Converting decimal numbers to fractions:
Examples:
3.14159265(1:ratio).=3/1 (0.14159265)
3.14159265(4:ratio).=16/5 (-0.05840735)
3.14159265(5:ratio).=22/7 (-0.00126449285714)
3.14159265(11:ratio).=355/113 (-0.00000027035398)
3.14159265(20:ratio).=96893/30842 (0.00000001657804)
replaces the decimal number 3.14159265 by the best possible approximate
fraction (ratio of two integers) by using accuracies 1,4,5,11 and 20.
The numbers in parentheses are errors.
In x(c:ratio)=n/m,
the best approximation n/m for x is found by minimising the function
f(x,m,n)=n+10^c*(m*x/n-1)^2
with respect to integers m and n. This technique is originally used
in S.Mustonen's studies for musical consonances and dissonances.
There x corresponds to a (impure) musical interval and the optimal
m and n values give the ideal ratio for the musical interval recognised
for a selected accuracy of the 'ear' c.
For example, the 'neutral third' 11/9=1.222222222
will be heard as an impure minor third (6/5) by a moderate ear c=4:
1.222222222(4:ratio)=6/5 (0.022222222)
but as an interval of its own (11/9) by a very accurate ear c=5:
1.222222222(5:ratio)=11/9 (-0.00000000022222)
C = More information on numerical conversions
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