Thursday, December 31, 2015

Key signatures in piano music: the underlying technology

Today, sheet music (at least, the major keys) went from magic to technology for me. Magic is spellcasting, is memorization. Technology is understanding, is derivation. Technology takes up so much less space in my head!

If you can read music enough to pick out a simple song but wonder why chords and their weird names seem so obvious to some people, this post is for you.

Those markings at the beginning of the line that show which notes are played as sharps (called a key signature) - I have been trying to memorize their names. This one is called D Major, and it means that all Fs and Cs are to be played as sharps, hit the black key to the right instead of the white key.

D Major signature
I know how to play music with C# and F#, but why on earth is this called D Major? And why is it a thing?

Today I read a few pages in a book about scales, and now I get it. It takes some history to understand.

See, a long time ago the Greeks made a lyre with four strings, and they tuned the strings into a tetrachord, four notes that sound good. On a piano, a tetrachord is made out of four notes with the following separations: a whole step, then a whole step, then a half step. A whole step goes two keys, counting white and black keys; a half step is one key. Like walking -- right foot, left foot makes a whole step. One easy tetrachord starts with middle C:
From C to D is a whole step because there is a black key in between. From E to F is a half-step because there is no key in between. The C tetrachord is C, D, E, F.

The same formula makes a tetrachord starting from any key. The tetrachord starting with D goes D, E, F#, G.
A whole step from D is easy, skip the black key and hit E, a white key. A whole step from E means skipping the next key (F, a white key) and hitting the key after that (F#, a black key). Then a half step means the very next key (G, a white key). This is where the F# in the key signature is coming from.

But wait! There's more!

Put two tetrachords together to get an Ionian scale, also called a major scale. The tetrachords are separated by a whole step. In C, the first tetrachord is C, D, E, F. Take a whole step to G and start the next tetrachord. It goes G, A, B, C.

Eight notes, the last one the same as the first (one octave higher), make a major scale. The keys have this pattern of separations between them, made up of the Ionian scale and tetrachord patterns. Each scale uses a little over half the keys on the keyboard, and ignores the rest. Songs in C major use all white keys and none of the black keys. You want anything else, gotta put a sharp or flat symbol in front of the note.

What does this mean for D? The D major scale starts with the D tetrachord and adds a second tetrachord: D, E, F#, G; A, B, C#, D.

C# and F#! There they are, the two black keys with numbers on them! The "normal" keys to play in the D scale include C# and F# (black keys), but never C or F (white keys). Putting the D Major key signature in front of the music means that all the keys in the D scale look like ordinary notes.
With the C# and F# handled by the key signature, any special marking (sharp, flat, or natural) points out a note that is unexpected, that does not fit in with the rest.

The same pattern works for other key signatures; try constructing the Ionian scale for G out of two tetrachords separated by a whole step. You'll find that the only black key used is F#, so this is G major:
G major key signature
These are historical explanations for the structure of major scales, of seven different notes (plus the same notes in other octaves) that sound good together. There are scientific explanations too, even ratios of wavelengths.

On the piano, this means only 7/12 of the keys are used in any song, ordinarily. Why have the other keys? They bring equality to the different notes: any key can start a scale, because the whole-steps and half-steps are there no matter where you start. C is not special, just convenient. The circle is complete. Actually 12 circles for the 12 keys in an octave. So many patterns are visible to me now!

Now I can name the key signatures and say why they have those names. I don't have to memorize them, because I can derive them. Next I can learn chords and why certain ones like C and F and G7 appear together frequently. All of this from two pieces of vocabulary and some counting. Goodbye magic, hello math.