scientific notation, means of expressing very large or very small numbers in a compact form that is easy to use in computations. In this notation, any number is expressed as a number between 1 and 10 multiplied by a power of 10 that indicates the correct position of the decimal point in the original number; numbers greater than 10 are expressed by positive powers of 10 and numbers less than 1 are expressed by negative powers of 10 (see exponent). For example, 43,700 is written in scientific notation as 4.37 × 10⁴ and 0.00526 as 5.26 × 10^-3. The larger the converted number, the more compactness is achieved: for example, the speed of light, about 30,000,000,000 cm per sec, becomes 3 × 10¹⁰ cm per sec. Calculations are greatly simplified by use of scientific notation: the first parts of a pair of numbers to be multiplied or divided are combined manually or by slide rule and the powers of 10 are added or subtracted in accordance with the rules for exponents. If the first part of the result is greater than 10, an adjustment is made. For example, in order to multiply 832,000 by 0.00035, one converts first to scientific notation as follows: (832,000)×(0.00035)=(8.32×10⁵)×(3.5×10^-4)=8.32×3.5×10⁵×10^-4=29.12×10¹=2.912×10² (in scientific notation) or 291.2 (in ordinary notation).

notation: see arithmetic and musical notation.

musical notation, symbols used to make a written record of musical sounds.

Two different systems of letters were used to write down the instrumental and the vocal music of ancient Greece. In his five textbooks on music theory Boethius (c.A.D. 470-A.D. 525) applied the first 15 letters of the alphabet to the notes in use at the end of the Roman period. Notation of Gregorian chant was by means of neumes, which are thought to have been derived from symbols used in the Greek language to indicate pitch inflection. Neumes were certainly in use by the 6th cent., although the earliest extant manuscripts containing them are fragmentary ones from the 8th cent. These neumes indicated only the grouping of sounds in a given melody, evidently to recall to a singer the approximate shape of a melody already learned by ear.

Heighted neumes, arranged above and below a line, made the intervals of a melody more discernible in 10th-century notation, and by the end of the 12th cent. the staff perfected by Guido d'Arezzo was in use. Guido placed letters on certain lines to indicate their pitch, and thereby the pitch of the remaining lines and spaces. The letters evolved into the clef signs used today. Guido also invented a system of naming scale degrees using the initial syllables of the lines of a Latin hymn (ut, re, mi, fa, sol, la). Originally used for teaching sight singing, these or their derivatives are also used in some languages for naming absolute pitches.

A staff of five lines for vocal music was adopted in France and one of six lines in Italy. Instrumental music employed staves of varying numbers of lines until the 16th cent., when the five-line staff became the standard. Signs for chromatic alteration of tones appear almost from the beginning and had assumed their present shapes by the end of the 17th cent. The essential problems in pitch notation, the use of both lines and spaces to indicate successive scale degrees and the use of extra symbols to indicate raising or lowering a tone by a half step, were solved comparatively rapidly.

However, the evolution of the rhythmic notation used today took much longer than that for pitch. Mensural notation, in which each note has a specific time value, became a necessity with the development of polyphony. At first, certain patternings of neumes were used to represent the various rhythmic modes; later, in his Ars cantus mensurabilis (c.1280), Franco of Cologne created a clear indication for each note of its exact rhythmic length and selected certain neumes to represent tones of long and short duration. In his system, the long value was in principle equal to three of the short values.

In the 14th cent. Philippe de Vitry, author of Ars nova, which expands the system of Franco, codified the ready availability of duple divisions of the long and short notes. At the various rhythmic levels of a given piece either a 2:1 or a 3:1 relationship was implied, and a system of signs and colored notes developed for indicating which relationships were in force or were being temporarily altered.

In the 15th cent. numbers with the appearance of fractions indicated that one proportionality of rhythmic values was temporarily being substituted for another. Modern signatures evolved from these numbers. Bar lines, expression signs, and Italian terms to indicate tempo and dynamics came into use in the 17th cent. With the adoption of equal temperament and the major and minor modes, signatures indicating a major key or its relative minor became conventional. They assumed their present form during the baroque period.

The advent of aleatory music has produced notation systems, varying from piece to piece, indicating only approximate pitch, duration, and dynamic relations. Notation for electronic music is still not standardized but generally uses traditional reference symbols (staff and clef signs) in conjunction with specially adapted pitch and rhythm notation.

For a system of notation of lute and keyboard music, see tablature. See also score.

Written, printed, or other visual representation of music. There are two basic approaches to notating music. Tablature (such as guitar chord diagrams) depicts the actions a performer is to take (in particular, showing where to put the fingers to produce a given sound). Symbolic notation describes the sounds themselves and includes methods that vary from assigning pitches different letters of the alphabet to representing a given combination of notes by a graphic sign. The Western notation system combines rhythmic notation (the appearance of a note indicates its duration) with pitch notation (the line or space on a staff where a note is placed indicates its pitch). Thus, a single symbol shows both pitch and duration, and a string of these symbols notates both melody and rhythm.

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Written recording of dance movements. The earliest notation, in the late 15th century, consisted of letter-symbols. Several attempts were made in later centuries to describe dance steps, but no unified system combined both rhythm and steps until the 1920s, when Rudolf Laban devised his system of Labanotation. In the 1950s, the competing system of Benesh notation, or “choreology,” devised by Rudolf and Joan Benesh, came into use.

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