Key Signatures and Scales
In our previous discussion about reading sheet music we went over what key signatures are, and showed how they could represent scales, as a recap, if we take key signature:
---9*
# 8*
#-----------------------------------------------------------------------------------6*---------------
4*
----------#---------------------------------------------------------3*-------------------------------
# 1*
----------------------------------------------------11*----------------------------------------------
9*
------------------------------------8*---------------------------------------------------------------
6*
--------------------4*-------------------------------------------------------------------------------
3*
----1*--
We deciphered that this could represent the 4* major scale, now you might have wondered why we chose to sharpen the notes that we did, for example this phenomena happens again when we sharpen the 5* to a 6*.
---9*
7*
#-----------------------------------------------------------------------------------6*---------------
4*
--------------------------------------------------------------------2*-------------------------------
0*
----------------------------------------------------11*----------------------------------------------
9*
------------------------------------7*---------------------------------------------------------------
6*
--------------------4*-------------------------------------------------------------------------------
2*
----0*--
which defines the 7* major scale.
But it doesn't always work, for example let's say we sharpen the 7* to be an 8* like this:
---9*
# 8*
------------------------------------------------------------------------------------5*---------------
4*
--------------------------------------------------------------------2*-------------------------------
0*
----------------------------------------------------11*----------------------------------------------
9*
------------------------------------8*---------------------------------------------------------------
5*
--------------------4*-------------------------------------------------------------------------------
2*
----0*--
There is no possible major or minor scale which can represent this collection of notes. This means that the notes which are being sharpened in key signatures have to be specific notes or else you might not get a valid major or minor key from it.
Instead of randomly sharpening random notes and seeing if it creates a valid scale, we will figure out the correct notes to modify through the circle of fourths.
We can first recall that all scales are embedded within the circle of fourths for example we can find the 1* major:
Which you should remember from our discussion on the circle of fourths. Now since these continuous sections of the circles form scales, then what happens if we do a small shift to one scale to get another?
We can see that the scale on the left is 0* major and the one on the right is 5* major, but the main thing to note is that these two scales don't differ very much at all, the only difference is that the 11* has gone away and the 10* has appeared.
So the notes in the 5* major scale would be the ones we would normally find in the 0* major scale except the 11* is now a 10*, in other words the 11* has been flattened to be a 10*, meaning that if we wanted to write this with a key signature it would look like this:
---9*
7*
------------------------------------------------------------------------------------5*---------------
4*
--------------------------------------------------------------------2*-------------------------------
0*
b---------------------------------------------------10*----------------------------------------------
9*
------------------------------------7*---------------------------------------------------------------
5*
--------------------4*-------------------------------------------------------------------------------
2*
----0*--
If we rotated the 5* major then we'd get the 10* major:
Notice that this time the 4* has gone away and the 3* has entered, or we could say that the 4* has turned into the 3* by going down by one step aka being flattened, so if we account for the flatting that happened when we went from 0* major to 5* major and this new flattening then we would say that 11* -> 10* and 4* -> 3*. So our key signature would look like:
---9*
7*
------------------------------------------------------------------------------------5*---------------
b 3*
--------------------------------------------------------------------2*-------------------------------
0*
b---------------------------------------------------10*----------------------------------------------
9*
------------------------------------7*---------------------------------------------------------------
5*
--------------------3*-------------------------------------------------------------------------------
2*
----0*--
Now that we've seen how one note leaves and another one enters as we rotate this section around the circle of fourths, then we can continue this pattern to figure out what the key signature would be for all the other scales.
So let's go back to our 10* major and rotate it once again, first of all the new scale is 3* major and 9* -> 8* since 9* is no longer there and 8* is. So we can deduce that the key signature of 3* major is
---8*
7*
------------------------------------------------------------------------------------5*---------------
b 3*
--------------------------------------------------------------------2*-------------------------------
0*
b---------------------------------------------------10*----------------------------------------------
b 8*
------------------------------------7*---------------------------------------------------------------
5*
--------------------3*-------------------------------------------------------------------------------
2*
----0*--
For good measure we can do this one more time, changing our 3* major into an 8* major and 2* -> 1* so we have:
---8*
7*
------------------------------------------------------------------------------------5*---------------
b 3*
--------------b-----------------------------------------------------1*-------------------------------
0*
b---------------------------------------------------10*----------------------------------------------
b 8*
------------------------------------7*---------------------------------------------------------------
5*
--------------------3*-------------------------------------------------------------------------------
1*
----0*--
In our initial paragraph we showed a diagram of 4* major using sharps, but so far whenever we have rotated the major segment clockwise around the circle notes have always gone down when they are modified. If that's true then let's see what happens when we go counterclockwise.
Thus rotating once more gives us 4* major with the sharps: 5* -> 6*, 0* -> 1*, 7* -> 8*, 2* -> 3* summing that up in the key signature, we get exactly that:
---9*
# 8*
#-----------------------------------------------------------------------------------6*---------------
4*
----------#---------------------------------------------------------3*-------------------------------
# 1*
----------------------------------------------------11*----------------------------------------------
9*
------------------------------------8*---------------------------------------------------------------
6*
--------------------4*-------------------------------------------------------------------------------
3*
----1*--
Take Away
Now that we have all this new information let's turn it into something usable. When we look at a key signature the number of sharps that appear is the number of times we've rotated counter clockwise in the circle of fourths, so assuming there were 3 sharps, then the scale is is (0*, 7*,) 2* major. If there were five flats, then on the contrary we rotate clockwise five times around the circle to get our scale (0*, 5*, 10*, 3*) 8* major.
Since moving around the circle is also the same as moving horizontally on the fretboard, we can associate flats as moving -> some number of times and associate sharps with moving <- some number of times, for example we start with 0* major, then if we look at the key signature and it has 2 sharps, then we add 2 to 0* (horizontal movement rule) to get 2* major. By doing this we can re-use information that we already know from the fretboard.
Even though we talk about flattening and sharpening notes, this is being done so that it remains a major/minor scale and since we work in anchored intervals we won't actually ever think of the notes we're playing, therefore the purpose of this article is to allow us to grapple with notation built for people who do play notes.
So for example when we see the following stave
---9*
# 8*
#-----------------------------------------------------------------------------------6*---------------
4*
----------#---------------------------------------------------------3*-------------------------------
# 1*
----------------------------------------------------11*----------------------------------------------
9*
------------------------------------8*---------------------------------------------------------------
6*
--------------------4*-------------------------------------------------------------------------------
3*
----1*--
What we do is first extract what scale it is, which we determined was 4* major and then anchor the stave like this:
---11
# 9
#-----------------------------------------------------------------------------------7----------------
0
----------#---------------------------------------------------------11-------------------------------
# 9
----------------------------------------------------7------------------------------------------------
5
------------------------------------4----------------------------------------------------------------
2
--------------------0--------------------------------------------------------------------------------
11
----9---
So the only thing that ever changes for us is the location of 0.