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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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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.



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.


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.


(computer science)
The use of symbols to denote quantities or operations.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.


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.
Copyright © 1981-2019 by The Computer Language Company Inc. All Rights reserved. THIS DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher.
References in periodicals archive ?
They tend to be cleverly efficient (minimizing the amount of space and writing used) but also opaque (the steps are not notationally expressive of their mathematical meaning).
This-will be notationally reflected by the bidirectional arrow, [??].
Notationally, we use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to denote equivalent, positive, and negative transformations respectively, where A is an unsupported construct, and B the rewritten native construct.
Whatever seeming success we imagine we have in capturing, notationally, the style of a Twyla Tharp or a Louis Armstrong is rather like the sense in which an improvisional act in ordinary human circumstances can be computationally simulated after the fact but without assurance that that `notation' can also be counted on to anticipate the stylistically `recurrent' (but perceptually distinctive) features of an artist's `signature' in further work.
Since classifications are notationally controlled vocabularies, these inevitably have limits.
Notationally, Fisher expresses the equation of exchange as:
First, the sections themselves are kept notationally discrete from one another: the ritornello, for example, is notated in the finished score so that it leads on directly from the end of the three-part verse - the vocal parts end with a semibreve so that the strings can complete the bar with an anacrusis of one minim - but in C616 this anacrusis is not subtracted from the end of the previous bar; similarly, the ending of the [cents] section of the bass verse is a full bar in C616, meaning that the anacrusis of 'The French unequal in the fight' is not taken into account.
Because pointers, heapallocated storage, and destructive updating are all mechanisms that introduce aliasing, the formal treatment of shape analysis is notationally somewhat formidable.
These ex-post tenure sequences are the following: individuals with one year or less of tenure who are in a different job one year later (notationally given as {1,1 }), individuals with one to two years of tenure who are in a different job one year later ({1,2,1}), individuals with two to three years of tenure who are in a different job one year later ({1,2,3,1}), and individuals who stay in their job for more than three years ({1,2,3,4+}).(16)
Our specification is notationally simpler and basically equivalent.
In so doing, we shall not distinguish notationally between the model elements and their counterparts in the homology Hopf ring.
3 Baumol and Oates offer a notationally compact model of PO in a many-consumer and many-producer economy (Baumol and Oates 1988, chap.