Re: [Harp-L] Why or how does the major scale equation work?

[Winslow Yerxa <winslowyerxa@xxxxxxxxx>]

The Major scale is built on the most elementary ratios that govern the physical behavior of sound waves, and that may have a lot to do with its widespread use.

Look at the harmonic series (also known as the overtone series).

If you play a string, then divide its length in half, you get a note one octave higher.  The rest of the notes in the scale are divisions of the octave that can be derived either from the generating note or from its strongest products.

We can get the strongest notes by dividing a string by various simple ratios, then transposing the notes down so they fit inside an octave. The simplest ratios are 1/2, 1/3, 1/4, and 1/5. These notes are called harmonics, or partials. They occur naturally in any sustained tone - fans, electric motors, airplane propellors, wolves howling, musical instruments, etc. Each note sung by a human or played on a musical instrument produces 
 harmonics. Each harmonic tends to decrease in intensity as the fraction gets smaller. For instance, most instruments sound the fundamental, or first partial (the ratio 1/1) most strongly, and we hear that as the note.

The next ratio after 1/2 (or octave) is 1/3. If the fundamental is C, this note will be G above the octave. 1/4 sill produce another octave, and 1/5 will produce E above that.

So now we have C, E, and G, or a major triad (three-note chord) built on C. We can transpose E and G down so that they occur close above C, and we can duplicate this structure in any octave.

But that gives us only three of the seven notes in the major scale. How do we get the others? We could continue up the harmonic series, but that produces so-so- results. Some of the intervals it produces only approximate major scale notes rather distantly, and they're unlikely to actually be heard.

Instead, let's look at the next-strongest note in the harmonic series after C: G. The 1/1 and 1/2 ratios both produce C, but the 1/3 ratio produces G. If we put C and G in the same octave with G above G we get a fifth (count up 1-2-3-4-5 from C to G). If you play these two notes together, you may recognize the characteristic hum they produce.

If we build a harmonic series on G, we get G, G, D, G, B, which gives us two new notes: D (the fifth, based on the 1/3 ratio) and B (the third, based on the 1/5 ratio). G, B, and D make a G major triad.

So now we have C, D, E, G, and B. We also have the I chord (C-E-G) and the V chord (G-B-D).

What about F and A?

Well, instead of going up a fifth from C, how about we go DOWN a fifth (5-4-3-2-1, C-B-A-G-F) to F. 

In the world of F, C is powerful, for the same reason that G is powerful in the word of C. Build the ratios of 1/2, 1/3, 1/4 and 1/5 on F and we get F, F, C, F, A. This boils down to F, A and C, which also forms an F major triad (which is the IV chord in C) and fills in the missing notes of the C major scale.

So you can see that the major scale is built on the most fundamental sonic ratios and is closely inked to the proverbial I, IV, and V chords.

There may be other equally compelling mathematical explanations for the widespread use of the major scale, but this one I think makes sense in relation to contemporary music.

I remember reading sometime in the 1970s that someone had deciphered some Babylonian (or maybe Sumerian) music written in cuneiform on ancient clay tablets. When they decoded it after much research, they were surprised to find that it appeared to correspond to the major scale instead of something more exotic.

Winslow

Winslow Yerxa

Author, Harmonica For Dummies ISBN 978-0-470-33729-5

--- On Fri, 11/20/09, Seth Galitzer <sethgali@xxxxxxxxx> wrote:

From: Seth Galitzer <sethgali@xxxxxxxxx>
Subject: Re: [Harp-L] Why or how does the major scale equation work?
To: "michael rubin" <michaelrubinharmonica@xxxxxxxxx>
Cc: "harp-l" <harp-l@xxxxxxxxxx>
Date: Friday, November 20, 2009, 11:12 AM

If I remember correctly, western music is based on the ancient Greek ideals of balance and harmony.  One web article I found attributed the divisions of the Major Scale to Pythagoras, which sounds credible.  I imagine if you were to compare the waveforms of the individual pitches you will find some congruity, which would contribute to why we think it "sounds good".  The Greek history also explains where the names of the modes comes from.  The same source goes on to say that the use of the Major Scale was formalized by the Church in the Middle Ages and Renaissance periods which has kept it in the forefront of modern Western music.

So why do we use it in this hemisphere?  Because our Greek ancestors decided the arrangement was the most harmonious to the human ear, and the Church agreed.  So far, Western musicians still think so (modern atonal composers notwithstanding).

Seth

michael rubin wrote:
> Early on  I teach my students about half steps and whole steps and
> that the do re mi fa so la ti do major scale can be played by
> beginning on any  note on the keyboard and then moving up a whole step
> followed by a whole, half, whole, whole, whole, half.
> 
> My new student asked me why or how does it work.  I did my best
> explanation but felt I came up short.  What say the physics guys or
> anyone with a good explanation?
> Thanks,
> Michael Rubin
> Michaelrubinharmonica.com

-- Seth Galitzer

The beatings will continue until morale has improved.
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