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From: floyd@burdvax.UUCP (Floyd Miller)
Newsgroups: net.music
Subject: Re: Scales Again
Message-ID: <939@burdvax.UUCP>
Date: Tue, 2-Aug-83 22:13:10 EDT
Article-I.D.: burdvax.939
Posted: Tue Aug  2 22:13:10 1983
Date-Received: Thu, 4-Aug-83 23:51:55 EDT
References: watdaisy.240
Lines: 46

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.