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In Tchaikovsky's Symphony No. 6 (Op.74), near the end of the first movement, at rehearsal "S" there are numbers written above the notes for the 1st violin part (ref. upper right in image). What do those numbers mean?

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I do know what numbers above notes mean in piano sheet music but not in an orchestral score. Also, I am aware that the "3" under the beams refers to triplets. Thanks!

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It's actually pretty much exactly what you see in piano sheet music! These are just fingerings to help the first violins execute that scalar run.

Note that, in contrast to piano fingerings, 1 indicates the index finger, 2 the middle finger, etc.

The next question may then be why the fingerings are only given for the first violins. I'm not entirely, sure but the run is the same in all voices, so I assume the fingerings would be, as well. Or perhaps the run just happens to be placed more easily in the other instruments. (But I'm no string player, beware of this explanation!)

PS: Great piece!

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    The fingering given for Vn 1 is different from the standard fingering for a chromatic scale, and seems to be designed to accentuate the 8th-note divisions of the run. As Tim says, the notes lie differently across the strings for the other instruments, and the larger size of the violas and cellos would also affect the fingering.
    – user19146
    Commented Jul 30, 2016 at 16:43
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    @alephzero: also, there's no way the violas would actually play a run like that chromatically as it's written... why would he bother giving them fingerings?
    – leftaroundabout
    Commented Jul 30, 2016 at 16:56
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    @leftaroundabout Do you mean it's impossible for the violas to play this run? I think I misunderstood you.
    – Richard
    Commented Jul 30, 2016 at 17:05
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    @leftaroundabout "there's no way the violas would actually play a run like that chromatically as it's written" - Why not? the viola part is marked "divisi". They are not being asked to play a chromatic scale in octaves.
    – user19146
    Commented Jul 30, 2016 at 17:05
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    Ah, you guys are no fun anymore.
    – leftaroundabout
    Commented Jul 30, 2016 at 17:25

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