The plucked string instrument provides a good opportunity
to experiment with tuning. The even-tempered tuning system we normally
use calculates intervals in terms of powers of 2^n/12. Thus to find
a frequency which is 1 semitone higher we multiply by 2^1/12, etc.
There are other views of tuning, many of which are opposed to this way
of thinking. Systems such as just tuning and
Pythagorean
tuning try to make intervals closer to ratios which are found at the
lower end of the harmonic series and are integer ratios. A
major third, for example, would be calculated by multiplying by 5/4, since
the interval between the 5th and 4th frequencies in the overtone series
is a major 3rd, and since it has an integer ratio it is often called a
pure major 3rd. There is an audible and obvious difference
between a frequency ratio of 5/4 (1.25) and 2^4/12 (1.25992).
Test it for yourself. Minor thirds are also quite different.
In tempered intervals you will be able to hear beating and this is often
how piano tuners adjust intervals so that they come out tempered.
If you were to tune by just perfect 5ths and 4ths, for example, to get the pitches
in a given octave, you would not come out in the right place after 12 steps,
and the difference is called the Pythagorean Comma, since it was discovered
by Pythagoras. You can test this for yourself as well.
Here are the intervals of a C major/minor scale tuned as integer ratios,
relative to C.
C 1/1
D 9/8
Eflat 6/5
E 5/4
F 4/3
G 3/2
Aflat 8/5
A 5/3
Bflat 7/4
B 15/8
The first step is to get a plucked string routine which can be tuned
precisely with an allpass filter. Once you've done this, do some
ear-training to learn to tell the difference between just and tempered
chords and intervals. The difference between a perfect and tempered
minor triad is quite noticeable.
Then write some music using just intonation.
To compute frequency ratios in Minc, use the pchnam() function.
Thus
newnote = (9/8) * pchnam("c") will give you a just major 2nd above middle
C.
Have fun. The computer is the only place where it is possible
to really experiment with tuning. You might want to try out a 13
note chromatic scale, for example by using ratios of 2^n/13.
You might find this page, the
Anatomy of an Octave, interesting.