Katana VentraIP

Music and mathematics

Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

While music theory has no axiomatic foundation in modern mathematics, the basis of musical sound can be described mathematically (using acoustics) and exhibits "a remarkable array of number properties".[1]

History[edit]

Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,[2] the Pythagoreans (in particular Philolaus and Archytas)[3] of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios,[4] particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers".[5]


From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being.[6] Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.[7]

Time, rhythm, and meter[edit]

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration – music would not be possible.[8] Modern musical use of terms like meter and measure also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and periodicity that is fundamental to physics.


The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).[9]

– this sample has half-step at 550 Hz (C in the just intonation scale), followed by a half-step at 554.37 Hz (C in the equal temperament scale).

Two sine waves played consecutively

– this sample consists of a "dyad". The lower note is a constant A (440 Hz in either scale), the upper note is a C in the equal-tempered scale for the first 1", and a C in the just intonation scale for the last 1". Phase differences make it easier to detect the transition than in the previous sample.

Same two notes, set against an A440 pedal

- Accomplished pianist and violinist.

Albert Einstein

(Simon & Garfunkel) – Masters in Mathematics Education, Columbia University

Art Garfunkel

(Queen) - BSc (Hons) in Mathematics and Physics, PhD in Astrophysics, both from Imperial College London.

Brian May

– PhD Mathematics, Imperial College London

Dan Snaith

- BA in mathematics and music from Cambridge.

Delia Derbyshire

(Coldplay) - Studied astronomy and mathematics at University College London.

Jonny Buckland

- Degree in music and MSc in mathematics.

Kit Armstrong

- Plays the tabla, won the Fields Medal in 2014.

Manjul Bhargava

(The Blasters) – Mathematics, University of California, Los Angeles

Phil Alvin

- Studied mathematics and philosophy at the University of Chicago.

Philip Glass

- BA mathematics from Harvard University.

Tom Lehrer

- Astronomer and played the oboe, violin, harpsichord and organ. He composed 24 symphonies and many concertos, as well as some church music.

William Herschel

- Five articles published in Mathematics Magazine 1951–6.

Jerome Hines

- Knuth is an organist and a composer. In 2016 he completed a musical piece for organ titled Fantasia Apocalyptica. It was premièred in Sweden on January 10, 2018

Donald Knuth

Dahlhaus, Carl. 1990. Wagners Konzeption des musikalischen Dramas. Deutscher Taschenbuch Verlag. Kassel: Bärenreiter.  9783423045384; ISBN 9783761845387.

ISBN

(1995) "Mozart 18, Beethoven 32: Hidden shadows of integers in classical music", pages 29 to 47 in History of Mathematics: States of the Art, Joseph W. Dauben, Menso Folkerts, Eberhard Knobloch and Hans Wussing editors, Academic Press ISBN 0-12-204055-4

Ivor Grattan-Guinness

Axiomatic Music Theory by S.M. Nemati

Music and Math by Thomas E. Fiore

Twelve-Tone Musical Scale.

Sonantometry or music as math discipline.

.

Music: A Mathematical Offering by Dave Benson

at Convergence

Nicolaus Mercator use of Ratio Theory in Music

Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.

The Glass Bead Game

.

Harmony and Proportion. Pythagoras, Music and Space

"Linear Algebra and Music"

— A complete table of note frequencies and ratios for midi, piano, guitar, bass, and violin. Includes fret measurements (in cm and inches) for building instruments.

Notefreqs

BBC Radio 4 discussion with Marcus du Sautoy, Robin Wilson & Ruth Tatlow (In Our Time, May 25, 2006)

Mathematics & Music

Measuring note similarity with positive definite kernels

Measuring note similarity with positive definite kernels