# Mathematics in music

Music is the art of arranging sounds in time to produce a composition through the elements of melody, harmony, rhythm, and timbre. It is one of the universal cultural aspects of all human societies. Music is indeed life-giving ambrosia whose lifting tones and rhythmic patterns erase all the fatigue and worries from our lives.

Have you ever wondered the reason behind the divinity of music or why some songs are more popular than others? The answer to all the queries regarding music lies in Mathematics. Music is a subjective experience. Some sound waves that reach the human ear are considered pleasant and, other sounds unpleasant or merely termed as noise. Music is the art of combining sounds with a view to beauty with expression and emotions. Mathematics is the basis of sound wave propagation and, a pleasant sound consists of harmony arising out of musical scales in terms of numerical ratios, particularly those of small integers. A suitable permutation and combination of some fundamental notes give rise to a piece of euphonious music that enthrals and transports one to a new enjoyable experience.

In Indian music, seven notes called 'Swaras' (S R G M P D N ) form the foundations of various Permutations and combinations. Theoretically, one can think of 7 factorial (7!), that is, 5040 possibilities, but only 72 of them have been analyzed and implemented for practical usage from a melody point of view.

On Proceeding to mathematical ratios, a musical interval is a mathematical ratio.  The simplest ratios, when measured on a monochord, produces harmonious musical intervals.

The most basic ratio or relationship is that of an octave which is 2:1. It is fundamentally the relationship of a vibration which is twice as much as another vibration. 2:1 is the fundamental principle of a harmonic relationship. If a string or a tube vibrates at a frequency f, then 2f is an octave higher. Similarly, f/2 is an octave lower. Every octave is twice the frequency of a previous octave. The octaves rise exponentially as 27.5, 55, 110, 220, 440, 880, 1760 and 3520.

For the Pythagoreans, initially calculated the ratio of an octave to be 12:6, which is the proportion of the primary octave 2:1. This ratio allowed for the calculation of the remaining intervals only using positive integers.

Music has so many more mathematical aspects in it. Learning music reinforces the parts of the brain used in math. Music has also been a source of research in numerous fields of mathematics such as Fourier transformation, set theory, number theory and abstract algebra.