³1username³2³2name³2text_default³2version³21³2ts³2980044208³2ip³224.20.108.34³2data³2newauthor³31³3minor³30³3text³3
There's a decent article in issue 4 of the Napalm e-zine
on Just Intonation and the western scale in general:

http://napalm.firest0rm.org/issue4.txt
³3summary³3*³2tscreate³2980044208³2keepts³2980050538³2id³2111³2host³2c373514-a.provo1.ut.home.com³2revision³21³1username³2³2name³2text_default³2version³21³2ts³2980050538³2ip³224.234.246.58³2data³2text³3There's a decent article in issue 4 of the Napalm e-zine
on Just Intonation and the western scale in general:

http://napalm.firest0rm.org/issue4.txt
----
Aw, c'mon: I don't want to read a whole article without first getting an idea of what "math of the Western musical scale" even ''means.''  C'mon, a paragraph about it, please?
³3minor³30³3newauthor³31³3summary³3*³2tscreate³2980044208³2keepts³2980063937³2id³21001³2host³2dhcp058.246.lvcm.com³2revision³22³1username³2³2name³2text_default³2version³21³2ts³2980063937³2ip³224.20.108.34³2data³2newauthor³31³3minor³30³3text³3There's a decent article in issue 4 of the Napalm e-zine
on Just Intonation and the western scale in general:

http://napalm.firest0rm.org/issue4.txt

It describes what makes a chord sound in tune to our ear, the
logarithmic nature of our perception of sound, and how we
went from just intonation to the less precise compromise of the
modern scale.  Pending their permission, I'll post the entire article text here.
³3summary³3*³2tscreate³2980044208³2keepts³2980198751³2id³2111³2host³2c373514-a.provo1.ut.home.com³2revision³23³1username³2³2name³2text_default³2version³21³2ts³2980198751³2ip³2128.187.101.144³2data³2text³3There's a decent article in issue 4 of the Napalm e-zine
on Just Intonation and the western scale in general:

http://napalm.firest0rm.org/issue4.txt

It describes what makes a chord sound in tune to our ear, the
logarithmic nature of our perception of sound, and how we
went from just intonation to the less precise compromise of the
modern scale.  With ajax's permission, 
I've posted the text of the entire article here.

Everything You Never Wanted To Know About Just Intonation
                         or
           Western Scales Ate My Brain

First, some introduction: according to napalm.firest0rm.org:80, "Napalm
is an e-zine devoted to computer security, with a healthy dose of music,
news, and ethics." Well, maybe we oughtta live up to that, so here's some
music theory in your ear. (Or your eye, unless you're text-to-speeching
this.)

Generally speaking, a scale is a progression of notes that takes you from
one octave to the next.  (Although it can actually be much larger than
just one octave, generally we don't do that.)  Notes have pitches, which
can be measured in oscillations per second, or hertz.  The human ear can
perceive from about 20Hz at the low end to around 20,000Hz at the high
end.  Now, it turns out that the human ear differentiates pitch
logarithmically, and that doubling any given pitch raises it an
octave.  Thus the human ear has a range of a little under ten octaves.

Now it also turns out that when the ear hears any two notes in
combination, the interval, whether major or minor, is perceived to be in
tune when the ratio between the two notes approaches a ratio of two low
integers.  The degenerate case would be a 1:1 ratio, which would be two
notes in unison; a 2:1 ratio, as above, is an octave.  Roughly, the
intervals for the twelve-tone scale used in western music are:

 1:1	unison 
 21:20	half step / minor second
 9:8	whole step / major second
 6:5	minor third
 5:4	major third
 4:3	perfect fourth
 7:5	tritone / augmented fourth / diminished fifth
 3:2	perfect fifth
 8:5	minor sixth
 5:3	major sixth
 9:5	dominant seventh / flat seventh
 17:9	major seventh / natural seventh
 2:1	octave

The note that a scale centers around is called the tonic.  A scale in
which the ratio of any note to the tonic is an integer ratio is called a
scale of just intonation.  These scales have a very natural-sounding
quality to them.

The problem is, they're murder to implement in any stopped or fretted
instrument.  The difficulty is subtle, but it means big headaches.  It's
hard to explain, though, so I'll give an example.

For purposes of tuning we need a reference pitch, something all the
instruments can agree on.  Usually a 440hz sine wave is used as the
reference pitch, as an A natural.  Now, according to our table above, we
can calculate the pitch of any other note by setting up a simple ratio
relationship.  For example, if I wanted to calculate the pitch of a
perfect fifth from an A440, I would write:

    (X / 440) = (3 / 2)

and solve for X.  Simple algebra, right?  In the above, X comes to 660.
Let's calculate two more:

    (X / 440) = (9 / 8); major second = 495
    (X / 440) = (5 / 4); major third = 550

Now, assume that what you know about scale theory is right and that the
second of a second (of a tonic) is the same note (and pitch) as the major
third of the tonic.  If that were true, then in:

    (X / 495) = (9 / 8)

X should equal 550.  This is the subtle bit, so make sure you understand
it.  The above equation takes the second of 440 (495) and finds the note a
second up from it (9 / 8).  I'll let you all punch that into xcalc and see
what happens.

Didn't work, did it?  You got 556.875, right?
This is the problem.  Any given scale of just intonation must be tuned to
a tonic, which is fine if you only want to play in one key.  However, you
have to retune the instrument every time you modulate keys.  As many
classical composers (and pop ballad writers) will tell you, this has a way
of limiting your expressive power.

So what's a keyboard manufacturer to do?  Tha answer is simple: make one
note in tune, and space all the other notes equally (logarithmically
equally, anyway).  This is what happens on most fretted instruments and
keyboard instruments.  Now, instead of calculating pitch with integer
ratios, we just plug an interval into the following equation:

    P = 440 * 2 ^ (n / 12)

where n is the number of half steps sharp you want to go (and hey, guess
what, negative numbers work as expected; (n == -3) finds the pitch of the
major sixth below A440).  This is called a scale of even (or equal)
temperament, since the distance to any other note is independent of (and
consistent across) key centers.

Now, waitasec, doesn't this put everything out of tune?  Well, yes it
does, but not by very much.  Observe:

 Note    Just Pitch      E.T. Pitch (approx.)    Error (%) 
 A       440             440                     0.0
 Bb      462             466.16                  +0.9
 B       495             493.88                  -0.2
 C       528             523.25                  -0.8
 C#      550             554.37                  +0.7
 D       586.6-          587.33                  +0.1
 D#      616             622.25                  +1.0
 E       660             659.26                  -0.1
 F       704             698.46                  -0.7
 F#      733.3-          739.99                  +0.9
 G       792             783.99                  -1.0
 Ab      831.1-          830.61                  -0.1

As you can see, we're never more than 1% out of pitch, which most people
can't hear.  However, you *are* out of tune inherently, so when your
instrument then goes further out of tune you sound *really* bad.  The
advantage, though, is that you get stopped instruments, which makes
composition and playing much easier.

(Of course, you'll notice I cheated, and listed a just intonation scale
that happens to be close to the 12-step equal temperament scale.  Sue
me.  There are also Indian raga scales and other non-standard scales that
are "just intonation" scales, but I wanted to present something
understandable.)

³3minor³30³3newauthor³31³3summary³3Added full article text from Napalm e-zine³2tscreate³2980044208³2keepts³2980540291³2id³2111³2host³2organ.cs.byu.edu³2revision³24