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see arithmeticarithmetic,
branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with numbers for practical purposes, e.g.
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 and musical notationmusical 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.
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a system of graphic signs for the written expression of music; the writing down of music.

Notation originated in antiquity. The ancient Greeks used a system of letters that indicated the pitch but not the duration of sounds. This system was followed until the tenth century, when the Greek letters were replaced by Latin ones. In the 20th century, Latin letters are still used to designate individual sounds and tonalities. In the Middle Ages neumatic notation was widely used. Special symbols called neumes were written above the texts to remind the singer of the chant melodies.

Later, horizontal lines, which made possible the more precise indication of the pitch of sounds, came into use. In the 11th century the Italian music theoretician Guido d’Arezzo introduced a system of four lines, the prototype of the modern staff. At the beginning of the lines he placed a letter indicating the pitch of the sounds represented. (This was the forerunner of the modern clefs.) Subsequently, the number of lines was increased to five, and the neumes were replaced by notes with square heads. This system, which was widely used to notate Gregorian chant, was called Choralnotation. The next development in notation was mensural notation, which fixed both the pitch and the duration of sounds. In addition to mensural notation, a system of letters or numbers, the tablatures, was used during the 15th through 17th centuries for the written representation of instrumental music. Like the modern number system that is designed to simplify the teaching of certain folk instruments, the tablatures indicated not the sounds but the strings and frets. In the 17th and 18th centuries chords were indicated by means of numbers placed over or under the bass notes (the figured bass).

From the late tenth century staffless znamennoe, or kriukovoe, notation was used in the Orthodox Church in ancient Rus’. In the 16th and 17th centuries there was a gradual shift from kriukovoe notation to the five-line staff.

Modern notation graphically indicates the pitch of the notes and their metric and rhythmic relationship. Together with the well-developed system of markings for tempo, dynamics, and expression, it makes possible a precise but not overly restrictive representation of a musical work. Although the performer does not depart from the written notes, he lends his own interpretation to a work. Exponents of 20th-century avant-garde musical tendencies have introduced new markings for the performer; however, these are not widely used. Some 20th-century composers have almost entirely abandoned the traditional forms of notation, providing in the written music only hints for the performer. There is a special system of notation for the blind.


Niurnberg, M. Notnaia grafika. Leningrad, 1953.
Wolf, J. Handbuch der Notationskunde, vols. 1–2. Leipzig, 1913–19.



(computer science)
The use of symbols to denote quantities or operations.


How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system.
References in periodicals archive ?
Depending upon the task complexity and the intended use of the composition, a child may wish to use graphic notation even if he or she is somewhat familiar with standard musical notation.
However, in a fascinating set of working notes to the piece (Treatise Handbook, 1971), Cardew specifies, "The sound should be a picture of the score, not vice versa." This inversion of the usual relationship between the language of a score and the music it represents reaches beyond Wittgensteinian complication toward a kind of conceptual music: Notation and composition not only determine each other in Treatise, they are one and the same.
In the edition of The Standard Course that year Curwen allowed the tonic sol-fa notation to stop being merely a mnemonic aid to sight singing from the staff and to become an end in itself.
Because of the recurrence of this critical theme, I began to realize that there may have been a problem in the notation. The original notation of the Gloria, for example, employed a half-note pulse that was, because of the visual appearance in notation, misunderstood as a quarter-note pulse.
Another frustrating (and possibly misleading) element of citation in the volume involves the actual bibliographic notations. Only the most recent editions of books are listed, and in most cases original dates of publication are not noted.
A notation at the top, "Waiver Request IRC Section 6011(e)(2);"
This volume and its accompanying CD carefully document six substantial pieces from the influential 1425 qin score Shenqi mipu ('Wondrous and Secret Notation') as interpreted by one of the great masters of recent years, Yao Bingyan (192083), Bell Yung's teacher, to whose memory the publication is eloquently dedicated.
SOMETIME around 1674 Louis XIV ordered Pierre Beauchamp, dance director at the Academie Royale de Musique (the Opera) and Louis's dancing master, to find a way to put dance on paper--that is, to develop a system of dance notation.(1) We do not know whether Louis gave a reason for this order, beyond the formulaic `Car tel est notre plaisir'.
We might put this structure into a simplified notation such as the following: "I {remember [perceiving] (x)}." The symbolism, like that of symbolic logic, would have the advantage of condensing a long verbal analysis into a simple notation and of reminding us of certain important structural elements, such as the fact that although there are two acts, one (the prior perception) nested inside the other (the present remembering), the whole memorial experience has only one target, one (x), the taxi being perceptually entered by me.
The programmer uses a programming notation with specific tasks or goals in mind, tasks that may well be complex and heterogeneous.