Our range of notes is now become rich. If we consider a note, a fourth above that note, a fifth above that lowest note, again there is an extra interval between that fourth and that fifth, a ratio which we have not seen yet. This interval is called a second, and is a whole tone. Its frequency ratio is given by the ratio of 3/2 (the fifth) to 4/3 (the fourth) and is the ratio of 9/8. This ratio of 9/8 is called the tone. It is not particularly harmonious. Two notes a Pythagorean tone apart do share a few harmonics, but not very many. (every ninth harmonic of the lower is shared with every eighth of the one a tone above).
How do we fit all this together? In our original investigation we found that
six tones fit together to make an octave. Do six Pythagorean tones fit together
to make an octave? If we go up six tones- ie go up in frequency 9/8 six times,
or
, we find that this is not exactly
an octave, it is slightly larger by about 1.3% or about a quarter of a
semi-tone. This difference between six Pythagorean tones and an octave is called
the Pythagorean comma, and its existence was recognized already by the
Pythagoreans. Just as and octave is not evenly divided by two prefect fifths
(they differ by a tone), six tones do not make up a octave (they differ by this
Pythagorean comma). One solution, used in some forms of music around the world
is to introduce a new subdivision equal to the comma, a micro-tone. But Western
music has not in general gone this route.