Aunt Betty’s Clock
An analysis of Aunt Betty’s circa-1900 clock. Note you’l need to “enable editing” in Word to open the embedded sound file.
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lstabile
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lstabile
An Architecture for the Execution of Applicative Languages (1979)
An old paper, unpublished. This is one of the nearly-final drafts.
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lstabile
Self-Descriptive Number Fixed-Point Function
A self-descriptive number in base 10 is a 10-digit number x, the digits of
x being numbered from most to least significant as d0,…,d9, such that di
represents the quantity of the digit i in the decimal representation of x. See Weisstein, Eric W. “Self-Descriptive Number.” From MathWorld, a Wolfram Web Resource. http://mathworld.wolfram.com/Self-DescriptiveNumber.htmlA number-theoretic function which is the base-10 fixed point of the self-describing number is presented in this paper.
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lstabile
Listening to the Riemann Zeta Function
In this paper methods are explored for listening to the Riemann zeta function and some spectral variants. Contains embedded sound files. Common Lisp code for emitting the sound files is here
Dave Wall and 
nerdlynotions 
Dave Wall 6:33 pm on June 11, 2018 Permalink |
Interesting analysis, but I didn’t see any mention of thermal effects. The length of the pendulum will vary with changes in ambient temperature, unless a compensating mechanism is used. (Some pendulum clocks use compound pendulum suspensions, where members expand/contract in opposition, nulling out any net change in pendulum length or period.) Does Aunt Betty’s Clock have any sort of thermal compensation mechanism.
The linear coefficient of thermal expansion (Cte) for common materials from which the pendulum shaft might be fabricated (brass, iron, steel) would be in the range of 10-20 ppm/C. Using round numbers, assuming a 100 mm pendulum length for a small case clock such as this, a 5C (9F) change in temperature would result in a change in pendulum length of ~ 0.005-0.010 mm. Following the logic in the analysis, this would result in an error of ~25-50 usec.
Similar to the periodic nature of certain errors cited in the analysis, this error would also be periodic, or at least pseudo-periodic, following day/night cycles and HVAC program cycles.
lstabile 7:48 pm on June 11, 2018 Permalink |
Yes, worth looking into. The pendulum is in fact two-piece, with one part being a flexible metal so that the length may be controlled. However I don’t know whether thermal compensation was also intended. The tick-tock recordings would need to be extended over days to provide the needed info and the ones I used were only 15 minutes or so each at the most. So that’a something to consider doing. While the size of an audio recording could be fairly large, that’s probably manageable. A greater challenge is attenuating and/or cancelling out background noise.
Dave Wall 4:02 pm on June 15, 2018 Permalink |
I’d like to add some comments relative to thermally induced time errors. Errors due to periodic or pseudo-periodic changes in the ambient temperature will typically exhibit a phase shift, dependent upon the thermal time constant. This, in turn, will be a function of factors such as the mass and specific heat capacity of the pendulum arm and the isolating/insulating properties of the clock case. A pendulum with a large thermal mass in a well insulated case could have a long thermal time constant relative to the ambient temperature change period. In addition to a large phase shift, this would also significantly dampen the magnitude of the induced time error.
This also got me thinking about other errors that would affect pendulum clocks, and since gravity is involved, it seems that tidal forces (which are also periodic) would have an affect. I haven’t researched this in any detail, but in Googling I stumbled across what looks like an interesting website – LeapSecond.com. It looks like it has a lot of interesting content on clocks and time measurement.
Finally, I’d like to mention a book I recently read – Einstein’s Clocks, Poincare’s Maps: Empires of Time by Peter Galison. It’s a history of time synchronization. One interesting period was related to determination of latitude. We are familiar with John Harrison and the Longitude Prize, but a major breakthrough came from the laying of sub-sea cables. Once distant locations were telegraphically connected, and methods were devised to measure and correct for propagation errors, it was possible to synchronize widely separated clocks. Comparing well synchronized time differences between astronomical observations at the two locations allowed for much more precise determination of longitude.
Another interesting item. The late 19th century/early 20th century was an important time for development of clock synchronization, propelled by the need for more tightly controlled railroad schedules. Switzerland was a hot bed of activity, and many elaborate electromechanical patents crossed Einstein’s desk during his years as a patent clerk in Bern, stimulating his own thinking about time synchronization in the formulation of Special Relativity.