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From: alves@castor.usc.edu (William Alves)
Newsgroups: comp.music
Subject: Re: theory behind the scales
Message-ID: <10737@chaph.usc.edu>
Date: 11 Jul 90 00:03:38 GMT
References: <8547@uhccux.uhcc.Hawaii.Edu>
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Organization: University of Southern California, Los Angeles, CA
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In article <8547@uhccux.uhcc.Hawaii.Edu> newsham@wiliki.eng.hawaii.edu (Timothy Newsham) writes:
>i am interested in finding out how our present (western) keyboards
>evolved and the theory behind them.  i know that the keyboard started
>out with the white keys and then keys where added (like Bb to avoid
>using tritones) but how did we originally arive at the 7 note scale
>that the modes are based on?  

This is not at all an easy question, and yet there are some intriguing
cross-cultural similarities. For example, systems with 5, 7, or 12 notes
per octave are relatively common in different cultures. Many have noted
that 5+7=12, and Barbour has pointed out that the most common microtonal
systems he has studied include 19, 31, and 50 notes per octave, forming
a "Fibonacci-like" sequence. Some have argued that tuning systems based
on successive tuning of 3/2's (fifths) come closest to completing the oc-
tave at those numbers. I don't think that is necessarily true, and Horn-
bostel's theory of tuning by "overblown-fifths" (slightly sharp successive
fifths supposedly created by overblowing a flute) has been blown way out
of context. Clearly there is no unified field theory of scales. Why the
7/12 system in the West? There's no one good answer that I'm aware of.

>i am interested at the foundations in
>the overtone series in particular.  i have also read about a theory
>that the minor triad is generated by undertones but i fail to find
>out how when i look through the undertone series. anyone have any 
>information on this or any good references for me to check out?

For some reason, the fact that the minor triad does not appear in successive
natural harmonics has been very troubling to some theorists seeking to
prove the god-given naturalness of the Western scale system. I think that
the theory you refer to may be that of Zarlino, who tried to argue that
if the succession major third->minor third occurs naturally then one need
only go the other way to get the minor third->major third relationship of
the minor triad. Partch called the former "otonalities" and the latter
"utonalities." In fact, neither has anything to do with "undertones." Hinde-
mith went through a convoluted system of "proving" that the 12-tone sys-
tem was the most natural extension of the harmonic series. In fact, he
just went the long way around to derive a 5-limit just intonation scale,
something that's been around at least since the early 16th century. He then
proceeds to throw out the tuning system in favor of equal temperament be-
cause it's "close enough".

All of this (with the possible exception of Partch) is theory after the fact.
I don't think there is One True Scale that music then adheres to. Rather, I
think that whether a composer or a culture decides to go with 12-tone equal
temperament, or 43-note 13-limit just intonation (Partch), or 6-tone quasi-
equal temperament (Thailand), or a whole multitude of heptatonic and penta-
tonic systems (Indonesia) has to do with the music and the instruments first.
In most cases, it is impossible to say what criteria were used or exactly how
they developed historically. Also keep in mind that often theory does not 
agree with practice. 

Bill Alves