''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, G ...

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 c ...

in 1965, commissioned by Radio Bremen 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 for ...

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

. 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 conce ...

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

, 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 ...

in order to create and analyze musical compositions. 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 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 Fo ...

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 Constantinopl ...

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 polyhedr ...

isomorphic to the rotations of a cube. 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 pr ...

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 ...

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 v ...

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