Cents without Logs

Most cheap hand calculators have only the four functions,
+, -, *, and /, though a few have a square root (SQR) key as well.
I've often wondered what the point of having this last key is as I would
expect that most people sufficiently sophisticated mathematically to
need a SQR key would probably need a log key too. After all, how often
do most people have to calculate geometric means, Std Dev's., diagonals
of rectanges, radii of circular areas, or RMS values? And if they did
these operations often, wouldn't they occasionally want other functions
like the sin, cos, tan, log, exp, etc. as well?
Anyway, I began to think how one might compute cents without a
log key and looked up Ellis's algorithm for integer cents in Helmholtz's
"On the Sensations of Tone " in appendix XX, section C. This algorithm is
really very simple: reduce the ratio to a single octave (remembering how
many octaves there were), reduce it to again if necessary so that it is
less than a 4/3 or 3/2 by multiplying by 3/4 or 2/3. Now take the
difference betweeen the Numerator and Denominator of the reduced ratio
and divide it by the sum of the N and D. Multiply this quantity times
the "Bimodulus" 3477 and add 1200 cents for each octave by which the ratio
was reduced and 498 or 702 if the octave reduced ratio was larger than a Fourth or Fifth. The final result should be accurate to a cent (there is
an minor 1 cent correction for reduced ratios between 450 and 498 cents).
In practice, it is simple and fast as the reductions may be largely done
by inspection.
Why does it work and why 3477? Well, there is an old and well-
known approximation to the natural log of (x+e)/(x-e), 2e/x, where e is
a small number. Let the reduced ratio be N/D (N>D), let x = N+D (the Sum
of N and D) and let e = N-D (the difference) and substitute into both
expressions. Then (x+e)/ (x-e) becomes (N+D+N-D) or 2N over (N+D-N+D) or
2D and Ln (2N/2D) = ln(N/D). This ln is approximated by 2*(N-D)/(N+D).
The cents value of N/E is defined as ln(N/D)*1200/ln(2) or
approximately 2*(N-D)/(N+D) *1200/Ln(2). The factor 1200/ln(2) is
1731.234+ and twice it would be 3462.5. Hence one would think that
formula for the approximate integer cents of small intervals would be (N-D)/(N+D)*3462. Ellis discovered empirically that 3477 works better,
giving nearly exact results for 5/4 and only slightly deviating ones for
larger and smaller intervals. 3462 is less convenient as it requires
small corrections. Reversing the procedure yields a close approximation to
the decimal form of the ratio if one is given an interval in cents.
While this technique is not a fundamental breakthrough, it is
an interesting reminder of just how clever the 19th century scholars
and scientists really were. The upshot is that if, alas, one is ever
stuck with only a four function calculator, one can still do tuning
theory.


--John

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