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Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as
tempo In musical terminology, tempo ( Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (ofte ...
,
chord progression In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural) is a succession of chords. Chord progressions are the foundation of harmony in Western musical tradition from the common practice ...
, form, and
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications 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
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. While music theory has no
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
foundation in modern mathematics, the basis of musical
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' b ...
can be described mathematically (using acoustics) and exhibits "a remarkable array of number properties".


History

Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
(in particular
Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pyt ...
and Archytas) of ancient Greece were the first researchers known to have investigated the expression of
musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...
s in terms of numerical
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s, particularly the ratios of small integers. Their central doctrine was that "all nature consists of harmony arising out of numbers". From the time of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, harmony was considered a fundamental branch of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, now known as
musical acoustics Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology (classification of the instruments), physiology, music theory, ethnomusicology, signal processing and instrument build ...
. Early
Indian Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...
and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and
rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...
s were fundamental not only to our understanding of the world but to human well-being.
Confucius Confucius ( ; zh, s=, p=Kǒng Fūzǐ, "Master Kǒng"; or commonly zh, s=, p=Kǒngzǐ, labels=no; – ) was a Chinese philosopher and politician of the Spring and Autumn period who is traditionally considered the paragon of Chinese sages. C ...
, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.


Time, rhythm, and meter

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of
pulse In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the n ...
repetition, accent,
phrase In syntax and grammar, a phrase is a group of words or singular word acting as a grammatical unit. For instance, the English expression "the very happy squirrel" is a noun phrase which contains the adjective phrase "very happy". Phrases can consi ...
and duration – music would not be possible. Modern musical use of terms like
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
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 Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...
that is fundamental to physics. The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).


Musical form

Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order. The common types of form known as
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...
and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.


Frequency and harmony

A
musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...
is a discrete set of pitches used in making or describing music. The most important scale in the Western tradition is the diatonic scale but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the octave. The octave of any pitch refers to a frequency exactly twice that of the given pitch. Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore, any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called doh or A or Sa, as the case may be). When expressed as a frequency bandwidth an octave A2–A3 spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave. Because we are often interested in the relations or
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s between the pitches (known as
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1), generally a note which functions as the tonic of the scale. For interval size comparison, cents are often used. :


Tuning systems

There are two main families of tuning systems: equal temperament and just tuning. Equal temperament scales are built by dividing an octave into intervals which are equal on a logarithmic scale, which results in perfectly evenly divided scales, but with ratios of frequencies which are
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s. Just scales are built by multiplying frequencies by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, which results in simple ratios between frequencies, but with scale divisions that are uneven. One major difference between equal temperament tunings and just tunings is differences in acoustical beat when two notes are sounded together, which affects the subjective experience of consonance and dissonance. Both of these systems, and the vast majority of music in general, have scales that repeat on the interval of every octave, which is defined as frequency ratio of 2:1. In other words, every time the frequency is doubled, the given scale repeats. Below are
Ogg Vorbis Vorbis is a free and open-source software project headed by the Xiph.Org Foundation. The project produces an audio coding format and software reference encoder/decoder (codec) for lossy audio compression. Vorbis is most commonly used in conjun ...
files demonstrating the difference between just intonation and equal temperament. You might need to play the samples several times before you can detect the difference. * Two sine waves played consecutively – 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). * Same two notes, set against an A440 pedal – this sample consists of a "
dyad Dyad or dyade may refer to: Arts and entertainment * Dyad (music), a set of two notes or pitches * ''Dyad'' (novel), by Michael Brodsky, 1989 * ''Dyad'' (video game), 2012 * ''Dyad 1909'' and ''Dyad 1929'', ballets by Wayne McGregor Other uses ...
". 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.


