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From: maverick@fir.berkeley.edu (Vance Maverick)
Newsgroups: comp.music
Subject: Re: theory behind scales
Message-ID: <26357@pasteur.Berkeley.EDU>
Date: 19 Jul 90 23:06:16 GMT
References: <8667@uhccux.uhcc.Hawaii.Edu>
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Reply-To: maverick@fir.berkeley.edu (Vance Maverick)
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X-Local-Date: 19 Jul 90 16:06:16 PDT

> i did find out what 4/3 came from (inversion of 3/4 or 3/2, the perfect
> fifth, so its a fifth below the tonic, i hope i got that right), but i am
> still baffled about the 5/3. inverting it u get 3/5. either way u dont
> get a power of 2 on the top or bottom.  how do u get this note?
> also, if u are accepting 4/3 what about 8/5?  ...etc?  anyone ever 
> exerimented with those?
>                                 -Tim

People have experimented with them all, and continue.  In a tonal
context, I think of 5/3 as "a major third above the fourth degree", i.e.
5/4 * 4/3.  This is not the only tonal "meaning" for the sixth degree,
though -- how about "a perfect fifth above the second degree", i.e. 3/2
* 9/8 = 27/16.  This is a major tone (9/8) above the fifth degree, not a
minor tone (10/9).  Any interval can be "decomposed" in this fashion;
8/5, for example, is "a minor third above the fourth degree" (6/5 * 4/3)
or "a major third below the octave" (2 / (5/4)), which as you see comes
to the same thing.  The just intonation people eat, sleep and breathe
this kind of arithmetic.  I'm working on a system which I hope will
enable ratio-based interval selection independent of explicit scale-building.

If you're in a position to experiment, check out the ratios involving 7
-- 7/4, for example, is the first candidate for a "flatted seventh
degree" one can draw directly from the harmonic series built on the
root, yet it sounds pretty strange used melodically in a tonal context. 
To my ear, 9/5 ("a minor third above the fifth degree") sounds more
normal, which is hardly to say better.

There's a neat HyperCard stack by Robert Rich (JI Calc, shareware from
Soundscape Productions, PO Box 8891. Stanford, CA 94309) which allows
you to twiddle ratios to your heart's content, building scales, playing
them over the Mac speaker, or dumping them to a MIDI synth.  Because of
the MIDI orientation, it assumes octave equivalence and twelve notes per
octave, but this is reasonable for most people's music.

Gerald Balzano wrote an article in Music Perception (spring? 1986) in
which he derived the rudiments of standard tonality from group-theory
properties of twelve-tone equal temperament.  Pretty implausible
historically, but I think he was being provocative to make a point --
that the degrees of the scale do a lot more than make pretty intervals
together, and that there are a lot more factors influencing the
construction of scales than the availability of perfect triads.