Music and mathematics

Relationships between 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".

Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the Pythagoreans (in particular Philolaus and Archytas) of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, 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, 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. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration – music would not be possible. 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).

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 and ternary ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.

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 numbers. Just scales are built by multiplying frequencies by rational numbers, 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 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". 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.

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 most common number is 12, which makes up the equal temperament chromatic scale. 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 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 keyboards. 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, 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 of Arabic music. 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, 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 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." Transformational theory is a branch of music theory developed by David Lewin. 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. The chromatic scale has a free and transitive action of the cyclic group , with the action being defined via transposition of notes. So the chromatic scale can be thought of as a torsor for the group.

Some composers have incorporated the golden ratio and Fibonacci numbers into their work.

The mathematician and musicologist Guerino Mazzola has used category theory (topos theory) for a basis of music theory, which includes using topology as a basis for a theory of rhythm and motives, and differential geometry as a basis for a theory of musical phrasing, tempo, and intonation.