Music and mathematics

It uses mathematics to study elements of music such as tempo, chord progression, form, and meter.

[1] 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.

[5] From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics.

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

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

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.

[10] 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.

Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.).

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).

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.

Below are Ogg Vorbis files demonstrating the difference between just intonation and equal temperament.

This was one of the scales Johannes Kepler presented in his Harmonices Mundi (1619) in connection with planetary motion.

American composer Terry Riley also made use of the inverted form of it in his "Harp of New Albion".

Western common practice music usually cannot be played in just intonation but requires a systematically tempered scale.

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.

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.

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.

Operations such as transposition and inversion are called isometries because they preserve the intervals between tones in a set.

For example, the pitch classes in an equally tempered octave form an abelian group with 12 elements.