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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 ?
This brought the reciting tone to b[flat], right in the middle of the compass of reciting tones, and made the ambitus of the second mode or tone notationally equivalent to the first mode and within the ambitus of Palestrina's tenor voice in normal clefs as shown in ex.
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One way to see this is to redo the proof with a weighted tree packing, but this becomes notationally messy.
Notationally, the arguments of functions u, [[Psi].
Throughout the paper, notationally, an asterisk denotes Foreign.
Model-theoretic and proof-theoretic weakest preconditions are distinguished notationally.
We shall say that if notationally equivalent pieces of music were somehow produced in 1800 and again in 1996, by distinct composers with different intentions and different stylistic reference points, we could well have two distinct musical works.
Finally, to keep matters notationally simple, we assume that capital depreciates completely in one period.
that a maxima preceding a long should be imperfected in preference to a trailing: maxima) and what seems to be the writer's declared preference for imperfecting the succeeding maxima (brought about: notationally by the punctus added to the first maxima).
So we are sometimes more swayed by its modern appearance than by what it will sound like, more by keeping it notationally clean than by what is really required to reflect the original sound in modern notation.
Notationally a single subscript indicates a single individual household and two subscripts a two individual household.
However these computations become notationally complex and so we present [H.