Just tunings

5-limit tuning Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...
, the most common form of just intonation, is a system of tuning using tones that are
regular number Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 ×&nb ...
harmonics of a single
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
. This was one of the scales Johannes Kepler presented in his
Harmonices Mundi ''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...
(1619) in connection with planetary motion. The same scale was given in transposed form by Scottish mathematician and musical theorist, Alexander Malcolm, in 1721 in his 'Treatise of Musick: Speculative, Practical and Historical', and by theorist Jose Wuerschmidt in the 20th century. A form of it is used in the music of northern India. American composer
Terry Riley Terrence Mitchell "Terry" Riley (born June 24, 1935) is an American composer and performing musician best known as a pioneer of the minimalist school of composition. Influenced by jazz and Indian classical music, his music became notable for ...
also made use of the inverted form of it in his "Harp of New Albion". Just intonation gives superior results when there is little or no
chord progression In a musical composition, a chord progression or harmonic progression (informally chord changes, used as a plural) is a succession of chords. Chord progressions are the foundation of harmony in Western musical tradition from the common practice ...
: voices and other instruments gravitate to just intonation whenever possible. However, it gives two different whole tone intervals (9:8 and 10:9) because a fixed tuned instrument, such as a piano, cannot change key. To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the tonic frequency. For instance, with a tonic of A4 (A natural above middle C), the frequency is 440  Hz, and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660 Hz. Pythagorean tuning is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is (9:8)2 = 81:64, rather than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is a secondary interval, being derived from two perfect fifths minus an octave, (3:2)2/2 = 9:8. The just major third, 5:4 and minor third, 6:5, are a
syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ...
, 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According to Carl , "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals." Western common practice music usually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities of
well temperament Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word ''wohltemperiert''. This word also appears in the title of ...
or be constructed as a
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. For instance, in 12-TET, the system of music most ...
, either some form of equal temperament or some other regular meantone, but in all cases will involve the fundamental features of
meantone temperament Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...
. For example, the root of chord ii, if tuned to a fifth above the dominant, would be a major whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a just subdominant degree of 4:3, however, the interval from the tonic would equal a minor whole tone (10:9). Meantone temperament reduces the difference between 9:8 and 10:9. Their ratio, (9:8)/(10:9) = 81:80, is treated as a unison. The interval 81:80, called the
syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ...
or comma of Didymus, is the key comma of meantone temperament.


Equal temperament tunings

In equal temperament, the octave is divided into equal parts on the logarithmic scale. While it is possible to construct equal temperament scale with any number of notes (for example, the 24-tone
Arab tone system The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament (24-TET), the distance between each successive note being a quarter tone (50 ...
), the most common number is 12, which makes up the equal-temperament
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise. For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of the
twelfth root of two The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semi ...
so that twelve of these equal half steps add up to exactly an octave. With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as
musical keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, sh ...
s. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world. Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The
19 equal temperament In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19  ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...
, first proposed and used by Guillaume Costeley in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal 12-semitone equal temperament at the cost of a flatter fifth. The overall effect is one of greater consonance. Twenty-four equal temperament, with twenty-four equally spaced tones, is widespread in the pedagogy and
notation In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
of
Arabic music Arabic music or Arab music ( ar, الموسيقى العربية, al-mūsīqā al-ʿArabīyyah) is the music of the Arab world with all its diverse music styles and genres. Arabic countries have many rich and varied styles of music and also man ...
. However, in theory and practice, the intonation of Arabic music conforms to rational ratios, as opposed to the irrational ratios of equally tempered systems. While any analog to the equally tempered quarter tone is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, or
neutral second In music theory, a neutral interval is an interval that is neither a major nor minor, but instead in between. For example, in equal temperament, a major third is 400 cents, a minor third is 300 cents, and a neutral third is 350 cents. A neutral ...
, frequently occur. These neutral seconds, however, vary slightly in their ratios dependent on maqam, as well as geography. Indeed, Arabic music historian
Habib Hassan Touma Habib Hassan Touma ( ar, حبيب حسن توما) (12 December 1934 – 10 August 1998) was a Palestinian composer and ethnomusicologist, born in Nazareth, who lived and worked for many years in Berlin, Germany. Touma authored a number of book ...
has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture."
53 equal temperament In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps (equal frequency ratios). Each step represents a frequency ratio of 2, or 22.6415& ...
arises from the near equality of 53 perfect fifths with 31 octaves, and was noted by
Jing Fang Jing Fang (, 78–37 BC), born Li Fang (), courtesy name Junming (), was born in present-day 東郡頓丘 (Puyang, Puyang, Henan) during the Han Dynasty (202 BC – 220 AD). He was a Chinese people, Chinese music theory, music theorist, ma ...
and
Nicholas Mercator Nicholas (Nikolaus) Mercator (c. 1620, Holstein – 1687, Versailles), also known by his German name Kauffmann, was a 17th-century mathematician. He was born in Eutin, Schleswig-Holstein, Germany and educated at Rostock and Leyden after which h ...
.


Connections to mathematics


Set theory

Musical set theory uses the language of mathematical
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 ...
in an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as transposition and inversion, one can discover deep structures in the music. Operations such as transposition and inversion are called
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
because they preserve the intervals between tones in a set.


