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Background: I am guitar player who do not now anything about tuning other instruments and classical music traditions

I really love sound of warming up (and final tuning) of an orchestra. They are always playing this or something very similar. What sounds orchestra is playing?

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When the concert master (first chair violinist) comes on stage, it is their job to tune the orchestra. The concert master plays a concert A for the orchestra. The sounds you then hear are those of each instrument tuning to the A given by the concert master.

Since each string instrument (violin, viola, cello, and double bass) has four strings, they must tune each of these strings.

The winds and brass as well tune to the concert A. Each of these instruments is unique, so depending on its particularities, these instruments may or may not tune to more than just the concert A. As a clarinetist, for instance, I like to tune not only my concert A, but also the the octave A's above and below, as well as the fifth below (concert E-natural). This is because the clarinet's tuning acts differently in different registers of the instrument.

There can be variations in this tuning tradition when there is a soloist playing with the orchestra. For example, if there is a piano concerto being performed, usually the concert master will play an A on the piano instead of on his/her violin.

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Good explanation, but symphony orchestras traditionally have the 1st chair oboe (or occasionally clarinet) give the tuning note. He or she is signaled to do so by the concertmaster after they take their bow. For a string orchestra without winds, you are correct--the concertmaster would provide the tone. –  NReilingh Dec 14 '11 at 2:23
    
NRellingh has it right - if there's an oboe in the orchestra, it will generally give the tuning note. Typically the winds will tune first, then after a brief pause, another A will be played and the strings will tune. I've seen junior orchestras even tune 3 times, having brass and woodwinds tune separately. The oboe is used to tune because the instrument itself is theoretically impervious to going out of tune. There's no tuning slide - tuning is controlled solely with the reed - thus, the principal oboist plays the note. –  wadesworld Dec 14 '11 at 8:02
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The reason for the unique sound quality of this is the layering of perfect intervals, essentially fifths. The string instruments, save for the double basses, have their strings tuned in fifths, that's what you mainly hear. Fifths are the canonical choice for comparison of different notes, since they are the secondmost pure interval after octaves, so you can hear very clearly when you're out of tune.

Maybe it's also in part because of this that pure perfect fifths are seldom used in classical music, they usually only come together with the less open-sounding thirds. They are, in fact, far more common in modern guitar play – not just in powerchords but also in modal chords. Stacking two fifths results in a sus2 chord, adding yet another fifth you also get a 6. No unusual in Jazzy guitar parts, but you seldom hear such chords arranged for string sections, so what you associate with them is just the tuning sound.

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