cbostrum@watdaisy.UUCP (Calvin Bruce Ostrum) (07/30/83)
I am getting confused about this alternate scale stuff. Can we have some
expert commentary? Particularly hazy points include the following:
It is a little off the mark to come down upon the "western system" as
"limited". It is not so ad hoc as some may think. Here is how we get it:
First we decide to have a discrete scale. Seems sensible to me but perhaps
the most arbitrary of all the assumptions this is based on.
Now, once we note that human hearing works to produce a
metric on tones that is a logarithmic function of frequency (does it?) we
get a scale where each note's frequency is a multiple of the previous ones.
(how many "nonwestern" scales actually fall outside this category essentially?)
This decision also produces the nice logical feature that the tone metrics
are not essentially related to the starting tone of the piece (ie it can
be transposed without losing any of the metric structure, although it may
lose qualities associated with absolute frequencies).
Next, if we agree that small integer ratios 2, 3/2, etc are very nice, we find
that the 12 tone scale is the very best scale with a manageable number of
tones. So it doesnt seem that ad hoc at all.
The argument above is based upon a lot of assumptions that I and probably
others would appreciate expert commentary on. The one I am particularly
interested in is the assumption that there is something intrinsically
aesthetic, from a sonic point of view, about octaves, fifths, etc. If so,
how is it related to the physical property that such waves have? I have
always been somewhat skeptical of this claim, because it seems to me that
only exact multiples would have special phsyical properties, whereas most of
us still fully appreciate the out of tuneness of a tempered scale, and many
of us still appreciate much worse.
Calvin Bruce Ostrum, Computer Science, University of Waterloo
...{decvax,allegra,utzoo}!watmath!watdaisy!cbostrum
ps: i posted this to net.music although it is a response to something
in net.math. what does this have to do with math? surely we computer types
know enough math that simple applied math problems can go in the group for
the area to which our math is being applied?floyd@burdvax.UUCP (Floyd Miller) (08/03/83)
I'm not an expert, but from a course I took years ago at Univ. of Mich.
called "The Physics of Music", I remember some discussion on musical scales.
The basis for most scales (especially the "western" 12 tone scale) is that
human hearing is very sensitive to certain intervals (differences in pitch
between two notes). This is pronounced when the two (or more) notes are
heard sameoltimeously. Not only are these intervals easily distinguishable,
but we are able to percieve when the tones are close to the interval (the
"beating" effect).
It seemed only natural to base a musical scale on these intervals since
most people could hear them and they sounded "nice". (these intervals
also happen to line up, more or less, on a logarithmic mathematical scale).
I don't remember all the details (I could look it up in the handouts I saved)
but the clearest interval is the octave (a 2:1 frequency ratio) and the western
scale has been based on this for a long time. An early scale was based on the
notes obtained by progressing at "fifth" intervals (ration of 3:2) and moving
each note by a factor of two to bring it into the base octave. A scale of
seven note resulted (the diatonic scale).
THe problem with that scale (also called the Pythagorean scale) is that it
ignores all the other natural intervals besides the octave and the fifth.
Many of the notes are close to, but not at, these other intervals. The
interval of a "third" is noticably "sour" as was demonstrated in one of the
lectures. An alternative scale, called the "Just" scale was developed using
the intervals of "octaves", "fifths" and "thirds" (5:4). Thus, the basic
unit of the "Just" scale is the major triad, a combination of three notes
whose frequencies are of the ratio 4:5:6 (containing the ratios of 5:4 & 3:2.
The "just" scale still has a serious problem: the interval from D to A
(asssuming the scale was based on C) is close to a perfect fifth but off by
enough to really louse up any song that tries to build chords on various
steps of the scale (most intersting music does).
Temperament is the process of modifying a scale by small amounts to achieve
a more even sounding scale. What is done to the "Just" scale is to spread
the difference between the interval D to A and a real fifth over the entire
scale to weaken its effect.
Thus, the "Equal Tempered Scale" in which only 3 intervals are exactly at the
mathematical and natural intervals, the octave, the fifth, and the fourth.
That's all I have time to type now.
<floyd>