This image file is a graphic representaion of one small portion of the wave form in Example 2. Notice the regularity of the wave shape.
MUSIC 171
TUNING SYSTEM LISTENING
Spend a little time with each of these examples, trying to hear
the differences between just intonation (based upon the physical
properties of the overtone series) and equal temperment (which
strives to equally divide the octave while still maintaining a
close kinship to the overtone series).
In this first example, you will hear a perfect fifth
("a" to "e") tuned first using just
intonation, then equal temperment, and finally with both sounding
at the same time. You should notice "beats," or a
fluctuation of intonation (and consequentally, volume) in the
final sound, but you will likely notice a similar result in the
middle sound as well.
Examples 2, 3, and 4 are simple an A-Major scale, but as
above, they use, in order, just intonation, equal temperment, and
both together.
Example 2 - Just Intonation
This image file is a graphic representaion of one small portion
of the wave form in Example 2. Notice the regularity of the wave
shape.
Example 3 - Equal Temperment
This image file is a graphic representaion of one small portion
of the wave form in Example 3. Notice the regularity of the wave
shape, just as in Example 2. For each single note, we are unable
to see a visible difference in the tuning systems.
Example 4 - Both Together
This image file is a graphic representaion of a small section (a
single note) of the combined waveforms. Notice that there is a
rise and fall in the waveform, a rise and fall that should be
perceptible on some scale degrees when the two notes are
"out of tune." The sum of the two waves are out of
phase, and the result is both audible and visible.
We can see the degree of fluctuation in the following example,
which depicts the entirety of Example 4.
If we change the timbre of one of the two systems, it is
easier to hear the difference in the two systems.
Example 5 - Both Together (square wave/sine wave)
Now listen to several chords formed using only notes drawn from
the A-Major scale. Again, both tuning systems, as well as a
combination, are represented for comparison. Notice the
difference, and also notice those chords where there are similar
fluctuations.
Example 6 - Just Intonation
Example 7 - Equal Temperment
Example 8 - Both Together
A small portion of Example 8 "looks" like this:
As in Example 5, try listening to both systems together, but with a different timbre:
Example 9 - Both Together (square wave/sine wave)
By Monday, I hope to have an explanation of the formulas posted for you to go along with this listening exercise.