Date: Wed, 3 Dec 1997 08:29:36 -0600 (CST)
From: Dave Rusin <rusin>
To: rishabh@pl.jaring.my
Subject: Re:  I NEED HELP!!!!!!.....can you please reply as sooon as possible?!!??!!??!?!

If the file  http://.../logs2  is beyond you, perhaps you should just look up
the "correct" frequencies of all 12 notes in an octave. You should observe
	1. The ratios of all consecutive pairs are equal
	2. The ratios of notes an octave apart are exactly 2:1
	3. The ratios of note in a "fifth" (e.g. G  to the C below it) are
		almost 3:2.
You need to learn a little about sound to understand that sound is vibration,
and that good sound is regular vibration, and in particular a nice chord
will have to involve vibrations whose frequencies are in the ratio of small
numbers. If you do all that, you'll understand why facts 2 and 3 are not
accidents but rather are part of the design of music.

Now see what would happen if you had to "invent" music. Pick one tone of
any frequency that you want to have in your music. Invent a rule that says
any time one frequency is used, so are all the frequencies which are
an octave above it and an octave below it; what tones are in your
music system now? Is that interesting enough to make music? Not to me.
So invent another rule that says any time a frequency is in the set to
be used, so are the ones which are exactly a "fifth" above and below it
(that is, if you use frequency  f,  you should use also the frequencies
(3/2)*f  and  (2/3)*f ). NOW what frequencies will you include? In particular,
how many tones do you get in an octave? (infinitely many!) Does this seem
good to you? -- seems like a mess to me!

So we scratch that last rule and instead add a rule that says, any time
we use a frequency  f  we should also use frequencies that are _pretty close_
to a "fifth" away, that is, if we use  f  we should also use  r*f  and  f/r
where  r  is some number which we will pick which will be pretty close to
1.5. The trick is in deciding what a good value for  r  is.  Classical
Chinese music used  r= 1.515716567. Try that and see what happens.
Even easier, I suppose, is the bugler's choice: try  r = 1.587401052  and
see what you get. The problem is that these aren't too close to  1.5. In
classical European music they use  r=1.498307077, which is so much closer
that it really sounds like a true fifth (r=1.5). You'll have to be a bit
more patient playing around with that number to discover what notes
are in your scale, but if you checked fact  1  at the beginning of this
letter, you should see that this ratio does appear in the list of
ratios of frequencies ( G to the C below it).

Of course you should ask how on earth these values of  r  are selected,
but you should have noticed (if you've actually _tried_ any of the
calculations I've suggested so far) that these particular  r  have th
property that  r^n = 2^m  for some  n  and  m. That's the key idea: the
decision for the number of tones per octave is based on what values of  n
(and  m)  will give you an  r  around  1.5.  With  8  tones per octave,
say, you'd have to settle for  r=1.414213562  or  r=1.542210825, neither
of which is even as good as the 5-tone scale, so you'd have more tones
per octave but a poorer sounding "fifth"; given the rules we invented
for music, that makes an 8-tone scale a less good choice than the simpler
5-tone scale.

So this should convince you that 12 tones per octave is a good choice,
given the two rules we invented. You can experiment with other numbers
of tones per octave; you'll see no  n  gives a better  r  than n=8
until you get to  n=12. And no n  gives a better  r  than that until  
n=29. You can pick out in advance what these "best" n's are, but that
takes a little more math -- that's what the log3/log2 stuff is in my
web site.

You said your report is due in two days. I'm afraid you may have waited
too long to get started if this material is too difficult for you! Good luck.

dave