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. There are two main families of tuning systems: equal temperament and just tuning. Main articles: Musical tuning and Musical temperament Each successive octave spans twice the frequency range of the previous octave. The third octave spans from 440В Hz to 880В Hz (span=440В Hz) and so on. The next octave will span from 220В Hz to 440В Hz (span=220В Hz). When expressed as a frequency bandwidth an octave A 2вЂ“A 3 spans from 110В Hz to 220В Hz (span=110В Hz).

all will be called doh or A or Sa, as the case may be). Therefore, any note and its octaves will generally be found similarly named in musical systems (e.g. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. The octave of any pitch refers to a frequency exactly twice that of the given pitch. A scale has an interval of repetition, normally the octave. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). 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.

( Ernst Chladni, Acoustics, 1802)ДЄ musical scale is a discrete set of pitches used in making or describing music. Musical form Д¬hladni figures produced by sound vibrations in fine powder on a square plate.

The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3). 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. Without the boundaries of rhythmic structure вЂ“ a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration вЂ“ music would not be possible. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection. 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. ДЇrom the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Their central doctrine was that 'all nature consists of harmony arising out of numbers'. 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.

7 Musicians who were or are also mathematicians.

Music math and me