AC, an 1/8 of an inch can be huge for pitch variation. Get off your duff and experiment. Chop chop!
Also, you should be able to easily source a belt from www.turntablebasics.com, just in time to still play those holiday classics.... Good luck and happy holidays.
Finally got around to doing some experimentin'...
Speed measurement std. deviation with the iphone app is
+ 0.15 RPM in quick mode, and about
+ 0.05 RPM in "exact" mode, so realistically a difference of ~ 0.2 RPM is significant.
Baseline (spinning, not playing a record) : 32.4 RPM 2.7% slow
Moving the motor 1/8" closer to the spindle--32.3 RPM (no difference)
Moving the motor 1/8" further from the spindle-- 32.4 RPM (no difference)
New belt (~0.030" thick vs. ~0.015" thick for the old belt): 34.1 RPM 2.4% fast

New belt playing a record: 33.8 RPM 1.5% fast

I seem to be able to hear a difference between the two belts, less the pitch than the pace. Regardless, both belts are within
+ 3% which the iPhone app rates at "good"