To understand how the notes are tuned in modern pianos, and why
this system of tuning has been adopted, you need to understand the
relationship of the intervals between the notes to the mathematical
concept of logarithms.
One of the things you are asked to do in this problem is to find
how many thirds there are in an octave using logarithms. Going back
in history, musicians would have used other methods to calculate
how many thirds in an octave because they did not know about
logarithms. The problem Euclid's Algorithm and Musical
is about a method they used.
There are many musical scales but in this problem we only consider
the 12 note piano scale.
An octave is the interval from any note to the next occurrence of
that note up or down the keyboard. There are 12 notes in an octave.
The black notes on the piano keyboard are sharps or flats, for
example the note between C and D is called C sharp or D flat.
The musical intervals correspond to the ratios of the lengths of
the strings which vibrate to produce the notes, for example the
octave is given by doubling (or halving) the length of the string.
If you halve the length of a string the pitch rises an octave, that
is from one C to the next C above it, or from one A to the next A
above it, and so on, so the associated ratio for this musical
interval is 2.
Similarly by taking 2/3 of the length of the string the pitch rises
by what musicians call a fifth, that is from C to G and taking 3/4
the length the pitch goes up by a fourth, from C to F say. Pleasant
harmonies arise from notes whose frequencies are related by simple
Taking eight-ninths of the original string gives the interval
called the tone (C to D above it or anything similar).
If the interval between two notes is a ratio of small integers the
notes sound good together. The pure intervals smaller than or equal
to an octave are: 2/1 - the octave; 3/2 - the perfect fifth; 4/3 -
the perfect fourth; 5/4 - the major third; 6/5 - the minor third;
5/3 - the major sixth and 8/5 - the minor sixth.
Although some of the intervals in the just intonation system are
perfect, other combinations of notes sound very bad. With equal
temperament, the intervals are never exact (except the octave), but
they are very close - always within about one percent or better.
The twelve-tone equal-tempered scale is the smallest equal-tempered
scale that contains all six of the pure intervals 3/2, 4/3, 5/4,
6/5, 5/3, 8/5 to a good approximation - within one percent.