Music and mathematics
Semitone Ratio Interval Natural Half Step
0 1:1 unison 480 0
1 16:15 minor semitone 512 16:15
2 9:8 major second 540 135:128
3 6:5 minor third 576 16:15
4 5:4 major third 600 25:24
5 4:3 perfect fourth 640 16:15
6 45:32 diatonic tritone 675 135:128
7 3:2 perfect fifth 720 16:15
8 8:5 minor sixth 768 16:15
9 5:3 major sixth 800 25:24
10 9:5 minor seventh 864 27:25
11 15:8 major seventh 900 25:24
12 2:1 octave 960 16:15
Tuning systems -- Just tunings
The first 16 harmonics , their names and frequencies , showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics . The first 16 harmonics , with frequencies and log frequencies . 5-limit tuning , the most common form of just intonation , is a system of tuning using tones that are regular number harmonics of a single fundamental frequency . This was one of the scales Johannes Kepler presented in his Harmonices Mundi ( 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 ' , [ 13 ] and by theorist Jose Wuerschmidt in the 20th century . A form of it is used in the music of northern India .
Music theory has no axiomatic foundation in modern mathematics, yet the basis of musical sound can be described mathematically (in acoustics) and exhibits a remarkable array of number properties. Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry. 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. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.