For reference, here's the comparison of the crossed and non-crossed version of the theme:
X:1
L:1/8
M:3/4
K:bm
%%score [{T1 T2} A B]
V:T1 clef=treble
V:T2 clef=treble
V:A clef=alto
V:B clef=tenor
% 1
[V:T1] !tenuto!B2 (e^G) (c3/2^E/2) | c4 r2 || !tenuto!f2 !tenuto!e!tenuto!d (!tenuto!c3/2!tenuto!B/2) | c4 r2
[V:T2] !tenuto!f2 (^Ad) (^E3/2B/2) | E4 r2 || !tenuto!B2 !tenuto!^A!tenuto!^G (!tenuto!^E3/2^!tenuto!E/2) | E4 r2
[V:A] !tenuto!D2 (FB,) (B,3/2G,/2) | ^A,4 r2 || !tenuto!^G2 !tenuto!F!tenuto!^E (!tenuto!B,3/2!tenuto!B,/2) | ^A,4 r2
[V:B] !tenuto!^G2 (C^E) (^G,3/2B,/2) | F,4 r2 || !tenuto!D2 !tenuto!C!tenuto!B, (!tenuto!^G,3/2!tenuto!=G,/2) | F,4
There are multiple plausible reasons why Tchaikovsky chose to use the crossed version for the beginning.
- As said by PiedPiper, Tchaikovsky would have assumed opposite-violins placement, which does give such interchanges a notable stereo effect. Even with an artificially hard-panned digital rendition over headphones it seems surprisingly hard to hear what's actually going on, however the non-crossing version does in comparison come over kind of flat and uninspired.
- One reason for this impression is likely that the voices are very parallel without this trick. Now, in a way that's definitely “mission accomplished” – it very much is an emotional descent. And Tchaikovsky was no stranger to prominently parallel lines, the most obvious example being the Serenade for Strings – but here it's quite a bit more extreme, in particular with three consecutive sevenths between viola and 1st violins together with fourths between cellos and violas. The crossing version doesn't remove that effect, but it brings it over more sophisticatedly than if every single instrument is literally moving downwards almost all the way.
- Within the individual voices, the resolutions are rather unsatisfying, in particular the E♯ in the 2nd violins would really rather resolve upwards. Well, in Tchaikovsky's version it does, albeit by a large jump.
- Without crossing, none of the accompaniment voices really have any movement between the dotted quaver and semiquaver. That means the legato slur is instead a pair of tied tenuto notes – still a single bow stroke, but it doesn't have quite the legato quality anymore or alternatively doesn't separate the notes at all.
Tchaikovsky definitely seems to have considered that (or maybe it was done later in typesetting), since the preceding notes are also tenuto and not slurred at all in the non-crossed version.
- Some people like to see cross symbolism as a death metaphor all over the Patétique.