From: Ronald Wong (ronwong@inreach.com)
Date: Fri Feb 18 2000 - 17:19:03 PST
Date: Fri, 18 Feb 2000 17:19:03 -0800 (PST) Message-Id: <l03102800b4d2361a5398@[209.209.18.100]> From: Ronald Wong <ronwong@inreach.com> Subject: Re: Math Music Scale(s)?
Just a footnote to Al Sefl's fine discussion regarding musical scales.
In musical circles, the scale that he referred to as SCIENTIFIC or JUST is
called a Diatonic Scale.
His scale, based on middle C starting at 256 Hz, can be used as a basis for
a whole series of diatonic scales. The 6 notes of the original C major
scale from D to B become the first notes for the subsequent scales. To
generate them, simply apply the same ratios used in generating the C major
scale to the first note of each of these new scales.
What you have will be a series of diatonic chromatic scales (sometimes
referred to as "natural" scales) and one of the benefits you get for all of
this trouble is truly harmonic cords. For instance, in any one of these
scales the major triads have a frequency ratio that are exactly 4:5:6 and,
as a result, they sound very harmonious when played. Modulations up the
scale (utilizing the sharps) sound different than modulations down the
scale (utilizing the flats) and music written with this in mind have a
broader range of expression.
Unfortunately, if you are making keyboard instruments, you find yourself
with an interesting problem. You need 18 keys per octave to do the job
right. In this scale, a C sharp is not the same as a D flat - and this is
true for similar combinations along the entire scale. With so many keys,
the reach from one octave to the next becomes problematic. If you start
with a reasonable span for one octave on the other hand, then designing a
keyboard that can produce so many notes within that span becomes a real
challenge.
Thus the compromise of equal temperament - and it IS a compromise. When you
play a major triad on a properly tuned piano (or any other instrument
constrained by design to equal temperament) the ratios are NOT exactly
4:5:6. Fortunately, the average person doesn't seem to notice the subtle
lack of harmony. For people with perfect pitch who can notice this, and
musicians in general, it's just a necessary compromise that comes with
having a practical keyboard instrument.
Given the fact that most composers today do their work using pianos, it's
not surprising that almost all the music being composed today tends to be
of equal temperament.
This wasn't true a few centuries ago and music written then took advantage
of the that fact.
It's important to keep in mind that there are musical instruments that are
NOT constrained to a scale of equal temperament.
Many non-keyboard instruments, like the violin or cello, can readily
produce both types of chromatic scales. Violinists can alter the equal
temperament for expressive reasons whenever they feel it's appropriate and
frequently do so. Not only is the harmony better but, as was mentioned
earlier, one can take advantage of the difference in how the music sounds
due to the differences in modulation when moving up the scale as opposed to
down. Here, a difference between an A sharp and a B flat is of some
significance (as any violin teacher will point out to their inattentive
student).
So much for a footnote - Sorry about that.
ron
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