Abstract algebra

Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
with 12 elements. It is possible to describe just intonation in terms of a free abelian group. Transformational theory is a branch of music theory developed by
David Lewin David Benjamin Lewin (July 2, 1933 – May 5, 2003) was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation", he did his most influential theoretical work on the development ...
. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves. Theorists have also proposed musical applications of more sophisticated algebraic concepts. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on a
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
. The
chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
has a free and transitive action of the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
\mathbb/12\mathbb, with the action being defined via transposition of notes. So the chromatic scale can be thought of as a
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non ...
for the group.


Numbers and series

Some composers have incorporated the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
and
Fibonacci numbers In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
into their work.


Category theory

The
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
and
musicologist Musicology (from Greek μουσική ''mousikē'' 'music' and -λογια ''-logia'', 'domain of study') is the scholarly analysis and research-based study of music. Musicology departments traditionally belong to the humanities, although some m ...
Guerino Mazzola Guerino Bruno Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as a writer. Education and career Mazzola obtained his PhD in mathematics at University of Zürich in 1971 under the supervision of Herbert Groß and ...
has used category theory (
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
) for a basis of music theory, which includes using
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
as a basis for a theory of
rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...
and motives, and differential geometry as a basis for a theory of
musical phrasing Musical phrasing is the method by which a musician shapes a sequence of notes in a passage of music to allow expression, much like when speaking English a phrase may be written identically but may be spoken differently, and is named for the i ...
,
tempo In musical terminology, tempo ( Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (ofte ...
, and intonation.


Musicians who were or are also mathematicians

*
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
- Accomplished pianist and violinist. *
Art Garfunkel Arthur Ira Garfunkel (born November 5, 1941) is an American singer, poet, and actor. He is best known for his partnership with Paul Simon in the folk rock duo Simon & Garfunkel. Highlights of Garfunkel's solo music career include one top-10 h ...
( Simon & Garfunkel) – Masters in Mathematics Education, Columbia University *
Brian May Brian Harold May (born 19 July 1947) is an English guitarist, singer, songwriter, and astrophysicist, who achieved worldwide fame as the lead guitarist of the rock band Queen. May was a co-founder of Queen with lead singer Freddie Mercury and ...
(
Queen Queen or QUEEN may refer to: Monarchy * Queen regnant, a female monarch of a Kingdom ** List of queens regnant * Queen consort, the wife of a reigning king * Queen dowager, the widow of a king * Queen mother, a queen dowager who is the mother ...
) - BSc (Hons) in Mathematics and Physics, PhD in Astrophysics, both from
Imperial College London Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cu ...
. *
Dan Snaith Daniel Victor Snaith (born March 29, 1978) is a Canadian composer, musician, and recording artist who has performed under the stage names Caribou, Manitoba, and Daphni. Career Snaith originally recorded under the stage name Manitoba, but afte ...
– PhD Mathematics, Imperial College London *
Delia Derbyshire Delia Ann Derbyshire (5 May 1937 – 3 July 2001) was an English musician and composer of electronic music. She carried out notable work with the BBC Radiophonic Workshop during the 1960s, including her electronic arrangement of the theme ...
- BA in mathematics and music from
Cambridge Cambridge ( ) is a College town, university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cam ...
. *
Jonny Buckland Jonathan Mark Buckland (born 11 September 1977) is an English-born Welsh musician and songwriter best known as the lead guitarist and co-founder of the rock band Coldplay. Raised in Pantymwyn, he began to play guitar from an early age, being ...
(
Coldplay Coldplay are a British rock band formed in London in 1997. They consist of vocalist and pianist Chris Martin, guitarist Jonny Buckland, bassist Guy Berryman, drummer Will Champion and creative director Phil Harvey. They met at University ...
) - Studied astronomy and mathematics at
University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
. *
Kit Armstrong Kit Armstrong ( zh, c=周善祥, p=Zhōu Shànxiáng, born March 5, 1992) is an American classical pianist, composer, and former child prodigy of British-Taiwanese parentage. Education Armstrong was born in Los Angeles into a non-musical famil ...
- Degree in music and MSc in mathematics. *
Manjul Bhargava Manjul Bhargava (born 8 August 1974) is a Canadian-American mathematician. He is the Brandon Fradd, Class of 1983, Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds A ...
- Plays the
tabla A tabla, bn, তবলা, prs, طبلا, gu, તબલા, hi, तबला, kn, ತಬಲಾ, ml, തബല, mr, तबला, ne, तबला, or, ତବଲା, ps, طبله, pa, ਤਬਲਾ, ta, தபலா, te, తబల ...
, won the Fields Medal in 2014. *
Phil Alvin Philip Joseph Alvin (born March 6, 1953) is an American singer and guitarist known primarily as the leader of the rock band The Blasters. His voice has been described as "robust...powerful...rich, resonant, ndsupremely confident." Biography Alvi ...
(
The Blasters The Blasters are an American rock band formed in 1979 in Downey, California, by brothers Phil Alvin (vocals and guitar) and Dave Alvin (guitar), with bass guitarist John Bazz and drummer Bill Bateman. Their self-described " American Music ...
) – Mathematics, University of California, Los Angeles * Philip Glass - Studied mathematics and philosophy at the
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
. *
Tom Lehrer Thomas Andrew Lehrer (; born April 9, 1928) is an American former musician, singer-songwriter, satirist, and mathematician, having lectured on mathematics and musical theater. He is best known for the pithy and humorous songs that he recorded in ...
- BA mathematics from
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
. *
William Herschel Frederick William Herschel (; german: Friedrich Wilhelm Herschel; 15 November 1738 – 25 August 1822) was a German-born British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Caroline ...
- Astronomer and played the oboe, violin, harpsichord and organ. He composed 24 symphonies and many concertos, as well as some church music. * Jerome Hines - Five articles published in ''
Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'' 1951-6. *
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
- 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


