During his lifetime, Xenakis was a vocal critic of modern Western music since the development of

polyphony Polyphony ( ) is a type of musical texture consisting of two or more simultaneous lines of independent melody, as opposed to a musical texture with just one voice ( monophony) or a texture with one dominant melodic voice accompanied by chord ...

for its diminished set of outside-time structures, especially when compared to

folk Folk or Folks may refer to: Sociology *Nation *People * Folklore ** Folk art ** Folk dance ** Folk hero ** Folk horror ** Folk music *** Folk metal *** Folk punk *** Folk rock ** Folk religion * Folk taxonomy Arts, entertainment, and media * Fo ...

and

Byzantine The Byzantine Empire, also known as the Eastern Roman Empire, was the continuation of the Roman Empire centred on Constantinople during late antiquity and the Middle Ages. Having survived the events that caused the fall of the Western Roman E ...

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 of the genre. ''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 Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...

octahedral group A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the rotations of a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

. The elements of the set are

permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...

and the

binary operator In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation o ...

composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

Layer 2

Layer 2 of ''Nomos Alpha''—in contrast to Layer 1—is not determined by

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, but is instead, a "continuous motion of registral evolution".DeLio. op. cit. 83

References

Sources

*DeLio, Thomas, ed. (1985). ''Contiguous Lines: Issues and Ideas in the Music of the '60's and '70's''. . *DeLio, Thomas. "The Dialectics of Structure and Materials: Iannis Xenakis' ''Nomos Alpha''". Cited in DeLio (1985). *Xenakis, Iannis (1971). ''Formalized Music'', pp. 219–236. Bloomington, Indiana: Indiana University Press.

External links

Nomos Alpha
performed by Moritz Müllenbach Solo cello pieces Compositions by Iannis Xenakis 1965 compositions Compositions that use extended techniques {{classical-composition-stub