Small Cremona pulley

My 1913 Cremona A-roll piano arrived with a 3/8" leather belt installed on the cone pulley transmission, which is much too big -- it barely stays in the intended grooves.  As far as I can determine by measurements and conversations with other Cremona owners, the cone pulleys were designed for 1/4" round leather belting.  Someday I want to make new cone pulleys to get a wider range of speed control, such as Tempo 65 to Tempo 95.  As it is now the A-roll won't move fast enough for some tunes.

The sketch above shows the geometry of a round belt riding in a standard Vee groove of 60 degrees total angle.  I assume that when the belt deforms slightly (it becomes slightly egg-shaped) that the contact point doesn't change significantly.  The simple formula that results says that the outside diameter (OD) equals the pitch diameter (PD) + 1.5 times the diameter of the round belt.  Conversely, one can measure the outside diameter to determine the pitch diameter.

I measured the pitch diameter of each groove of the small pulley shown above by inserting 1/4" shafting (actually 1/4' drill bits) and holding them parallel which I measured the outside diameter (OD) with vernier calipers.  I also measured the larger pulley which drives the takeup spool.  I recorded this data:

Cone pulley with 1/4" round belt:
  groove  O.D.    P.D.
    "G"     2.69    2.312
    "H"     2.52    2.15
    "J"      2.36    1.99
    "K"    2.25    1.88
    "L"     2.04    1.67

   "A"    4.25     3.88
   "B"    4.34     3.97
   "C"    4.43     4.06
   "D"    4.53     4.16
   "E"    4.64     3.88

For fastest play speed the belt rides in grooves G and A; the pitch diameter ratio A/G is 3.88/2.31 = 1.68
At slowest play speed the belt rides in grooves L ad E, and the pitch diameter ratio is E/L = 2.32

The ratio of these two numbers is the total speed range ratio: 2.32/1.68 = 1.38 which is, for example, the ratio of Tempo 55 to Tempo 76.

[to be continued]