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Fingering
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Fingering is the choice of which fingers and hand positions to use when playing certain instruments. For example, when fingering the saxophone, chords or melodies can often be played with a variety of different assignments of fingers to played keys. Fingering, in this context, is the choice of which finger to use for which key, for each note. Fingering typically changes throughout a piece; the challenge of choosing good fingering for a piece is to make the hand movements as comfortable as possible without changing hand position too often.

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Encyclopedia
Fingering is the choice of which fingers and hand positions to use when playing certain instruments. For example, when fingering the saxophone, chords or melodies can often be played with a variety of different assignments of fingers to played keys. Fingering, in this context, is the choice of which finger to use for which key, for each note. Fingering typically changes throughout a piece; the challenge of choosing good fingering for a piece is to make the hand movements as comfortable as possible without changing hand position too often. A fingering can be the result of the working process of the composer, who puts it into his manuscript, an editor, who adds it into the printed score, or the performer, who puts his own fingering in the score or in his performance.
Fingering also applies to other instruments, such as woodwind instruments and stringed instruments. Depending on the instrument, not all the fingers may be used. For example, saxophonists do not use the right thumb and string instruments (usually) only use the fingers.
Notational methods
Small numbers are often written next to selected notes on musical scores as a fingering guide, with the numbers 1 to 5 representing the fingers of the appropriate hand, with the thumb being counted as 1, counting outward toward the little finger of each hand. If a fingering number is preceded by an en dash (e.g. --1) it indicates that the finger is to be slid or slipped from the previous position; though this technique is not as clean as standard position fingering, it utilizes the possibility of faster and/or more complex passages.
String instruments
On string instruments fingers are numbered from 1 to 4 where the thumb isn't counted because it doesn't play on a string, and a 0 indicates an open string. In those cases on string instruments where the thumb is used (such as high notes on a cello), it is represented by a symbol the shape of a 0 with a vertical stem below (somewhat similar to O, for instance).
The classical guitar also has a fingering notation system for the plucking hand, known as pima (or less commonly pimac), where p=pulgar (thumb), i=indio (index finger), m=medio (middle finger), a=anular (ring finger) and, very rarely, c=chiquita (little finger). It is usually only notated in scores where a passage is particularly difficult, or requires specific fingering for the plucking hand. Otherwise, plucking-hand fingering is generally left to the discretion of the guitarist.
Keyboard instruments
On keyboard instruments all fingers are used. So there are numbers from 1 (thumb) to five.
The numbers are related to the fingers themselves, not to the hand position on the keyboard.
After Cristofori invented the pianoforte from the harpsichord in 1700 and it become popular in the decades after 1740, eventually replacing the harpsichord, the piano technique developed strongly (parallel with the piano builders´ progress that also piano pedagogy and as part of it piano fingering changed.
Keyboard instruments: Piano
There are only few publications about piano fingering.
It is mentioned by Carl Philipp Emanuel Bach (son of Johann Sebastian Bach ) in his book "Versuch über die wahre Art, das Clavier zu spielen" ("trying to write about the true way of playing the piano") he dedicated several paragraphs to this topic, see the German original: .
The british pianist Tobias Matthay wrote a small book "Principles of Fingering".
The german pianist Uli Molsen wrote a "Fingersatz-Kurs". The german pianist Wolfgang Ellenberger published an eBook about a systematical nomenclature of piano fingering including a "fingering edition" with different piano pieces.
History
Johann Sebastian Bach introduced an innovation in fingering for the organ and the clavier. (A similar, although according Bach's son Carl Philipp Emanuel Bach less radical, innovation was introduced by François Couperin, at roughly the same time in 1717, in his book L'art de toucher le clavecin.) Prior to Bach, playing rarely involved the thumb. Bach's new fingering retained many features of the conventional fingering up until that point, including the passing of one finger under or over another (Playing many of Bach's works requires such fingering, especially passing the third finger over the fourth or the fourth finger over the fifth.), but introduced the far greater use of the thumb. Modern fingering also uses the thumb to a similar extent, and involves the passing of the thumb under the other fingers, but does not, as Bach's did, generally involve the passing of any other fingers over or under one another.
In the 1980s Lindley and Boxall have shown that the above relies solely on C.P.E. Bach testimony: all the extant fingerings from J.S.Bach and his circle use the ancient methods, with very limited use of the thumb. More recently it has been shown that all his harpsichord works and most of the organ works as well are playable with the old technique.
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