Tuesday, March 10, 2009

The Virtue of Random Error and Its Exploitation by Early Black Forest Clockmakers

In a previous post, I summarized the initial results of an analysis being conducted in cooperation with Dr. Martin Wolk regarding the placement of pins and small indentations on the the outer diameter of Angelus count wheels of Beha cuckoo and monk clocks. As the purpose of the pinned positions are well understood to trigger the Angelus ringing automation in the clock, our analysis focused upon less understood intent of the non-functional (unpinned) positions and the horological logic of their pattern.

Our basic hypothesis is that, in the cottage Black Forest clock industry of the mid- to late-nineteenth century, there may have been an attempt to gain economies of scale in manufacturing by fabricating clock parts that could be used in multiple applications and clocks. Accordingly, the pattern of the indentations in the Angelus count wheel might represent a series of pre-registered positions that could be drilled, and pinned, to create a wide range of time intervals over which horological-registered events (i.e. a monk ringing the Angelus, a music box playing, etc.) could be activated. Thus, a "standard" count wheel could be customized via the setting of pin positions to fabricate a multiplicity of different clocks.

According to our initial hypothesis, if one lets x represent the smallest interval between two consecutive marked positions on the Angelus count wheel, it appears that there are two intervals with 1x spacing, eight intervals with a 2x spacing, six intervals with a 3x spacing and six intervals with a 4x spacing. Thus, there are a total of 60x intervals (2*1x + 8*2X + 6*3x + 6*4x) about the circumference of the Angelus count wheel. Since there are sixty teeth on the Angelus count wheel and the Angelus count wheel rotates once every 24 hours, each x interval must then correspond to 0.40 hours or 24 minutes. Thus the intervals defined by successive markings on the Angelus count wheel would correspond to 0.40 hours (for each of the 1x intervals), 0.80 hours (for each of the 2x intervals), 1.20 hours (for each of the 3x intervals) and 1.60 hours (for each of the 4x intervals). Starting from position 1 and moving counter-clockwise, the interval sequence (in hours) is approximately …0.8, 0.8, 08, 0.4, 0.8,1.6,1.6, 1.6, 1.2, 1.2, 1.2, 0.8, 0.8, 0.8, 0.4, 0.8, 1.6, 1.6, 1.6, 1.2, 1.2, 1.2…

Given this sequence, pins may then set via combinations and permutations to create all of the following time intervals (in hours): 0.40, 0.80, 1.20, 1.60, 2.00, 2.40, 2.80, 3.20, 3.60, 4.00, 4.40, 4.80, 5.20, 5.60, 6.00, 6.40, 6.80, 7.20, 7.60, 8.00, 8.40, 8.80, 9.20, 9.60, 10.00, 10.40,10.80, 11.20, 11.60, 12.00, 12.40, 12.80, 13.20, 13.60, 14.00, 14.40, 14.80, 15.20, 15.60, 16.00, 16.40, 16.80, 17.20, 17.60, 18.00, 18.80, 19.20, 19.60, 20.00, 20.40, 20.80, 21.20, 21.60, 22.00, 22.40, 22.80, 23.20, 23.60, 24.00. Additional flexibility in these intervals could obviously be gained by modifying the rotation frequency of this count wheel.

On this basis, we initially concluded that this sequence of pre-registered positions on the count wheel would have provided the nineteenth century Black Forest clockmaker with a wide range of closely spaced time intervals for customizing animation, music, and other events registered to time. Whereas it was clear that the spacings of the holes varied somewhat from a uniform x (or multiple of x) spacing, our assumption was that this variation was unintentional and reflected the limits of the technology available to a rural cottage clock industry in the mid- to late-nineteenth century.

While this premise we believe to be still basically true, examination and analysis of additional examples of Angelus count wheels for Beha cuckoo and monk clocks now leads us to speculate that it was the intent of Black Forest clockmakers to consciously introduce random error into the spacings between the holes to enable a more complete spectrum of time intervals to be derived from the same number of hole drillings. Introduction of this random error into the spacings between the pre-drilled holes would thus enable the clock maker to use the pre-drilled positions to activate a horological complication at a greater number of distinct (and hence more closely spaced) intervals over the period of the count wheels rotation than if the pre-drilled holes were regularly spaced.

To illustrate this effect let us assume that instead of the existence of a uniform x (or multiple of x) interval separating the pre-drilled positions we have an interval x, or multiple of x, to which each interval is added a random error y (y may be either positive or negative so the actual interval may in fact be larger or smaller than x or the corresponding multiple of x). For comparison purposes, let us now take the exact spacings (as determined from the sector angles defined by the hole positions, see below illustration) between the various pre-drilled holes present in the above Angelus count wheel (that is, assuming the spacings are what they are measured to be and not a hypothetical target x or multiple of x) and generate the analogous series of time intervals derived from the various combinations and permutations.

Whereas, our initial analysis using the regular spacing intervals results in 61 different possible time intervals between 0.4 and 24.0 hours with a regular 0.4 hour spacing, using the experimentally determined irregular intervals results in 181 different time intervals between 0.6 and 24.00 hours with spacing intervals varying from 0.07 to 0.61 hours. Clearly, the effect of non-uniform hole positioning has resulted in almost three times the number of distinct intervals that can be used for timing horological events. This is graphically depicted in the below histogram that illustrates the number and distribution of the time intervals for these two cases. In this histogram, the bin increment is 0.1 hour and each frequency count represents a unique time interval (a frequency count of "2" means that there are two distinct time intervals in a common bin).

The irony of this result is that relaxation of the clockmaker's accuracy and precision of pre-drilled pin position placements not only results in increased potential future utility but would have been expected to lower his manufacturing cost as well. Who says there isn't a free lunch....

We are indebted to our fellow Black Forest clock collectors Dr. Wilhelm Schneider, Mr. Mark Singleton and Mr. Dean Sarnell for the generosity of their time, information and helpful suggestions, without which this analysis would not have been possible.

Again, this is an still evolving theory so all comments, questions and rude remarks are welcome :-)


1 comment:

  1. John-

    You got a nice clock, and clearly spent a great deal of time and thought on this...Wow Im impressed!

    There may be a simple solution.

    What about duplicating a wheel off another...???

    I know where there are at least 5 932's off the top of my head.

    The other 932's should be a perfect match.

    There is another right here in the States, that a good friend of mine owns... That I am sure this could be arranged.