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Theory of Scales

 

A scale (in this article) divides an octave into steps, either half steps or whole steps. The chromatic scale divides the octave into the largest possible group of steps, 12.

 

The whole tone scale divides the octave into the smallest possible group of steps, 6.

 

The octave can’t be divided into groups of steps smaller than 6. If you try, you’ll find you have to put in a skip somewhere. Pentatonic scales don’t count as scales in this article.

 

I am interested in exploring the universe between 12 and 6 tone scales. The scales most people are familiar with are the 7 tone major and minor scales. If we look at the major scale, we find it has 5 whole steps and 2 half steps.

 

C    D    E    F    G    A    B    C

 

The steps in the major scale are arranged WWHWWWH

 

If we continue the scale past one octave we see the repeating pattern:

WWHWWWHWWHWWWHWWHWWWH.... etc

 

If we look at the modes of the major scale

WHWWWHW        dorian

HWWWHWW        phrygian

WWWHWWH        lydian

WWHWWHW        mixolidian

WHWWHWW        aeolian

HWWHWWW        locrian

 

and extend them past one octave

 

WHWWWHW    WHWWWHW    WHWWWHW

    HWWWHWWHWWWHWWHWWWHWW

WWWHWWH    WWWHWWH    WWWHWWH

    WWHWWHWWWHWWHWWWHWWHW

    WHWWHWWWHWWHWWWHWWHWW

HWWHWWW    HWWHWWW    HWWHWWW

 

We can see the same repeating pattern....

 

WHWWWHW    WHWWWHW    WHWWWHW

     HWWWHWWHWWWHWWHWWWHWW

WWWHWWH    WWWHWWH    WWWHWWH

     WWHWWHWWWHWWHWWWHWWHW

     WHWWHWWWHWWHWWWHWWHWW

HWWHWWW    HWWHWWW    HWWHWWW

 

...of half steps separated by three whole steps then two whole steps. Of course, this is a rather trivial finding, since we already know that the modes are the same as the major scale, started from the various scale steps. What is interesting to me is that we can use the above exploration as the basis for exploring other scales. It also shows how different permutations of whole and half can be thought of as essentially modes of the same scale.

 

A seven-tone scale must always have 2 half steps and 5 whole steps. This is because we need 12 half steps in order to divide the octave, and we have allotted ourselves exactly 7 steps (no more, no less) in the scale. If we only used whole steps we would reach the octave in 6 steps instead of 7. If we use only half steps, we stop at the interval of a perfect fifth, far short of an octave. So we will have to use a combination of whole and half steps that add up to 7.

 

1) W + H = 7

 

In addition the total number of half steps must span an octave. Since a whole step is 2 half steps, we have....

 

2) 2W + H = 12

 

Subtracting equation 1) from equation 2)

 

{  2W + H = 12 }

{-  W + H  = 7 }

{     W = 5    }

 

And substituting W into either equation gives us H = 2

 

So we will always have 5 whole steps and 2 half steps in a seven tone scale.

 

How can these be steps be laid out in ways that do not duplicate the major scale or one of its modes? The possibilities are somewhat limited. The way to think about the answer is to think about how many whole steps come between the two half steps and how many are left out.

 

We can make a scale with only one whole step between the two half steps. There are then 4 whole steps needed to fill out the octave.

 

HWHWWWW

We can easily see that this scale and its modes will not duplicate the major scale modes.

 

HWHWWWWHWHWWWWHWHWWWW

 

vs

 

HWWHWWW    HWWHWWW    HWWHWWW

 

However once we add a second whole step in between the two half steps, we do duplicate the major scale pattern.

 

HWWHWWW

 

Putting three whole steps in between simply moves the major scale pattern out to one of its modes.

 

HWWWHWW

 

Putting 4 whole steps in between simply moves the first scale we created out to one of its modes.

 

HWWWWHW

 

Putting 5 notes in between is the most we can do, since we only have 5 whole steps in a seven-tone scale to work with.

 

HWWWWWH

 

This is the same as zero whole steps between the bookend half steps.

 

ie HHWWWWWHHWWWWWHHWWWWW.... etc.

 

So we have exhausted the possibilities and we see that there are only three primal patterns of seven tone scales.

 

HWWHWWW

HWHWWWW

HHWWWWW

 

Of these, it must be admitted, the standard HWWHWWW is still the most varied and interesting. Since it contains both the major and the minor tonalities. The two new scales that we’ve just derived, given that they clump more whole tones together,  sound more like the whole scale tone. In particular, HHWWWWW is basically a whole tone scale with one of its steps chromatically divided.

 

Nevertheless, we can have fun with these new scales. In particular, they are note-for-note substitutes for the major/minor scales you find in most older music. Therefore you can get some interesting music by substituting new for old in pre-existing music.

 

The modes of the chromatic and whole tone scale are themselves chromatic and whole tone scales. Whatever note you start on, you will produce a sequence of 12 half steps or 6 whole steps. So each mode of a chromatic scale sounds just like (and is!) a chromatic scale, and each mode of a whole tone scale sounds just like (and is) a whole tone scale.

 

The chromatic and whole tone scales bookend the types of scales we can have. Between them we have 7, 8, 9 , 10, 11 tone scales. We’ve already explored the 7 tone scales, just one level above the homogeneity of the 6 tone (whole tone) scale. Let’s proceed now from 12 steps down to 11 steps.

 

The 11 step scale divides the octave into 11 steps. As was the case with the 7 tone scale, these can’t all be whole steps as they would overshoot the octave, and they can’t all be half steps because they wouldn’t reach the octave. So the 11 steps have to be some combination of whole and half steps.

 

1) W + H = 11, with the same constraint holding as before

2) 2W + H = 12

 

Subtracting 1 from 2, we have

 

{  2W + H = 12 }

{ - W + H = 11 }

{   W     = 1  }

 

So we see that the 11 tone scale has 1 whole step and 11 half steps. There is also only one primal form of this scale, since every arrangement of whole steps around the half step reverts to a mode of the same pattern HWWWWWWWWWW.  This is actually more properly thought of as a tonality rather than a scale. As a scale is isn’t very interesting, sounding almost indistinguishable from the chromatic scale.

 

The 10 step scale must satisfy the equations -

 

1) W + H = 10

2) 2W + H = 12

 

or W=2, H=10.

 

We can see the following pattern of increasing whole steps as we go from chromatic to whole tone scales

 

12  -> W=0

11 -> W=1

10 -> W=2

9 -> W=3

8 -> W=4

7 -> W=5

6 -> W=6

 

In contrast to the examples we’ve looked at so far, there are 5 primal 10 tone scales....

 

WWHHHHHHHH

WHWHHHHHHH

WHHWHHHHHH

WHHHWHHHHH

WHHHHWHHHH

 

Each with ten modes

 

10 primal Nine tone scales (W=3, H=6)

 

WWWHHHHHH

WWHWHHHHH

WWHHWHHHH

WWHHHWHHH

WWHHHHWHH

WWHHHHHWH

 

WHWHWHHHH

WHWHHWHHH

WHWHHHWHH

 

WHHWHHWHH

 

Each with 9 modes

 

And 10 primal 8 tone scales W=4, H=6

 

WWWWHHHH

WWWHWHHH

WWWHHWHH

WWWHHHWH

 

WWHWWHHH

WWHHWWHH

WWHWHWHH

WWHWHHWH

WWHHWHWH

 

WHWHWHWH - this last one being the diminished scale