See also

*
Computational musicology Computational musicology is an interdisciplinary research area between musicology and computer science. Computational musicology includes any disciplines that use computers in order to study music. It includes sub-disciplines such as mathematical m ...
* Equal temperament * Euclidean rhythms (traditional musical rhythms that are generated by
Euclid's algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...
) * Harmony search * Interval (music) *
List of music software This is a list of software for creating, performing, learning, analyzing, researching, broadcasting and editing music. This article only includes software, not services. For streaming services such as iHeartRadio, Pandora, Prime Music, and Spotify, ...
*
Mathematics and art Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art mathematical beauty, motivated by beauty. Mathematics can be discerned in arts such as Music and mathematics, music, dance, painting, Mathema ...
* Musical tuning *
Non-Pythagorean scale Robert Peter Schneider (born March 9, 1971) is an American musician and mathematician. He is the lead singer, songwriter, guitarist and producer of rock/pop band the Apples in Stereo and has produced and performed on albums by Neutral Milk Ho ...
*
Piano key frequencies This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (refe ...
*
Rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...
*''
The Glass Bead Game ''The Glass Bead Game'' (german: link=no, Das Glasperlenspiel, ) is the last full-length novel by the German author Hermann Hesse. It was begun in 1931 in Switzerland, where it was published in 1943 after being rejected for publication in German ...
'' *
3rd bridge The 3rd bridge is an extended playing technique used on the electric guitar and other string instruments that allows a musician to produce distinctive timbres and overtones that are unavailable on a conventional string instrument with two bri ...
(harmonic resonance based on equal string divisions) *
Tonality diamond In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
*
Tonnetz In musical tuning and harmony, the (German for 'tone network') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the ''Tonnetz'' can be used to show traditi ...
* Utonality and otonality


References

* * Ivor Grattan-Guinness (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 Eberhard Knobloch (born 6 November 1943, in Görlitz) is a German historian of science and mathematics. Career From 1962 to 1967 Knobloch studied classics and mathematics at the University of Berlin and the Technical University of Berlin, a ...
and
Hans Wussing Hans-Ludwig Wußing (October 15, 1927 in Waldheim – April 26, 2011 in Leipzig) was a German historian of mathematics and science. Life Wussing graduated from high school, and from 1947 to 52 studied mathematics and physics at the Universit ...
editors,
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ...


External links


''Axiomatic Music Theory'' by S.M. Nemati''Music and Math'' by Thomas E. FioreSonantometry or music as math discipline.Music: A Mathematical Offering by Dave Benson

Nicolaus Mercator use of Ratio Theory in Music
a
Convergence''The Glass Bead Game''
Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.


"Linear Algebra and Music"Notefreqs
— 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.
Mathematics & Music
BBC Radio 4 discussion with Marcus du Sautoy, Robin Wilson & Ruth Tatlow (''In Our Time'', May 25, 2006)
Measuring note similarity with positive definite kernels
Measuring note similarity with positive definite kernels {{DEFAULTSORT:Music And Mathematics Mathematics and art Mathematics and culture