''Nomos Alpha'' ( el, Νόμος α΄) is a piece for solo

cello The cello ( ; plural ''celli'' or ''cellos'') or violoncello ( ; ) is a bowed (sometimes plucked and occasionally hit) string instrument of the violin family. Its four strings are usually tuned in perfect fifths: from low to high, C2, G2, ...

composed by

Iannis Xenakis Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde ...

1965 Events January–February * January 14 – The Prime 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 sworn in for a full te ...

, commissioned by Radio

Bremen Bremen (Low German also: ''Breem'' or ''Bräm''), officially the City Municipality of Bremen (german: Stadtgemeinde Bremen, ), is the capital of the German state Free Hanseatic City of Bremen (''Freie Hansestadt Bremen''), a two-city-state con ...

for cellist

Siegfried Palm Siegfried Palm (25 April 1927 – 6 June 2005) 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 f ...

, 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 radicals ...

, 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 Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concer ...

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, and

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...

in order to create and analyze musical compositions. Along with symbolic music, Xenakis is known for his development of

stochastic music Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...

. 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 chords, ...

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 music *** Folk metal *** Folk punk *** Folk rock ** Folk religion * Folk taxonomy Arts, entertainment, and media * Folk Plus or ...

and the

Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantino ...

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

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 polyhed ...

isomorphic In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...

to the rotations of a

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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the onl ...

. The elements of the set are

permutations In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

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, an internal binary o ...

is composition.

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 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 ...

, 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