Here are two further methods for determining the Tempo boost ratio
resulting from the effective increase in diameter of the take-up spool.

1. A direct reading gauge on the inside of a take-up spool flange.
Please refer to the attached split image, showing initial condition
(a single turn on the spool) and the condition partway through playing.

The design of the gauge is straightforward.  In my case the take-up
spool diameter is 50 mm, a radius of 25 mm.  I placed the first mark
5 mm from spool.  The Tempo boost at this mark is 30 (the new radius)
divided by the base radius: 30/25 = 1.2.  The next mark is at 10 mm,
to be labelled 35/25 = 1.4.  I was pressed for space so the final mark
is at 37.5 mm corresponding to a boost of 37.5/25 = 1.5.

This method has the advantages that it can be read at any time during
playing and it does not require knowledge of the roll length or paper
thickness.

2. The second method borrows from the techniques introduced by Harald
Mueller.  The area of the annulus of the paper on the take-up spool at
completion will equal the area of the initial annulus of paper on the
host spool.

If we equate these areas we arrive at:
D^2 - d^2 = H^2 - h^2  (Equation 1)
where h, the inner diameter of the host roll, is known.  (For pianola
rolls it is 22 mm or 7/8 inch.)

H, the outer diameter of the host spool, is readily measured.  D and d
are the take-up spool outer and inner diameters.

We wish to determine D/d, as it is the Tempo boost.

From (1), D^2 = d^2 + H^2 ­h^2
So, D^2/d^2 = 1 + H^2/d^2 -h^2/d^2  (Equation 2)
and from Eqn. 2 the Tempo boost, D/d, at completion will be the square
root of (1 + H^2/d^2 -h^2/d^2).

This is quite sound, but inelegant.  An advantage: if we take h and d
as knowns, then for any given roll the method requires only a single
measurement, H.

On an associated matter I would be interested to hear if it is generally
accepted (or not) that pianola rolls are punched with Tempo compensation
included.

Jase Haysom
Victoria, Australia
m.haysom@cartography.id.au.geentroep [delete ".geentroep" to reply]

 [ A simple Tempo boost indicator
 [ http://www.mmdigest.com/Attachments/19/10/31/191031_095332_tempo_boost_gauge.png 

 [ Only a small percentage of all pianola rolls include Tempo
 [ compensation.  Welte-Mignon rolls (all forms) and Ampico rolls
 [ after 1926 have compensation.  I don't know about Aeolian rolls.
 [ -- Robbie