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'Tuning and Ratio' printed from https://nrich.maths.org/
Robert managed to solve this
toughnut:
Find the ratio corresponding to B.
Also compare the D-A ratio with the ideal fifth (3/2):
"B's ratio is calculating by use of
the fact that the interval from E to B is a perfect fifth, and a
perfect fifth's ratio is 3/2. Also E's ratio is 5/4. Therefore B =
15/8, and:
$\frac{A}{D}=\frac{5/3}{9/8}=\frac{40}{27}=80/54$
The ideal fifth ratio=81/54.Therefore
if we were to tune two adjacent keys to two different A's, the
first being in the ratio A/D, and the second a perfect fifth above
D (in the ratio 3/2). The second A would be higher than the first
by a ratio of 81/80."
Find also exactly how many
major tones/thirds there are in an octave.
To do this for the tonesRobert took logs of
the equation.
We are trying to find $x$ such that $\left({9\over 8}\right)^x
= 2$ then $$x \log{9\over 8} = \log 2,\ x= {\log 2\over \log 9/8} =
5.8849492$$ to 8 significant figures.
Similarly for thirds:
To find the number of thirds in an octave we are looking for
the value of $y$ such that $\left({5\over 4}\right)^y = 2$. Observe
that: $$\left({5\over 4}\right)^2 = 1.5625,\ \left({5\over
4}\right)^3 = 1.953125,\ \left({5\over 4}\right)^4 = 2.4414063$$
and hence $3< y< 4$ and $y\approx 3.1$. Using logs
$$y\log{5\over 4} = \log 2,\ y= {\log 2\over \log 5/4} =
3.1062837$$ to 8 significant figures.
In order to convert the Pythagorean & Just Intonation
ratios for intervals, into the standard units of tuning ie cents,
One must solve the following equation and then multiply the
solution by 100 to end up with cents:
$(2^{1/12})^n$=R, where R = The Pythagorean or Just Intonation
ratio for an interval.
Doing this, Robert obtained the
solution:
|
C |
D |
E |
F |
G |
A |
B |
C |
| Equal tempered scale |
0 |
200 |
400 |
500 |
700 |
900 |
1100 |
1200 |
| Pythagorean scale |
0 |
204 |
408 |
498 |
702 |
906 |
1110 |
1200 |
| Just intonation |
0 |
204 |
386 |
498 |
702 |
884 |
1088 |
1200 |
Well Done!