Gene Ward Smith
Encyclopedia
Gene Ward Smith is an American
mathematician
and music theorist. In mathematics he has worked in the areas of Galois theory
and Moonshine theory. In music theory, he is noted for a number of innovations in the theory of musical tuning
, such as the introduction of multilinear algebra
and for being the first to write music in a number of exotic intonation systems. A boyhood friend of Steven Spielberg
, a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.
. For any integer n not divisible by eight, this constructs a polynomial which, upon specializing the values, gives all of the cyclic extensions of any given base field with characteristic
prime to n. This can then be extended to metacyclic extensions, such as dihedral group
s.
Smith was also a member of the Amdahl Six group which held the record for the largest known prime number from 1989-1992.
s as a way of classifying regular temperament
s, and of dealing with the problem of torsion
. In this system, a temperament is specified by means of a wedgie, which technically may be identified as a point on a Grassmannian
.
Smith has long been drawing attention to the relationship between equal divisions of the octave and the Riemann zeta function.
Smith was among the first to consider extending the Tonnetz
of Leonhard Euler
beyond the 5-limit
and hence into higher dimensional lattices
. In three dimensions, the hexagonal lattice
of 5-limit harmony extends to a lattice of type A3 ~ D3.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and music theorist. In mathematics he has worked in the areas of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
and Moonshine theory. In music theory, he is noted for a number of innovations in the theory of musical tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
, such as the introduction of multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
and for being the first to write music in a number of exotic intonation systems. A boyhood friend of Steven Spielberg
Steven Spielberg
Steven Allan Spielberg KBE is an American film director, screenwriter, producer, video game designer, and studio entrepreneur. In a career of more than four decades, Spielberg's films have covered many themes and genres. Spielberg's early science-fiction and adventure films were seen as an...
, a few of his biographical details appear incidentally in the biography of Spielberg by Joseph McBride.
Mathematics
In mathematics, Smith's most notable achievement is the construction of what has been called the Smith generic cyclic polynomialGeneric polynomial
In Galois theory, a branch of modern algebra, a generic polynomial for a finite group G and field F is a monic polynomial P with coefficients in the field L = F of F with n indeterminates adjoined, such that the splitting field M of P has Galois group G over L, and such that every extension K/F...
. For any integer n not divisible by eight, this constructs a polynomial which, upon specializing the values, gives all of the cyclic extensions of any given base field with characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
prime to n. This can then be extended to metacyclic extensions, such as dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
s.
Smith was also a member of the Amdahl Six group which held the record for the largest known prime number from 1989-1992.
Music theory
Smith introduced wedge productExterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
s as a way of classifying regular temperament
Regular temperament
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios...
s, and of dealing with the problem of torsion
Torsion (abstract algebra)
In abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free.- Definition :...
. In this system, a temperament is specified by means of a wedgie, which technically may be identified as a point on a Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
.
Smith has long been drawing attention to the relationship between equal divisions of the octave and the Riemann zeta function.
Smith was among the first to consider extending the Tonnetz
Tonnetz
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....
of Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
beyond the 5-limit
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
and hence into higher dimensional lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
. In three dimensions, the hexagonal lattice
Hexagonal lattice
The hexagonal lattice or equilateral triangular lattice is one of the five 2D lattice types.Three nearby points form an equilateral triangle. In images four orientations of such a triangle are by far the most common...
of 5-limit harmony extends to a lattice of type A3 ~ D3.
See also
- Robert Holford Macdowell Bosanquet
- Shohe TanakaShohé Tanakawas a Japanese physicist, music theorist, and inventor. He graduated from Tokyo University in 1882 as a science student. On an imperial scholarship, he was sent to Germany for doctoral studies in 1884, together with Mori Ōgai...
- Adriaan FokkerAdriaan FokkerAdriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony Fokker...
- Erv WilsonErv WilsonErvin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...
- Paul ErlichPaul ErlichPaul Erlich is a guitarist and music theorist living near Boston, Massachusetts. He is known for his seminal role in developing the theory of regular temperaments, including being the first to discover pajara temperament and its decatonic scales in 22-ET. He holds a Bachelor of Science degree in...
External links
- Mirror of Smith's personal website
- Music by Gene Ward Smith in the Internet ArchiveInternet ArchiveThe Internet Archive is a non-profit digital library with the stated mission of "universal access to all knowledge". It offers permanent storage and access to collections of digitized materials, including websites, music, moving images, and nearly 3 million public domain books. The Internet Archive...