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Old 12-11-2014, 05:26 PM   #1
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Major Tetrachords, Part 2: Understanding Key Signatures with Sharps

In Part 1 we saw that it's possible to consider the major scale as consisting of two major tetrachords separated by a whole-step. Now in Part 2 let's see how this alternate view can give us a rock-solid understanding of the key signatures that use sharps.

This might be dangerous, but I'm going to go out on a limb here and assume that everyone reading this post knows what a key signature does or, at the very least, has seen some of them.

Intuitively, one might think that the simplest key signature (no flats, no sharps) would identify the major key named using the first letter of the alphabet, "A". But no -that distinction belongs to the major key named using the third letter, "C". I have no clue why this is so, but there it is. And in the interest of reducing word count, every use of the word "tetrachord" in this post will refer to the major tetrachord.

If we start at this musical tone "C" and construct two tetrachords separated by a whole-step we produce this sequence:

1. C
2. D
3. E
4. F
5. G
6. A
7. B
8. C

If you play this sequence at a piano you'll notice that every key you press is white, signifying that they are all "natural" tones. So our first key signature therefore consists of no flats and no sharps, telling us that the tones of the C major scale are all naturals.

As a review, we see that C-D is a whole-step, D-E is a whole-step, and E-F is the half-step that completes the first tetrachord.

The whole-step F-G separates the two tetrachords.

Continuing, G-A is a whole-step, A-B is a whole-step, and B-C is the half-step that completes the second tetrachord and brings us to the C one octave above our starting point.

Now let's label the C-D-E-F grouping the "lower tetrachord" and the G-A-B-C grouping the "upper tetrachord". Here is where our the key-signature understanding truly begins...

If we now set the C-D-E-F lower tetrachord aside and replace it with the G-A-B-C upper tetrachord, we have the foundation for a major scale built on the tone "G". Let's do it...

1. G
2. A
3. B
4. C
5. D
6. E
7. F
8. G

Checking our work, we find that G-A is a whole-step, A-B is a whole-step, and B-C is the half-step completing the lower tetrachord. So far, so good.

Continuing, C-D is the whole step separating the two tetrachords, with D being the starting point of the new upper tetrachord.

From D-E is a whole step, E-F is a half-step, and F-G is a whole-step.

Oops. Our upper tetrachord does not conform to the pattern. We need a whole-step between tones two and three of the upper tetrachord (tones six and seven of the major scale), and we need a half-step between tones three and four of the upper tetrachord (tones seven and eight of the major scale). How can we fix this?

It turns out that if we can somehow manage to raise that seventh tone F by one half-step, we can repair both incorrect intervals at the same time.

And that is precisely what that sharp (F#) in the G major key signature does. Applying that F# produces this sequence:

1. G
2. A
3. B
4. C
5. D
6. E
7. F#
8. G

G-A is a whole-step, A-B is a whole-step, B-C is a half-step completing the lower tetrachord, C-D is the whole-step link, D-E is a whole-step, E-F# is a whole-step, and F#-G is the half-step completing the upper tetrachord and bringing us home to the octave G.

Let's repeat the process. Set aside the G-A-B-C lower tetrachord and replace it with its D-E-F#-G upper tetrachord to establish a new lower tetrachord starting on the tone D:

1. D
2. E
3. F#
4. G
5. A
6. B
7. C
8. D

The same interval errors we saw when starting on G appear between tones six and seven and between seven and eight when we start on D.

Fortunately, adding a sharp to raise tone seven by one half-step again corrects both errors. And that is why the key signature of D major contains two sharps: F# and C#. Here is the correct form of the D major scale:

1. D
2. E
3. F#
4. G
5. A
6. B
7. C#
8. D

You probably see where this is going. As we continue to replace the lower tetrachord with the upper tetrachord in order to start our next major scale on the first tone of the new lower tetrachord, we'll continue to encounter the interval errors between tones six and seven and tones seven and eight. We'll also be able to correct those errors by raising the seventh tone of the sequence by one half-step using a sharp and then adding that sharp to the key signature.

Eventually we'll arrive at the C# major scale, whose key signature contains seven sharps: F#, C#, G#, D#, A#, E# and B#. This is the maximum number of sharps available to us in our system of notation, and these sharps always appear in this order in the signature. The keys identified by the addition of each new sharp to the signature are G, D, A, E, B, F# and C#, respectively.

Remember this factoid: Sharps Sharpen the Seventh. This simply means that the sharp you just added to the key signature to correct the interval errors between tones six and seven, and seven and eight, represents the seventh tone of the major scale. As an example, if you just added an A# to the signature, you know that you have created a B major scale, because A# is the seventh tone of that scale. Put another way, the sharp farthest to the right in the key signature represents the seventh tone of that major scale.

If you'll take the time to build major scales by manipulating major tetrachords in this way for yourself, I believe you'll come away with an unshakable understanding of the key signatures that use sharps.

Next up: manipulating major tetrachords to produce keys with signatures that use flats.
All things are difficult before they are easy.
- Dr. Thomas Fuller (British physician, 1654-1734)
Originally Posted by Freepower
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